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| \begin{document} |
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|
| \title{Free energy of the gas of spin 1/2 fermions beyond the second order and the Stoner phase transition} |
| \author{\em Oskar Grocholski$~\!^1$ and Piotr H. |
| Chankowski$~\!^2$\footnote{Emails: |
| oskar.grocholski@cea.fr chank@fuw.edu.pl}\\ |
| $^1$IRFU, CEA, Universit\'e Paris-Saclay, F-91191 Gif-sur-Yvette, France\\ |
| $^2$Faculty of Physics, University of Warsaw,\\ |
| Pasteura 5, 02-093 Warsaw, Poland |
| } |
| \maketitle |
| \abstract{In the previous work we have developed a systematic thermal |
| (imaginary time) perturbative expansion and applying it to the relevant |
| effective field theory computed, up to the second order in the interaction, |
| the free energy $F$ of the diluted gas of (nonrelativistic) spin $1/2$ |
| fermions interacting through a spin-independent repulsive two-body potential. |
| Here we extend this computations to higher orders: assuming that the |
| only relevant parameter specifying the interaction potential is the $s$-wave |
| scattering length $a_0$, we compute the complete order $(k_{\rm F}a_0)^3$ |
| ($k_{\rm F}$ is the Fermi wave vector) contribution to the system's free |
| energy as a function of the numbers $N_+$ and $N_-$ of spin up and spin |
| down fermions (i.e. as a function of its polarization) and |
| the temperature $T$. We also extend the computation beyond a fixed order |
| by resumming the contributions to $F$ of two infinite sets of Feynman |
| diagrams: the so called particle-particle rings and the particle-hole rings. |
| We find that including the second one of these two contributions |
| has a dramatic consequence for the transition of the system from the |
| paramagnetic to the ferromagnetic phase (the so called Stoner phase |
| transition): in this approximation the phase transition simply disappears.} |
| \vskip0.1cm |
|
|
| \noindent{\em Keywords}: Diluted gas of interacting fermions, effective field |
| theory, itinerant ferromagnetism, phase transitions. |
|
|
| \newpage |
|
|
| \noindent{\large\bf 1. Introduction} |
| \vskip0.3cm |
|
|
| \noindent In the recent years some progress has been achieved in the computation |
| of equilibrium zero temperature properties of the gas of $N$ (nonrelativistic) |
| fermions interacting with each other through a spin independent repulsive |
| potential $V_{\rm pot}(|{\mathbf x}_i-{\mathbf x}_j|)$. It was mainly related to |
| the application to this classic \cite{Lenz,Stoner,HuangYang57,Kesio,Pathria} |
| many-body quantum mechanics/statistical physics problem of general methods (see |
| e.g. \cite{KolczastyiSka}) of the effective field theory. In this approach, |
| initiated in the seminal paper \cite{HamFur00} (see also \cite{HamFur02}), the |
| original spatially nonlocal |
| potential $V_{\rm pot}$ assumed to be characterized by a length scale $R$ is |
| replaced by the a priori infinite series of local interactions (written here |
| using the standard second quantization formalism - see e.g. \cite{FetWal}) |
| \begin{eqnarray} |
| \hat V_{\rm int}=C_0\!\int\!d^3{\mathbf x}~\! |
| (\hat\psi_+^\dagger\hat\psi_+)(\hat\psi_-^\dagger\hat\psi_-) |
| +\hat V_{\rm int}^{(C_2)}+\hat V_{\rm int}^{(C_2^\prime)}+\dots, |
| \label{eqn:Vint} |
| \end{eqnarray} |
| of decreasing length dimension. The coefficients (couplings) $C_0$, $C_2$, |
| $C_2^\prime,\dots$ of the interaction (\ref{eqn:Vint}) can be then directly |
| determined in terms |
| of the quantities - the scattering lengths $a_0, a_1,\dots$ and the effective |
| radii $r_0,\dots$ - parametrizing the general expansion (in powers of the |
| relative momentum) of the amplitude of the elastic scattering of two particles. |
| Trading the (bare) couplings of (\ref{eqn:Vint}) for these measurable |
| quantities which characterize the underlying potential $V_{\rm pot}$ of the |
| binary interactions has also the effect of removing ultraviolet (UV) |
| infinities engendered by the locality of (\ref{eqn:Vint}). The simplifications |
| brought in by this approach allowed first to easily reproduce \cite{HamFur00} |
| those terms of the perturbative expansion of the ground state energy $E$ |
| of the system of spin $s$ fermions with equal densities of different spin |
| projections which in the past were obtained by more traditional (and requiring |
| considerably more work) methods of many-body quantum mechanics |
| and to extend \cite{WeDrSch} this computation |
| to the fourth order in the systematic, organized by the power counting rules |
| \cite{HamFur00}, |
| expansion in powers (in higher orders modified also by logarithms) of the |
| dimensionless product $k_{\rm F}R$ of the system's overall Fermi wave vector |
| \begin{eqnarray} |
| k_{\rm F}=(6\pi^2n/g_s)^{1/3}~\!,\label{eqn:kFDef} |
| \end{eqnarray} |
| (here $n=N/V$ is the overall density of the gas of spin $s$ fermions and |
| $g_s=2s+1$) and the characteristic length $R$. The same approach allowed also |
| to compute the ground state energy of the system of spin $1/2$ fermions for |
| different densities of fermions with the |
| up and down spin projections \cite{CHWO1} recovering the old result of Kanno |
| \cite{KANNO} (obtained analytically for the hard core interaction potential) |
| and to easily extend it \cite{PECABO1} to fermions having spin $s$ greater |
| than 1/2 (and, therefore, more possible spin projections). Finally in the works |
| \cite{CHWO3,CHWO4} and \cite{PECABO2} this computation has been extended up to |
| the third order in the systematic expansion in powers of $k_{\rm F}R$. |
|
|
| These results allowed to investigate more quantitatively the phase transition, |
| called also the Stoner transition, to the ordered state in which the densities |
| $N_+/V$ and $N_-/V$ of fermions of opposite spin projections in the case of |
| $s=1/2$ are not equal and the system exhibits a nonzero polarization |
| $P\equiv(N_+-N_-)/N$ which, according to the standard qualitative argument |
| based on the positivity of the repulsive interaction energy and the Pauli |
| exclusion principle, for sufficiently strong repulsion and/or sufficiently |
| high overall density, should be energetically (at zero temperature) favored |
| over the state with equal densities. In this regime the system should, |
| therefore, exhibit the property called itinerant ferromagnetism. It has been |
| found that the clear first order character of this transition |
| predicted by the second order of the perturbative expansion (in |
| agreement with the general quantitative arguments given long time ago in |
| \cite{BeKiVoj}) - at sufficiently large values of the ``gas parameter'' |
| $k_{\rm F}a_0$ two symmetric minima of $E(P)$ are formed away from $P=0$ |
| separated from the one at $P=0$ by a finite barrier and at |
| $(k_{\rm F}a_0)_{\rm cr}=1.054$ they become the global minima |
| ($|P_{\rm cr}|=0.58$) - gets appreciably weakened ($|P_{\rm cr}|$ shifts |
| significantly towards $P=0$ and the barrier becomes much lower) if the complete |
| order $(k_{\rm F}a_0)^3$ contribution is taken into account \cite{CHWO3,CHWO4}. |
| This seemed to support the results of the work \cite{He1} in which a certain |
| class of contributions to $E$ (arising from the so-called particle-particle |
| ring diagrams) has been resummed to all orders in |
| $k_{\rm F}a_0$ finding that in this approximation the |
| transition is continuous - the new minima start to continuously move |
| away from $P=0$ as the gas parameter crosses some critical value |
| $(k_{\rm F}a_0)_{\rm cr}\approx0.8$ ($0.858$ if only 1 hole-hole $N-1$ |
| particle-particle parts of the $N$-th order particle-particle ring |
| diagrams and $0.79$ if the complete $N$-th order particle-particle ring |
| diagrams are resummed; see also \cite{He2} for a refinement of this approach). |
| It should be added that the predictions of \cite{He1}, which qualitatively are |
| supported also by the results of a different approach \cite{HEIS}, |
| seem to agree quite well with the results |
| obtained with the help of the Monte Carlo simulations \cite{QMC10}. |
|
|
| However if all scattering lengths $a_\ell$ and effective radii $r_\ell$ are of |
| the same order of magnitude, $\sim\!R$, the complete (according to the power |
| counting rules \cite{HamFur00} which apply, strictly speaking, only to this |
| case) order $(k_{\rm F}R)^3$ contribution to the ground state energy depends |
| also on $a_1$ and $r_0$ and it has been shown in \cite{PECABO2} that in this |
| case the character of the phase transition (at zero temperature) predicted |
| by this approximation depends on the relative magnitudes and signs of the |
| parameters $a_1$, $r_1$ and $a_0>0$ (that is, on the more detailed |
| characteristics of the underlying potential $V_{\rm pot}$). The computation in |
| which to all orders |
| resummed are only (some) contributions depending on the powers of $k_{\rm F}a_0$ |
| corresponds rather to the situation encountered in physics of dilute atomic |
| gases in which the $s$-wave scattering length $a_0$ (it can be of either sign) |
| is made positive and very large compared to the remaining parameters |
| ($|a_0|\gg R\sim |a_1|,|r_1|,\dots$) by exploiting properties of the Feshbach |
| resonances (see e.g. \cite{ChiGriJuTie}). In this case however the underlying |
| interaction of fermions (atoms) is attractive and the scattering length $a_0$ |
| is positive in the regime in which bound states composed of two fermions |
| of opposite spins can form. The true ground state of the system is then |
| very different than the one of nointeracting atoms (they may not be |
| adiabatically connected to one another in the thermodynamic limit implicit in |
| the field theory approach) which |
| is used in perturbative computations performed within the effective field |
| theory approach. Although the formulae for the system's energy density $E/V$ |
| obtained perturbatively (or with the help of a |
| resummation) by expanding around the ground state of noninteracting fermions |
| can seem to imply the transition to the |
| ordered state, they apply at best to a metastable (from the thermodynamic |
| point of view) state of the system and in real experiments the |
| transition (which in principle could occur in a |
| metastable state \cite{Pippard}) |
| to the ferromagnetic state has in fact not been observed |
| \cite{ItFMObs,ItFMNotObsT,ItFMNotObsE} due to the too rapid formation |
| (at very low temperatures at which these experiments were carried out) |
| of atomic dimers (bosons). |
| It can be also remarked that in the situation in which the underlying |
| interaction is attractive (despite giving rise to a |
| positive $s$-wave scattering length) |
| the mentioned qualitative argument for the occurrence of the transition |
| no longer applies and the expectation that the transition should occur |
| is mainly based on the textbook mean field correction to the energy |
| density \cite{Kesio,Pathria} |
| (equivalent to the first order correction of the perturbative expansion) |
| which depends only on the $s$-wave scattering length $a_0$. |
|
|
| Computation of the ground state energy density $E/V$ as a function of |
| the densities of fermions with different spin projections allows only to |
| investigate the equilibrium properties of the system of interacting |
| fermions at zero (or very low) temperatures. It is however of interest to |
| determine its behavior also at nonzero temperatures. (In the context of |
| the physics of atomic gases it is physically clear that formation of |
| atomic dimers, which at very low temperatures makes observation of the |
| phase transition to the ordered phase impossible, at higher temperatures |
| should be less important.) This requires computing |
| one of the thermodynamic potentials of the system of interacting fermions. |
| So far such a computation of the free energy $F$ has been |
| done \cite{DUMacDO} only up to the second order, i.e. up to terms of order |
| $(k_{\rm F}a_0)^2$, using the old-fashioned thermal perturbation theory |
| (see e.g. \cite{LL}, par. 32) based on the ordinary second order |
| Rayleigh - Schr\"odinger perturbative expression for energy levels |
| entering the statistical sum. |
| In \cite{CHGR} we have recovered this second order expression for $F$ using |
| the systematic thermal perturbative expansion (exploiting the imaginary time |
| formalism \cite{FetWal}) and reproduced the thermal characteristics |
| of the Stoner phase transition it predicts (pointing out however |
| problems - not discussed in \cite{DUMacDO} - with accurate numerical |
| determination of the critical values |
| of the polarization), but have encountered |
| a technical problem which prevented us to immediately extend the computation |
| to higher orders. Here we show how this problem can be resolved and |
| applying the developed systematic thermal expansion to the first |
| term of the effective field theory interaction (\ref{eqn:Vint}) we |
| derive the formulae allowing to compute numerically |
| the complete order $(k_{\rm F}a_0)^3$ corrections to |
| the free energy $F(T,V,N_+,N_-)$. With additional work, including the |
| contributions of the next two terms of (\ref{eqn:Vint}) it would, of course, |
| be possible to compute the complete order $(k_{\rm F}R)^3$ correction |
| to the free energy. |
|
|
| Instead of completing the order $(k_{\rm F}R)^3$ corrections to the free energy |
| which would allow to investigate in more details the thermal profile of the |
| Stoner phase transition to the ordered state induced by truly repulsive spin |
| independent two-body potentials $V_{\rm pot}$ which necessarily give rise to |
| the parameters $a_1$ and $r_0$ of comparable magnitude to that of $a_0$ |
| (all $\sim R)$, we in this paper profit from the possibility provided by the |
| simpler structure of the terms of the expansion generated by the imaginary |
| time formalism and resum to all orders in $k_{\rm F}a_0$ not only the |
| contributions to the temperature-dependent free energy $F$ of the |
| particle-particle ring diagrams (done for zero temperature in the papers |
| \cite{He1,He2,He3}) but also of the particle-hole ring diagrams. |
|
|
| Including in the free energy $F(T,V,N,P)$ only the resummed contribution |
| of the particle-particle ring diagrams we recover for $T=0$ the results of the |
| works \cite{He1,He3} and can show how they are modified at nonzero temperatures |
| not exceeding the Fermi temperature (we expect that the results obtained within |
| the effective field theory should be valid for temperatures in this range). |
| However as we show, inclusion of the contribution of the resummed |
| particle-hole diagrams changes the situation drastically: |
| the phase transition to the ordered state simply disappears (the minimum of |
| $F$ is at $P=0$ for all values of the parameter $k_{\rm F}a_0$ and all |
| temperatures). This is a somewhat surprising result |
| and we comment on its possible meaning in the Conclusions. |
|
|
| \vskip0.5cm |
|
|
| \noindent{\large\bf 2. The formalism} |
| \vskip0.3cm |
|
|
| \noindent The natural statistical formalism in which to compute equilibrium |
| properties of the gas of fermions the interaction of which preserve their spins |
| and therefore the numbers $N_\pm$ of spin up and spin down particles, is the |
| Grand Canonical Ensemble with two independent chemical potentials $\mu_\pm$. |
| The relevant statistical operator is then |
| \begin{eqnarray} |
| \hat\rho={1\over\Xi_{\rm stat}}~\!e^{-\beta\hat K}~\!, |
| \end{eqnarray} |
| where $\beta\equiv1/k_{\rm B}T$, with $T$ the temperature and $k_{\rm B}$ |
| the Boltzmann constant, and |
| \begin{eqnarray} |
| \hat K=\hat H_0-\mu_+\hat N_+-\mu_-\hat N_-+\hat V_{\rm int}\equiv |
| \hat K_0+\hat V_{\rm int}~\!.\label{eqn:KandK0} |
| \end{eqnarray} |
| The associated partition function |
| $\Xi_{\rm stat}(T,V,\mu_+,\mu_-)={\rm Tr}(e^{-\beta\hat K})$ gives the |
| thermodynamical potential $\Omega(T,V,\mu_+,\mu_-)=-Vp(T,\mu_+,\mu_-) |
| =-k_{\rm B}T\ln\Xi_{\rm stat}(T,V,\mu_+,\mu_-)$. In the second quantization |
| formalism \cite{FetWal} the operator $\hat K_0$ of the considered system |
| of fermions has the form\footnote{To simplify the formulae the symbol |
| $\int_{\mathbf p}$ stands for the integral with respect to the measure |
| $d^3{\mathbf p}/(2\pi)^3$.} |
| \begin{eqnarray} |
| \hat K_0=\sum_{\sigma=\pm}\!\int_{\mathbf p}(\varepsilon_{\mathbf p}-\mu_\sigma) |
| a^\dagger_\sigma({\mathbf p})a_\sigma({\mathbf p})~\!, |
| \end{eqnarray} |
| with $\varepsilon_{\mathbf p}=\hbar^2{\mathbf p}^2/2m_f$. Standard systematic |
| thermodynamical perturbative expansion \cite{FetWal,CHGR} gives the potential |
| $\Omega$ in the form of the series in powers of the interaction |
| $\hat V_{\rm int}$ |
| \begin{eqnarray} |
| \Omega=\Omega^{(0)}-{1\over\beta}\sum_{N=1}^\infty{(-1)^N\over N!} |
| \!\int_0^\beta\!d\tau_N\dots\!\int_0^\beta\!d\tau_1~\!{\rm Tr}\!\left( |
| \hat\rho^{(0)}{\rm T}_\tau[\hat V_{\rm int}^I(\tau_N)\dots\hat V_{\rm int}^I(\tau_1)] |
| \right)^{\rm con}.\label{eqn:OmegaPertExpansion} |
| \end{eqnarray} |
| Here $\hat V_{\rm int}^I(\tau)=e^{\tau\hat K_0}\hat V_{\rm int}e^{-\tau\hat K_0}$ is the |
| interaction operator in the (imaginary time) interaction picture, ${\rm T}_\tau$ |
| is the chronological ordering and $\hat\rho^{(0)}$ is the statistical operator |
| of the noninteracting system. The superscript ``con'' means that only connected |
| contributions (Feynman diagrams) should be taken into account. The first term |
| in (\ref{eqn:OmegaPertExpansion}) is the textbook \cite{Kesio,Pathria} grand |
| thermodynamical potential of the nointeracting system |
| \begin{eqnarray} |
| \Omega^{(0)}(T,V,\mu_+,\mu_-)=-{V\over\beta}\sum_{\sigma=\pm}\int_{\mathbf p} |
| \!\ln\!\left(1+e^{-\beta(\varepsilon_{\mathbf p}-\mu_\sigma)}\right). |
| \label{eqn:Omega0Textbook} |
| \end{eqnarray} |
| Owing to the thermal analog of the Wick formula (see e.g. \cite{FetWal}) |
| computation of the |
| successive terms $\Omega^{(N)}$ of the expansion (\ref{eqn:OmegaPertExpansion}) |
| reduces to drawing all possible connected Feynman diagrams with $N$ |
| interaction vertices arising from $\hat V_{\rm int}$ joined by oriented lines |
| and integrating over positions ${\mathbf x}$ and ``times'' $\tau$ labeling |
| these vertices the corresponding products of free the thermal propagators |
| \begin{eqnarray} |
| -{\cal G}^{(0)}_{\sigma_2\sigma_1}(\tau_2-\tau_1,{\mathbf x}_2-{\mathbf x}_1) |
| ={1\over\beta}\sum_{n\in{\mathbb Z}}\int_{\mathbf p}e^{-i\omega_n^{\rm F}(\tau_2-\tau_1)} |
| ~\!e^{i{\mathbf p}\cdot({\mathbf x}_2-{\mathbf x}_1)}\left( |
| -\tilde{\cal G}^{(0)}_{\sigma_2\sigma_1}(\omega_n^{\rm F},{\mathbf p})\right), |
| \end{eqnarray} |
| the Fourier transform $-\tilde{\cal G}^{(0)}_{\sigma_2\sigma_1}$ of which have |
| the form \cite{FetWal} |
| \begin{eqnarray} |
| -\tilde{\cal G}^{(0)}_{\sigma_2\sigma_1}(\omega_n^{\rm F},{\mathbf p}) |
| ={-\delta_{\sigma_2\sigma_1}\over |
| i\omega_n^{\rm F}-(\varepsilon_{\mathbf p}-\mu_\sigma)}~\!, |
| \end{eqnarray} |
| associated with (oriented) lines connecting vertices of the diagram. |
| The resulting ``momentum'' space Feynman rules are almost identical with the |
| ordinary ones except that the integrations over frequencies (energies) are |
| replaced by summations over the (fermionic) Matsubara frequencies |
| $\omega_n^{\rm F}=(\pi/\beta)(2n+1)$, $n\in{\mathbb Z}$. |
|
|
| Applying this formalism with the interaction operator $\hat V_{\rm int}$ given |
| by the first term of (\ref{eqn:Vint}) one finds that the order $C_0$ term |
| of the expansion (\ref{eqn:OmegaPertExpansion}) is simply given by |
| \begin{eqnarray} |
| \Omega^{(1)}=C_0V{\cal G}_{++}(0,{\mathbf 0})~\!{\cal G}_{--}(0,{\mathbf 0})~\!, |
| \label{eqn:Omega(1)} |
| \end{eqnarray} |
| with |
| \begin{eqnarray} |
| {\cal G}_{\pm\pm}(0,{\mathbf 0})=\int_{\mathbf p} |
| \left[1+e^{\beta(\varepsilon_{\mathbf p}-\mu_\pm)}\right]^{-1}~\!.\label{eqn:G(0,0)} |
| \end{eqnarray} |
| Higher order contributions to the potential $\Omega$ can also be systematically |
| computed. If $\hat V_{\rm int}$ in (\ref{eqn:KandK0}) were the true, spatially |
| nonlocal, two-body interaction (corresponding to a two potential |
| $V_{\rm pot}(|{\mathbf x}_i-{\mathbf x}_j|)$, where ${\mathbf x}_i$ are the |
| positions of fermions), the successive terms of the expansion |
| (\ref{eqn:OmegaPertExpansion}) would be (ultraviolet) finite. If |
| $\hat V_{\rm int}$ is the local interaction (\ref{eqn:Vint}) of the effective |
| theory, the successive terms of the expansion (\ref{eqn:OmegaPertExpansion}) |
| involve ultraviolet divergences and have to be regularized. As in our previous |
| works we will employ for this purpose the cutoff $\Lambda$ on the wave vectors |
| of virtual particles. Finite (in the limit $\Lambda\rightarrow\infty$) |
| contributions to the potential $\Omega$ (a physical quantity) are then |
| obtained by systematically expressing the (bare) couplings of $\hat V_{\rm int}$ |
| in terms of other measurable (physical) quantities. As it is customary, and in |
| line (at least when the gas is very diluted) with the physical intuition that |
| properties of the gas are mainly determined by elastic two-body collisions of |
| its constituents, one expresses the couplings $C_0$, $C_2$, etc. of |
| (\ref{eqn:Vint}) in terms of the measurable quantities related to such a |
| scattering process, namely in terms of the scattering lengths $a_0$, $a_1$, |
| effective ranges $r_0$, etc. \cite{HamFur00}. In this work we will only need |
| to express the coupling $C_0$ in this way; the relevant formula obtained by |
| matching the amplitude of the elastic fermion-fermion scattering computed |
| perturbatively using the first term of the interaction (\ref{eqn:Vint}) |
| onto the general form of the same amplitude parameterized by $a_0$, |
| $a_1,\dots$ and $r_0,\dots,$ reads |
| \cite{CHWO1,CHWO3,WeDrSch} |
| \begin{eqnarray} |
| C_0={4\pi\hbar^2\over m_f}~\!a_0\left(1+{2\over\pi}~\!a_0\Lambda |
| +{4\over\pi^2}~\!a^2_0\Lambda^2\dots\right) |
| \equiv C_0^{\rm R}\left(1+{2\over\pi}~\!a_0\Lambda |
| +{4\over\pi^2}~\!a^2_0\Lambda^2\dots\right).\label{C0intermsofCORen} |
| \end{eqnarray} |
| \vskip0.1cm |
|
|
| From the thermodynamic point of view much more convenient to work with than |
| the potential $\Omega$ is the free energy $F$ which canonically depends on the |
| variables $T$, $V$ and the particle numbers $N_\pm$ which are given by |
| the derivatives |
| \begin{eqnarray} |
| N_\pm=-(\partial\Omega/\partial\mu_\pm)_{T,V}~\!.\label{eqn:NsFromOmega} |
| \end{eqnarray} |
| In principle, in each order of the expansion (\ref{eqn:OmegaPertExpansion}) |
| to construct the free energy one should invert the relations |
| (\ref{eqn:NsFromOmega}) |
| to obtain the chemical potentials as functions of the particle numbers $N_+$ |
| and $N_-$ and insert these in the formula $F=\Omega+\mu_+N_++\mu_-N_-$. Thus |
| the values of the chemical potentials $\mu_+$ and $\mu_-$ change with each |
| successive order of the expansion and the procedure of constructing the free |
| energy looks rather cumbersome. It turns out, however, that in the systematic |
| expansion this procedure simplifies considerably: it amounts in effect to using |
| the chemical potentials $\mu_\pm^{(0)}$ determined by inverting the formula |
| (\ref{eqn:NsFromOmega}) with $\Omega$ replaced by $\Omega^{(0)}$ given by |
| (\ref{eqn:Omega0Textbook}) and omitting |
| in the expansion (\ref{eqn:OmegaPertExpansion}) those diagrams which give |
| vanishing contribution in computing the corrections $\Delta E$ to the ground |
| state energy $E=E^{(0)}+\Delta E$ of the system of interacting particles using |
| the ordinary Dyson expansion of the formula |
| \cite{HamFur00,WeDrSch,CHWO1,CHWO3,CHWO4} ($T$ stands here for time, not |
| for the temperature) |
| \begin{eqnarray} |
| \Delta E=\lim_{T\rightarrow\infty}{i\hbar\over T}~\! |
| \langle0|{\rm T}_t\exp\!\left({1\over i\hbar}\!\int_{-T/2}^{T/2}\!dt~\! |
| V_{\rm int}^I(t)\right)\!|0\rangle~\!,\label{eqn:CorrectionsToE} |
| \end{eqnarray} |
| in which $|0\rangle$ is the ground state of the noninteracting system of |
| $N=N_++N_-$ fermions. In |
| the case of the interaction proportional to $C_0$ this has been explicitly |
| demonstrated in \cite{CHGR} up to the third order of the perturbative expansion. |
| This prescription is obviously consistent with the fact that in the zero |
| temperature limit the corrections to the free energy obtained from the |
| thermodynamic expansion should go over into the corresponding corrections to |
| the ground state energy given by (\ref{eqn:CorrectionsToE}). |
| \vskip0.1cm |
|
|
| The interaction of the system of spin $1/2$ fermions with the external magnetic |
| field ${\cal H}$ represented by the operator (the magnetic moment is here |
| included in ${\cal H}$) |
| \begin{eqnarray} |
| \hat V_{\rm int}^{({\cal H})}=-{\cal H}\int_V\!d^3{\mathbf x}\left( |
| \hat\psi^\dagger_+\hat\psi_+-\hat\psi^\dagger_-\hat\psi_-\right), |
| \end{eqnarray} |
| can be also easily taken into account in this formalism by including it in the |
| free Hamiltonian $\hat H_0$ which amounts to shifting the |
| chemical potentials $\mu_\pm\rightarrow\tilde\mu_\pm\equiv\mu_\pm\pm{\cal H}$ |
| in $\hat K_0$ given by (\ref{eqn:KandK0}). The free energy is then given |
| as the series |
| \begin{eqnarray} |
| F(T,V,{\cal H},N_+,N_-)=F^{(0)}+F^{(1)}+F^{(2)}+\dots, |
| \end{eqnarray} |
| in which |
| \begin{eqnarray} |
| F^{(0)}(T,V,{\cal H},N_+,N_-)=\Omega^{(0)}(T,N,\tilde\mu_+^{(0)},\tilde\mu_-^{(0)}) |
| +(\tilde\mu_+^{(0)}-{\cal H})N_++(\tilde\mu_+^{(0)}+{\cal H})N_-~\!, |
| \label{eqn:F(0)Term} |
| \end{eqnarray} |
| and $F^{(N)}(T,V,{\cal H},N_+,N_-) |
| =\Omega^{(N)}(T,N,\tilde\mu_+^{(0)},\tilde\mu_-^{(0)})$ for $N=1,2,\dots,$ |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| where, as explained above, in computing $\Omega^{(1)}$, $\Omega^{(2)}$ etc. |
| one should take into account only those diagrams of the expansion |
| (\ref{eqn:OmegaPertExpansion}) which give nonzero contributions to $\Delta E$. |
| The (shifted) chemical potentials $\tilde\mu_+^{(0)}$, $\tilde\mu_-^{(0)}$ |
| are given by |
| \begin{eqnarray} |
| \tilde\nu_\pm^{(0)}\equiv\tilde\mu_\pm^{(0)}/k_{\rm B}T= |
| f^{-1}\!\left((1\pm P)\left({\varepsilon_{\rm F}(n)\over |
| k_{\rm B}T}\right)^{3/2}\right),\label{eqn:nu(0)Determination} |
| \end{eqnarray} |
| where $\varepsilon_{\rm F}(n)\equiv\hbar^2k_{\rm F}^2/2m_f$ and |
| $f^{-1}(x)$ is the inverse of the monotonic function (mapping |
| ${\mathbb R}$ onto ${\mathbb R}_+$) defined by the integral |
| \begin{eqnarray} |
| f(\nu)={3\over2}\!\int_0^\infty\!d\xi~\!{\xi^{1/2}\over1+e^{\xi-\nu}}~\!. |
| |
| \end{eqnarray} |
|
|
| If the computation of $F$ is restricted to the order $C^{\rm R}_0$ (i.e. to the |
| order $k_{\rm F}a_0$) correction given, as follows from the formulated |
| prescription and the result (\ref{eqn:Omega(1)}) and (\ref{eqn:G(0,0)}), by |
| \begin{eqnarray} |
| F^{(1)}=VC_0(N_+/V)(N_-/V)~\!,\label{eqn:F(1)} |
| \end{eqnarray} |
| the condition for the minimum of $F$ with respect to $N_+$ and $N_-$ (at fixed |
| $N_++N_-=N$) which determines the system's polarization $P$ takes the form |
| ($t\equiv k_{\rm B}T/\varepsilon_{\rm F}$, $h\equiv{\cal H}/\varepsilon_{\rm F}$) |
| \begin{eqnarray} |
| {8\over3\pi}~\!(k_{\rm F}a_0)~\!P+2h=t |
| \left[f^{-1}\!\left({1+P\over t^{3/2}}\right)- |
| f^{-1}\!\left({1-P\over t^{3/2}}\right)\right].\label{eqn:meanFieldCond} |
| \end{eqnarray} |
| If the asymptotic expansion |
| \begin{eqnarray} |
| f^{-1}(x)=x^{2/3}\left[1-(\pi^2/12)x^{-4/3}-(\pi^4/80)x^{-8/3} |
| -(247\pi^6/25920)x^{-4}+\dots\right], |
| \end{eqnarray} |
| valid for $x\gg1$ (obtained by inverting the Sommerfeld expansion \cite{LL} |
| of the function $f(\nu)$) is used, the formula (\ref{eqn:meanFieldCond}) |
| reproduces the textbook \cite{Kesio,Pathria} low temperature equilibrium |
| condition (equivalent to the condition $\mu_+=\mu_-$) and leads |
| to the well known prediction that the Stoner phase transition to the ordered |
| state is continuous with divergent magnetic susceptibility characterized |
| by the critical exponent $\gamma=1$ and a finite discontinuity of the heat |
| capacity. (In fact, this continuous character of the transition is accidental: |
| in the same approximation the transition is of first order in the system |
| of spin $s>1/2$ fermions and/or if the space dimension is not 3.) If the |
| correction $F^{(2)}$ is included, the transition becomes first order, |
| at least at sufficiently low temperatures \cite{DUMacDO,CHWO1,PECABO1} |
| in agreement with the arguments given in the past in \cite{BeKiVoj}. However |
| the computation of the complete order $(k_{\rm F}a_0)^3$ correction to the |
| ground state energy $E$ performed in \cite{CHWO3,CHWO4,PECABO2} showed that |
| the first order character of the transition at zero temperature, very clear |
| in the second order approximation, is strongly weakened, at least as long |
| as the contributions proportional to $k_{\rm F}r_0$ and $k_{\rm F}a_1$ |
| (which, if the underlying interaction potential $V_{\rm pot}$ is ``natural'', |
| i.e. if all $a_\ell$, $r_\ell$, etc. are of the same order of magnitude, |
| are of the same order, $(k_{\rm F}R)^3$, as the $(k_{\rm F}a_0)^3$ correction) |
| are not taken into account \cite{PECABO2}. Below we extend the |
| existing computations in two ways: we compute the complete (proportional |
| to $(C_0^{\rm R})^3$, i.e. to $(k_{\rm F}a_0)^3$) temperature dependent |
| third order corrections to the free energy $F$ and, moreover, we show how to |
| perform the ressumation of two infinite subsets of temperature dependent |
| corrections to $F$ of which the first one is the finite temperature |
| generalization of the subset of diagrams taken into account in (the last |
| section of) ref. \cite{He1}. |
| \vskip0.5cm |
|
|
| \noindent{\large\bf 3. Order $(k_{\rm F}a_0)^2$ and order $(k_{\rm F}a_0)^3$ |
| particle-particle corrections to $F$} |
| \vskip0.3cm |
|
|
| \noindent We begin by recalling the computation of the |
| order $(C_0^{\rm R})^2$ term $F^{(2)}$ performed in \cite{CHGR}. In agreement |
| with the formulated prescription it is given by |
| the single Feynman diagram shown in Figure \ref{fig:ElementaryLoops}. The |
| corresponding analytical expression can be obtained by convoluting either two |
| $A$-''blocks'' or two $B$-''blocks'' shown in the same Figure: |
| \begin{eqnarray} |
| F^{(2)}=-{1\over2}~\!C_0^2V~\!{1\over\beta}\sum_{l\in\mathbb{Z}}\! |
| \int_{\mathbf q}[A(\omega_l^B,\mathbf{q})]^2 |
| =-{1\over2}~\!C_0^2V~\!{1\over\beta}\sum_{l\in\mathbb{Z}}\! |
| \int_{\mathbf q}[B(\omega_l^B,\mathbf{q})]^2.\label{eqn:F2InTermsOFAblocks} |
| \end{eqnarray} |
| To make the formulae resulting from convoluting $A$-blocks more transparent |
| it will be convenient to introduce the following notation: |
| \begin{eqnarray} |
| &&N_{--}^{\mathbf p}\equiv n_+({\mathbf p})~\!n_-({\mathbf q}-{\mathbf p})~\!, |
| \nonumber\\ |
| &&N_{++}^{\mathbf p}\equiv[1-n_+({\mathbf p})]~\! |
| [1-n_-({\mathbf q}-{\mathbf p})]~\!,\nonumber\\ |
| &&n_\pm({\mathbf p}) |
| =\left[1+\exp\{\beta(\varepsilon_{\mathbf p}-\tilde\mu_\pm^{(0)})\} |
| \right]^{-1},\label{eqn:Defs}\\ |
| &&\{{\mathbf p}\}\equiv n_+({\mathbf p})+n_-({\mathbf q}-{\mathbf p})-1~\!, |
| \nonumber\\ |
| &&[{\mathbf p}]\equiv\varepsilon_{\mathbf p}-\tilde\mu_+^{(0)} |
| +\varepsilon_{{\mathbf q}-{\mathbf p}}-\tilde\mu_-^{(0)}~\!.\nonumber |
| \end{eqnarray} |
| At zero temperature $N_{--}^{\mathbf p}$ and $N_{++}^{\mathbf p}$ reduce |
| respectively to |
| $\theta(p_{{\rm F}+}-|{\mathbf p}|)\theta(p_{{\rm F}-}-|{\mathbf q}-{\mathbf p}|)$ |
| and $\theta(|{\mathbf p}|-p_{{\rm F}+}) |
| \theta(|{\mathbf q}-{\mathbf p}|-p_{{\rm F}-})$, |
| hence the subscripts $--$ and $++$. It is also easy to check that |
| \begin{eqnarray} |
| \{{\mathbf p}\}=N_{--}^{\mathbf p}-N_{++}^{\mathbf p} |
| =N_{--}^{\mathbf p}\left(1-e^{\beta[{\mathbf p}]}\right).\label{eqn:Ids} |
| \end{eqnarray} |
| The form of the distribution functions $n_\pm({\mathbf p})$ plays the |
| role only in the second one of these two identities. |
|
|
| \begin{figure}[] |
| \begin{center} |
| |
| \begin{picture}(370,40)(5,0) |
| \ArrowArc(30,20)(25,70,290) |
| \DashArrowArc(30,20)(25,290,70){2} |
| |
| \DashArrowArc(50,20)(25,110,250){2} |
| \ArrowArc(50,20)(25,250,110) |
| \Vertex(40,-2.5){2} |
| \Vertex(40,42.5){2} |
| |
| \Text(155,30)[]{$A(\omega_{l+1}^B,\mathbf{q})=$} |
| \Vertex(190,30){2} |
| \ArrowArc(210,35)(20,195,345) |
| \DashArrowArcn(210,25)(20,165,15){2} |
| \Vertex(230,30){2} |
| \Text(215,50)[]{$^{q-k,~l-n}$} |
| \Text(215,8)[]{$_{k,~n}$} |
| |
| \Text(295,30)[]{$B(\omega_l^B,\mathbf{q})=$} |
| \Vertex(330,30){2} |
| \ArrowArc(350,35)(20,195,345) |
| \DashArrowArc(350,25)(20,15,165){2} |
| \Vertex(370,30){2} |
| \Text(355,50)[]{$^{k,~n+l}$} |
| \Text(355,8)[]{$_{k+q,~n}$} |
| |
| \end{picture} |
| \end{center} |
| \caption{The order $C_0^2$ diagram contributing to the thermodynamic potential |
| $F$ of the gas of spin $1/2$ fermions and two ``elementary'' one-loop |
| diagrams ($A$- and $B$-''blocks'') |
| out of which the second order and those higher order (in the $C_0$ |
| coupling) contributions which are taken into account in this work |
| are composed. Solid and dashed lines denote propagators of fermions |
| with the spin projections $+$ and $-$, respectively.} |
| \label{fig:ElementaryLoops} |
| |
| \end{figure} |
|
|
| In the introduced notation the $A$-block (obtained in \cite{CHGR}) takes the |
| form |
| \begin{eqnarray} |
| A(\omega_l^B,\mathbf{q})=\int_{\mathbf p}\!{\{{\mathbf p}\}\over |
| i\omega_l^B-[{\mathbf p}]}~\!,\label{eqn:AblockExplicit} |
| \end{eqnarray} |
| After the sum in (\ref{eqn:F2InTermsOFAblocks}) over the bosonic Matsubara |
| frequencies $\omega^B_l=(\pi/\beta)l$ is performed |
| using the standard formulae \cite{FetWal,CHGR} one gets |
| \begin{eqnarray} |
| {F^{(2)}\over V}=-{1\over2}~\!C_0^2\!\int_{\mathbf q}\!\int_{{\mathbf p}_1}\! |
| \int_{{\mathbf p}_2}\!{\{{\mathbf p}_1\}\{{\mathbf p}_2\} |
| \over[{\mathbf p}_1]-[{\mathbf p}_2]} |
| \left({1\over1-e^{\beta[{\mathbf p}_1]}}-{1\over1-e^{\beta[{\mathbf p}_2]}}\right). |
| \label{eqn:F2SymmetricForm} |
| \end{eqnarray} |
| Since the two terms are formally identical (after making in the integrals in |
| one of the terms the interchange ${\mathbf p}_1\leftrightarrow{\mathbf p}_2$), |
| one arrives, using (\ref{eqn:Ids}), at the final form of $F^{(2)}/V$: |
| \begin{eqnarray} |
| {F^{(2)}\over V}=C_0^2\!\int_{\mathbf q}\!\int_{{\mathbf p}_1}\!\int_{{\mathbf p}_2}\! |
| N_{--}^{{\mathbf p}_1}~\!{1\over[{\mathbf p}_1]-[{\mathbf p}_2]} |
| -C_0^2\!\int_{\mathbf q}\!\int_{{\mathbf p}_1}\!\int_{{\mathbf p}_2}\! |
| N_{--}^{{\mathbf p}_1}~\! |
| {\{{\mathbf p}_2\}^{\rm sub}\over[{\mathbf p}_1]-[{\mathbf p}_2]}~\!, |
| \label{eqn:F(2)inTermsOfC0} |
| \end{eqnarray} |
| in which |
| $\{{\mathbf p}\}^{\rm sub}\equiv\{{\mathbf p}\}+1 |
| = n_+({\mathbf p})+n_-({\mathbf q}-{\mathbf p})$ |
| and the integrals should be understood in the Principal Value sense. Notice |
| that the denominators $[{\mathbf p}_1]-[{\mathbf p}_2]$ do not depend on the |
| chemical potentials. This profiting from the symmetry of the two terms of |
| (\ref{eqn:F2SymmetricForm}), seemingly not problematic, has indeed |
| no consequences here but, as will be shown, in higher orders if applied |
| blindly would lead to incorrect results. |
|
|
| The first term in (\ref{eqn:F(2)inTermsOfC0}) is divergent. The change of the |
| variables |
| ${\mathbf q}=2{\mathbf s}$, ${\mathbf p}_1={\mathbf s}-{\mathbf t}_1$, |
| ${\mathbf p}_2={\mathbf s}-{\mathbf t}_2$ (the Jacobian equals 8) makes the |
| innermost integral elementary and allows to write it in the form |
| \begin{eqnarray} |
| {16\pi^2\hbar^4\over m^2_f}~\!a^2_0 |
| \left(1+{4\over\pi}~\!a_0\Lambda+\dots\right)\! |
| \int_{\mathbf s}\!\int_{{\mathbf t}_1}\! |
| 8~\!n_+({\mathbf s}-{\mathbf t}_1)~\!n_-({\mathbf s}+{\mathbf t}_1)~\! |
| {m_f\over2\pi^2\hbar^2} |
| \left(-\Lambda+{{\mathbf t}^2_1\over\Lambda}+\dots\right),\nonumber |
| \end{eqnarray} |
| after using (\ref{C0intermsofCORen}). |
|
|
| Expressing $C_0$ similarly in the second term of the formula |
| (\ref{eqn:F(2)inTermsOfC0}) and in $F^{(1)}$ given by (\ref{eqn:F(1)}), |
| one finds that the divergent terms of order $a_0^2\Lambda$ cancel out and |
| \begin{eqnarray} |
| {F^{(1)}+F^{(2)}\over V}=C_0^{\rm R}\!\int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\! |
| n_+({\mathbf k}_1)~\!n_-({\mathbf k}_2) -(C_0^{\rm R})^2\int_{\mathbf q}\! |
| \int_{{\mathbf p}_1}\!\int_{{\mathbf p}_2}\!N_{--}^{{\mathbf p}_1}~\! |
| {\{{\mathbf p}_2\}^{\rm sub}\over[{\mathbf p}_1]-[{\mathbf p}_2]}\nonumber\\ |
| -{16\hbar^2\over\pi m_f}~\!a_0^3\!\int_{\mathbf s}\!\int_{{\mathbf t}_1} |
| 8~\!n_+({\mathbf s}-{\mathbf t}_1)~\!n_-({\mathbf s}+{\mathbf t}_1)~\! |
| (\Lambda^2-2{\mathbf t}_1^2)\phantom{aaaaaaaaa}~\!\label{eqn:F1AndF2}\\ |
| -{64\pi\hbar^4\over m_f^2}~\!a_0^3\Lambda\! |
| \int_{\mathbf q}\!\int_{{\mathbf p}_1}\!\int_{{\mathbf p}_2}\!\ |
| N_{--}^{{\mathbf p}_1}~\! |
| {\{{\mathbf p}_2\}^{\rm sub} |
| \over[{\mathbf p}_1]-[{\mathbf p}_2]}+{\cal O}(1/\Lambda)~\!. |
| \phantom{aaaaaaaa}~\!\nonumber |
| \end{eqnarray} |
| The first two terms constitute the complete, finite contribution to $F/V$ up |
| to the order $(C_0^{\rm R})^2$; the remaining terms are formally of higher order |
| and can be considered only after including other third and higher order |
| contributions. |
|
|
| In \cite{CHGR} it has been found that it is convenient to evaluate |
| (the finite part of) $F^{(2)}/V$ by substituting ${\mathbf p}_1={\mathbf k}_1$, |
| ${\mathbf q}={\mathbf k}_1+{\mathbf k}_2$, ${\mathbf p}_2={\mathbf p}$ |
| (the Jacobian is 1), replacing (by another change of the integration variable) |
| $n_-({\mathbf k}_1+{\mathbf k}_2-{\mathbf p})$ |
| with $n_-({\mathbf p})$ and then performing explicitly the integral over the |
| cosine of the angle between ${\mathbf p}$ and ${\mathbf k}_1+{\mathbf k}_2$. |
| This allows to represent the order $(k_{\rm F}a_0)^2$ contribution to |
| $F$ in the form |
| \begin{eqnarray} |
| {F^{(2)}\over V}=C_0^{\rm R}\!\int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\! |
| n_+({\mathbf k}_1)~\!n_-({\mathbf k}_2)~\!L({\mathbf k}_1,{\mathbf k}_2)~\!. |
| \label{eqn:F(2)final} |
| \end{eqnarray} |
| The (dimensionless function) $L({\mathbf k}_1,{\mathbf k}_2)$ is given |
| by the single integral |
| \begin{eqnarray} |
| L({\mathbf k}_1,{\mathbf k}_2)= |
| -{C_0^{\rm R}m_f\over(2\pi)^2\hbar^2|{\mathbf k}_1+{\mathbf k}_2|} |
| \!\int_0^\infty\!dp~\!p~\![n_+(p)+n_-(p)]\ln\! |
| \left|{(p-\Delta_+)(p-\Delta_-)\over(p+\Delta_+)(p+\Delta_-)}\right|, |
| \label{eqn:Lfunction} |
| \end{eqnarray} |
| in which |
| \begin{eqnarray} |
| \Delta_\pm={1\over2}|{\mathbf k}_1+{\mathbf k}_2|\pm{1\over2} |
| |{\mathbf k}_1-{\mathbf k}_2|~\!. |
| \end{eqnarray} |
| In \cite{CHGR} we have checked that in the zero temperature limit, in which the |
| Fermi distribution functions $n_+({\mathbf p})$ and $n_-({\mathbf p})$ are |
| replaced by the step functions $\theta(p_{{\rm F}+}-|{\mathbf p}|)$ and |
| $\theta(p_{{\rm F}-}-|{\mathbf p}|)$, this formula reproduces |
| numerically the second order |
| correction to the ground state energy computed first (analytically) by |
| Kanno \cite{KANNO} and then recovered (semi-analytically) in several |
| works (e.g. in \cite{CHWO1,PECABO1}) for all values of the polarization $P$. |
| We have also analyzed the free energy $F$ with the corrections $F^{(1)}$ and |
| $F^{(2)}$ included and recovered, up to uncertainties following from the finite |
| precision of the (rather complicated) numerical evaluation of the relevant |
| multiple integrals the characteristics of the phase transition to the |
| ordered state (for temperatures $T<\varepsilon_{\rm F}/k_{\rm B}$) first obtained |
| in \cite{DUMacDO}. |
| |
| |
| \vskip0.1cm |
|
|
| \begin{figure}[] |
| \begin{center} |
| |
| \begin{picture}(240,80)(5,0) |
| \ArrowArc(40,40)(40,330,90) |
| \ArrowArc(40,40)(40,90,210) |
| \ArrowArc(40,40)(40,180,360) |
| \Vertex(74,20){2} |
| \Vertex(6,20){2} |
| \Vertex(40,80){2} |
| \DashArrowArcn(40,-53)(80,115,65){2} |
| \DashArrowArcn(120,86)(80,235,185){2} |
| \DashArrowArcn(-40,86)(80,355,305){2} |
| |
| \ArrowArc(200,40)(40,330,90) |
| \ArrowArc(200,40)(40,90,210) |
| \ArrowArc(200,40)(40,180,360) |
| \Vertex(234,20){2} |
| \Vertex(166,20){2} |
| \Vertex(200,80){2} |
| \DashArrowArc(200,-53)(80,65,115){2} |
| \DashArrowArc(280,86)(80,185,235){2} |
| \DashArrowArc(120,86)(80,305,355){2} |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| |
| \end{picture} |
| \end{center} |
| \caption{The particle-particle and the particle-hole diagrams contributing |
| in the order $C^3_0$ to the thermodynamic potential $F$.} |
| |
| |
| \label{fig:C0cubeMercedes} |
| |
| \end{figure} |
|
|
| If only the interaction term proportional to $C_0$ in (\ref{eqn:Vint}) is taken |
| into account, there are two Feynman diagrams contributing to the free energy |
| $F$ in the third order. The first one, shown in the left panel of Figure |
| \ref{fig:C0cubeMercedes}, is termed the particle-particle ring diagram. Its |
| contribution $F^{(3)pp}$ is given by the convolution of three $A$-blocks |
| \begin{eqnarray} |
| {F^{(3)pp}\over V}={1\over3}~\!C_0^3~\!{1\over\beta}\sum_l |
| \int_{\mathbf q}\![A(\omega_l^{\rm B},\mathbf{q})]^3~\!.\label{eqn:F(3)pp} |
| \end{eqnarray} |
| After decomposing the product of three $A$-blocks into simple fractions, |
| performing the summation over the bosonic Matsubara frequencies |
| $\omega_l^{\rm B}$ and then using the identities (\ref{eqn:Ids}) one arrives at |
| \begin{eqnarray} |
| {F^{(3)pp}\over V}={1\over3}~\!C_0^3\!\int_{\mathbf q}\! |
| \int_{{\mathbf p}_1}\!\int_{{\mathbf p}_2}\!\int_{{\mathbf p}_3}\! |
| \left(N_{--}^{{\mathbf p}_1}~\! |
| {\{{\mathbf p}_2\}\over[{\mathbf p}_1]-[{\mathbf p}_2]}~\! |
| {\{{\mathbf p}_3\}\over[{\mathbf p}_1]-[{\mathbf p}_3]} |
| +{\rm two~other~terms}\right),\label{eqn:SymmetricFormOfF3pp} |
| \end{eqnarray} |
| where ``two other terms'' means the terms in which the role of ${\mathbf p}_1$ |
| is played by ${\mathbf p}_2$ and ${\mathbf p}_3$. It is good to make at this |
| point a contact with the contribution of this third order particle-particle |
| ring diagram to the ground state energy density $E/V$ obtained in |
| \cite{CHWO3,CHWO4} (and in \cite{PECABO2}) to which the expression |
| (\ref{eqn:SymmetricFormOfF3pp}) should reduce in the zero temperature limit, |
| that is when the Fermi distribution functions are replaced by the theta |
| functions. The contribution to $E/V$ of the particle-particle diagram was in |
| \cite{CHWO3,CHWO4} given by two terms (Eq. (21) in \cite{CHWO4}) |
| whereas here it is given by the single |
| term (three seemingly identical terms). The equivalence of the two approaches |
| is ensured by the algebraic, i.e. independent of |
| the precise forms of $N_{--}^{\mathbf p}$ and $[{\mathbf p}]$ (recall that |
| $\{{\mathbf p}\}=N_{--}^{\mathbf p}-N_{++}^{\mathbf p}$), identity which results |
| from the symmetrization: |
| \begin{eqnarray} |
| N_{--}^{{\mathbf p}_1}~\! |
| {N_{--}^{\mathbf{p}_2}-N_{++}^{\mathbf{p}_2}\over[\mathbf{p}_1]-[\mathbf{p}_2]}~\! |
| {N_{--}^{\mathbf{p}_3}-N_{++}^{\mathbf{p}_3}\over[\mathbf{p}_1]-[\mathbf{p}_3]} |
| +{\rm two~other~terms}\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaa} |
| \label{eqn:AlgId}\\ |
| =\left(N_{--}^{\mathbf{p}_1}~\! |
| {N_{++}^{\mathbf{p}_2}\over[\mathbf{p}_1]-[\mathbf{p}_2]}~\! |
| {N_{++}^{\mathbf{p}_3}\over[\mathbf{p}_1]-[\mathbf{p}_3]} |
| +N_{++}^{\mathbf{p}_1}~\!{N_{--}^{\mathbf{p}_2}\over[\mathbf{p}_1]-[\mathbf{p}_2]}~\! |
| {N_{--}^{\mathbf{p}_3}\over[\mathbf{p}_1]-[\mathbf{p}_3]}\right) |
| +{\rm two~other~terms}.\nonumber |
| \end{eqnarray} |
| |
| After using the symmetry, |
| i.e. taking only the content of the bracket and multiplying it by 3, |
| it allows to rewrite the expression for $F^{(3)pp}/V$ as the sum of two |
| terms which in the $T=0$ limit precisely reduce to the two terms, $G_1$ |
| and $G_2$, which in \cite{CHWO3,CHWO4} contributed to $E/V$. |
|
|
| Naively, as all the three terms of (\ref{eqn:SymmetricFormOfF3pp}) seem also |
| identical, one is tempted to compute only one of them and multiply the result |
| by three. $F^{(3)pp}/V$ would be in this way given by a single four-fold |
| integral. This, as we have found, would lead to an incorrect result which in |
| the zero temperature limit would not agree with the contribution of the |
| particle-particle diagram to $E/V$ (this is precisely the problem which |
| did no allow us to immediately extend the computation reported in \cite{CHGR}). |
|
|
| To understand the problem it is instructive to consider the triple integral |
| \begin{eqnarray} |
| \int_0^1\!dx\!\int_0^1\!dy\!\int_0^1\!dz\left({1\over(x-y)(x-z)} |
| +{1\over(y-x)(y-z)}+{1\over(z-x)(z-y)}\right).\label{eqn:Integral} |
| \end{eqnarray} |
| The integrand is algebraically zero and the result of the integration should |
| be zero too. Yet the integrand has (spurious) singularities and the integrals |
| in (\ref{eqn:Integral}), similarly as the ones encountered in the computation |
| of $F^{(3)}$, should be understood in the Principal Value |
| sense. If one naively says that the integrals of the three terms are equal |
| and evaluates only one of them (multiplying it by 3) one will get |
| \begin{eqnarray} |
| 3\int_0^1\!dx\!\left({\rm P}\!\int_0^1\!dy~\!{1\over x-y}\right)^2 |
| =3\int_0^1\!dx~\!\ln^2{1-x\over x}=3~\!{\pi^2\over3}.\nonumber |
| \end{eqnarray} |
| The correct result (zero) is obtained if one first regularizes the integrand |
| of (\ref{eqn:Integral}) by |
| setting $x\rightarrow x+i\epsilon$, $y\rightarrow y+2i\epsilon$, |
| $z\rightarrow z+3i\epsilon$ (the sign of $\epsilon$ is irrelevant; |
| the integrand is still algebraically zero |
| but its singularities are now off the integration axes). It is then |
| straightforward to find that the application of the Sochocki formula |
| $1/(x\pm i0)=P(1/x)\mp i\pi\delta(x)$ to the regularized integral |
| (\ref{eqn:Integral}) leads to |
| (the terms linear in the Dirac deltas neatly cancel out) |
| \begin{eqnarray} |
| 3\int_0^1\!dx\!\left({\rm P}\!\int_0^1\!dy~\!{1\over x-y}\right)^2 |
| +\int_0^1\!dx\!\left(i\pi\!\int_0^1\!dy~\!\delta(x-y)\right)^2=0~\!.\nonumber |
| \end{eqnarray} |
| If the same procedure is applied to (\ref{eqn:SymmetricFormOfF3pp}) one gets |
| \begin{eqnarray} |
| {F^{(3)pp}\over V}=C_0^3\!\int_{\mathbf q}\!\int_{{\mathbf p}_1}\!~ |
| N_{--}^{{\mathbf p}_1}\!\left[\left({\rm P}\!\int_{{\mathbf p}_2}~\! |
| {\{{\mathbf p}_2\}\over[{\mathbf p}_1]-[{\mathbf p}_2]}\right)^2 |
| +{1\over3}\left(i\pi\!\int_{{\mathbf p}_2}\!\{{\mathbf p}_2\}~\! |
| \delta([{\mathbf p}_1]-[{\mathbf p}_2])\right)^2\right]. |
| \label{eqn:F3ppCorrect} |
| \end{eqnarray} |
|
|
| One can now check that in the sum $F^{(1)}+F^{(2)}+F^{(3)pp}$ all the divergences |
| (up to the order $(k_{\rm F}a_0)^3$) cancel out (as will be seen, the |
| contribution $F^{(3)ph}$ of the other diagram of Figure \ref{fig:C0cubeMercedes}, |
| which completes the order $(k_{\rm F}a_0)^3$ contribution to $F^{(3)}$, is finite; |
| this also follows from the computations of the |
| order $(k_{\rm F}a_0)^3$ corrections to $E/V$ performed in \cite{CHWO3,CHWO4}). |
| Writing $\{{\mathbf p}_i\}=-1+\{{\mathbf p}_i\}^{\rm sub}$ in the |
| first term in the square bracket in (\ref{eqn:F3ppCorrect}) allows to single |
| out the divergent part of $F^{(3)}/V$. It is given by (to this order one can |
| set in (\ref{eqn:F3ppCorrect}) $C_0=C_0^{\rm R}$; we also suppress the symbol |
| $P$ of the Principal Value) |
| \begin{eqnarray} |
| {F^{(3)pp}_{\rm div}\over V}={64\pi^3\hbar^6\over m_f^3}~\!a_0^3 |
| \!\int_{\mathbf q}\!\int_{{\mathbf p}_1} |
| N_{--}^{{\mathbf p}_1}\!\left(\int_{{\mathbf p}_2}~\! |
| {1\over[{\mathbf p}_1]-[{\mathbf p}_2]}\right)^2 |
| \phantom{aaaaaaaaaaa}\nonumber\\ |
| -2~\!{64\pi^3\hbar^6\over m_f^3}~\!a_0^3\!\int_{\mathbf q}\!\int_{{\mathbf p}_1} |
| N_{--}^{{\mathbf p}_1}\!\int_{{\mathbf p}_2}~\! |
| {\{{\mathbf p}_2\}^{\rm sub}\over[{\mathbf p}_1]-[{\mathbf p}_2]} |
| \int_{{\mathbf p}_3}~\!{1\over[{\mathbf p}_1]-[{\mathbf p}_3]}~\!. |
| \end{eqnarray} |
| Making now in the first term the change of the variables |
| ${\mathbf q}=2{\mathbf s}$, ${\mathbf p}_1={\mathbf s}-{\mathbf t}_1$, |
| ${\mathbf p}_2={\mathbf s}-{\mathbf t}_2$ (the Jacobian is 8) |
| and performing the innermost integral (over $d^3{\mathbf t}_2$) one finds |
| that it precisely cancels the entire middle line of (\ref{eqn:F1AndF2}). |
| Moreover, after making similar changes of the variables in the |
| last line of (\ref{eqn:F1AndF2}) and in the last term of |
| $F^{(3)pp}_{\rm div}/V$ they too mutually cancel out. |
| The remaining contribution of the left diagram |
| of Figure \ref{fig:C0cubeMercedes} is, therefore, given |
| by (\ref{eqn:F3ppCorrect}) with |
| $\{{\mathbf p}_2\}$ in the first term (but not in the second one!) |
| replaced by $\{{\mathbf p}_2\}^{\rm sub}$ and $C_0$ replaced by $C_0^{\rm R}$. |
| Making as previously the change ${\mathbf p}_1={\mathbf k}_1$, |
| ${\mathbf q}={\mathbf k}_1+{\mathbf k}_2$, ${\mathbf p}_2={\mathbf p}$ |
| of the integration variables |
| one can represent the contribution of the particle-particle |
| order $(k_{\rm F}a_0)^3$ diagram to $F/V$ in the form |
| \begin{eqnarray} |
| {F^{(3)pp}_{\rm fin}\over V}=C_0^{\rm R}\!\int_{{\mathbf k}_1}\!\int_{{\mathbf k}_1}\! |
| n_+({\mathbf k}_1)~\!n_-({\mathbf k}_2)\left[L^2({\mathbf k}_1,{\mathbf k}_2) |
| +{1\over3}\left(iL_\delta({\mathbf k}_1,{\mathbf k}_2)\right)^2\right], |
| \label{eqn:F(3)ppfinal} |
| \end{eqnarray} |
| where the function $L({\mathbf k}_1,{\mathbf k}_2)$ is given by |
| (\ref{eqn:Lfunction}) while the dimensionless function |
| $L_\delta({\mathbf k}_1,{\mathbf k}_2)$ is given by the finite integral |
| \begin{eqnarray} |
| L_\delta({\mathbf k}_1,{\mathbf k}_2) |
| =\pi~\!{C_0^{\rm R}m_f\over(2\pi)^2\hbar^2|{\mathbf k}_1+{\mathbf k}_2|}\! |
| \int_{p_{\rm min}}^{p_{\rm max}}\!dp~\!p~\![n_+(p)+n_-(p)-1]~\!, |
| \label{eqn:Ldeltafunction} |
| \end{eqnarray} |
| in which $p_{\rm min}=|\Delta_-|$ and $p_{\rm max}=\Delta_+$ are determined by |
| the condition that the zero of the argument of the Dirac delta treated as a |
| function of the cosine of the angle between ${\mathbf p}$ and |
| ${\mathbf k}_1+{\mathbf k}_2$ lies between $-1$ and $+1$. |
| Since $\varepsilon_{\mathbf p}$ depends on $p^2$, |
| the function $L_\delta({\mathbf k}_1,{\mathbf k}_2)$ can, of course, be |
| obtained in a closed analytic form. |
| The expression (\ref{eqn:F(3)ppfinal}) is finite (as indicates its superscript) |
| the ultraviolet convergence\footnote{The singularity introduced by the |
| factor $1/|{\mathbf k}_1+{\mathbf k}_2|^2$ is superficial because |
| for ${\mathbf k}_1+{\mathbf k}_2={\mathbf0}$ vanishes also |
| the logarithm in (\ref{eqn:Lfunction}) while in (\ref{eqn:Ldeltafunction}) |
| $p_{\rm min}=p_{\rm max}$.} |
| of the integrations being secured by the exponential suppression provided by |
| the Fermi distribution functions $n_\pm$. |
| We have also checked that the expression |
| (\ref{eqn:F(3)ppfinal}) |
| evaluated for $T=0$ (so that the Fermi distribution functions can be replaced |
| by the step functions) reproduces numerically in the entire range of |
| polarizations $P$ the contribution to the ground state energy density |
| of the third order particle-particle diagram |
| of Figure \ref{fig:C0cubeMercedes} obtained in \cite{CHWO3,CHWO4}. |
| \vskip0.5cm |
|
|
|
|
| \noindent{\large\bf 4. Resummation of the contributions of the particle-particle |
| diagrams} |
| \vskip0.3cm |
|
|
| \noindent It turns out that the contribution to the free energy $F$ of the |
| infinite series of Feynman diagrams composed of $N$-fold rings of the |
| particle-particle $A$-blocks of Figure \ref{fig:ElementaryLoops} |
| can be summed in a closed form. Consider first the order |
| $(k_{\rm F}a_0)^N$ term of this series (the |
| factor $(-1)^{N+1}$ is the same as in (\ref{eqn:OmegaPertExpansion}) - there |
| are as many rearrangements of the $\hat\psi_+$ operators as of the $\hat\psi_-$ |
| ones; the factor $1/N!$ in (\ref{eqn:OmegaPertExpansion}) is reduced to |
| $1/N$ as there are $(N-1)!$ identical diagrams) |
| \begin{eqnarray} |
| {F^{(N)pp}\over V}=(-1)^{N+1}~\!{C_0^N\over N}~\!{1\over\beta}\sum_l |
| \int_{\mathbf q}[A(\omega_l^{\rm B},{\mathbf q})]^N~\!. |
| \end{eqnarray} |
| Decomposing the product of the integrands of the $N$ $A$-blocks |
| (\ref{eqn:AblockExplicit}) using the identity |
| \begin{eqnarray} |
| \prod_{i=1}^N{1\over x-a_i}=\sum_{n=1}^N\left(\prod_{j\neq n}^N{1\over a_n-a_j} |
| \right){1\over x-a_n}~\!, |
| \end{eqnarray} |
| and performing then the summation over the Matsubara frequencies one gets |
| the integrand of the $(N+1)$-fold integral in the form |
| \begin{eqnarray} |
| \{{\mathbf p}_1\}\dots\{{\mathbf p}_N\} |
| \sum_{n=1}^N\left(\prod_{j\neq n}^N{1\over[{\mathbf p}_n]-[{\mathbf p}_j]}\right) |
| {1\over1-e^{\beta[{\mathbf p}_n]}}~\!, |
| \end{eqnarray} |
| and finally, after using the relations (\ref{eqn:Ids}), |
| $F^{(N)pp}/V$ takes the form |
| \begin{eqnarray} |
| {F^{(N)pp}\over V}=(-1)^{N+1}~\!{C_0^N\over N}\!\int_{\mathbf q}\! |
| \int_{{\mathbf p}_1}\!\dots\!\int_{{\mathbf p}_N}\sum_{n=1}^N |
| N_{--}^{{\mathbf p}_n}\left(\prod_{j\neq n}^N |
| { \{ {\mathbf p}_j \}\over[{\mathbf p}_n]-[{\mathbf p}_j] |
| +i(n-j)\epsilon}\right), |
| \end{eqnarray} |
| in which, in order to regularize the integrals, the substitution |
| $[{\mathbf p}_l]\rightarrow[{\mathbf p}_l]+il\epsilon$ has been made. Using the |
| Sochocki formula this can be then rewritten (assuming that $\epsilon>0$ |
| - it will |
| be seen that the result does not depend on the sign of $\epsilon$) in the form |
| \begin{eqnarray} |
| {F^{(N)pp}\over V}=(-1)^{N-1}~\!{C_0^N\over N}\!\int_{\mathbf q} |
| \sum_{n=1}^N\int_{{\mathbf p}_n}\!\!N_{--}^{{\mathbf p}_n} |
| \left\{\prod_{j=1}^{n-1}\int_{{\mathbf p}_j}\!\left( |
| {\{{\mathbf p}_j\}\over[{\mathbf p}_n]-[{\mathbf p}_j]} |
| -i\pi~\!\{{\mathbf p}_j\}~\!\delta([{\mathbf p}_n]-[{\mathbf p}_j])\right) |
| \right.\nonumber\\ |
| \left.\times\prod_{j=n+1}^N\int_{{\mathbf p}_j}\!\left( |
| {\{{\mathbf p}_j\}\over[{\mathbf p}_n]-[{\mathbf p}_j]} |
| +i\pi~\!\{{\mathbf p}_j\}~\! |
| \delta([{\mathbf p}_n]-[{\mathbf p}_j])\right)\right\},\nonumber |
| \end{eqnarray} |
| in which the integrals of the factors |
| $\{{\mathbf p}_j\}/([{\mathbf p}_n]-[{\mathbf p}_j])$ are understood in the |
| Principal Value sense. The experience with the order $(k_{\rm F}a_0)^2$ and |
| $(k_{\rm F}a_0)^3$ contributions teaches that removing divergences amounts |
| simply to replacing $\{{\mathbf p}_j\}$ by $\{{\mathbf p}_j\}^{\rm sub}$ in |
| the first terms of the integrands of the integrals over ${\mathbf p}_j$-s |
| (but not in the delta-terms) and $C_0^N$ in front by $(C_0^{\rm R})^N$. This |
| would be obvious had the dimensional regularization been used to handle |
| ultraviolet divergences - |
| |
| by definition it |
| sets the integrals like $\int_{\mathbf p}({\rm const})$ to zero and, as is well |
| known (see e.g. \cite{HamFur00}), $C_0=C_0^{\rm R}$ to all orders, if such a |
| regularization is used. |
| After the change of the variables ${\mathbf p_n}={\mathbf k}_1$, |
| ${\mathbf q}={\mathbf k}_1+{\mathbf k}_2$ |
| the order $(k_{\rm F}a_0)^N$ particle-particle diagram contribution to $F$ |
| can be then neatly written in the form |
| \begin{eqnarray} |
| {F^{(N)pp}\over V}={C_0^{\rm R}\over N}\! |
| \int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\!n_+({\mathbf k}_1)~\!n_-({\mathbf k}_2) |
| \sum_{n=1}^N(L+iL_\delta)^{n-1}(L-iL_\delta)^{N-n}~\!. |
| \end{eqnarray} |
| Summing the (finite) geometric series then gives |
| \begin{eqnarray} |
| {F^{(N)pp}\over V}={C_0^{\rm R}\over N}\! |
| \int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\!n_+({\mathbf k}_1)~\!n_-({\mathbf k}_2) |
| ~\!{(L+iL_\delta)^N-(L-iL_\delta)^N\over2iL_\delta}~\!.\label{eqn:FNpp} |
| \end{eqnarray} |
| This is real and independent of the sign of $L_\delta$ which reflects the fact |
| that in the prescription allowing to properly handle the $P$-value integrals |
| the sign of $\epsilon$ is arbitrary; in particular it has nothing to do with |
| the prescription $+i0^+$ for standard Feynman propagators in the real time |
| formalism. As can be easily checked, for $N=2$ and $N=3$ (\ref{eqn:FNpp}) |
| reproduces the results (\ref{eqn:F(2)final}) and (\ref{eqn:F(3)ppfinal}), |
| respectively. It is also clear that for $N=1$ the mean field result |
| (\ref{eqn:F(1)}) is recovered. |
| Finally, summation over $N$ can also be done\footnote{We use the formula |
| arctan$~\!t=(1/2i)\ln[(1+it)/(1-it)]$ and the expansion of the logarithm |
| in powers of $t$.} |
| and leads to the expression |
| \begin{eqnarray} |
| {F^{pp}\over V}=C_0^{\rm R}\int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\! |
| n_+({\mathbf k}_1)~\!n_-({\mathbf k}_2) |
| ~\!{{\rm arctan}(L_\delta/(1-L))\over L_\delta}~\!.\label{eqn:FppSummed} |
| \end{eqnarray} |
| \begin{figure} |
| \centerline{\hbox{ |
| \psfig{figure=updated_F_difference_PP_T_00.eps,width=9.cm,height=7.0cm} |
| \psfig{figure=updated_F_difference_PP_T_02.eps,width=9.cm,height=7.0cm} |
| }} |
| \caption{The difference $(F(P)-F(0))^{pp}/N$ (in units |
| $(3/5)\varepsilon_{\rm F}$) for $T=0$ and |
| $T=0.2~\!T_{\rm F}\equiv0.2~\!\varepsilon_{\rm F}/k_{\rm B}$ as a function of the |
| polarization |
| $P=(N_+-N_-)/N$ for different values of the gas parameter $k_{\rm F}a_0$.} |
| \label{fig:FresPPonlyT00and02} |
| \end{figure} |
| The zero temperature analog of this formula (i.e. representing the contribution |
| of the particle-particle ring diagrams to the ground state energy density |
| $E/V$) has been for $P=0$ first given by Kaiser \cite{KAJZERKA1} who in |
| deriving it relied on combinatoric arguments. The formula which is |
| the zero temperature analog of (\ref{eqn:FppSummed}) for arbitrary polarization |
| $P$ has been then written (by invoking the Kaiser's reasoning) |
| down in \cite{He1,He3} (see also \cite{KAJZERKA2}). |
|
|
| \begin{figure} |
| \centerline{\hbox{ |
| \psfig{figure=updated_F_difference_PP_T_03.eps,width=9.cm,height=7.0cm} |
| \psfig{figure=updated_F_difference_PP_T_05.eps,width=9.cm,height=7.0cm} |
| }} |
| \caption{As in Figure \ref{fig:FresPPonlyT00and02} but for |
| $T=0.3~\!T_{\rm F}$ and $T=0.5~\!T_{\rm F}$.} |
| \label{fig:FresPPonlyT03and05} |
| \end{figure} |
|
|
| The numerical procedure for evaluating the expression (\ref{eqn:F(2)final}) |
| described in details in \cite{CHGR} (the main trick is to construct - for given |
| values of $t=k_{\rm B}T/\varepsilon_{\rm F}$ and $P$, which together, through |
| (\ref{eqn:nu(0)Determination}), determine the chemical potentials - the |
| interpolations of the functions of the parameter $\Delta$ into which the |
| function (\ref{eqn:Lfunction}) can be decomposed) can be used to evaluate also |
| the expression (\ref{eqn:FppSummed}). Figures \ref{fig:FresPPonlyT00and02} and |
| \ref{fig:FresPPonlyT03and05} show for four different values of the temperature |
| and several values of the gas parameter $k_{\rm F}a_0$ the difference |
| $(F(P)-F(0))^{pp}/V$ (in units $(k^3_{\rm F}/3\pi^2)(3/5)\varepsilon_{\rm F}$) |
| obtained by adding to the zeroth order term |
| |
| |
| (\ref{eqn:F(0)Term}) given explicitly by the formula (25) in \cite{CHGR} |
| the contribution |
| (\ref{eqn:FppSummed}). In agreement with the result obtained in the last |
| section of the work \cite{He1}, one can observe that for $T=0$ the minimum of |
| $(F(P)-F(0))^{pp}/V$ starts to move away from $P=0$ for $k_{\rm F}a_0=0.79$ |
| indicating the continuous transition to the ordered state. As can be expected, |
| with increasing temperature this critical value of the expansion parameter |
| shifts towards larger values (0.85, 0.92 and 1.12 for $T=0.2~\!T_{\rm F}$, |
| $0.3~\!T_{\rm F}$ and $0.5~\!T_{\rm F}$, respectively). One can also see |
| that the minimum is back at $P=0$ for $k_{\rm F}a_0>0.96$ (at $T=0$) - this is |
| the first order ``reentrant'' transition to the paramagnetic state observed in |
| \cite{He2,He3} which is the consequence of the existence of the maximum of |
| the energy density (at $T=0$) for $P=0$ treated as a function of $k_{\rm F}a_0$ |
| shown by the dashed (blue) lines in the left panel of Figure \ref{fig:FakFT00}. |
| (From the physical point of view this reentrant transition is largely |
| irrelevant as it occurs for the values of the gas parameter for which the |
| formation of dimers prevails and the free energy of the metastable state is no |
| longer physical.) This maximum of the energy density (indeed seen in |
| experiments with cold gases \cite{MaxInEexp}) which occurs close to the Feshbach |
| resonance on its so-called BEC side (i.e. for large positive scattering |
| length $a_0$) and the existence of which for higher temperatures (for which |
| there is no phase transition) has been given a theoretical explaination (using a |
| completely different approach) in \cite{SheHo} results here, as has been shown |
| in \cite{He1,He2}, from the appearance for $k_{\rm F}a_0>1.34$ of the simple |
| pole in the ``in-medium'' particle-particle elastic scattering amplitude |
| which can be interpreted as being due to the ``in-medium'' positive energy |
| bound state of two fermions (of opposite spin projections). |
| The dashed (blue) lines on the right panel of this figure and in Figures |
| \ref{fig:FakFT02} and \ref{fig:FakFT05} illustrate how the contribution |
| (\ref{eqn:FppSummed}) to the free energy and its maximum |
| change with the polarization and the temperature. |
|
|
| As the right panel of Figure \ref{fig:FresPPonlyT00and02} and Figure |
| \ref{fig:FresPPonlyT03and05} show, with increasing temperature the reentrant |
| transition occurs for higher ($1.06$ and $1.46$ for $T=0.2~\!T_{\rm F}$ and |
| $T=0.3~\!T_{\rm F}$, respectively and yet higher for $T=0.5~\!T_{\rm F}$) values |
| of the expansion parameter. The maximal depth of the minimum (at which $P\neq0$) |
| of $F$ first slightly decreases with the growing temperature (up to |
| $T\approx0.2~\!T_{\rm F}$) and then increases with it. Similarly $P=1$ is for |
| temperatures up to $T\approx0.2~\!T_{\rm F}$ reached only for $k_{\rm F}a_0$ |
| values approaching the one at which the reentrant transition takes place but |
| for higher temperatures it is reached well before it. |
| \vskip0.5cm |
|
|
| \noindent{\large\bf 5. The particle-hole diagrams} |
| \vskip0.3cm |
|
|
| \noindent At the order $(k_{\rm F}a_0)^3$ to the free energy contributes also |
| the second diagram shown in Figure \ref{fig:C0cubeMercedes}. The corresponding |
| analytical expression is given by the convolution |
| \begin{eqnarray} |
| {F^{(3)ph}\over V}={C_0^3\over3}~\!{1\over\beta}\sum_l\!\int_{\mathbf q} |
| [B(\omega^{\rm B}_l,{\mathbf q})]^3~\!,\label{eqn:F(3)phOriginal} |
| \end{eqnarray} |
| of three $B$-blocks which have the form \cite{CHGR} |
| \begin{eqnarray} |
| B(\omega_l,{\mathbf q})=\int_{\mathbf p}{\{{\mathbf p}\}\over i\omega^{\rm B}_l |
| -[{\mathbf p}]}~\!,\label{eqn:BblockExplicit} |
| \end{eqnarray} |
| analogous to (\ref{eqn:AblockExplicit}) but with now different meaning |
| of the symbols $\{{\mathbf p}\}$ and $[{\mathbf p}]$: |
| \begin{eqnarray} |
| &&N^{\mathbf p}_{+-}\equiv[1-n_+({\mathbf q}+{\mathbf p})]~\!n_-({\mathbf p})~\!, |
| \nonumber\\ |
| &&N^{\mathbf p}_{-+}\equiv n_+({\mathbf q}+{\mathbf p})~\! |
| [1-n_-({\mathbf p})]~\!,\label{eqn:BblockIds}\\ |
| &&\{{\mathbf p}\}\equiv n_+({\mathbf q}+{\mathbf p})-n_-({\mathbf p}) |
| =-N^{\mathbf p}_{+-}(1-e^{\beta[{\mathbf p}]})~\!,\nonumber\\ |
| &&[{\mathbf p}]\equiv\varepsilon_{\mathbf p}-\tilde\mu_-^{(0)}- |
| \varepsilon_{{\mathbf q}+{\mathbf p}}+\tilde\mu_+^{(0)}~\!, |
| \nonumber |
| \end{eqnarray} |
| After performing in (\ref{eqn:F(3)phOriginal}) the summation over the bosonic |
| Matsubara frequencies and using (\ref{eqn:BblockIds}) one arrives at |
| \begin{eqnarray} |
| {F^{(3)ph}\over V}=-{C_0^3\over3}\!\int_{\mathbf q}\!\int_{{\mathbf p}_1}\! |
| \int_{{\mathbf p}_2}\!\int_{{\mathbf p}_3}\!\left(N_{+-}^{{\mathbf p}_1}~\! |
| {\{{\mathbf p}_2\}\over[{\mathbf p}_1]-[{\mathbf p}_2]}~\! |
| {\{{\mathbf p}_3\}\over[{\mathbf p}_1]-[{\mathbf p}_3]}+{\rm two~other~terms} |
| \right),\label{eqn:F(3)phSymmetric} |
| \end{eqnarray} |
| where ``two other terms'' means terms in which the role of $[{\mathbf p}_1]$ |
| is played by $[{\mathbf p}_2]$ and $[{\mathbf p}_3]$. One can again make |
| contact with the order $(k_{\rm F}a_0)^3$ contribution of the particle-hole |
| diagram of Figure \ref{fig:C0cubeMercedes} to the ground state |
| energy density $E/V$ computed in \cite{CHWO3,CHWO4} where it was given |
| as a sum of two functions $K_1$ and $K_2$ (Eq. (17) in \cite{CHWO3}), by using |
| the algebraic identity |
| \begin{eqnarray} |
| N_{+-}^{{\mathbf p}_1}~\! |
| {N_{-+}^{{\mathbf p}_2}-N_{+-}^{{\mathbf p}_2}\over[{\mathbf p}_1]-[{\mathbf p}_2]}~\! |
| {N_{-+}^{{\mathbf p}_3}-N_{+-}^{{\mathbf p}_2}\over[{\mathbf p}_1]-[{\mathbf p}_3]} |
| +{\rm two~other~terms}\phantom{aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa} |
| \label{eqn:AlgIdB}\\ |
| =N_{+-}^{{\mathbf p}_1}~\! |
| {N_{-+}^{{\mathbf p}_2}\over[{\mathbf p}_1]-[{\mathbf p}_2]}~\! |
| {N_{-+}^{{\mathbf p}_3}\over[{\mathbf p}_1]-[{\mathbf p}_3]} |
| +N_{-+}^{{\mathbf p}_1}~\! |
| {N_{+-}^{{\mathbf p}_2}\over[{\mathbf p}_1]-[{\mathbf p}_2]}~\! |
| {N_{+-}^{{\mathbf p}_3}\over[{\mathbf p}_1]-[{\mathbf p}_3]} |
| +{\rm two~other~terms},\nonumber |
| \end{eqnarray} |
| (it is in fact the identity (\ref{eqn:AlgId}) but written with different |
| symbols). Using the symmetry of this expression allows to write the expression |
| for $F^{(3)ph}/V$ as the sum of two terms which in the zero temperature limit |
| reproduce the two terms, $K_1$ and $K_2$, which in \cite{CHWO3,CHWO4} |
| represented the contribution of the second diagram of Figure |
| \ref{fig:C0cubeMercedes} to $E/V$. |
|
|
| \begin{figure} |
| \centerline{\hbox{ |
| \psfig{figure=F_apF_T_00_P_00.eps,width=9.cm,height=7.0cm} |
| \psfig{figure=F_apF_T_00_P_075.eps,width=9.cm,height=7.0cm} |
| }} |
| \caption{Dependence on the gas parameter $k_{\rm F}a_0$ of the resumed |
| contributions of the particle-particle diagrams given by the expressions |
| (\ref{eqn:FppSummed}) (dashed blue lines), of the particle-hole diagrams |
| given by (\ref{eqn:FphSummed}) (dotted red lines) and of their sum (solid |
| green lines) for zero temperature and two values of the polarization $P$.} |
| \label{fig:FakFT00} |
| \end{figure} |
|
|
| In order to properly treat the singularities in (\ref{eqn:F(3)phSymmetric}) |
| we again make the substitutions |
| $[{\mathbf p}_l]\rightarrow[{\mathbf p}_l]+il\epsilon$ which allow to profit |
| from the symmetry of the three terms of (\ref{eqn:F(3)phSymmetric}). |
| One then, similarly as in the case of $F^{(3)pp}$, obtains |
| \begin{eqnarray} |
| {F^{(3)ph}\over V}=-C_0^3\!\int_{\mathbf q}\!\int_{{\mathbf p}_1}\! |
| N_{+-}^{{\mathbf p}_1}\!\left[\left({\rm P}\!\int_{{\mathbf p}_2} |
| {\{{\mathbf p}_2\}\over[{\mathbf p}_1]-[{\mathbf p}_2]}\right)^2 |
| +{1\over3}\left(i\pi\!\!\int_{{\mathbf p}_2}\!\{{\mathbf p}_2\}~\! |
| \delta([{\mathbf p}_1]-[{\mathbf p}_2])\right)^2\right].~ |
| \label{eqn:F3phCorrect} |
| \end{eqnarray} |
| Making now the changes of the variables: first |
| ${\mathbf p}_1+{\mathbf q}={\mathbf k}_1$, ${\mathbf p}_1=-{\mathbf k}_2$, |
| and then in the $n_+$ terms of $\{{\mathbf p}_2\}$ the change |
| ${\mathbf p}_2+{\mathbf k}_1+{\mathbf k}_2=-{\mathbf p}$ |
| and ${\mathbf p}_2={\mathbf p}$ in the $n_-$ terms of $\{{\mathbf p}_2\}$, |
| and then taking explicitly the integral over the cosine of the angle |
| between ${\mathbf p}$ and ${\mathbf k}_1+{\mathbf k}_2$ |
| the expression for $F^{(3)ph}/V$ can be written in the form |
| \begin{eqnarray} |
| {F^{(3)ph}\over V}=-C_0^{\rm R}\!\int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\! |
| [1-n_+({\mathbf k}_1)]~\!n_-({\mathbf k}_2) |
| \left[K^2({\mathbf k}_1,{\mathbf k}_2) |
| +{1\over3}\left(iK_\delta({\mathbf k}_1,{\mathbf k}_2)\right)^2\right], |
| \label{eqn:F(3)phFinal} |
| \end{eqnarray} |
| in which the dimensionless functions $K$ and $K_\delta$ are given by the |
| integrals |
| \begin{eqnarray} |
| K({\mathbf k}_1,{\mathbf k}_2) |
| ={C_0^{\rm R}m_f\over(2\pi)^2\hbar^2|{\mathbf k}_1+{\mathbf k}_2|} |
| \left(\int_0^\infty\!dp~\!p~\!n_+(p) |
| \ln\!\left|{p-\Delta_1\over p+\Delta_1}\right|\right.\phantom{aaa}\nonumber\\ |
| \left.+\int_0^\infty\!dp~\!p~\!n_-(p) |
| \ln\!\left|{p-\Delta_2\over p+\Delta_2}\right|\right),\label{eqn:Kfunction} |
| \end{eqnarray} |
| \begin{eqnarray} |
| K_\delta({\mathbf k}_1,{\mathbf k}_2) |
| =\pi~\!{C_0^{\rm R}m_f\over(2\pi)^2\hbar^2|{\mathbf k}_1+{\mathbf k}_2|} |
| \left(\int_{|\Delta_1|}^\infty\!dp~\!p~\!n_+(p) |
| -\int_{|\Delta_2|}^\infty\!dp~\!p~\!n_-(p)\right),\label{eqn:Kdeltafunction} |
| \end{eqnarray} |
| in which |
| \begin{eqnarray} |
| \Delta_1\equiv{{\mathbf k}_1\!\cdot\!({\mathbf k}_1+{\mathbf k}_2)\over |
| |{\mathbf k}_1+{\mathbf k}_2|}~\!,\phantom{aaa} |
| \Delta_2\equiv{{\mathbf k}_2\!\cdot\!({\mathbf k}_1+{\mathbf k}_2)\over |
| |{\mathbf k}_1+{\mathbf k}_2|}~\!. |
| \end{eqnarray} |
| Again, the limits of the integrals in $K_\delta$ are determined by the condition |
| that the zeroes of the arguments of the Dirac deltas treated as functions of |
| the cosine of the angle between ${\mathbf p}$ and |
| ${\mathbf k}_1+{\mathbf k}_2$ lie between $-1$ and $+1$. And again |
| the function $K_\delta({\mathbf k}_1,{\mathbf k}_2)$ can be written |
| down in a closed analytical form - see below. |
|
|
| \begin{figure} |
| \centerline{\hbox{ |
| \psfig{figure=F_apF_T_02_P_00.eps,width=9.cm,height=7.0cm} |
| \psfig{figure=F_apF_T_02_P_075.eps,width=9.cm,height=7.0cm} |
| }} |
| \caption{As in Figure \ref{fig:FakFT00} but for $T=0.2~\!T_{\rm F}$.} |
| \label{fig:FakFT02} |
| \end{figure} |
|
|
| The expression (\ref{eqn:F(3)phFinal}) is finite although this is not |
| immediately obvious: while the integrals defining the functions $K$ and |
| $K_\delta$ are clearly (ultraviolet) finite owing to the presence of the |
| distribution functions $n_+(p)$ and $n_-(p)$, the integral over |
| $d^3{\mathbf k}_1$ has no such an exponentially suppressing factor. In |
| addition, the presence of the factors $1/|{\mathbf k}_1+{\mathbf k}_2|$ |
| in front of the functions $K$ and $K_\delta$ seem to imply the potential |
| presence of a |
| singularity in the integral over the directions of ${\mathbf k}_1$ (for |
| $|{\mathbf k}_1|=|{\mathbf k}_2|$). To prove the |
| finiteness of (\ref{eqn:F(3)phFinal}) we will first analyze the behavior of the |
| difference $K^2-K_\delta^2/3$ in the limit $|{\mathbf k}_1|\rightarrow\infty$. |
| To this end it is helpful to split the functions $K$ and $K_\delta$ into |
| $K_++K_-$ and $K_{\delta+}+K_{\delta-}$ (the subscripts $\pm$ corresponds to the |
| $n_\pm$ distribution functions). |
| The hardest to see is the convergence of the integral over $d^3{\mathbf k}_1$ |
| involving the factor $K_-^2-K^2_{\delta-}/3$. The dangerous term |
| $n_-({\mathbf k}_2)[K_-^2-K^2_{\delta-}/3]$ comes from the term |
| \begin{eqnarray} |
| \int_{\mathbf q}\!\int_{{\mathbf p}_1}\!\int_{{\mathbf p}_2}\!\int_{{\mathbf p}_3} |
| \left\{{n_-({\mathbf p}_1)~\!n_-({\mathbf p}_2)~\!n_-({\mathbf p}_3)\over |
| ([{\mathbf p}_1]-[{\mathbf p}_2])([{\mathbf p}_1]-[{\mathbf p}_3])} |
| +{\rm two~other~terms}\right\},\nonumber |
| \end{eqnarray} |
| in (\ref{eqn:F(3)phSymmetric}). This, however is algebraically zero (just as |
| the integrand of (\ref{eqn:Integral})) and, therefore, the factor |
| $K_-^2-K^2_{\delta-}/3$ must be zero too. As any inaccuracy in the numerical |
| evaluation of the integrals (\ref{eqn:Kfunction}) and |
| (\ref{eqn:Kdeltafunction}) could lead to a nonzero difference |
| $K_-^2-K^2_{\delta-}/3$ and, therefore, to a (fake) nonconvergence of the |
| integration over $d^3{\mathbf k}_1$, in the term with unity arising from |
| $[1-n_+({\mathbf k}_1)]$ we simply replace $K^2-K^2_{\delta}/3$ by |
| $K_+^2+2K_+K_--(K_{\delta+}^2+2K_{\delta+}K_{\delta-})/3$. The rest of the terms |
| are separately integrable in the limit $|{\mathbf k}_1|\rightarrow\infty$: |
| \begin{eqnarray} |
| K_{\delta+}\propto{1\over|{\mathbf k}_1+{\mathbf k}_2|}~\! |
| \ln\!\left(1+e^{-\beta(\hbar^2\Delta_1^2/2m-\tilde\mu^{(0)}_+)}\right) |
| \approx{1\over|{\mathbf k}_1+{\mathbf k}_2|}~\! |
| e^{-\beta(\hbar^2\Delta_1^2/2m-\tilde\mu^{(0)}_+)}~\!,\nonumber |
| \end{eqnarray} |
| because $\Delta^2_1$ grows like ${\mathbf k}_1^2$ as |
| $|{\mathbf k}_1|\rightarrow\infty$. This secures the convergence of the |
| integrals of the terms $K_{\delta+}^2$ and $2K_{\delta+}K_{\delta-}$. |
| Similarly, it can be estimated that the integral over $p$ in |
| $K_+$ behaves as $1/\Delta_1$ when $|{\mathbf k}_1|\rightarrow\infty$. |
| Since each of the $K_\pm$ functions has the factor |
| $1/|{\mathbf k}_1+{\mathbf k}_2|$ in front of it, the term $(K_+)^2$ |
| behaves for $|{\mathbf k}_1|\rightarrow\infty$ as |
| $1/({\mathbf k}_1^2+{\mathbf k}_1\cdot{\mathbf k}_2)^2$ and this secures |
| the convergence of the integration over $d^3{\mathbf k}_1$. The term |
| $2K_+K_-$, instead, behaves only as |
| $1/({\mathbf k}_1^2+{\mathbf k}_1\!\cdot\! |
| {\mathbf k}_2)|{\mathbf k}_1+{\mathbf k}_2|$, |
| but the integration over the |
| cosine of the angle between ${\mathbf k}_1$ and ${\mathbf k}_2$ kills |
| the term of order $1/|{\mathbf k}_1|^3$ and the remaining terms are |
| already integrable. This only power-like suppression of the integration |
| of the term $K_+^2+2K_+K_-$ makes, however, numerical evaluation of the |
| particle-hole contribution to the free energy more difficult and, therefore, |
| potentially less accurate than the evaluation of the particle-particle one. |
| As to the potentially singular factors $1/|{\mathbf k}_1+{\mathbf k}_2|$, |
| one should first notice that the original expressions (\ref{eqn:F(3)phOriginal}) |
| and (\ref{eqn:BblockExplicit}) as well as similar formulae giving the |
| contributions of the $N$-th order particle-hole rings do not contain such |
| singularities. They are due to the symmetrizations needed to arrive at the final |
| formulae and must, therefore, cancel out (like the spurious singularities |
| of the integrand in (\ref{eqn:Integral})) even if it is not directly evident. |
| In the third order finitness of (\ref{eqn:F(3)phFinal}) can be seen as |
| follows: since the integrals in the definitions of the |
| $K_\pm$ and $K_{\delta\pm}$ functions are finite |
| in the limit ${\mathbf k}_1+{\mathbf k}_2\rightarrow{\mathbf0}$, the |
| singularities of (\ref{eqn:F(3)phFinal}) have essentially the form |
| $1/({\mathbf k}_1^2+{\mathbf k}_2^2+2|{\mathbf k}_1||{\mathbf k}_2|\xi)$ where |
| $\xi$ is the cosine of the angle between ${\mathbf k}_1$ and ${\mathbf k}_2$. |
| In the third order contribution considered here, |
| after integration over $\xi$ they give rise to terms |
| $\ln\!||{\mathbf k}_1|-|{\mathbf k}_2||$ which are |
| integrable.\footnote{In fact for $|{\mathbf k}_1|=|{\mathbf k}_2|$ |
| the factors $\Delta_1$ and $\Delta_2$ behave as $\sqrt{1+\xi}$ and |
| it can be checked (numerically) that the integrals in the functions $K_+$ |
| and $K_-$ vanish then as $\sqrt{1+\xi}$, so these functions have therefore |
| finite limits. The singularities reside only in the terms |
| involving the functions $K_{\delta\pm}$.} |
| In the numerical evaluation of the integrals in (\ref{eqn:F(3)phFinal}) |
| and in (\ref{eqn:FphSummed}) it is however sufficient to impose a cutoff |
| $|{\mathbf k}_1+{\mathbf k}_2|>\kappa$ and check that the results stabilize |
| as $\kappa$ approaches zero. |
| Thus the expression (\ref{eqn:F(3)phFinal}) is finite and we have checked that |
| evaluated for $T=0$ (so that the Fermi distribution functions can be replaced |
| by the step functions) it reproduces numerically in the entire range of |
| polarizations $P$ the contribution to the ground state energy density |
| of the third order particle-hole diagram |
| of Figure \ref{fig:C0cubeMercedes} obtained in \cite{CHWO3,CHWO4}. |
| \vskip0.1cm |
|
|
| \begin{figure} |
| \centerline{\hbox{ |
| \psfig{figure=F_apF_T_05_P_00.eps,width=9.cm,height=7.0cm} |
| \psfig{figure=F_apF_T_05_P_075.eps,width=9.cm,height=7.0cm} |
| }} |
| \caption{As in Figure \ref{fig:FakFT00} but for $T=0.5~\!T_{\rm F}$.} |
| \label{fig:FakFT05} |
| \end{figure} |
|
|
| Using the same tricks as previously the contribution to the free energy |
| of the infinite series of Feynman diagrams composed of $N$-fold rings of the |
| particle-hole $B$-blocks of Figure \ref{fig:ElementaryLoops} can be |
| summed in a closed form. The order $(k_{\rm F}a_0)^N$ term of this series is |
| \begin{eqnarray} |
| {F^{(N)ph}\over V}=-(-1)^{N+1}~\!{C_0^N\over N}\!\int_{\mathbf q}\! |
| \int_{{\mathbf p}_1}\!\dots\!\int_{{\mathbf p}_N}\sum_{n=1}^N |
| N_{+-}^{{\mathbf p}_n}\left(\prod_{j\neq n}^N |
| {\{ {\mathbf p}_j \}\over[{\mathbf p}_n]-[{\mathbf p}_j] |
| +i(n-j)\epsilon}\right), |
| \end{eqnarray} |
| (apart from the extra minus originating from the minus sign in the identity |
| $\{{\mathbf p}\}=-N^{\mathbf p}_{+-}(1-e^{\beta[{\mathbf p}]})$, the origin of the |
| rest of the prefactor is the same as in the case of $F^{(N)pp}$) and, repeating |
| the steps one arrives at the formal sum |
| \begin{eqnarray} |
| \sum_{N=1}^\infty{F^{(N)ph}\over V} |
| =-~\!C_0^{\rm R}\int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\! |
| [1-n_+({\mathbf k}_1)]~\!n_-({\mathbf k}_2) |
| ~\!{{\rm arctan}(K_\delta/(1-K))\over K_\delta}~\!.\label{eqn:FphSumFormal} |
| \end{eqnarray} |
| From this formal sum one has to subtract two first terms of the series: |
| there is no order $C_0^{\rm R}$ particle-hole diagram and the order |
| $(C_0^{\rm R})^2$ term is already |
| (recall that the order $C_0^2$ contribution of the diagram shown in Figure |
| \ref{fig:ElementaryLoops} can be written either as a convolution of two |
| $A$-blocks or of two $B$-blocks shown in the same Figure), in the properly |
| renormalized form, included in (\ref{eqn:FppSummed}). Thus the final |
| form of the ressumed contributions of the particle-hole ring diagrams is |
| \begin{eqnarray} |
| {F^{ph}\over V}=-C_0^{\rm R}\int_{{\mathbf k}_1}\!\int_{{\mathbf k}_2}\! |
| [1-n_+({\mathbf k}_1)]~\!n_-({\mathbf k}_2)\left[ |
| {{\rm arctan}(K_\delta/(1-K))\over K_\delta}-1-K\right].\label{eqn:FphSummed} |
| \end{eqnarray} |
|
|
| One has to comment again on the finiteness of the expression |
| (\ref{eqn:FphSummed}). The singularities related to the factors |
| $1/|{\mathbf k}_1+{\mathbf k}_2|$ are now harmless because the function $K$ |
| is, as remarked, finite in the limit |
| ${\mathbf k}_1+{\mathbf k}_2\rightarrow{\mathbf0}$ and the |
| singular function $K_\delta$ is now in the denominator and under the arctan |
| function. As to the ultraviolet finiteness |
| of (\ref{eqn:FphSummed}), the integral of the factor $K$ explicitly subtracted |
| in the square brackets is ultraviolet divergent being simply equivalent to the |
| divergent expression (\ref{eqn:F(2)inTermsOfC0}). To ensure proper cancelation |
| of this term in (\ref{eqn:FphSummed}) for large |
| $|{\mathbf k}_1|$ and/or $|{\mathbf k}_2|$ we expand the arctan function |
| in $K_\delta$ and $K$ up to the sixth order (we have checked that taking |
| more terms of the expansion does not change the result appreciably). The |
| ultraviolet divergent terms linear in $K$ then disappear and, moreover, |
| the expansion allows to implement the discussed trick with replacing |
| $K^2-K_\delta^2/3$ by $K^2_++2K_+K_--(K^2_{\delta+}+2K_{\delta+}K_{\delta-})$. |
| \vskip0.1cm |
|
|
| In \cite{CHWO4} we have compared the order $(k_{\rm F}a_0)^3$ contributions to |
| the ground state energy (i.e. to the free energy $F$ for zero temperature) of |
| the particle-particle and of the particle-hole diagrams (shown in Figure |
| \ref{fig:C0cubeMercedes}) in the entire range of the polarization $P$ and |
| found that the second one is not much smaller than the first one. |
| In Figures \ref{fig:FakFT00}, \ref{fig:FakFT02} and \ref{fig:FakFT05} |
| we compare the magnitudes of the resummed contributions (\ref{eqn:FppSummed}) |
| and (\ref{eqn:FphSummed}) to the free energy as functions of $k_{\rm F}a_0$ |
| for three different temperatures |
| and two representative values of the system's polarization $P$. We |
| also plot the sum of these two contributions. It again follows that for |
| $k_{\rm F}a_0\sim1$ the resummed contribution of the particle-hole diagrams |
| is not much smaller than that of the particle-particle ones |
| and is of the opposite sign. It is also clear that |
| in this most important region the sum of the two contributions is significantly |
| distorted compared to the resummed contribution of the particle-particle |
| diagrams. This raises the question what impact the resummed contribution |
| of the particle-hole ring diagrams has on the results reported in the |
| papers \cite{He1,He2,He3}. |
|
|
|
|
| \begin{figure} |
| \centerline{\hbox{ |
| \psfig{figure=updated_F_difference_total_T_00.eps,width=9.cm,height=7.0cm} |
| \psfig{figure=updated_F_difference_total_T_02.eps,width=9.cm,height=7.0cm} |
| }} |
| \caption{The difference $(F(P)-F(0))^{pp+ph}/N$ (in units |
| $(3/5)\varepsilon_{\rm F}$ for $T=0$ and |
| $T=0.2~\!T_{\rm F}$ as a function of the polarization |
| $P=(N_+-N_-)/N$ for different values of the gas parameter $k_{\rm F}a_0$.} |
| \label{fig:FresTotalT00and02} |
| \end{figure} |
|
|
| Figures \ref{fig:FresTotalT00and02} and \ref{fig:FresTotalT03and05}, |
| analogous to Figures \ref{fig:FresPPonlyT00and02} and |
| \ref{fig:FresPPonlyT03and05} illustrate the consequences of adding the |
| resummed contribution (\ref{eqn:FphSummed}) of the particle-hole |
| ring diagrams to the free energy. The result is dramatic: the phase transition |
| to the ordered state discussed in \cite{He1,He2,He3} completely disappears! |
| Thus, even if the selection of the particle-particle ring diagrams as |
| giving the dominant contribution can be (partially) justified by invoking the |
| arguments, given in \cite{Steele}, based on using $1/2^{D/2}$ where $D$ is the |
| number of space dimensions as the expansion parameter, they in practice do not |
| turn out to be really dominant: the contribution of other subsets of diagrams |
| (the number of such subsets beginning at a given order of the expansion |
| grows with the order number) can, as our results show, change qualitatively |
| the behavior of the thermodynamical potentials of the system of |
| fermions close to the Feshbach resonance. |
|
|
| \vskip0.5cm |
|
|
|
|
| \noindent{\bf\large 7. Conclusions} |
| \vskip0.3cm |
|
|
| \noindent We have applied the systematic thermal (imaginary time) perturbative |
| expansion to the effective (low energy) field theory to compute the free |
| energy of the gas of interacting (nonrelativistic) spin $1/2$ fermions |
| for arbitrary values of the gas polarization and temperatures not exceeding the |
| Fermi temperature. We have shown how to circumvent the technical problem which |
| previously prevented us from immediately extending such a computation beyond |
| the second order in the gas parameter $k_{\rm F}a_0$ and have given explicit |
| formulae for the order $(k_{\rm F}a_0)^3$ contributions to the system's free |
| energy. It turned out that the analytical part of this computations |
| is more transparent and easier than the corresponding |
| direct computations of the ground state energy based on the formula |
| (\ref{eqn:CorrectionsToE}) which gives only the zero temperature limit of the |
| results obtained with the help of the thermal expansion. (Of course, numerical |
| evaluation of the resulting expressions for a nonzero temperature is |
| considerably more involved than for $T=0$). |
|
|
| \begin{figure} |
| \centerline{\hbox{ |
| \psfig{figure=updated_F_difference_total_T_03.eps,width=9.cm,height=7.0cm} |
| \psfig{figure=updated_F_difference_total_T_05.eps,width=9.cm,height=7.0cm} |
| }} |
| \caption{As in Figure \ref{fig:FresTotalT00and02} but for |
| $T=0.3~\!T_{\rm F}$ and $T=0.5~\!T_{\rm F}$.} |
| \label{fig:FresTotalT03and05} |
| \end{figure} |
|
|
|
|
| To obtain the complete order $(k_{\rm F}R)^3$ contribution to the free energy |
| of the gas of fermions interacting through a truly repulsive spin-independent |
| two-body potential (characterized by a length scale $R$) one would have to add |
| the contributions arising from the operators of lower length |
| dimension in the effective theory interaction term because |
| such potentials naturally give the $p$-wave scattering length $a_1$ and |
| the $s$-wave effective range $r_0$ comparable to the $s$-wave scattering |
| length $a_0$. Instead of doing this, in this work we have profited from the |
| simple structure of the contributions of the particle-particle and |
| particle-hole ring diagrams and managed to give simple formulae for their |
| contributions to the free energy resummed to all orders in the gas parameter |
| $k_{\rm F}a_0$. These formulae apply therefore rather to cold gases of |
| fermionic atoms (interacting through attractive poentials) |
| close to the Feshbach resonance where their $s$-wave |
| scattering length $a_0$ is made positive and much larger than the remaining |
| scattering lengths and effective radii. Using these formulae we have |
| first checked |
| that including only the contributions of the particle-particle rings we |
| reproduce for zero temperatures all the results obtained in \cite{He1}. |
| In particular we confirm that in this approximation at zero temperature the |
| transition to the ordered phase occurs for $k_{\rm F}a_0=0.79$ |
| and that it is continuous. These results seem to agree |
| well with the results of the dedicated Monte Carlo computations. |
| Our formula would, however, allow to study also the thermal profile of the |
| transition. |
|
|
| However we have found that the phase transition to the ordered state |
| completely disappears |
| after including into the free energy the resummed contribution of the |
| particle-hole ring diagrams - the minimum of the free energy is always for |
| zero polarization. |
| This may indicate at least that the agreement of the critical value of |
| the gas parameter $k_{\rm F}a_0$ found in the papers \cite{He1,He2,He3} |
| with its value obtained from the Monte Carlo simulations (done with |
| attractive potentials tuned so that $a_0$ is positive and large) |
| is just accidental. It may however also indicate that |
| there is indeed no transition (in the metastable state) to the |
| spin ordered phase |
| if the true interaction is attractive (but $a_0$ is made positive and |
| large by approaching a Feshbach resonance). The |
| vanishing at zero temperature of the inverse spin susceptibility |
| ($1/\chi\propto(\partial^2F/\partial P^2)_{P=0}$) for some |
| value of the gas parameter found in the Monte Carlo simulations |
| (which in the case of attractive potentials are not as clean as in the |
| repulsive case because of the need to exclude - in the finite volume, that |
| is without taking the thermodynamic limit - an overlap with the true |
| ground state of the considered system of atoms) \cite{QMC10} from which one |
| infers its existence and its continuous character would then be |
| misleading (the second order approximation to $F$ predicts |
| analytically vanishing of the |
| inverse spin susceptibility at $k_{\rm F}a_0=1.058$ whereas the transition |
| is first order and occurs for $k_{\rm F}a_0=1.054$ \cite{He1}). |
| If this hypothesis is true it would provide yet |
| another (in addition to the formation of atomic dimers) reason for the |
| failure to simulate the intinerant ferromagnetism with the help of cold atoms. |
| In an case it is clear that more theoretical studies are needed to clarify |
| the situation. |
| \vskip0.5cm |
|
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