Full-law convergence and condensate structure in the supercritical Berlin–Kac spherical model with multiple energy minima

For the complex Berlin–Kac spherical model at supercritical density, when the periodic lattice energy function (dispersion relation) has finitely many non-degenerate global minima, Lukkarinen (2020) proved that the measure is close to a factorized measure with an independent condensate and a Gaussian normal fluid, but only at the level of local observables: the Wasserstein bound $W_2(\mu_0,\mu_1)\le C_2 L^{d/2}\varepsilon_L^{1/4}$ diverges for every condensate-mode split constructed so far, and the number and distribution of condensate modes can change with the lattice size (parity and subsequence effects). The central open problem is to determine the sharp (necessary and sufficient) scaling conditions on the dispersion relation and on the selection of condensate modes under which the full configuration law converges, i.e. $W_2(\mu_0,\mu_1)\to 0$ in the unscaled $L^2$ configuration norm — or to establish that no admissible choice of condensate-mode splits makes the Wasserstein distance vanish for the class of dispersion relations considered. Two further limitations documented in the same paper directly condition this question: the known split constructions are worst-case and not always optimal, and the condensate multiplicity can change with the lattice size (parity and subsequence effects).

Background

The Berlin–Kac spherical model was introduced in 1952 as an exactly solvable surrogate for the Ising model: instead of Ising spins it uses continuous degrees of freedom subject to a global 'spherical' constraint that fixes the total squared amplitude, which makes the thermodynamics exactly computable. Here we consider its complex-field version on a $d$-dimensional periodic lattice $\Lambda_L$ with $V=L^d$ sites: the field $\phi=(\phi_x)_{x\in\Lambda_L}\in\mathbb{C}^{\Lambda_L}$ carries a fixed total mass $N_L[\phi]=\sum_x|\phi_x|^2=\rho V$ (the spherical constraint) and is weighted by a translation-invariant quadratic energy $H_L$. Because $H_L$ is diagonalized by the discrete Fourier transform, it is characterized by a dispersion relation $\omega:\mathbb{T}^d\to\mathbb{R}$, the energy per Fourier mode as a function on the $d$-dimensional unit torus $\mathbb{T}^d$ of wave numbers, so that $H_L=\sum_{k\in\Lambda^*}\omega(k)|\hat\phi_k|^2$ on the dual lattice $\Lambda^*=\{n/L : n\in\{0,\dots,L-1\}^d\}$, the grid of allowed Fourier wave numbers. Physically, $\phi$ is a classical wave amplitude; the measure is the canonical equilibrium ensemble of a harmonic lattice with fixed wave action, mathematically equivalent to the canonical ensemble of the ideal Bose gas.

For $d\ge 3$, the normal-fluid modes can carry at most a finite critical mass density $\rho_\infty=\int_{\mathbb{T}^d}dk/(\omega(k)-\omega_{\min})$ (where $\omega_{\min}$ is the minimum of $\omega$ and $dk$ is normalized Lebesgue measure on the torus); for a supercritical density $\rho>\rho_\infty$ the excess mass $\Delta=\rho-\rho_c(L)$ must condense into the Fourier modes of near-minimal energy, in complete analogy with Bose–Einstein condensation. When $\omega$ has a single non-degenerate minimum (e.g. nearest-neighbour interactions), the condensate is the single ground-state mode. For finitely many non-degenerate minima the situation is richer: earlier work on the ideal Bose gas had already shown 'generalized' condensation (van den Berg and Lewis, 1982) and coexistence of two kinds of condensation (Bru and Zagrebnov, 2000), while a fixed one-particle spectral gap forces single-state condensation in interacting gases (Lauwers, Verbeure and Zagrebnov, 2003). The present problem comes from Lukkarinen's 2020 analysis of exactly this multi-minima case for the spherical model.

Lukkarinen's setup: for each lattice size $L$, split the dual lattice into two nonempty sets — condensate candidates $\Lambda_0^*$ (of cardinality $V_0$) and normal-fluid modes $\Lambda_+^*$ — separated by an energy interval: writing $\omega_0=\min_{k\in\Lambda^*}\omega(k)$ and $e_k=\omega(k)-\omega_0$, all condensate-candidate modes have $e_k\le a_L$ and all normal-fluid modes have $e_k\ge b_L$ for some thresholds $0\le a_L<b_L$ — which gives the split a relative energy gap $\delta_L^{-1}$ with $\delta_L=\max_{k\in\Lambda_0^*}e_k/\min_{k\in\Lambda_+^*}e_k<1$ (smaller $\delta_L$ means better separation), and let $\mu_0$ denote the true spherical-model measure and $\mu_1$ the factorized supercritical measure of the source's main theorem (eq. (2.10) of arXiv:1806.01806): the product of a condensate measure on $\Lambda_0^*$ carrying the excess mass $\Delta=\rho-\rho_c(L)$ per site, with condensate weight $\mathrm{e}^{-E_0[\Phi]\,(1-\rho_c/\Delta)}\prod_{k\in\Lambda_+^*}\left(1-E_0[\Phi]L^{-d}/(e_k\Delta)\right)^{-1}\delta(\rho_0[\Phi]-\Delta)$, where $E_0[\Phi]=\int_{\Lambda_0^*}dk\,e_k|\Phi_k|^2$ and $\rho_0[\Phi]=\frac1V\int_{\Lambda_0^*}dk\,|\Phi_k|^2$, and of an independent normal fluid, the latter being a Gaussian field at criticality (the critical Gaussian free field, i.e. the massless Gaussian field whose covariance is the inverse of the lattice energy operator). The source also considers the simpler variant $\mu_1'$ (eq. (2.12)) in which the condensate amplitudes are uniformly distributed on a sphere in the $2V_0$-dimensional real space, and calls both $\mu_1$ and $\mu_1'$ factorized supercritical measures; the bound and admissibility conditions quoted below are the source's theorem for $\mu_1$, which is the comparison measure used throughout this problem. The quality of the split is measured by

$$\varepsilon_L := \max\left(2\delta_L,\; \frac{1}{V^2\rho_c^2}\sum_{k\in\Lambda_+^*}\frac{1}{e_k^2}\right),$$

where $\rho_c=\rho_c(L)$ is the critical norm density, i.e. the expected mass density carried by the critical Gaussian normal fluid, $\rho_c(L)=\int_{\Lambda_+^*}dk/(\omega(k)-\omega_0)$ — integrals over subsets of the dual lattice use the source's discrete convention $\int_{\Lambda^*}dk\,g(k):=\frac1V\sum_{k\in\Lambda^*}g(k)$ (eq. (2.4)) — and $\rho_c(L)$ converges to $\rho_\infty$ as $L\to\infty$. The main theorem (Theorem 1 of the published version) proves a bound on the 2-Wasserstein distance $W_2(\mu_0,\mu_1)$ — the optimal-transport distance defined as the infimum, over all couplings of the two measures, of $(\mathbb{E}\|\Phi-\Phi'\|^2)^{1/2}$ in the unscaled $L^2$ configuration norm — of the form $W_2(\mu_0,\mu_1)\le C_2 L^{d/2}\varepsilon_L^{1/4}$, with $C_2=2^4(\rho/\Delta)^{V_0/2}\sqrt{(\rho+\Delta)V_0}$. A companion lemma (Lemma 1) shows that for every $C^2$ dispersion relation with finitely many non-degenerate minima (invertible Hessians) admissible splits exist with $V_0\le M_0$ and $\varepsilon_L=O(\delta_L+L^{-2\kappa})$, where $\kappa<d/2$ ($\kappa<1$ in $d=3$) is a free parameter. Since $\varepsilon_L$ decays at best like a power smaller than $L^{-d}$, the prefactor $L^{d/2}$ wins and the bound grows like a positive power of $L$: the author states explicitly that no convergence result follows in the unscaled $L^2$ norm. Only for local correlation functions does an additional $L^{-d/2}$ cancellation make the approximation error vanish, as $L^{-p'}$ with $p'=(d/2-1)/(2M_0+1)$. Two further limitations are documented in the same paper: the split constructions of Lemma 1 are worst-case and 'not always optimal', losing sharpness for strongly anisotropic Hessians; and the condensate structure itself can depend on the lattice size — explicit examples include a dispersion relation with $2^d$ torus minima, $\omega(k)=\sum_i\sin^2(2\pi k_i)$, whose condensate is single-mode for odd $L$ but $2^d$-fold for even $L$; a dispersion relation with a unique torus minimum $\zeta$ (e.g. $\omega(k;\zeta)=\sum_i\sin^2(\pi(k_i-\zeta_i))$ with $\zeta_i=\tfrac12$) which has $2^d$ lattice minima, and hence $2^d$ condensate modes, for odd $L$ but a single condensate mode for even $L$; subsequence effects along $L=2^N$ when an irrational minimum location lies off the dual lattice; and a two-component condensate behaviour persisting for all sufficiently large odd $L$ in an anisotropic example. The measure was also motivated as the weak-coupling limit of canonical equilibrium measures of the discrete nonlinear Schrödinger equation (DNLS, the lattice analog of the nonlinear Schrödinger equation with an on-site nonlinearity), where the supercritical regime is precisely the case excluded by existing kinetic-theory results.

Problem Statement

Consider the complex Berlin–Kac spherical model on the periodic lattice $\Lambda_L$, $V=L^d$, $d\ge 3$, with measure

$$\mu_0^{(L)}[d\phi] \propto e^{-H_L[\phi]}\,\delta\!\left(N_L[\phi]-\rho V\right)\prod_{x\in\Lambda_L}d\phi_x^*d\phi_x, \qquad N_L[\phi]=\sum_x|\phi_x|^2,$$

where $H_L$ is a translation-invariant quadratic energy with dispersion relation $\omega:\mathbb{T}^d\to\mathbb{R}$ (in Fourier variables $H_L=\sum_{k\in\Lambda^*}\omega(k)|\hat\phi_k|^2$). Assume $\omega$ is $C^2$ with finitely many non-degenerate global minima (invertible Hessians at each minimum) and that the density $\rho$ is supercritical, $\rho>\rho_\infty=\int_{\mathbb{T}^d}dk/(\omega(k)-\omega_{\min})$. For each $L$, let $(\Lambda_0^*(L),\Lambda_+^*(L))$ be a split of the dual lattice into two nonempty sets — condensate candidates (of cardinality $V_0$) and normal-fluid modes — separated by an energy interval: writing $\omega_0=\min_{k\in\Lambda^*}\omega(k)$ and $e_k=\omega(k)-\omega_0$, all condensate-candidate modes satisfy $e_k\le a_L$ and all normal-fluid modes satisfy $e_k\ge b_L$ for some thresholds $0\le a_L<b_L$, and the split has relative energy gap $\delta_L^{-1}$ with $\delta_L=\max_{k\in\Lambda_0^*}e_k/\min_{k\in\Lambda_+^*}e_k<1$. Let $\mu_1^{(L)}$ be the corresponding factorized supercritical measure of the source's main theorem (eq. (2.10) of arXiv:1806.01806; Theorem 2.2 in the arXiv version, Theorem 1 in the published version): the product of a critical Gaussian normal fluid on $\Lambda_+^*$ and a condensate measure on $\Lambda_0^*$ carrying the excess mass $\Delta=\rho-\rho_c(L)$ per site, with condensate weight $\mathrm{e}^{-E_0[\Phi]\,(1-\rho_c/\Delta)}\prod_{k\in\Lambda_+^*}\left(1-E_0[\Phi]L^{-d}/(e_k\Delta)\right)^{-1}\delta(\rho_0[\Phi]-\Delta)$, where $E_0[\Phi]=\int_{\Lambda_0^*}dk\,e_k|\Phi_k|^2$ and $\rho_0[\Phi]=\frac1V\int_{\Lambda_0^*}dk\,|\Phi_k|^2$, and integrals over subsets of the dual lattice use the discrete convention $\int_{\Lambda^*}dk\,g(k):=\frac1V\sum_{k\in\Lambda^*}g(k)$; here $\rho_c(L)$ is the critical norm density — the expected mass density carried by the critical Gaussian normal fluid, $\rho_c(L)=\int_{\Lambda_+^*}dk/(\omega(k)-\omega_0)$, with limit $\rho_\infty$ as $L\to\infty$. (The source's simpler variant $\mu_1'$ with a condensate uniformly distributed on a sphere, eq. (2.12), is a different factorized supercritical measure and is not the comparison measure used here.) Define $\varepsilon_L:=\max(2\delta_L,\,V^{-2}\rho_c^{-2}\sum_{k\in\Lambda_+^*}e_k^{-2})$; a split is called admissible if it satisfies the standing smallness conditions of the source's Theorem 1, namely $\delta_L\le 1/2$ and $\varepsilon_L\le\Delta^2/(2^5V_0^2\rho^2)$.

The open problem is to pin down the scaling conditions under which the full configuration law converges, i.e. $W_2(\mu_0^{(L)},\mu_1^{(L)})\to 0$ as $L\to\infty$, in the unscaled $L^2$ configuration norm and without invoking the $L^{-d/2}$ cancellation available for local observables; here a scaling condition is a criterion relating the local behaviour of $\omega$ near its minima (decay of $\omega$, Hessian anisotropy) to the $L$-scaling of the split parameters ($V_0$, $\delta_L$, and hence $\varepsilon_L$). A complete resolution has either of two forms: constructive — for one explicitly parametrized family of the dispersion relations considered above (for instance a single fixed $\omega$ carried through the limit $L\to\infty$, or a family parametrized by a fixed anisotropy ratio), exhibiting an admissible split and coupling with $W_2\to 0$ and proving, within that same family, the matching necessary conditions — that no weaker $L$-scaling of the split parameters suffices for convergence — so that the condition is sharp (necessary and sufficient) there; or negative — establishing that no admissible choice of splits makes the Wasserstein distance vanish for the class of dispersion relations considered above; the known admissible splits give at best $\varepsilon_L=O(L^{-2\kappa})$ with $\kappa<d/2$, while the bound $W_2\le C_2L^{d/2}\varepsilon_L^{1/4}$ would require roughly $\varepsilon_L=o(L^{-2d})$ for convergence.

Two further limitations documented in the source paper bear directly on this question but are not themselves required deliverables of it: the split constructions of its Lemma 1 are worst-case and 'not always optimal', losing sharpness for strongly anisotropic Hessians, so sharper dispersion-specific criteria for the minimal condensate energy gap and for the uniform bound $M_0$ on the number of condensate modes ($V_0\le M_0$ for all $L$) remain to be found; and the condensate multiplicity and the distribution of the supercritical mass can depend on the lattice size (parity classes; subsequences such as $L=2^N$ when a minimum lies off the dual lattice), so the subsequential limiting behaviour of the constrained measure conditions any full-sequence limit.

The extension of the constructive coupling and factorization approach to weakly nonlinear or interacting lattice models (e.g. discrete nonlinear Schrödinger canonical ensembles at finite coupling) is an acknowledged further direction of the source paper, but it is a distinct research program and is not part of the completion boundary of this problem.

References

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