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+filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf,len=493
+page_content='Many-body resonances in the avalanche instability of many-body localization  Hyunsoo Ha,1 Alan Morningstar,1, 2 and David A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Huse1  1Department of Physics, Princeton University, Princeton, New Jersey 08544, USA  2Department of Physics, Stanford University, Stanford, California 94305, USA  (Dated: January 13, 2023)  Many-body localized (MBL) systems fail to reach thermal equilibrium under their own dynamics,  even though they are interacting, nonintegrable, and in an extensively excited state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' One instability  towards thermalization of MBL systems is the so-called “avalanche”, where a locally thermalizing  rare region is able to spread thermalization through the full system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The spreading of the avalanche  may be modeled and numerically studied in finite one-dimensional MBL systems by weakly coupling  an infinite-temperature bath to one end of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We find that the avalanche spreads primarily  via strong many-body resonances between rare near-resonant eigenstates of the closed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Thus  we find and explore a detailed connection between many-body resonances and avalanches in MBL  systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Introduction— Many-body localized (MBL) systems  are a class of isolated many-body quantum systems that  fail to thermalize due to their own unitary dynamics,  even though they are interacting, nonintegrable and ex-  tensively excited [1–7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This happens for one-dimensional  systems with short-range interactions in the presence  of strong enough quenched randomness, which yields a  thermal-to-MBL phase transition of the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In  the MBL phase, there are an extensive number of emer-  gent localized conserved operators [8–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  One instability of the MBL phase which is believed to  play a central role in the asymptotic, long-time, infinite-  system MBL phase transition is the avalanche [12–14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Rare locally thermalizing regions necessarily exist, how-  ever sparse they may be, due to the randomness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Starting  from such a local thermalizing region, this “thermal bub-  ble” spreads through the adjacent typical MBL regions  until the relaxation rate of the adjacent spins becomes  smaller than the many-body level spacing of the ther-  mal bubble, in which case the spreading of this avalanche  halts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' If the strength of the randomness is insufficient,  the relaxation rate remains larger than the level spacing  and the avalanche does not stop: the full system then  slowly thermalizes and is no longer in the MBL phase (al-  though it is likely in a prethermal MBL regime [15, 16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The avalanche has been numerically simulated in  small-sized  systems  [15,  17–21]  and  experimentally  probed [22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Recent work shows that the instability of  MBL to avalanches occurs at much stronger randomness  than had been previously thought [15, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This leaves a  large intermediate prethermal-MBL regime in the phase  diagram between the onset of MBL-like behavior in small  samples (or correspondingly short times) and the asymp-  totic MBL phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Clear numerical evidence  has been obtained for many-body resonances being an  important part of the physics in the near-thermal part  of this regime [15, 16, 23–29], while no such evidence for  the expected thermalizing rare regions has been found  yet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In the part of this intermediate prethermal MBL  regime that is farther from the thermal regime, it re-  Bath (T=  )  0  After Long time  (a)  (c)  (b)  n  n  n  Slowest Mode (  )  near-resonance  dominant decay  B  A  B  A  1  2  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  L  Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Schematic illustrations showing (a) the avalanche  model, (b) the long time dynamics governed by the slowest  mode, and (c) the dominant decay processes involving four  eigenstates with a near-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' (a) We connect the bath  in the weak-coupling limit with the one-dimensional MBL sys-  tem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Specifically, we analyze the decay of the slowest mode  (ˆτS), which is localized near the end of the system farthest  from the bath;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' ˆτS is a “localized integral of motion” in the  MBL phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' (b) Thermalization at the latest times is gov-  erned by ˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' (c) Schematic decay of ˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' A large fraction of  the probability current in the decay of ˆτS passes through four  eigenstates associated with a rare near-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  mains unclear what is the primary mechanism that leads  to thermalization for samples larger than those that can  be diagonalized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In this work, we explore how an avalanche spreads  through typical MBL regions for systems that are near  the avalanche instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We do not simulate the rare  region that initiates the avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Instead, we assume  a large avalanche is spreading and we model that as an  infinite-temperature bath (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 1) weakly coupled to  one end of our MBL spin chain [15, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We find that  particular many-body near-resonances of the closed sys-  tem play a key role in facilitating the spreading of the  avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  These many-body near-resonances are the  arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='04658v1  [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='stat-mech]  11 Jan 2023  2  dominant process by which the bath at one end of the  chain thermalizes the spins at the other end of the chain  and thus propagates the avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Model— Our model consists of a chain of L spin-  1/2 degrees of freedom (or qubits).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The dynamics of  the closed system is given by the random-circuit Floquet  MBL model studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' [15], which has unitary Flo-  quet operator ˆUF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The disorder strength in this model  is given by the parameter α, with the MBL regime be-  ing at large α, while the thermal regime is at small α  (see the Appendix for the full description of this model).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  To investigate avalanche spreading, we weakly connect  an infinite-temperature Markovian bath to spin L at the  right end of the system [15, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The quantum state of  this open system is the density matrix ˆρ(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In our open-system Floquet model, the bath is repre-  sented by the super-operator Sbath that acts once each  time period:  Sbath[ˆρ] =  ˆρ  1 + 3γ +  γ  1 + 3γ  3  �  j=1  ˆEj ˆρ ˆE†  j ,  (1)  where ( ˆE1, ˆE2, ˆE3) = ( ˆXL, ˆYL, ˆZL) are the jump opera-  tors acting on the last spin at site L (connected to the  bath).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We will take the weak coupling limit γ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The  open-system Floquet super-operator Speriod that takes  our system through one full time-period is  Speriod[ˆρ(t)] = Sbath[ ˆUF ˆρ(t) ˆU †  F ] = ˆρ(t + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (2)  The time evolution of the system’s state is given by  ˆρ(t) = ˆI/2L + p1e−r1tˆτ1 +  �  k≥2  pke−rktˆτk,  (3)  where e−rk is the kth largest eigenvalue of Speriod with  eigenoperator ˆτk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Note that the largest (0th) eigenvalue  is 1 and is nondegenerate when γ > 0, with eigenoper-  ator proportional to the identity, which is the long-time  steady state of this system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The mode with the slowest  relaxation for γ > 0 is ˆτS := ˆτ1, which relaxes with rate  rS := Re(r1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The relaxation rate rS is proportional to  γ, and we work to first order in γ [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Relaxation of the slowest mode— In the weak-  coupling limit, one can obtain ˆτS as a superposition of  the diagonal terms |n⟩ ⟨n|, where |n⟩ are the eigenstates  of the closed system such that ˆUF |n⟩ = eiθn |n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' When  γ = 0, then |m⟩ ⟨n| are eigenoperators of Speriod with  eigenvalues ei(θm−θn), so all diagonal terms |n⟩ ⟨n| are  degenerate, with rk = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Therefore, in the γ ≪ 1 limit,  one can obtain ˆτS in degenerate perturbation theory by  diagonalizing the (super)operator S[ˆρ] := 1  3  �3  j=1 ˆEj ˆρ ˆE†  j  in this degenerate subspace [20], where the matrix ele-  ments are  Smn = ⟨m| S[|n⟩ ⟨n|] |m⟩ = 1  3  3  �  j=1  | ⟨m| ˆEj |n⟩ |2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (4)  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='5  Probability Density (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=')  4  L=  MBL  12  Thermal  Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Probability distribution over samples of the ratio  R = Dµν/(2LΓ) (see text).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In the thermal regime (dark red to  yellow), the ratio exponentially decays with increasing L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In  comparison, R barely drifts with L for the MBL case (blue).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We observe that R never exceeds the value 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='5 (gray dashed  line), which is explained with the minimal model in the main  text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In this figure, we used α = 30 and α = 1 for MBL and  thermal regimes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Note that in the degenerate subspace, S is a symmet-  ric stochastic matrix with real eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In partic-  ular, ˆτS = �  n cn |n⟩ ⟨n| where ⃗c is the eigenvector of  S with the smallest spectral gap Γ from the steady  state (�  m Snmcm = (1 − Γ)cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The relaxation rate is  rS = 3γΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We normalize ˆτS so Tr{ˆτ 2  S} = �  n |cn|2 = 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Naively, the slowest mode is the local integral of motion  (LIOM) that is farthest from the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' More precisely,  as we show in the Appendix, the slowest mode, ˆτS, is  a traceless superposition of projectors on to the eigen-  states of the closed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Among such operators it is  the one with the smallest weight of Pauli strings with  non-identity at site L (connected to the bath).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' It is a  LIOM that is indeed localized far from the bath, but it  is different in detail from the ℓ-bits and LIOMs discussed  in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' [8–10, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 1(b), the latest time dynamics  are determined by ˆτS, with ˆρ(t) ≃ ˆI/2L + pSe−rStˆτS for  any initial conditions that contain ˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This assumes a  nonzero gap between (r1/γ) and (r2/γ), which is indeed  the case for all samples examined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We can view the slow  relaxation of ˆτS in terms of probability currents that flow  between the eigenstates of the isolated system, leading to  the final ˆρ = ˆI/2L equilibrium where all eigenstates have  equal weight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Specifically, we may quantify the contribu-  tion Dmn of the pair of eigenstates m, n to the relaxation  of ˆτS as  Dmn := Smn(cm − cn)2 ≥ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (5)  One can show (see Appendix) that the relaxation rate of  ˆτS is given by the sum of the contributions from all pairs  3  of eigenstates of the closed system:  �  m<n  Dmn = Γ Tr{ˆτ 2  S} = 2LΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (6)  We find the pair of eigenstates |µ⟩, |ν⟩ of the closed sys-  tem that gives the strongest contribution to the relax-  ation of ˆτS by finding the pair with the largest Dmn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We  quantify the fraction of the full relaxation that is due  to this pair by the ratio R ≡ Dµν/(2LΓ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The distribu-  tion of R over disorder realizations and for different sys-  tem sizes is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 2, both for the thermal regime  and for the MBL regime near the avalanche instability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In the thermal regime, R decreases exponentially with  increasing L, indicating that many pairs of eigenstates  are contributing similar amounts to the relaxation of the  slowest mode, which is as should be expected for this  thermal regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In the MBL regime, on the other hand, we find that  this pair of eigenstates contributes an order-one fraction  of the relaxation of ˆτS, and this fraction does not decrease  substantially with increasing system length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This is  consistent with previous works [15, 16, 31, 32] that found  extremely broad distributions for the matrix elements of  local operators among eigenstates of the closed system  within the MBL regime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' On looking at the relaxation  more thoroughly, we find that in this MBL regime the  strongest contribution to the relaxation actually involves  a set of 4 eigenstates of the closed system, at least two  of which are involved in a many-body near-resonance, as  we will now describe in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Near-resonant eigenstate set— Deep in the MBL  regime, at inverse interaction α near the estimated  avalanche instability, we typically find that the strongest  contributions to the relaxation of the slow mode ˆτS come  from a set of 4 eigenstates of the closed system, which  include a near-resonant pair of eigenstates |A⟩, |B⟩ (we  explain what “near-resonant” means below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The other  two eigenstates involved are obtained by “flipping” the  ℓ-bit ˆτL adjacent to the bath:  |A′⟩ = ˆτ x  L |A⟩ ,  |B′⟩ = ˆτ x  L |B⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (7)  In some cases, |A′⟩ and |B′⟩ are also near-resonant, as we  discuss in the Appendix with details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The polarization  cA = ⟨A|ˆτS |A⟩ of the slow mode differs substantially  between the “A” eigenstates (|A⟩ and |A′⟩) and the “B”  eigenstates, while it is essentially unchanged by flipping  ˆτL, so cA ∼= cA′ and cB ∼= cB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Thus it is the transitions  between {A, A′} eigenstates and {B, B′} eigenstates that  relax ˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' If we “de-mix” [15] the near-resonance to make  a more localized pair of orthonormal states |a⟩, |b⟩, we  obtain:  |A⟩ ∼= |a⟩ − ϵ |b⟩ ,  |B⟩ ∼= |b⟩ + ϵ |a⟩ ,  (8)  where |a⟩ and |b⟩ differ by spin flips at a nonzero fraction  of all sites, including the sites that are most polarized in  (a)  (b)  Z  A  B  B  A  XY  A  B  B  A  XY case  Z case  (2/3)  XY  (2/3)  (4/3)  Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Two distinct cases for the set of 4 important  eigenstates of ˆUF that are most important for the avalanche  to propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The near-resonant pair {A, B} are coupled with  ˆ  XL and ˆYL jump operators to B′ and A′, respectively, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=',  they have anomalously large matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In what we call  the “Z case” (b), the near-resonant pair involves a spin flip  next to the bath (site L), and the bath can additionally couple  the resonant pair with the ˆZL jump operator with matrix  element SAB twice as large as SA′B and SAB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The two states  with the largest Dµν are colored red.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In the XY case (a), this  pair is not the near-resonant pair, and {µ, ν} = {A, B′};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' while  they are the same in the Z case, so {µ, ν} = {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In some  samples that are of the XY case, eigenstates |A′⟩ and |B′⟩  are also near-resonant (not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  ˆτS, so this is a long-range many-body resonance, in that  sense, and it includes a “flip” of ˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The spin that is  flipped between |a⟩ and |b⟩ that is closest to the bath is  at site x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' As we show in the Appendix, for the largest  L that we can access, the most probable location of x is  near x/L = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In the regime near the estimated avalanche instability,  the “mixing” ϵ in the near-resonance is typically exponen-  tially small in L, although it is exponentially larger than  the mixing between other typical pairs of eigenstates that  differ over a similar distance range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This is why we call  this a “near-resonance”: the mixing is relatively strong,  partly due to the two eigenstates being near degeneracy  in the spectrum of UF , but it is not fully resonant, since  ϵ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In many samples, |A′⟩ and |B′⟩ are not near-resonant  and are well-approximated by |A′⟩ ∼= ˆXL |a⟩ and |B′⟩ ∼=  ˆXL |b⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' As a consequence of the near-resonance between  |A⟩ and |B⟩, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=', the relatively large ϵ compared to other  pairs of states, there are anomalously large matrix ele-  ments | ⟨A| ˆXL |B′⟩ |2 ∼= | ⟨B| ˆYL |A′⟩ |2 ∼= ϵ2 of the two  jump operators ˆE1 = ˆXL and ˆE2 = ˆYL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' These drive two  particularly large contributions, DAB′ and DBA′ (and  their transposes), to the total relaxation rate of ˆτS be-  cause SAB′ ∼= SBA′ ∼= 2ϵ2/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In addition, when the near-resonance involves flipping  the polarization of site L next to the bath (so x = L),  there is another anomalously large matrix element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This  time it is the matrix element | ⟨A| ˆZL |B⟩ |2 ∼= 4ϵ2 of the  jump operator ˆE3 = ˆZL between the two near-resonant  4  eigenstates themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This results in SAB ∼= 4ϵ2/3 and  another large contribution DAB ∼= 2DAB′ ∼= 2DBA′ (and  its transpose) to the total relaxation rate of ˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Thus there are two cases for Dµν, the largest contribu-  tion to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 6, corresponding to whether or not the reso-  nance involves flipping the spin nearest to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The  bath can drive relaxation of the slowest mode through  a resonance indirectly (with X and Y jump operators),  i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=', the pairs of eigenstates involved are not the near-  resonant pair, and sometimes directly (with Z jump op-  erator).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We depict the two cases in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 3(a,b) and call  them the “XY ” and “Z” cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Our claim is that the  relaxation of the slowest mode, and thus the spread of  the avalanche, proceeds via these dominant processes in-  volving 4 eigenstates that include a particularly strong  near-resonance, as explained in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This struc-  ture implies that the largest contribution Dµν comes from  either the Z jump operator or the X and Y jump op-  erators of the bath, depending on which of the above  cases is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In the Z case, which is x = L, the  largest contribution (A-B pair) is accompanied by two  XY contributions of half the magnitude (A-B′ and B-A′  pairs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In the XY case, which is x < L, there are two  roughly equal largest contributions (A-B′ and B-A′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In  both cases, the largest contribution Dµν doesn’t exceed  half of the total relaxation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The phenomenology of the  near-resonant eigenstate set explained in this section is  corroborated by numerical observations in the next sec-  tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' An extension of this simple description is discussed  in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Numerical Observations— We numerically deter-  mine the eigenstates of ˆUF , the slowest mode of S, and  the contributions Dmn to its relaxation rate associated  with probability currents between eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We exam-  ine the pairs of eigenstates that contribute the most (|µ⟩  and |ν⟩), identify the 4 important eigenstates discussed  earlier, and quantify how they are related to each other  to verify that the near-resonant set of eigenstates does  indeed have that structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Recall that in our model, the polarization has a pre-  ferred spin direction: Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Deep in the MBL regime, the  local ℓ-bit operators typically are very close to the lo-  cal single-spin Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Therefore, we identify  |γ′⟩ ≡ τ x  L |γ⟩ for γ ∈ {µ, ν} simply by finding the eigen-  state with largest |⟨γ| ˆXL |n⟩| among all eigenstates |n⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  To determine which jump operator from the bath dom-  inantly couples |µ⟩ and |ν⟩ we define a tool  Zjump(m, n) :=  | ⟨m| ˆZL |n⟩ |2  �3  j=1 | ⟨m| ˆEj |n⟩ |2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (9)  Note that ˆE3 = ˆZL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We find that Zjump(µ, ν) is ex-  tremely close to 0 or 1 in any one sample, as shown in  the inset of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 4(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This separation is captured by  the minimal model since Zjump(µ, ν) ∼= 0 (or 1) is asso-  ciated to the XY (or Z) case where the pair {µ, ν} isn’t  (a)  (b)  (c)  (d)  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='5  1  0  0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='5  1  0  1  0  1  XY and Z  AB’ and A’B  AB  only Z  XY case  system size (L)  system size (L)  Z case  jump  jump  Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Consistency with the near-resonant eigenstate  set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' All data here are for α = 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' (a) The near-resonance  between {A, B} causes the AB′ and A′B matrix elements to  be roughly equal, as described in the minimal model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The  distribution of the ratio of SAB′ and SA′B is indeed peaked  near 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The inset shows Zjump of AB′ and A′B are peaked at  zero, implying that these pairs are coupled with the ˆ  XL and  ˆYL jump operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' (b) When the near-resonant pair involves  the spin flip of the last site (Z case), the bath can directly  couple |A⟩ and |B⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The distribution of the shown ratio of  matrix elements demonstrates that SAB ≃ 2SAB′ ≃ 2SA′B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The inset shows Zjump of AB is peaked at 1, implying A and  B are coupled with ˆZL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The bottom two panels show the  demixing angles for (c) the XY case and (d) the Z case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The  points correspond to AB (red dot), A′B (gray ×), AB′ (gray  dot), and A′B′ (blue dot).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The near-resonant pair identified  as in the main text (AB) has the largest demixing angles for  all cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The calculations in (a) and (b) are done with L = 11  and 104 disorder realizations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (or is) the near-resonant pair {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We therefore iden-  tify the near-resonant pair for each sample based on the  minimal model as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In the case that |µ⟩ and |ν⟩—  the pair with the largest Dmn—are “connected” with Z  (Zjump ∼= 1), these states are identified as |A⟩ and |B⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Otherwise, if they are connected with X and Y instead  (Zjump ∼= 0), we associate {|A⟩ , |B⟩)} to {|µ′⟩ , |ν⟩} or  {|µ⟩ , |ν′⟩}, whichever pair has the smaller quasienergy  splitting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We first checked that cγ ∼= cγ′ indeed holds for γ ∈  {A, B}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This relation is satisfied because Sγ′γ is of or-  der one, so the slow mode ˆτS contains negligible pop-  ulation difference between γ and γ′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Also, we checked  that |cµ − cν| is of order one, which should be true as  5  we picked the pair with the largest Dµν = Sµν(cµ − cν)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Therefore, the matrix elements Smn (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 4) are a good  proxy for Dmn (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 5) within this set of important eigen-  states, so we compare the matrix elements going forward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We compare these matrix elements to confirm the re-  sults are consistent with the minimal model described  in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 3(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' As we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 4(a), SAB′ ∼= SBA′  holds (and their transposes), and XY jump operators  couple them (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Furthermore, in the case that spin  L is flipped between |A⟩ and |B⟩, we additionally observe  SAB ∼= 2SAB′ ∼= 2SBA′ as presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In this  case |A⟩ and |B⟩ are coupled with ˆZL (inset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We note that the near-resonant eigenstate set of the  previous section explains why R < 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' When the near-  resonance doesn’t involve a spin flip next to the bath,  SAB′ = SA′B, so DAB′ = DA′B and no single pair con-  tributes more than 1/2 of the total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  If the resonance  flips the last spin, DAB = 2DAB′ = 2DA′B, so again the  maximum contribution can’t be greater than half of the  total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We finally tested our picture using the “demixing” pro-  cedure introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In that procedure, one  calculates the most localized basis of a subspace spanned  by two eigenstates, and thus also the corresponding ba-  sis rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The basis rotation corresponds to a location  on a Bloch sphere;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' the polar angle, called the “demixing  angle”, is a measure of the resonance strength between  the two eigenstates (ϵ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 4(c,d),  the demixing angle between the eigenstates we identified  as |A⟩ and |B⟩ is by far the largest among other pairs in  our eigenstate set, consistent with the idea that that pair  is indeed a near-resonance that enables the bath to relax  the slowest mode, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=', the avalanche to propagate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Conclusion— Assuming the avalanche has proceeded  for a sufficiently large distance, we model the putative  thermal bubble as an infinite Markovian bath with infi-  nite temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In this model, we discovered the exis-  tence of dominant processes in the avalanche, involving  only a few pairs of eigenstates of the closed system, in-  cluding a strong near-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The avalanche proceeds  through these rare eigenstate pairs, leveraging many-  body near-resonances to relax the spins some distance  away along the chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The inner structure of the domi-  nant set of eigenstates is dictated by whether or not the  associated resonance involves flipping a spin at the site  next to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This sets what jump operators effec-  tively use the resonance present in the closed system to  spread the avalanche.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We presented a minimal model in-  volving two near-resonant eigenstates and two additional  auxiliary states to explain how this works in detail, and  verified our picture with numerical observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Our  work advances the understanding of the avalanche in-  stability of many-body localization and provides a de-  tailed connection to rare many-body resonances present  in MBL systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Acknowledgements— We thank Sarang Gopalakr-  ishnan, Vedika Khemani and Jacob Lin for discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The work at Princeton was supported in part by NSF  QLCI grant OMA-2120757.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' was supported in part  by the Stanford Q-FARM Bloch Postdoctoral Fellowship  in Quantum Science and Engineering and the Gordon  and Betty Moore Foundation’s EPiQS Initiative through  Grant GBMF8686.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Simulations presented in this work  were performed on computational resources managed and  supported by Princeton Research Computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Appendix  Random Floquet Model  1  2  L  L-1  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Random Floquet Model introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Details of the model are described in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We use the Floquet circuit model of Morningstar, et  al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' [15] and depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The Floquet unitary  operator UF is obtained by applying a layer of one-site  gates Ud followed by L−1 two-site gates—one per bond—  denoted by Uu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The onsite gates are first sampled from  the 2 × 2 circular unitary ensemble (CUE) and then di-  agonalized, so the localizing disorder of the model is in  the Z direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The two-site gates ui act on sites i and  i + 1 and can be written as  ui = exp  � i  αMi  �  (10)  where Mi are sampled from the 4 × 4 Guassian unitary  ensemble (GUE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 1/α controls the interaction strength so  that the effect of the disorder gets stronger as α increases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The order of applying the ui’s is determined randomly for  each sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Slowest Mode  As explained in the main text, one can obtain the slow-  est mode ˆτS in the weak-coupling limit by diagonalizing  6  (a)  (b)  (c)  Disorder  Strength  site ( )  site ( )  site ( )  Pauli string weight (  )  Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Pauli String weight wn for (a) τ1, (b) ¯τ1, and (c) the slowest mode τS, averaged (after taking the logarithm) for 103  samples with system size L = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  the superoperator S[ˆρ] :=  1  3  �  µ ˆEµˆρ ˆE†  µ, in the degen-  erate subspace of Floquet eigenstates |n⟩ ⟨n|, where the  matrix elements are  Snm = ⟨m| S[|n⟩ ⟨n|] |m⟩ = 1  3  �  µ  | ⟨m| ˆEµ |n⟩ |2  (11)  and ˆEµ = ( ˆXL, ˆYL, ˆZL) are the jump operators at the  spin next to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In particular, ˆτS = �  n cn |n⟩ ⟨n|  where ⃗cn is the eigenvector of Snm with the smallest spec-  tral gap Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' As the slowest mode is constructed from the  degenerate subspace of |n⟩ ⟨n|, it is a conserved quantity  (a LIOM) of the purely unitary dynamics given by UF .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We derive useful features of the slowest mode ˆτS below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' For any Pauli-string operator P ′ acting  on the (L − 1) spins away from the bath, and ˆEµ =  ( ˆXL, ˆYL, ˆZL) acting on the last spin next to the bath, the  following holds:  S[P ′ ⊗ IL] = P ′ ⊗ IL,  S[P ′ ⊗ ˆEµ] = −1  3P ′ ⊗ ˆEµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (12)  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The super-operator S only acts on site L, such  that for an arbitrary matrix M ∈ M(2,2), S[P ′ ⊗ M] =  P ′ ⊗  �  1  3  �  µ ˆEµM ˆE†  µ  �  .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 12 follows by substituting IL  and ˆEµ in M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The slowest mode is the conserved quantity  of the closed system (which is not an identity) with the  smallest weight from the Pauli strings with non-identity  at site L (connected to the bath).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Consider a conserved quantity written as ˆτ =  �  n cn |n⟩ ⟨n| where cn ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The normalization condi-  tion is �  n c2  n = 2L as we defined in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We  can write the slowest mode in the Pauli-string basis like  ˆτS = �  P ¯cP P with P being the Pauli-strings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The coef-  ficients are ¯cP = Tr(ˆτSP)/2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In the Pauli-string basis,  the normalization condition is rewritten as �  P ¯c2  P = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The total weight of the Pauli-strings with non-identity at  site L is  wL =  �  P ′  ¯c2  P ′⊗ ˆ  XL +  �  P ′  ¯c2  P ′⊗ ˆYL +  �  P ′  ¯c2  P ′⊗ ˆ  ZL  =  �  µ  �  P ′  ¯c2  P ′⊗ ˆ  Eµ  (13)  where P ′ is a Pauli-string defined on L-1 sites and ˆEµ =  ( ˆXL, ˆYL, ˆZL) are the Pauli operators from site L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Next, we define a functional R(⃗c) := �  n,m cnSnmcm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Below, we show how the functional R is related to wL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Tr (ˆτS[ˆτ])) = Tr  ��  l  cl |l⟩ ⟨l|  �  n  cnS[|n⟩ ⟨n|]  �  = Tr  ��  l  cl |l⟩ ⟨l|  �  n,m  cnSnm |m⟩ ⟨m|  �  =  �  n,m  cnSnmcm = R  (14)  We can rewrite the same equation under the Pauli-string  basis, ˆτ = �  P ¯cP P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Here, we use Lemma 1 introduced  earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Tr (ˆτS[ˆτ])) =  �  P ′  ¯c2  P ′⊗I − 1  3  �  µ  �  P ′  ¯c2  P ′⊗ ˆ  Eµ  = 1 − 4  3  �  µ  �  P ′  ¯c2  P ′⊗ ˆ  Eµ  = 1 − 4  3wL  (15)  From the above two equations, we derive the relation  R = 1 − (4/3)wL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  7  In the final step, let’s define f := �  n c2  n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Using the  Lagrange multiplier method, functional R is local ex-  tremum under the constraint f = 2L when ∇R−λ∇f = 0  holds, which is equivalent to the eigenvalue problem:  ∀n,  �  m Snmcm = λcn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In this condition, we obtain  R = λ, and the weight of the Pauli-strings with non-  identity at site L is wL = (3/4)(1−λ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Therefore, finding  ˆτ which minimizes wL is equivalent to finding the eigen-  vector of the super-operator S with the second largest  eigenvalue (largest eigenvalue is one which corresponds  simply to an identity matrix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Hence, the conserved  quantity with the smallest wL except the identity is the  slowest mode ˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The slowest mode is traceless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' From the definition of the slowest mode, it is  an eigenmatrix of superoperator S with an eigenvalue  1 − Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Therefore, Tr(ˆτS)  =  Tr(S[ˆτS])/(1 − Γ)  =  Tr( 1  3  �  µ ˆEµˆτS ˆE†  µ)/(1 − Γ) = Tr(ˆτS)/(1 − Γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  As 0 <  Γ < 1, it should be traceless Tr(ˆτS) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We numerically compared the slowest mode ˆτS with  the LIOM (τ1) and ℓ-bit (¯τ1) introduced in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The explicit forms of τ1 and ¯τ1 are  τ1 ≡  �  n  ⟨n| ˆZ1 |n⟩ |n⟩ ⟨n|  (16)  ¯τ1 ≡  �  n  sgn(⟨n| ˆZ1 |n⟩) |n⟩ ⟨n| ,  (17)  where |n⟩s are the eigenfunctions of the closed system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Note that τ1 and ¯τ1 are the conserved quantities that  are localized near site 1, farthest from the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In par-  ticular, we calculate the Pauli string weight wi(ˆτ) for  ˆτ ∈ {τ1, ¯τ1, ˆτS}, which is the total weight of the Pauli  string bases with the last non-identity matrix placed at  i-th site.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' These Pauli bases are products of identity ma-  trices for sites farther than i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We compare the Pauli string weight in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' For suf-  ficiently large disorder strength α, all three operators are  localized far from the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' It is clear that the slowest  mode is different from τ1 and ¯τ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Compared to the other  two quantities, the slowest mode has the least weight at  the last site, as we proved in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The differ-  ence manifests when we compare the three operators for  a single sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' While τ1 and ¯τ1 always have maximum  weight on the first site, the slowest mode often has max-  imum weight other than the first site but is still far away  from the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' What constrains the slowest mode is not  the first site to have the maximum weight but to have  the least weight on the last site next to the bath, which  was indeed true for every sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Derivation of Equation 6  As explained in the main text, the slowest mode is  ˆτS = �  n cn |n⟩ ⟨n| where ⃗cn is the eigenvector of a sym-  metric Markovian matrix Snm with the smallest spec-  tral gap Γ (�  m Snmcm = (1 − Γ)cn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The operator  ˆτS is normalized such that ∥ˆτS∥2  1 = �  n c2  n = 2L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  In  our model, the relaxation rate of the slowest mode is  rS = 3γΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We assume a gap between the smallest and  second-smallest spectral gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  With this assumption,  the density matrix at the latest time is asymptotically  ˆρ(t) ≃ I/2L + ϵe−rStˆτS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Therefore, ˆρ(t) thermalizes and  converges to ˆρ(∞) = I/2L asymptotically at the latest  time with the rate:  d  dt (∥ˆρ(t) − ˆρ(∞)∥1)2  ≃ d  dt  �  ∥ϵe−rStˆτS∥1  �2  = d  dt  �  |ϵ|2 e−2rSt ∥ˆτS∥2  1  �  = d  dt  �  |ϵ|2 e−2rSt 2L�  = −2rS |ϵ|2 e−2rSt 2L  = −3γ |ϵ|2 e−2rSt 2L+1 Γ  (18)  Also, one can view ˆτS as specifying a structure of  probability currents that flow between eigenstates of the  isolated system during the late-time approach to equi-  librium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Specifically, if we define a current jnm :=  Snm(cn − cm), then  d  dt ˆρmm(t) = 3ϵγe−rSt �  n jnm holds  at late times, which implies that jnm obtained from ˆτS  determines the structure of the late time thermalization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  One can estimate how ˆρ(t) converges to ˆρ(∞) with the  inner structure of ˆτS:  d  dt (∥ˆρ(t) − ˆρ(∞)∥1)2  ≃ d  dt  ��  m  |ˆρmm(t) − 1/2L|2  �  = 6γ|ϵ|2 �  m  e−rStcn  �  e−rSt �  n  jnm  �  = 3γ|ϵ|2e−2rSt �  n,m  (cn − cm)jmn  = −3γ|ϵ|2e−2rSt �  n,m  (cn − cm)2Snm  = −3γ|ϵ|2e−2rSt �  n,m  Dnm  (19)  Here, we used the fact that at the latest time, only the  diagonal terms in ˆρ(t) remain (under the weak coupling  limit γ ≪ 1, the slowest mode consists of only the diago-  nal terms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Dnm quantifies the contribution of a pair |n⟩  and |m⟩ to the relaxation of the slowest mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  8  Comparing the two equations above, one can derive  the following:  �  n,m  Dnm =  �  n,m  (cn − cm)2Snm = 2L+1Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (20)  This can also be directly derived by:  �  n,m  Dnm =  �  n,m  (cn − cm)2Snm  =  �  n,m  (c2  n + c2  m)Snm − 2  �  n  cn  ��  m  Snmcm  �  = 2L+1 − 2L+1(1 − Γ)  = 2L+1Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (21)  From the above derivations, the largest contribution  Dµν is related with Γ by  Dµν = R  �  n<m  Dnm = 2LRΓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (22)  Therefore, one can naively derive the relation between  the thermalization rate and the matrix element Sµν be-  tween the most important pair of states with an effective  O(1) value of R as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  rS = 3γΓ = 3γ(cµ − cν)2  R  2−LSµν  (23)  As the observed ratio R and cµ −cν are both O(1) values  for MBL regimes, we estimate a relation rS ∼ 2−LSµν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Detuning of near-resonance by flipping ˆτL  In this section, we examine to what extent there is  also a near-resonance between eigenstates |A′⟩ and |B′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  These two eigenstates differ from the near-resonant states  |A⟩ and |B⟩ by the flipping of ℓ-bit ˆτL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  This change  may strongly detune the near-resonance, which is what  happens for samples where the near-resonance involves  flipping spins that are close to or include spin L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' But  in other samples, the near-resonance only involves spin  flips that are rather far from spin L, so flipping ˆτL has a  rather small effect on the near-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We denote the  location of the spin-flip that is closest to L as x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The minimal wavefunction model introduced in the  main text (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 8) neglected the possibility of a near-  resonance between (A′, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Revising our minimal model  by including and “de-mixing” the possible near-resonance  between (A′, B′), one can write:  |A⟩ = |a⟩ − ϵ |b⟩  |B⟩ = |b⟩ + ϵ |a⟩  |A′⟩ = |a′⟩ − ϵ′ |b′⟩  |B′⟩ = |b′⟩ + ϵ′ |a′⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  (24)  The quasi-energy splittings of these two near-resonances  are ∆ = |θA − θB| and ∆′ = |θA′ − θB′|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' we have labelled  them so that ∆ < ∆′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' We can partially quantify how  much flipping ˆτL alters the near-resonance by comparing  ϵ′ and ∆′ to ϵ and ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The distributions of ϵ′/ϵ and ∆′/∆ for different system  sizes are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 7(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Both distributions show  long tails to |ϵ′| ≪ |ϵ| and ∆′ ≫ ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This shows that in a  substantial fraction of the samples the near-resonance is  strongly detuned by flipping ˆτL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 7(d,e) shows that  this mostly happens due to the near-resonance extending  to near site L, so x/L is close to or equal to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  There are also many samples that have |ϵ|, |ϵ′|, and  |ϵ − ϵ′| all of the same order, as indicated by the peaks  near ϵ′/ϵ ∼= ∆′/∆ ∼= 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 7(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' These are the cases  where flipping ˆτL does alter the near-resonance, but not  by a particularly large or small amount, which mostly  happens for smaller x/L [see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 7(d,e)].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The cases where flipping ˆτL has very little effect on  the near resonance should have ϵ′/ϵ and ∆′/∆ very close  to one, so would make a very sharp peak in those distri-  butions, which is not seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 7(a,b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This suggests  that only a small fraction of samples are in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  For such samples SAB′ is small due to a near cancella-  tion between the contribution from |a⟩ coupling to |a′⟩  and that from |b⟩ coupling to |b′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' This near-cancellation  will not happen if we replace |A⟩ with |a⟩ and look in-  stead at SaB′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Thus in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 7(c) we show the distribu-  tion of SAB′/SaB′ (∼ |(ϵ − ϵ′)/ϵ′|2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The much weaker  tail in this distribution to very small SAB′/SaB′ are the  samples where the near-resonance is only very weakly af-  fected by flipping ˆτL, while the much stronger tail to very  large SAB′/SaB′ are those more common samples where  this flip strongly detunes the near-resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 7(f)  shows the median SAB′/SaB′, with the expected trend of  stronger detuning (larger SAB′/SaB′) as x/L increases  and the near-resonance thus extends closer to the flipped  ˆτL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Range of Near-Resonance  The previous sections described how the bath uses  near-resonances to flip the spins at the other end of an  MBL chain and thermalize the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Therefore, it is  perhaps natural to expect the near-resonance involved to  be an end-to-end resonance such that the last spin-flip  between |A⟩ and |B⟩ should not be far away from the  bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Strikingly, however, the distance from the bath to  the last spin-flip appears to increase in proportion to the  system length L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In particular, we compare the spin po-  larization on every site for the near-resonant eigenstate  pair (A, B) and find the location x ≤ L where the last  spin flip occurs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' These two eigenstates are directly cou-  pled with the ˆZx operator (| ⟨A| ˆZx |B⟩ |2 ∼= 4ϵ2), and in  the special case when x = L, the jump operator from the  bath can directly couple the near-resonant pair (which  we defined as the Z case).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  9  Distribution (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=')  Distribution (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=')  Distribution (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=')  (a)  (b)  (d)  (e)  (c)  (f)  4  L=  13  median (  )  median (  )  median (  )  Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Comparing (A, B) and (A′, B′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' All data here are for α = 30 and L ∈ [4, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' Distributions of (a) ϵ′/ϵ, (b) ∆′/∆,  and (c) SAB′/SaB′ show there are many samples satisfying |ϵ′| ≪ |ϵ| or |ϵ| ∼ |ϵ′| ∼ |ϵ − ϵ′|, but cases with |ϵ − ϵ′| ≪ |ϵ| are  less common (see text of the Appendix).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The median of (d) ϵ′/ϵ, (e) ∆′/∆, and (f) SAB′/SaB′ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' the location of the last  spin-flip (x/L) of the near-resonance are illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The detuning due to flipping ˆτL gets more significant as the last spin-flip  gets closer to the bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  L= 4  Normalized Last Spin-Flip Position (x/L)  13  Distribution (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=')  Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Distribution of the last spin flip location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  We  normalized the location with system size (x/L).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  The normalized position of the last spin flip (x/L) is  illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' 8 with different colors for different sys-  tem sizes (L ∈ [4, 13]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' It is clear that as the system size  gets larger, the curve peaks when it is smaller than 1 (the  peak is near x/L ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='8 < 1), indicating that an extensive  number of sites are in between the bath and the last spin  flip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  As we explained in the previous section, the min-  imal model including the possible near-resonance be-  tween (A′, B′) predicts the matrix element proportional  to |ϵ−ϵ′|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' The value of x that yields the largest matrix el-  ement is determined by two competing factors: First, as x  becomes smaller, the resonance’s range becomes shorter  and fewer degrees of freedom are involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  Therefore  the resonance becomes stronger, with a typically larger  ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' However, decreasing x by too much also causes the  resonance to decouple from the bath: flipping the ℓ-bit  beside the bath may then produce a near copy of the res-  onance and result in small |ϵ − ϵ′| even though |ϵ| and  |ϵ′| are large, which will result in that resonance not be-  ing useful to the bath in relaxing the slowest mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content=' In  the other direction, as x becomes larger the resonance  becomes longer-ranged and thus weaker (|ϵ| and |ϵ|′ be-  come smaller), but it also couples more strongly to the  bath so that |ϵ − ϵ′| ∼= |ϵ| ≫ |ϵ′|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
+page_content='  10  Therefore, in most samples for large L the bath uses  near-resonances with x/L < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/99E3T4oBgHgl3EQfrQpl/content/2301.04658v1.pdf'}
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