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Phan Thành Nam

Publications and source records attributed to Phan Thành Nam.

At least 19 recordsLinked to original sources
arXiv2026

$\mathcal{P}(Φ)_2$ Theory from many-body quantum Gibbs states

We derive the $\mathcal{P}(Φ)_2$ measure on the two-dimensional unit torus as the rigorous limit of many-body quantum Gibbs states. In particular, the quantum model corresponding to the $Φ^{2p}_2$ measure is formulated in the grand-canonical ensemble with a general symmetric $p$-body interaction potential which is not required to have a factorized form. Unlike in the $Φ^4_2$ case investigated by Fröhlich--Knowles--Schlein--Sohinger in \cite{FKSS25}, in the treatment of higher-order interactions, Wick renormalization generates a full hierarchy of lower-order interactions which we organize systematically using a graphical formalism. One key ingredient of our analysis is a logarithmic stability estimate for both the nonlocal Hartree functional and the many-body Hamiltonian that is uniform in the interaction range, allowing us to use the positivity of the leading $p$-body interaction to control all lower-order terms generated by the Wick counterterms.

math-ph↗
arXiv2026

Dynamical classical-field limit of Bosonic Gibbs states: Renormalized Hartree NLS correlations in 2D and 3D

In this paper we derive the time-dependent correlation functions of the renormalized Hartree NLS equation on the torus $\mathbb T^d$, $d=2,3$, from the corresponding bosonic many-body Gibbs dynamics. In contrast with the 1D problem studied earlier by Fröhlich, Knowles, Schlein and Sohinger, in higher dimensions the scaled quantum particle number is not uniformly bounded and the relevant classical fields require Wick renormalization. Our proof combines convergence of the quantum generators in weighted Hilbert spaces with uniqueness for positive solutions of the limiting Liouville equation that are dominated by the Gibbs measure.

math-ph↗
arXiv2026

Derivation of the focusing $Φ^6_1$ measure in the optimal mass regime from many-body quantum Gibbs states

We derive the focusing $Φ^6_1$ measure on the torus $\mathbb{T}$ as the high-temperature/mean-field limit of many-body quantum Gibbs states with an attractive three-body interaction. The main difficulty in the focusing setting is to relate the classical mass cutoff to the quantum particle-number cutoff. Our result reaches the optimal mass threshold for the classical field identified by Oh, Sosoe, and Tolomeo (2022), and thereby extends the earlier work of Rout and Sohinger (2025). At the critical threshold, the short-range interaction is allowed to shrink to a Dirac delta function on a logarithmic scale in the temperature parameter. Strictly below the threshold, the same convergence holds with a polynomial dependence on the temperature. Moreover, we establish a quantum-level phase transition at the same mass threshold. The proof develops the variational framework of Lewin, Nam, and Rougerie (2015) in the focusing setting and relies on two new ingredients: a non-factorized trial state construction and a delicate tail estimate for the interacting lower symbol. These allow us to control the localization and relative entropy errors caused by the particle-number cutoff, as well as the contribution of the focusing exponential weight.

math-ph↗
arXiv2026

On the asymptotic behavior at the kinetic time of a weakly interacting Fermi gas

This paper is devoted to the dynamics of a weakly interacting Fermi gas at the kinetic time regime $t\sim λ^{-2}$ where $λ\ll 1$ is the strength of the interaction potential. We prove that if the initial state is close to equilibrium, then the two-point time correlation function of the many-body quantum dynamics can be computed effectively. In fact, we show that its leading order behavior is determined completely by the collisional frequency of the Boltzmann-Nordheim collision operator at equilibrium. This settles a prediction by Lukkarinen-Spohn, and thus gives a justification of the quantum Boltzmann equation from many-body quantum mechanics.

math-ph↗
arXiv2026

The Correlation Energy of the Electron Gas in the Mean-Field Regime

We prove a rigorous lower bound on the correlation energy of interacting fermions in the mean-field regime for a wide class of singular interactions, including the Coulomb potential. Combined with the upper bound obtained in \cite{ChrHaiNam-23b}, our result establishes an analogue of the Gell-Mann--Brueckner formula $c_{1}ρ\log\left(ρ\right)+c_{2}ρ$ for the correlation energy of the electron gas in the high-density limit. Moreover, our analysis allows us to go beyond mean-field scaling while still covering the same class of potentials.

math-ph↗
arXiv2026

Derivation of Gibbs measure from Gibbs state with the fractional Bessel interaction in Two Dimensions

We derive the classical Gibbs measure on $\mathbb{T}^2$ associated with the fractional Bessel interaction potential $\widehat{v}_β(k)=\langle k\rangle^{-β}$ from a renormalized grand-canonical quantum Bose gas with the same interaction. Our result covers the whole range $\frac32<β\leq2$, where $\widehat{v}_β(k)$ is not summable and the quantum model cannot be written in the usual density-square form, as the associated self-energy diverges. We therefore need to renormalize the zero mode by a centered number-fluctuation term and then develop a detailed analysis for the high-frequency remainders. All this allows us to implement a low-frequency localization and obtain the convergence of the quantum relative free energy to the classical fractional-Bessel free energy, as well as the convergence of the reduced density matrices to the limiting Gibbs measure.

math-ph↗
arXiv2026

Eigenvalue asymptotics of Müller minimizers for atoms and molecules

We study the spectral properties of minimizers of the Müller functional for atoms and molecules with $N$ electrons and total nuclear charge $Z$. We prove that under some suitable assumptions on $Z$ and $N$, the $k$-th eigenvalue of a Müller minimizer $γ_*$ behaves as $A_* k^{-8/3}$ when $k\to \infty$, with a constant $A_*>0$ determined explicitly by the density of $γ_*$. In particular, in the atomic case $V=Z|x|^{-1}$ our assumption holds if $Z$ is sufficiently large and $N\le Z- C_0 Z^{1/3}$. While our proof is inspired by Sobolev's work on the asymptotic behavior of the one-particle density matrix of Schrödinger ground states, the analysis in Müller theory requires several new ingredients concerning both the singular behavior of the integral kernel of the minimizers near the diagonal and the decay properties at infinity.

math-ph↗
arXiv2026

The Huang-Yang formula for the low-density Fermi gas: upper bound

We study the ground state energy of a gas of spin $1/2$ fermions with repulsive short-range interactions. We derive an upper bound that agrees, at low density $ρ$, with the Huang-Yang conjecture. The latter captures the first three terms in an asymptotic low-density expansion, and in particular the Huang-Yang correction term of order $ρ^{7/3}$. Our trial state is constructed using an adaptation of the bosonic Bogoliubov theory to the Fermi system, where the correlation structure of fermionic particles is incorporated by quasi-bosonic Bogoliubov transformations. In the latter, it is important to consider a modified zero-energy scattering equation that takes into account the presence of the Fermi sea, in the spirit of the Bethe-Goldstone equation.

math-ph↗
arXiv2026

Dispersive estimates and long-time validity for Bogoliubov dynamics of interacting Bose gases

We consider the Bogoliubov approximation for the many-body quantum dynamics of weakly interacting Bose gases and establish a uniform-in-time validity of the Bogoliubov theory. The proof relies on a detailed analysis of the dispersive behavior of the symplectic Bogoliubov dynamics, which allows for a rigorous derivation of the Bogoliubov theory as an effective description of quantum fluctuations around the Bose-Einstein condensate on all time scales.

math-ph↗
arXiv2026

Pseudo-relativistic fermionic systems with attractive Yukawa potential

We study the Hartree-Fock and Hartree-Fock-Bogoliubov theories for a large fermionic system with the pseudo-relativistic kinetic energy and an attractive Yukawa interaction potential. We prove that the system is stable if and only if the total mass does not excess a critical value, and investigate the existence and properties of ground states in both sub-critical and critical mass regimes.

math-ph↗
arXiv2025

Finite-Rank Optimizers for the mass--supercritical Lieb--Thirring and Hardy--Lieb--Thirring Inequalities

We establish the existence of finite-rank operators for an interpolation version of the Lieb--Thirring inequality in the mass--supercritical case, thereby extending a result of Hong, Kwon, and Yoon in 2019 to the full parameter regime. Our method also applies to the Hardy--Lieb--Thirring inequality, where the existence of optimizers faces additional difficulties due to the singularity of the inverse-square potential.

math-ph↗
arXiv2025

Mass-Critical Neutron Stars in the Hartree-Fock and Hartree-Fock-Bogoliubov Theories

We investigate the ground states of neutron stars and white dwarfs in the Hartree-Fock (HF) and Hartree-Fock-Bogoliubov (HFB) theories. It is known that the system is stable below a critical mass, which depends on the gravitational constant, while it becomes unstable if the total mass exceeds the critical mass. We prove that if the total mass is at the critical mass, then the HFB minimizers do not exist for any gravitational constant, while the HF minimizers exist for every gravitational constant except for a countable set, which is fully characterized by the Gagliardo-Nirenberg inequality for orthonormal systems. Our results complement the existence results in the sub-critical mass case established in [E. Lenzmann and M. Lewin, Duke Math. J., 2010].

math.AP↗
arXiv2025

The Gibbs state of the mean-field Bose gas

We consider the homogeneous mean-field Bose gas at temperatures proportional to the critical temperature of its Bose-Einstein condensation phase transition. We prove a trace norm approximation for the grand canonical Gibbs state in terms of a reference state, which is given by a convex combination of products of coherent states and Gibbs states associated with certain temperature-dependent Bogoliubov Hamiltonians. The convex combination is expressed as an integral over a Gibbs distribution of a one-mode $Φ^4$-theory describing the condensate. This result justifies an analogue of Lee and Yang's extension of Bogoliubov theory to positive temperatures, and it allows us to derive various limiting distributions for the number of particles in the condensate, as well as precise formulas for the one- and two-particle density matrices of the Gibbs state. Key ingredients of our proof, which are of independent interest, include two novel abstract correlation inequalities. The proof of one of them is based on an application of an infinite-dimensional version of Stahl's theorem.

math-ph↗
arXiv2025

$Φ^4_3$ Theory from many-body quantum Gibbs states

We derive the $Φ^4_3$ measure on the torus as a rigorous limit of the quantum Gibbs state of an interacting Bose gas. To be precise, starting from many-body quantum mechanics, where the problem is linear and regular but involving non commutative operators, we justify the emergence of the $Φ^4_3$ measure as a semiclassical limit which captures the formation of Bose--Einstein condensation just above the critical temperature. We employ and develop several tools from both stochastic quantization and many-body quantum mechanics. Since the quantum problem is typically formulated using a nonlocal interaction potential, our first key step involves approximating the $Φ^4_3$ measure through a Hartree measure with nonlocal interaction, achieved by developing new techniques in paracontrolled calculus. The connection between the quantum problem and the Hartree measure emerges through a variational interplay between classical and quantum models.

math-ph↗
arXiv2025

Minimizing sequences of Sobolev inequalities revisited

We give a new proof of the compactness of minimizing sequences of the Sobolev inequalities in the critical case. Our approach relies on a simplified version of the concentration-compactness principle, which does not require any refinement of the Sobolev embedding theorem.

math.AP↗
arXiv2025

The Huang-Yang conjecture for the low-density Fermi gas

Our work establishes a three-term asymptotic expansion of the ground state energy of a dilute gas of spin $1/2$ fermions with repulsive short-range interactions, validating a formula predicted by Huang and Yang in 1957. The formula is universal in the sense that it holds for a large class of interaction potentials and depends on those only via their scattering length. We have recently proved an upper bound on the ground state energy of the desired form, and the present work completes the program by proving the matching lower bound.

math-ph↗
arXiv2025

Positive-density ground states of the Gross-Pitaevskii equation

We consider the nonlinear Gross-Pitaevskii equation at positive density, that is, for a bounded solution not tending to 0 at infinity. We focus on infinite ground states, which are by definition minimizers of the energy under local perturbations. When the Fourier transform of the interaction potential takes negative values we prove the existence of a phase transition at high density, where the constant solution ceases to be a ground state. The analysis requires mixing techniques from elliptic PDE theory and statistical mechanics, in order to deal with a large class of interaction potentials.

math-ph↗