diff --git "a/8tFST4oBgHgl3EQfaTh3/content/tmp_files/load_file.txt" "b/8tFST4oBgHgl3EQfaTh3/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/8tFST4oBgHgl3EQfaTh3/content/tmp_files/load_file.txt" @@ -0,0 +1,954 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf,len=953 +page_content='Tunable BCS-BEC crossover,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' reentrant,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' and hidden quantum phase transitions in two-band superconductors with tunable valence and conduction bands Giovanni Midei1 and Andrea Perali2 1School of Science and Technology,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Physics Division,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' University of Camerino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Via Madonna delle Carceri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 62032 - Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Italy 2School of Pharmacy,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Physics Unit,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' University of Camerino,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Via Madonna delle Carceri,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 62032 - Camerino (MC),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Italy Two-band electronic structures with a valence and a conduction band separated by a tunable en- ergy gap and with pairing of electrons in different channels can be relevant to investigate the proper- ties of two-dimensional multiband superconductors and electron-hole superfluids,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' as monolayer FeSe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' recently discovered superconducting bilayer graphene,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' and double-bilayer graphene electron-hole sys- tems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This electronic configuration allows also to study the coexistence of superconductivity and charge density waves in connection with underdoped cuprates and transition metal dichalcogenides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' By using a mean-field approach to study the system above mentioned, we have obtained numerical results for superconducting gaps, chemical potential, condensate fractions, coherence lengths, and superconducting mean-field critical temperature, considering a tunable band gap and different filling of the conduction band, for parametric choice of the pairing interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' By tuning these quantities, the electrons redistribute among valence and conduction band in a complex way, leading to a new physics with respect to single-band superconductors, such as density induced and band-selective BCS-BEC crossover, quantum phase transitions, and hidden criticalities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' At finite temperature, this phenomenon is also responsible for the non-monotonic behavior of the superconducting gaps resulting in a superconducting-normal state reentrant transition, without the need of disorder or magnetic effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' INTRODUCTION Multi-band and multi-gap superconductivity is a com- plex quantum coherent phenomenon with peculiar fea- tures that cannot be found in single-band and single- gap superconductors [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The increased number of de- grees of freedom in the condensate state allows for novel quantum effects which are unattainable otherwise, for in- stance enriching the physics of the BCS-BEC crossover [2–5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Proximity to the crossover regime of the BCS-BEC crossover in multi-band superconductors having deep and shallow bands can determine a notable increase of su- perconducting gaps and critical temperature (Tc) [6–9], associated with an higher mean-field Tc, together with optimal conditions for the screening of superconduct- ing fluctuations [10–12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Furthermore, the interplay of low-dimensional two-band systems allows for screening of fluctuations in systems composed by coupled quasi-2D bands or even in the vicinity of a van Hove singularity (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=', in the case of quasi-1D), enabling shrinking of the pseudo-gap phase and robust high-critical temperatures [13–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Motivated by high temperature superconductivity and anomalous metallic state properties in underdoped cuprates, interest has grown in the pseudogap physics, in which a blurred gap persists in the normal state near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' There are different models and explana- tions for this pseudogap, the simplest one being a smooth crossover from the BCS regime towards a Bose-Einstein condensation regime in which bound pairs form first at higher temperatures, and then below a critical temper- ature Tc they condense, with the pseudogap being the excitation energy of the quasi-molecular pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Another explanation relevant for underdoped cuprates is the pres- ence of other mechanisms different from pair fluctuations, such as charge density waves (CDWs) [16–19] and their fluctuations that can modify the energy spectrum with opening of (pseudo)gaps and at the same time mediate Cooper pairing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thus, systems in which CDWs and su- perconductivity coexist are of primary interest to study the BCS-BEC crossover when an energy gap separates the electronic spectrum in two bands, determining a va- lence and a conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In addition to underdoped cuprates, an interesting ex- ample is given by the transition metal dichalcogenide (TMD) family, MX2, where M = Ti, Nb, Mo, Ta and X = S, Se, which exhibits a rich interplay between super- conductivity and CDW order [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In these materials, su- perconductivity occurs in an environment of pre-existing CDW order [21, 22], making them an ideal platform to study many-body ground states and competing phases in the 2D regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The relationship between CDW and superconductivity in such systems is still under investi- gation [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In general, their mutual interaction is competitive, but evidence to the contrary, indicating a cooperative interplay, has also been reported in angle- resolved photoemission spectroscopy (ARPES) studies [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Among them, bulk Niobium diselenide (2H-NbSe2) undergoes a CDW distortion at T=30 K and becomes su- perconducting at 7 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' References [25, 26] reported that Tc lowers to 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='9 K in 2H-NbSe2 single-layers and that the CDW measured in the bulk is preserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Theoretical sup- port is given by Chao-Sheng Lian et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' [27]: they demon- strate enhanced superconductivity in the CDW state of monolayer tantalium diselenide (TaSe2) with DFT cal- culations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In contrast with 2H-NbSe2, they report that arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='13795v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='supr-con] 31 Jan 2023 2 as TaSe2 is thinned to the monolayer limit, its super- conducting critical temperature rises from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='14 K in the bulk to 2 K in the monolayer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Another appealing super- conducting material is the monolayer FeSe grown on a SrTiO3 substrate, which exhibits a huge increase of Tc up to 100 K [28] and it is characterized by a valence and a conduction band structure near the Fermi level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Fur- thermore, very recently 2D superconductivity has been found in bilayer graphene systems, in which conduction and valence bands are separated by a small energy band- gap (0 ÷ 100 meV) that can be precisely tuned by an external electric field [29] (for a review see [30]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Cou- pling a monolayer of WSe2 with bilayer graphene has been found to enhance superconductivity by an order of magnitude in Tc and superconductivity emerges already at zero magnetic field [31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Finally, it turns out that the two-band superconducting system considered in this work is in close correspondence with two-band electron- hole superfluids in double bilayer graphene [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Therefore, the growing experimental realization of 2D superconductors with valence and conduction bands sep- arated by a tunable energy gap and electron-hole super- fluidity in multilayer systems motivated us to investigate the BCS-BEC crossover in this kind of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The de- tailed analysis of this configuration is lacking in the liter- ature to the best of our knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' A pioneering work on a related system with valence and conduction parabolic bands has been done by Nozi`eres and Pistolesi [33] to study the phase transition from a semiconducting to a superconducting state and the consequent (pseudo)gap opening, in the specific case of equal pairing strengths for all interaction channels considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In our work we consider a superconductor with two tight-binding bands with different intra-band and pair-exchange couplings, in order to probe the possibility to have coexisting Cooper pairs of different average sizes [34] in the valence and con- duction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, for most of multi-band supercon- ductors the tuning of intra-band and pair-exchange inter- actions is rather challenging and their properties cannot be studied easily in a continuous way across the BCS- BEC crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' As shown in this work, a different way to explore the BCS-BEC crossover in such systems can be achieved by tuning the energy gap between the valence and the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In fact, since the number of particles in the single bands is not conserved, when the energy band gap is modified the number of holes and of electrons forming Cooper pair respectively in the valence and in the conduction bands changes, allowing for the occurrence of a density induced multi-band BCS-BEC crossover [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This redistribution of charges between the valence and the conduction band leads also to novel and interesting quantum phase transitions (QPTs) from a superconducting to an insulating state, or hidden crit- icalities evidenced by the analysis of the order parame- ter coherence lengths [36, 37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' At finite temperature, a new type of reentrant superconducting to normal state transition has been also found and characterized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The results reported and discussed in this work demonstrate the richness of the proposed valence and conduction band configuration to generate and tune new types of crossover phenomena and quantum phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The manuscript is organized as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In section II we describe the model for the physical system considered and the theoretical approach for the evaluation of the superconducting state properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In section III we report our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The conclusions of our work will be reported in Section IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' MODEL SYSTEM AND THEORETICAL APPROACH We consider a two-dimensional (2D) two-band super- conductor with a valence and a conduction electronic band in a square lattice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The valence and the conduction bands are modelled by a tight-binding dispersion given, respectively, by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' (1) and (2): ε1(k) = 2t[cos(kxa) + cos(kya)] − 8t − Eg (1) ε2(k) = −2t[cos(kxa) + cos(kya)] (2) where t is the nearest neighbour hopping parameter as- sumed to be the same for both bands, a is the lattice parameter and the wave-vectors belong to the first Bril- louin zone − π a ≤ kx,y ≤ π a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Eg is the energy band-gap between the conduction and the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The band dispersions are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In order to study the superconducting state properties of our system, we as- sume that Cooper pairs formation is due to an attractive interaction between opposite spin electrons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The two- particle interaction has been approximated by a separa- ble potential Vij(k, k′) with an energy cutoff ω0, which is given by: Vij(k, k′) = −V 0 ijΘ � ω0 − |ξi(k)| � Θ � ω0 − |ξi(k′)| � (3) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Electronic band structure of the two-band 2D system considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Eg is the energy gap between the valence (i = 1) and the conduction (i = 2) band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' CONDUCTION BAND 82 = - 2t(cos(akx) + cos(ak,) E VALENCEBAND 81 = 2t(cos(akx) + cos(ak,) - 8t - Eg3 where V 0 ij > 0 is the strength of the potential in the different pairing channels and i, j label the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' V 0 11 and V 0 22 are the strength of the intra-band pairing inter- actions (Cooper pairs are created and destroyed in the same band).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' V 0 12 and V 0 21 are the strength of the pair- exchange interactions (Cooper pairs are created in one band and destroyed in the other band, and vice versa), so that superconductivity in one band can induce super- conductivity in the other band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The same energy cutoff ω0 of the interaction for intra-band and pair-exchange terms is considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Through out this work, ω0 is con- sidered an energy scale larger than the total bandwidth of our system to model an effective pairing of electronic origin, or a contact attractive potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This is a key as- sumption to make possible for the system to explore the entire BCS-BEC crossover [38].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The terms corresponding to Cooper pairs forming from electrons associated with different bands (inter-band or cross-band pairing) are not considered in this work (see [39]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ξi(k) = εi(k) − µ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' (3) is the energy dispersion for the band i with re- spect to the chemical potential µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconducting state of the system and its evolution with relevant sys- tem parameters is studied at a mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The BCS equations for the superconducting gaps have to be coupled with the density equation which fixes the chemi- cal potential, since the self-consistent renormalization of the chemical potential is a key feature to account for the BCS-BEC crossover physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Zero and finite tempera- ture cases have been considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The BCS equations for the superconducting gaps in the two-band system at a given temperature T are ∆1(k) = − 1 2Ω � k′ � V11(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆1(k′) E1(k′) tanh E1(k′) 2T + V12(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆2(k′) E2(k′) tanh E2(k′) 2T � (4) ∆2(k) = − 1 2Ω � k′ � V22(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆2(k′) E2(k′) tanh E2(k′) 2T + V21(k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' k′)∆1(k′) E1(k′) tanh E1(k′) 2T � (5) where Ei(k) = � ξi(k)2 + ∆i(k)2 is the dispersion of single-particle excitations in the superconducting state and Ω is the area occupied by the 2D system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ℏ = 1 and kB = 1 throughout the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconduct- ing gaps have the same energy cutoff of the separable interaction: ∆i(k) = ∆iΘ � ω0 − |ξi(k)| � (6) The total electron density of the two-band system is fixed and given by the sum of the single-band densities, ntot = n1 + n2, that can vary instead.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The electronic density ni in the band (i) at temperature T is given by, ni = 2 Ω � k � vi(k)2f � − Ei(k) � + ui(k)2f � Ei(k) �� (7) where f(E) is the Fermi-Dirac distribution function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The BCS coherence weights vi(k) and ui(k) are: vi(k)2 = 1 2 � 1 − ξi(k) � ξi(k)2 + ∆i(k)2 � (8) ui(k)2 = 1 − vi(k)2 (9) For the valence band the definition of the condensate fraction is the ratio of the number of Cooper pairs in the valence band to the number of holes in the valence band,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' αh 1 = � k � u1(k)v1(k) �2 � k u1(k)2 (10) For the conduction band instead,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' the expression already used in the one-band case is generalized to the number of Cooper pairs divided by the total number of carriers in the conduction band αe 2 = � k � u2(k)v2(k) �2 � k v2(k)2 (11) The intra-pair coherence length ξpairi has the same form for both the valence and the conduction bands,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' that is ξ2 pairi = � k ��∇ � ui(k)vi(k) ���2 � k � ui(k)vi(k) �2 (12) Regarding the superconducting order parameter coher- ence length,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' two characteristic length scales in the spatial behavior of superconducting fluctuations are expected,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' since the system is made up by two partial condensates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the pair-exchange interaction is not present, these two lengths are simply the order parameter coherence lengths of the condensates of the valence ξc1 and of the conduction ξc2 band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the pair-exchange interac- tions is different from zero, one has to deal with coupled condensates, and these length scales cannot be attributed to the single bands involved, describing instead the col- lective features of the whole two-component condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange interactions mix the superconducting order parameters of the initially non-interacting bands, that acquire mixed character.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The soft, or critical, co- herence length ξs diverges at the phase transition point, while the rigid, or non-critical, coherence length ξr re- mains finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Following the approach in [37],' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' these char- acteristic length scales are given by ξ2 s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='r = G(T) ± ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='G2(T) − 4K(T)γ(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2K(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(13) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='where ξs corresponds to the solution with the plus and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='ξr to the one with the minus sign and ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='G(T) = (V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='12)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='˜g1(T)β2(T) + ˜g2(T)β1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11˜g1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22β2(T)+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22˜g2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11β1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(14) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='K(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11˜g1(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 − V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22˜g2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='12)2˜g1(T)˜g2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(15) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='γ(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='11V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='22 − (V 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='12)2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='β1(T)β2(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='˜gi(T) = gi(T) − 3νi(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∆i(T) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(17) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='gi(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='ξi(k) tanh ξi(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='νi(T) = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='− 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂|∆i|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='Ei(k) tanh ξi(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∆i=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(19) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='βi(T) = − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='k ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='∂q2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='l ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='ξi(k) + ξi(k − q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='tanh ξi(k) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='+ tanh ξi(k − q) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2T ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='q=0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='(20) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='where l refers to the Cartesian axis in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In order to describe the physics of the quantum phase transition, the values of the coherence lengths at zero temperature have been approximated by choosing a low enough temperature so that the superconducting gaps and the chemical potential retain the same behavior of the zero temperature case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The energies are normalized in units of the hopping t and the dimensionless couplings λii are defined as λii = NV 0 ii, where N = 1/4πa2t is the density of states at the top / bottom of the valence / conduction band, that coincide since the density of states is not modified by the concavity of the band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra- pair coherence lengths ξpairi are normalized using the average inter-particle distance in the normal state li = 1/√πni, where ni is the density in the band i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This quantities differ by a factor of √ 2 by the inverse of the respective Fermi wave-vector KF i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The soft ξs and the rigid ξr coherence lengths are normalized with respect to the lattice constant a, since in the two-band case they cannot be attributed to any of the two bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' RESULTS In this section we study the properties of the super- conducting ground state and give a full characteriza- tion of the BCS–BEC crossover in the two-band system considered in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' First, we study the zero tem- perature superconducting gaps in the conduction (∆2) and in the valence (∆1) band through the BCS-BEC crossover, for the case of unbalanced intra-band couplings (λ11 ̸= λ22).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The results are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2, in which the superconducting gaps are reported as functions of the energy band-gap Eg, for different values of the total density a2ntot and for different pair-exchange couplings λ12 = λ21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case of an empty conduction band 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='016 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='58 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='59 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='61 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='62 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 3 (a) Δ2 / t (b) a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) Δ1 / t Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t QCP QCP QCP QCP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Superconducting gaps ∆2/t opening in the conduc- tion band (a)-(b) and in the valence band ∆1/t (c)-(d) as functions of the band-gap energy Eg/t for an energy cutoff of the attractive interactions ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band cou- plings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconducting gaps are reported for different values of the total density a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' and a completely filled valence band, corresponding to a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00, a quantum phase transition (QPT) to the normal state takes place at a specific quantum critical point (QCP), that occurs when Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the car- rier concentration in the conduction band is non-zero, the phase transition becomes a crossover and superconduc- tivity extends for all values of the band gap Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, the system presents different behaviors if the value of the band gap is smaller or larger of E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For finite doping, the valence band contributes very weakly to the super- conducting state of the system for Eg > E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this regime the bands are almost decoupled and the super- conducting gaps does not depend on Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, in the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2(c) since the pair-exchange couplings are weak the conduction band cannot sustain the super- conductivity in the valence band and ∆1 is suppressed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thus, continuously tuning Eg to higher values will result in ∆1 << ∆2 so that there is only one significant super- 5 conducting gap and one significant condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the other case instead (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 2(d)), the pair-exchange cou- plings are stronger and ∆1 is not much suppressed with respect to its initial value, since in these cases the su- perconductivity in the valence band is sustained by the condensate of the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Another interesting feature of this system is that ∆1 is enhanced for lower values of the total density as long as Eg < E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When Eg > E∗ g instead, the opposite situ- ation occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The value of E∗ g at which this behavior takes place depends on the level of filling of the conduc- tion band, shifting to the left when higher total densities are considered, and on the pair-exchange couplings that shifts E∗ g to the right when larger interactions strength are considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The reason behind the behavior of the 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 (a) a2 n2 e (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) a2 n1 h Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Electron density a2ne 2 (a)-(b) in the conduction band and hole density a2nh 1 (c)-(d) in the valence band as functions of the band-gap Eg/t for different values of the total density a2ntot, normalized to the area of the unit cell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' superconducting gaps can be found by looking at the den- sities of particles forming Cooper pairs, which are elec- trons in the conduction band and holes in the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' While the total density is fixed, the density in each band can vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this way, the density of particles in the conduction band n2 is no longer controlled only by doping as for a single band system, there are instead ad- ditional particles excited from the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Never- theless, for larger values of Eg the gain in the interaction energy due to superconductivity is much smaller than the kinetic energy cost for transferring electrons from the valence band to the conduction band, so that very few electrons (compared to the total density of electrons in the valence band) are excited into the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This behavior is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' As one can see for a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 the hole density in the valence band and the electron density in the conduction band coincide and are monotonically decreasing, both of them vanishing at the QCP Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This is a sign that superconductivity is due to holes in the valence band and to electrons in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the other cases the hole density in the valence band is almost zero for Eg > E∗ g, while the electron density in the conduction band is approaching the asymptotic value given by the total density minus the density of the filled valence band a2n2 = a2ntot − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='7 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (a) µ / t Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (b) Eg / t a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Chemical potential µ/t as a function of the band- gap Eg/t for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001),(b) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The chemical potential µ is reported for different total densities a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The black and the magenta dashed lines correspond to the bottom of the conduction band and the top of the valence band, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 4 the chemical potential is reported as a function of Eg, for different total densities a2ntot and for different pair-exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For higher values of the total density and of the pair-exchange couplings the chemical potential shift toward higher energies, due to the larger number of electrons in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In particu- lar, when Eg is increased, in the low density regime the chemical potential starts deep inside the valence band and then enters the gap between the two bands, mean- ing that the condensate in the valence band spans a wide region of the BCS-BEC crossover, while the conduction band is always located in the BEC side of the crossover regime or in the BEC regime, depending on whether the chemical potential lies inside the conduction band or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When Eg > E∗ g the chemical potential acquires a flat de- pendence and is not modified by Eg, in a similar way to what happens to the superconducting gaps and the den- sities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5 the condensate fraction is shown as a func- tion of Eg, for different a2ntot and for different pair- exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The usual choice of the boundaries between the different pairing regimes has been adopted: for α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 the superconducting state is in the weak- coupling BCS regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 < α < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 the system is in the crossover regime;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' for α > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 the system is in the strong-coupling BEC regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Consistently with the information obtained from the chemical potential, in the low density regime the condensate in the valence band ex- plores the entire BCS-BEC crossover by varying Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For the considered pair-exchange interactions in (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5(c)) 6 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (a) α2 e (b) a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) α1 h Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t Crossover BEC BCS BEC Crossover BCS BEC Crossover BCS BCS Crossover BEC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Condensate fractions in the conduction band αe 2 (a)- (b) and in the valence band αh 1 (c)-(d) as functions of the band-gap Eg/t for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The condensate fractions are reported for different total densities a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thin grey dashed lines correspond to α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 from bottom to top.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' the valence band condensate is in the BCS regime for small Eg, while for larger pair-exchange interactions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 5(d)) is in the crossover regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the energy gap or the total density increases, the valence band condensate enters the BEC regime, with the hole condensate fraction αh 1 approaching unity, indicating that the remaining few holes are all in the condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The situation in the con- duction band is different, since due to the strong intra- band coupling the condensate is always located in the BEC side of the crossover regime or in the BEC regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 both the condensate fractions suddenly drop to zero when Eg = E∗ g due to the quantum phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 6 the intra-pair coherence length is reported as a function of Eg, for different a2ntot and for different pair- exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Since for low densities and small pair-exchange couplings the valence band condensate is in the BCS regime (6(a)) when Eg is small, ξpair1 assumes initially larger values with respect to the average inter- particle distance l1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For larger Eg the system enters the BEC regime and ξpair1 becomes much smaller than the average inter-particle distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The valence band con- densate goes from the crossover to the BEC regime in a small range of band gap values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This behavior is observed also for larger values of the total density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The conduction band instead, due to the strong intra-band coupling re- tains a small value of the intra-pair coherence length with respect to the the average inter-particle distance l2 for all the considered values of the system density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this way we found Cooper pairs of different size coexisting in the system for low density and low pair-exchange couplings 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='3 (a) ξpair2 / l2 (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (c) ξpair1 / l1 Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 2 (d) Eg / t a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Intra-pair coherence length ξpair2/l2 for the Cooper pairs of the conduction band (a)-(b) and intra-pair coherence length ξpair1/l1 for the Cooper pairs of the valence band (c)- (d) as functions of the band-gap Eg/t for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair- exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-pair coherence lengths ξpairi/li are reported for different a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' values, in the regime of small Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For the zero doping case the intra-pair coherence length is defined only for Eg < E∗ g, since in this regime the system is not super- conducting and a intra-pair coherence length cannot be defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The fact that the intra-pair coherence length is approaching zero at the QCP in the BEC regime is dif- ferent from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' [34], where giant Cooper pairs are found in the vicinity of the QCP in the BCS side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this case instead, what we have found is equivalent to the finite- density to zero-density QCP of tightly bound molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Namely, near the present QCP in the BEC side the pair size is so small that pairs behave as point-like bosons and the system can be described by its bosonic counterpart [40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7 the order parameter coherence coherence length is reported as a function of Eg, for different a2ntot and for different pair-exchange couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 the soft or critical coherence length ξs diverges when the band gap reaches the critical value Eg = E∗ g, since the system undergoes a quantum phase transition to the insulating state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the other cases a2ntot ̸= 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00, the soft coherence length ξs is not diverging, since no quantum phase transition occurs in the system for any Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In par- ticular, in the cases of a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 and a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 the soft coherence length ξs shows a maximum in correspon- dence of the respective Eg = E∗ g, showing its memory about the quantum phase transition of the valence band condensate, which takes place when the pair-exchange interactions are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The increase of λ12 = λ21 sup- presses the maximum, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7(a) and (b), since the band-condensates become more coupled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the 7 0 2 4 6 (a) ξs / a (b) a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='07 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='26 a2 ntot=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 (c) ξr / a Eg / t 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 (d) Eg / t HC QPT QPT HC HC HC HC HC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Soft ξs (a)-(b) and rigid ξr (c)-(d) order parame- ter coherence length, normalized to the lattice constant a, as functions of the band-gap Eg/t between the two bands at tem- perature T/t = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00065 and for ω0/t = 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a),(c) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='001), (b),(d) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The coherence lengths ξs,r are reported for different values of the total density a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the case a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00 (orange dashed line) ξr has been rescaled by a factor of 7 (c) and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 (d) to make the plot more visible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' case of a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='35 instead, since the valence band is never superconducting for any Eg when the band- condensates are decoupled, there is no quantum phase transition and no peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The rigid coherence length ξr in- stead remains finite for all Eg and for all a2ntot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Anyway, we find the memory of the quantum phase transition that takes place when the conduction band is empty and the valence band is filled (anntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this case in fact, also the conduction band returns to the normal state at Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Indeed, for zero pair-exchange couplings, the rigid coherence length ξr reduces to the coherence length of the conduction band ξ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Even though for finite pair- exchange coupling the coherence length is non-diverging, it encodes the memory of the quantum phase transition of the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Also the maximum value of the rigid coherence length ξr is suppressed by the increase of λ12 = λ21 in this case, as shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 7(c) and (d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' We consider now finite temperature effects on the critical energy band gap for the case of no doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The super- conducting gaps as functions of temperature for different band gaps are reported in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The superconducting gaps present a non-monotonic behavior, that is very dif- ferent from the temperature dependence of the gaps in conventional superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The strong enhancement of ∆2 at finite temperature is due to the thermal excita- tion of the electrons from the valence band to the con- duction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This behavior becomes more pronounced for larger Eg, especially in the case of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 8(c) in which the system is initially in the normal state for tempera- tures close to zero, and then becomes superconducting for 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 2 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='5 (a) (b) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (c) Δ / t (d) 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (e) T / Tc 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='8 1 (f) T / Tc Eg/t = 0 Eg/t = 2 Eg/t = 3 NS SC Δ2 Δ2 Δ2 Δ1 Δ1 Δ1 Δ1 Δ1 Δ1 Δ2 Δ2 Δ2 NS NS NS SC FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Superconducting gaps ∆2/t opening in the conduc- tion band and in the valence band ∆1/t as functions of tem- perature T, normalized with respect to the critical tempera- ture Tc, for a2ntot = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='00.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are (λ12 = λ21): (a), (c), (e) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='03), (b), (d), (f) (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' larger temperatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This superconducting-normal state reentrant transition that we have found in our two-band system is based on a different mechanism with respect to the reentrant transitions observed in superconductors containing magnetic elements [41] or in granular super- conducting systems [42–45]: in the former it is attributed to the competition of magnetic ordering and supercon- ductivity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' while in the latter is attributed to tunneling barriers effect,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' while in our valence-conduction bands sys- tem the thermal excitation of electrons from the valence into the conduction band play a crucial role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9 we report the phase diagram T vs Eg for our system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9 the branch of the phase transition from the su- perconducting to the normal state corresponding to the reentrant behavior results from the second solution at lower temperatures of the linearized self-consistent equa- tions for the superconducting gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' From the left panel of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9 it is clear how the reentrant transition is more pro- nounced when the intra-band couplings are unbalanced (λ22 ≃ 3λ11 in the figure), while the reentrance is reduced when the intra-band couplings have similar values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This effect occurs in a less evident manner also when the pair- exchange couplings are increased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Therefore, the most relevant parameter to control the reentrance phenomenon is the intra-band coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' CONCLUSIONS We have studied the superconducting properties of a two-band system of electrons, interacting through a sep- 8 λ11 → λ22 λ22 → λ11 λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75 λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1 1 0 1 2 3 4 T / t Eg / t 0 1 2 3 4 Eg / t SC NS SC NS λ12 ↑→ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Phase diagrams in the temperature versus energy band gap plane, for the zero doping case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the left panel the red dashed line is for λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23, λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='4, the green dashed line is for λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23, λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75 and the blue dashed line is for λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='65, λ22 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The pair-exchange couplings are the same for all curves, λ12 = λ21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In the right panel the pair-exchange couplings from left to right are: λ12 = λ21 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='03, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='1, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='2, while the intra-band couplings are λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='23 and λ11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' arable attractive potential with a large energy cutoff and multiple pairing channels, at a mean-field level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The su- perconducting state properties are studied by varying the energy gap between the bands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' We have considered dif- ferent levels of filling for the conduction band, while the valence band is always completely filled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the band- gap is modified, the density of electrons in the two bands changes, allowing for the occurrence of a density-induced BCS-BEC crossover.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When the pair-exchange couplings are small, the condensate in the valence band remains su- perconducting but with a strongly suppressed supercon- ducting gap ∆1 for Eg > E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Therefore, in the regime of small pair-exchange coupling, after E∗ g, there is only one significant superconducting gap and one significant condensate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Interestingly, in this case the soft coherence length present a peak as a memory of the quantum phase transition that the valence band condensate undergoes in absence of pair exchanges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This peak is more pronounced if the pair-exchange couplings are sufficiently weak and disappears for higher values of the pair-exchange cou- plings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For higher values of λij, superconductivity in the valence band is sustained by the condensate in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Furthermore, in this regime we have found that superconductivity is enhanced in the valence band for increasing doping as long as Eg < E∗ g, while for Eg > E∗ g superconductivity is enhanced for lower doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' We have also found that superconductivity may occur even when no free carriers exist in the conduction band in the normal state at T = 0, as soon as the gain in super- conducting energy exceeds the cost in producing carriers across the band gap Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' If the binding energy is larger than the energy band-gap, the system becomes unstable under the formation of Cooper pairs and superconduc- tivity emerges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' However, there exists a critical value of the energy band gap E∗ g in correspondence of which the process of creating Cooper pairs is not energetically fa- vorable anymore, at this point a quantum phase transi- tion occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This quantum phase transition is confirmed by the soft coherence length, which is diverging in corre- spondence of the critical band gap Eg = E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Thus, the ground state is superconducting if Eg < E∗ g, insulating if Eg > E∗ g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' At finite temperature, the value of E∗ g is larger than its zero temperature value, because the elec- trons are thermally excited from the valence band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This situation is responsible for the non-monotonic behavior of the superconducting gap opening in the conduction band, which is enhanced at low temperatures because of the electrons that jump from the valence band into the conduction band due to thermal excitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' When there is a finite doping in the system, the sharp phase transi- tion becomes a smooth crossover and superconductivity extends for all Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' In this case, for Eg > E∗ g the va- lence band contributes very weakly to the superconduct- ing state, since the hole density becomes almost zero in this regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' To conclude, we have found that the system explores dif- ferent regimes of the BCS-BEC crossover by tuning the energy band-gap and the total density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The valence-band condensate spans the entire BCS-BEC crossover for low enough density by varying the band-gap Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' For larger values of the total density, the condensate of the valence band is very dilute and results in the BEC regime for any Eg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' The condensate of the conduction band instead re- sides in the BEC side of the crossover or completely inside the BEC regime, due to the strength of the intra-band coupling of electrons in the conduction band.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This pic- ture of the BCS-BEC crossover for the system has been found by analyzing the consistent behavior of the chemi- cal potential, the condensate fractions and the coherence lengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Finally, in the case of zero doping and at finite temperature, an interesting new type of reentrant super- conducting to normal state transition has been numer- ically discovered for unbalanced intra-band couplings, showing that in this configuration superconductivity is assisted instead of being suppressed by increasing tem- perature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This happens because the electrons in the va- lence band are able to jump into the conduction band even for larger values of the zero temperature critical band gap, due to thermal excitation, making the super- conducting state available for a wider range of Eg when the temperature is higher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' ACKNOWLEDGMENTS We are grateful to Tiago Saraiva (HSE-Moscow) and Hiroyuki Tajima (University of Tokyo) for interesting dis- cussions and a critical reading of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' acknowledges INFN for financial support of his Ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' grant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' This work has been partially supported by PNRR MUR project PE0000023-NQSTI.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='.]in, and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Chu, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' B 29, 1493 (1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' [44] U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='Welp, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Kwok, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Crabtree, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Claus, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Vandervoort, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Dabrowski, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Mitchell, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Richards, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Mark, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content='G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Hinks, Physica C 156, 27 (1988).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' [45] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Chudinov, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Mancini, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Minestrini, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Natali, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Stizza, and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Bozhko, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Condens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'} +page_content=' Matter 14, 193 (2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/8tFST4oBgHgl3EQfaTh3/content/2301.13795v1.pdf'}