Abstract
We provide a rigorous derivation of nonlinear Gibbs measures in two and three space dimensions, starting from many-body quantum systems in thermal equilibrium. More precisely, we prove that the grand-canonical Gibbs state of a large bosonic quantum system converges to the Gibbs measure of a nonlinear Schrödinger-type classical field theory, in terms of partition functions and reduced density matrices. The Gibbs measure thus describes the behavior of the infinite Bose gas at criticality, that is, close to the phase transition to a Bose–Einstein condensate. The Gibbs measure is concentrated on singular distributions and has to be appropriately renormalized, while the quantum system is well defined without any renormalization. By tuning a single real parameter (the chemical potential), we obtain a counter-term for the diverging repulsive interactions which provides the desired Wick renormalization of the limit classical theory. The proof relies on a new estimate on the entropy relative to quasi-free states and a novel method to control quantum variances.
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Notes
Also known as canonical correlation or Bogoliubov scalar product.
Note that the expectation \(\langle \mathrm {d}\Gamma (Pe ^{ik \cdot x}P) \rangle _0\) in \(\Gamma _0\) is the same as that in \((\Gamma _0)_P\).
Our convention in this section is that all microscopic density matrices have a tilde, whereas the macroscopic ones do not.
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Acknowledgements
Insightful discussions with Jürg Fröhlich, Markus Holzmann, Antti Knowles, Benjamin Schlein, Robert Seiringer, Vedran Sohinger, Jan Philip Solovej, Laurent Thomann, Daniel Ueltschi and Jakob Yngvason are gratefully acknowledged. This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 Research and Innovation Programme (Grant agreements MDFT No 725528 and CORFRONMAT No 758620), and Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy (EXC-2111-390814868).
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Appendices
Appendix A: The counter-term problem
In this appendix we discuss the counter-term problem in detail.
1.1 A.1. Hartree versus reduced Hartree energy functional
We recall that to any one-body density matrix \(\gamma \geqslant 0\) one can associate a unique Gaussian state
on the Fock space which has the one-particle density matrix \(\Gamma ^{(1)}=\gamma \) [8, 156]. The unique corresponding one-body operator a is given by
Then its energy terms and entropy can be expressed as in (3.21), resulting the Hartree free energy
Thus if we are interested in equilibrium states minimizing the free energy, in the quasi-free class this leads to the following variational problem
When \(\widehat{w}\geqslant 0\), the functional \(\mathcal {F}^\mathrm{H}[\gamma ]\) turns out to be strictly convex. Hence, with the confining potential V it admits a unique minimizer \(\gamma ^\mathrm{H}\), that defines a unique corresponding quasi-free state in Fock space \(\Gamma ^\mathrm{H}\) (the proof is the same as that of Lemma 3.2). The optimal density matrix solves the nonlinear equation
where \(\rho ^\mathrm{H}(x)=\gamma ^\mathrm{H}(x;x)\) is the density and \(X^\mathrm{H}\) is the exchange operator with integral kernel \(X^\mathrm{H}(x;y)=w(x-y)\gamma ^\mathrm{H}(x;y)\).
In the limit \(\lambda \rightarrow 0\) with \(T=1/\lambda \), the quasi-free state \(\Gamma ^\mathrm{H}\) is rather badly behaved. Its density \(\rho ^\mathrm{H}\) diverges very fast. However, it turns out that, although \(\rho ^\mathrm{H}(x)\) depends on x, its growth as \(\lambda \rightarrow 0\) is more or less uniform in x and can be captured by
provided that
Recall that \(\lambda \varrho _0^\kappa (\lambda )\) diverges in dimension \(d\geqslant 2\) but it does not depend on x by translation invariance of \(-\Delta + \kappa \). On the other hand, the exchange term \(\lambda X^\mathrm{H}\) typically stays bounded, for instance in the Hilbert-Schmidt norm.
This suggests to simplify things a little bit by removing the exchange term as we did in the paper, that is, to consider the simplified minimization problem associated with the reduced Hartree or, simply, mean-field free energy (3.22) which, we recall, is given by
By doing so we will pick as reference state a quasi-free state which is not the absolute minimizer of the true quantum free energy in the quasi-free class. However, manipulating states depending only on a potential simplifies the analysis. Fortunately, it turns out to be sufficient for our purpose, as justified in Lemma 3.2, which we now prove.
1.2 A.2. Proof of Lemma 3.2
Under our assumptions, the first eigenvalue of the Friedrichs realization of \(h=-\Delta +V\) is positive. In addition, we have
by the Golden–Thompson inequality, see [150, Section 8.1]. The same properties hold if we shift V by \(\nu \in {\mathbb {R} }\) and only keep the positive part. Then we obtain
for all \(\gamma \geqslant 0\). The last expression is the minimum free energy associated with \(-\Delta /2+(V-\nu )_+\).
In order to prove that the functional \(\mathcal {F}^\mathrm{MF}\) is bounded from below, it therefore remains to show that
is bounded from below, uniformly in \(\rho \geqslant 0\), under the assumption that \(\widehat{w}\geqslant 0\) and \(w\not \equiv 0\) (if \(w\equiv 0\) then the Lemma holds true trivially). Since \(w\in L^1({\mathbb {R} }^d)\), we can find a \(k_0\in {\mathbb {R} }^d\) and a small radius \(r>0\) such that the continuous non-negative function \(\widehat{w}\) is at least equal to \(\widehat{w}(k_0)/2\) on \(B(k_0,r)\). We then choose \(\varphi \) in the Schwartz class such that \(\varphi >0\) and \(\widehat{\varphi }\geqslant 0\) with \(\mathrm{supp}(\widehat{\varphi })\subset B(k_0,r)\). Since \(V\geqslant 0\) and \(V\rightarrow +\infty \) at infinity, the function \((V-\nu )_-\) has compact support and is bounded by \(\nu _+\). Therefore, we have
for \(C= \left| \left| \varphi ^{-1}(V-\nu )_- \right| \right| _{L^\infty ({\mathbb {R} }^d)}<\infty \). After completing the square and using \(\widehat{w}\geqslant 0\), we then deduce
Combining with (A.5) we find as stated that
for all \(\nu \in {\mathbb {R} }\).
Let us now prove the existence of a minimizer. Writing
where we have displayed the parameter \(\nu \) for convenience, we obtain that minimizing sequences \(\{\gamma _n\}\) for \(\mathcal {F}^\mathrm{MF}_\nu \) are necessarily bounded in the trace-class. In particular, \(\Vert \gamma _n\Vert \) is also bounded. In addition, the inequality (A.5) implies that \({{\,\mathrm{\mathrm{Tr}}\,}}(-\Delta )\gamma _n\) is bounded. From the Hoffmann-Ostenhof inequality [88]
and the Sobolev inequality, we deduce that \(\rho _{\gamma _n}\) is bounded in \(L^{p^*/2}({\mathbb {R} }^d)\) where \(p^*=+\infty \) in dimension \(d=1\), \(p^*<\infty \) arbitrarily in dimension \(d=2\) and \(p^*=2d/(d-2)\) in dimensions \(d\geqslant 3\). Hence, up to extraction of a subsequence, we have \(\gamma _n\rightharpoonup \gamma \) weakly-\(*\) in \(\mathfrak {S}^1\) and \(\rho _{\gamma _n}\rightharpoonup \rho _{\gamma }\) weakly in \(L^1({\mathbb {R} }^d)\cap L^{p^*/2}({\mathbb {R} }^d)\). Using Fatou’s lemma and the concavity (hence weak upper semi-continuity) of the entropy, we obtain that \(\gamma \) is a minimizer for \(\mathcal {F}_\nu ^\mathrm{MF}\). The nonlinear equation (3.24) follows from classical arguments. Then, according to (A.1), \(V_\lambda \) must solve (3.20) because the minimizer \(\gamma ^\mathrm{MF}\) is the one-body density matrix of the quasi-free state associated with \(\mathrm {d}\Gamma (-\Delta + V_\lambda )\). \(\square \)
1.3 A.3. Comments on Theorem 3.3
Let us briefly discuss Theorem 3.3. In [60, Section 5], the existence of the solution \(V_\lambda \) to (3.20) was established by means of a fixed-point argument (which requires that \(d\leqslant 3\) and that \(\kappa \) is sufficiently large). The fixed point is performed in the (complete) metric space
for the unknown \(u=V_\lambda -\kappa \) and provides the Hilbert-Schmidt convergence
Our notation here is slightly different from [60] as we shift potentials by a constant. Moreover, since \(V_\lambda -\kappa \in B(V)\) we have
There remains to discuss the nonlinear equation (3.28) for \(V_0\), which we can rewrite in the form
Here we just need to pass to the limit in the similar equation at \(\lambda >0\)
Since we know that \(V_\lambda /V\rightarrow V_0/V\) in \(L^\infty \), we have \(V_\lambda \rightarrow V_0\) in \(L^\infty _\mathrm{loc}\) and it suffices to prove the convergence of the density on the right side, which we denote for simplicity
In [60, Eq. (5.21)] it is shown that
Hence from the dominated convergence theorem and the assumptions on w, it suffices to prove that
almost everywhere. Applying again [60, Eq. (5.21)] we find that
which tends to 0 in \(L^\infty _\mathrm{loc}\) since \((V_\lambda -V_0)/V\rightarrow 0\). Hence we can replace \(V_\lambda \) by \(V_0\) throughout. Next we write, following [60, Eq. (5.16)],
Using that \(V_0\geqslant \kappa \), we have the pointwise bound on the operator kernels
by the same argument as in Lemma 11.2 and in [60, Eq. (5.17)]. Using [60, Lemma 5.4] we see that we get a convergent domination. So by the dominated convergence theorem, the strong local convergence of \(\rho ^{V_0}_\lambda \) follows from that of the kernels
In fact this convergence is strong in \(L^2({\mathbb {R} }^d\times {\mathbb {R} }^d)\) since the corresponding operators converge in the Hilbert-Schmidt norm, by [60, Lemma C.1]. Passing to the limit, this proves the strong local convergence
where in the last equality we have used the resolvent formula. The uniform bound (A.7) then allows to pass to the limit in the equation for \(V_\lambda \) and obtain (A.6). \(\square \)
Appendix B: Interpretation in terms of the phase transition of the Bose gas
Here we reformulate and discuss our main results (Theorems 3.1 and 3.4) in microscopic variables and clarify the link with the phase transition of the infinite Bose gas.
1.1 B.1. Homogeneous case
We start with the homogeneous case which is the usual setting in which the thermodynamics of the free Bose gas is formulated, see for instance [30, Section 5.2.5], [161, Sections 2.5.19–20] and [165, Chapter 4].
Let us consider the non-interacting Bose gas on the large torus \(L\mathbb {T}^d\) of side length L. In the grand canonical setting we choose a chemical potential \(\widetilde{\nu }<0\) and a temperature \(T>0\). Our system is then represented by the Gaussian quantum state in Fock space, associated with the one-particle operator \((-\Delta _L-\widetilde{\nu })/T\) where \(-\Delta _L\) is the L-periodic Laplacian. Its one-body density matrix isFootnote 3
and the number of particles per unit volume is given by
The critical density for Bose–Einstein condensation is obtained in the limit \(\widetilde{\nu }/T\rightarrow 0^-\) and it equals
The grand-canonical model in infinite space stops to exist at \(\widetilde{\nu }=0\), where the one-body density matrix converges weakly to
The corresponding infinite Gaussian state (properly defined over the \(C^*\)-algebra of Canonical Commutation Relations [30]) has the number of particles per unit volume equal to \(\rho _c(T)\).
The phenomenon of Bose–Einstein Condensation (BEC) is better understood in the canonical setting with N particles, going back to the thermodynamic limit. In dimensions \(d\geqslant 3\), fixing the density \(N/L^d=\rho >\rho _c(T)\) one obtains a density matrix which has a two-scale behavior [30, Sec. 5.2.5]. It contains a rank-one part with the diverging eigenvalue \(L^d(\rho -\rho _c(T))\) and constant eigenfunction \(f_0(x)=L^{-d/2}\), living at the macroscopic scale (the Bose–Einstein condensate). When this rank-one operator is removed, the rest of the density matrix converges weakly to \(\widetilde{\Gamma }_\infty ^{(1)}\) in (B.2).
To understand the behavior of the system just before the phase transition, we have to look at the simultaneous limit \(L\rightarrow \infty \) with \(\widetilde{\nu }\rightarrow 0^-\), see [161, Sec. 2.5.20.3]. At the macroscopic scale (that is, after rescaling length by L), the (rescaled) one-particle density matrix equals
and thus we see that the natural scaling for \(\widetilde{\nu }\) is
with a fixed \(\kappa >0\), in all dimensions \(d\geqslant 1\). We are therefore exactly in the setting studied in this paper and in [104] with the choice
The density matrix converges to
strongly in the Schatten space \(\mathfrak {S}^p(L^2(\mathbb {T}^d))\) for all \(p>d/2\) (\(p\geqslant 1\) if \(d=1\)), where \(\mu _\kappa \) is the classical Gaussian measure with covariance \((-\Delta +\kappa )^{-1}\). Equivalently,
where \(\mu _{\kappa ,T}\) is the Gaussian measure with covariance \(T(-\Delta +\kappa )^{-1}\). Similar properties hold for higher density matrices.
Our conclusion is that, close to its phase transition, the free Bose gas is properly described by the classical Gaussian measure \(\mu _{\kappa ,T}\) on \(\mathbb {T}^d\) (macroscopic scale) where \(\kappa \) describes the speed at which the chemical potential \(\widetilde{\nu }\) approaches 0 via (B.3) or, equivalently, at which the corresponding density approaches the critical density \(\rho _c(T)\). This elementary fact is rarely mentioned in textbooks on Bose gases.
The speed at which the density approaches \(\rho _c(T)\) is computed by using the following
Lemma B.1
(Particle number of the free Bose gas) In dimensions \(d\geqslant 1\), we have for every fixed \(\kappa >0\)
with the positive decreasing function
The proof is provided below in “Appendix B.3”. The integral in the first equality of (B.6) is the Fourier transform of \((2\pi )^{-d/2}(|k|^2+\kappa )^{-1}\), that is, the Klein-Gordon Green function. In dimensions \(d\leqslant 3\), we can also write by Poisson’s formula
but the sum is not absolutely convergent in dimensions \(d\geqslant 4\). Since we have the expansions
we find from (B.5) that the density approaches the critical density \(\rho _c(T)\) from below as
Note that \(\varphi _3(\kappa )\leqslant \sqrt{\kappa }/(4\pi )\) which is its behavior when \(\kappa \rightarrow 0^+\).
So far our discussion applies to any dimension \(d\geqslant 1\). Next we discuss the inclusion of interactions for \(d\leqslant 3\). In our work the potential w is introduced at the macroscopic level. Re-expressed in microscopic variables and taking \(T=1\) for simplicity, we obtain the microscopic n-particle Hamiltonian
The interaction has the very small intensity \(L^{-4}\) but varies on length scales comparable with the size of the box. The form of the microscopic Hamiltonian (B.9) is the same in all dimensions. The thermodynamic limit \(L\rightarrow \infty \) of this model at fixed \(\widetilde{\nu }<0\) is the same as the non-interacting case. This is because we have the lower bound
due to the fact that \(\widehat{w}\geqslant 0\) (for an upper bound on the free energy, use the non-interacting state). We conclude that w does not, to leading order, change the phase diagram as compared to the non-interacting case. The effect of w is only visible when zooming just before the phase transition. From Theorem 3.1, when the chemical potential goes to zero as
then the behavior close to the transition is described by the nonlinear Gibbs measure \(\mu \) at the macroscopic scale, which depends on w and \(\kappa \) solving
More physical interactions are much bigger and have a much shorter range. Although a universal behavior can still be expected at the phase transition, the phase diagram depends on w at leading order and a mathematical treatment seems out of reach with the present techniques. A simpler behavior is however expected in the dilute regime \(\rho \rightarrow 0\) with \(\rho \sim T^{d/2}\) (Gross-Pitaevskii regime [48]). In dimension \(d=3\) and at our macroscopic scale, the Gross-Pitaevskii limit corresponds to replacing \(\lambda w\) by \(\lambda w_\lambda \) with \(w_\lambda (x)=\lambda ^{-3} w(x/\lambda )\) in our many-particle Hamiltonian. In this case one would expect the phase transition to be described by the (appropriately renormalized) nonlinear Gibbs measure \(\mu \) over the torus \(\mathbb {T}^3\), with w replaced by the Dirac delta \(8\pi a \delta _0\) where a is the scattering length of w [48, 115, 116]. Proving such a result seems a formidable task.
1.2 B.2. Trapped gases
The theory of Bose–Einstein condensation for trapped gases is analogous to the homogeneous case, but the formulas are slightly different, see for instance [9, 161, Sec. 2.5.15] and [49]. Here we only discuss the case \(V(x)=|x|^s\) for simplicity. At the microscopic scale the one-body Hamiltonian takes the form
where L is now a parameter used to open the trap whenever the number of particles grows. At fixed \(\widetilde{\nu }<0\), the number of particles in the non-interacting Gaussian state is given by
In the first equality we have rescaled lengths by the factor \(L^{(2+s)/s}\), which places the system in a conventional semi-classical limit with effective parameter \(\hbar =L^{-(2+s)/s}\rightarrow 0\), hence the second limit. Computing in the same manner the average against |x| one sees that the gas is extended at the length scale
Dividing by the effective volume \((\ell _\mathrm{gas})^d\), the density obtained in the thermodynamic limit is therefore proportional to
Bose–Einstein condensation is obtained as before when \(\widetilde{\nu }/T\rightarrow 0^-\), with the critical density
This is finite for all \(s>0\) in dimensions \(d\geqslant 2\). In dimension \(d=1\), \(\rho '_c(T)\) is finite for \(s<2\) and infinite otherwise.
As before the Bose–Einstein condensate emerges in the limit \(L\rightarrow \infty \) in the canonical setting, when \(N>\rho '_c(T)(\ell _\mathrm{gas})^d\). The corresponding condensate wavefunction is the first eigenfunction of \(-\Delta +L^{-2-s}|x|^s\). This is nothing but that of \(-\Delta +|x|^s\) dilated to the scale L. Therefore, in this system the BEC length scale is L and it is always smaller than the natural extension length \(\ell _\mathrm{gas}\) of the cloud in (B.12), at a fixed temperature \(T>0\). The two coincide only in a sharp container (\(s=+\infty \)).
The Gaussian Gibbs measure based on \(h=-\Delta +|x|^s\) emerges at the BEC length scale L, whenever the chemical potential is chosen as
At this scale the one-particle Hamiltonian just becomes \((-\Delta +|x|^s+\kappa )/L^2\) so that \(\lambda =1/(TL^2)\) like in the homogeneous case. The arguments from [60] and “Appendix A.3” in the non-interacting case give
for every \(x\in {\mathbb {R} }^d\). Here the density on the second line plays the role of \(\varphi _d\) in the homogeneous case. In particular we obtain all the same formulas as in (B.8) with \(\varphi _d(\kappa )\) replaced by
In the inhomogeneous case, interactions were introduced at the scale L of the BEC and the interpretation is the same as in the homogeneous case.
1.3 B.3. Proof of Lemma B.1
We write
and obtain
We use Poisson’s formula
for \(f(x)=e^{-n\lambda |x|^2}-e^{-n\lambda |\cdot |^2}*\chi (x)\) where \(\chi =(2\pi )^{-d}{\mathbb {1} }_{(-\pi ,\pi )^d}\) and find
This is a Riemann sum which converges to
where the right side is the Fourier transform of \(k\mapsto (2\pi )^{-d/2}(|k|^2+\kappa )^{-1}\), see [114]. \(\square \)
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Lewin, M., Nam, P.T. & Rougerie, N. Classical field theory limit of many-body quantum Gibbs states in 2D and 3D. Invent. math. 224, 315–444 (2021). https://doi.org/10.1007/s00222-020-01010-4
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DOI: https://doi.org/10.1007/s00222-020-01010-4