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Zhituo Wang

Publications and source records attributed to Zhituo Wang.

At least 19 recordsLinked to original sources
arXiv2023

Phase Transitions in the Hubbard Model on the Square Lattice

We study the low temperature properties of the two-dimensional weakly interacting Hubbard model on $\ZZZ^2$ with renormalized chemical potential $μ=2-μ_0$, $μ_0=10^{-10}$ fixed, in which case the Fermi surface is close to a perfect square. Using fermionic functional integrals, cluster expansions and rigorous renormalization group analysis, we prove that the perturbation series for the two-point Schwinger function is analytic in the coupling constant $ł$ in the domain $ł\in\RR_T=\{ł\in\RRR,\vertλ\log^2(μ_0T/C_1)|\le C_2\}$ for any fixed temperature $T>0$, suggesting that there is a phase transition with critical temperature $T_c= \frac{C_1}{\m_0}\exp{(-C^{1/2}_2|λ|^{-1/2})}$. Here $C_1, C_2$ are positive constants independent of $T$ and $ł$. We also prove that the second derivative of the momentum space self-energy function w.r.t. the external momentum is not uniformly bounded, suggesting that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. This result can be viewed as a first step towards rigorous study of the Fermi liquid-non Fermi liquid crossover phenomenon.

math-ph↗
arXiv2023

Honeycomb Hubbard Model at van Hove Filling

This paper is devoted to the rigorous study of the low temperature properties of the two-dimensional weakly interacting Hubbard model on the honeycomb lattice in which the renormalized chemical potential $μ$ has been fixed such that the Fermi surface consists of a set of exact triangles. Using renormalization group analysis around the Fermi surface, we prove that this model is {\it not} a Fermi liquid in the mathematically precise sense of Salmhofer. The main result is proved in two steps. First we prove that the perturbation series for Schwinger functions as well as the self-energy function have non-zero radius of convergence when the temperature $T$ is above an exponentially small value, namely ${T_0\sim \exp{(-C|λ|^{-1/2})}}$. Then we prove the necessary lower bound for second derivatives of self-energy w.r.t. the external momentum and achieve the proof.

math-ph↗
arXiv2021

On the Sector Counting Lemma

In this short note we prove a sector counting lemma for a class of Fermi surface on the plane which are $C^2$-differentiable and strictly convex. This result generalizes the one proved in \cite{FKT} for the class of $C^{2+r}$-differentiable, $r\ge3$, strictly convex and strongly asymmetric Fermi surfaces, and the one proved in \cite{FMRT} and \cite{BGM1}, for the class of $C^2$-differentiable, strictly convex and central symmetric Fermi surfaces. This new sector counting lemma can be used to construct interacting many-fermion models for the doped graphene, in which the Fermi surface is extended and quasi-symmetric.

math-ph↗
arXiv2019

Non-Fermi Liquid Behaviors in the Hubbard model on the Honeycomb lattice

In this paper we study the low temperature ($T\rightarrow 0$) behaviors of the weakly interacting Hubbard model on the honeycomb lattice with a bare chemical potential, which takes values in a small neighborhood of the renormalized chemical potential $μ$, which is fixed to be $1$. We prove that the two-point Schwinger's function is an analytic function of the coupling constant $λ$ in the domain $|λ|\cdot|\log T|^2<1$. But the ground state of this model is not a Fermi liquid, because analytic properties of self-energy don't satisfy the Salmhofer's criterion on the Fermi liquids at finite temperature.

math-ph↗
arXiv2018

Constructive Renormalization of the $2$-dimensional Grosse-Wulkenhaar Model

We study a quartic matrix model with partition function $Z=\int d\ M\exp{\rm Tr}\ (-ΔM^2-\fracλ{4}M^4)$. The integral is over the space of Hermitian $(Λ+1)\times(Λ+1)$ matrices, the matrix $Δ$, which is not a multiple of the identity matrix, encodes the dynamics and $λ>0$ is a scalar coupling constant. We proved that the logarithm of the partition function is the Borel sum of the perturbation series, hence is a well defined analytic function of the coupling constant in certain analytic domain of $λ$, by using the multi-scale loop vertex expansions. All the non-planar graphs generated in the perturbation expansions have been taken care of on the same footing as the planar ones. This model is derived from the self-dual $ϕ^4$ theory on the 2 dimensional Moyal space, also called the 2 dimensional Grosse-Wulkenhaar model. This would also be the first fully constructed matrix model which is non-trivial and not solvable.

math-ph↗
arXiv2017

Quantum Quench dynamics in Non-local Luttinger Model: Rigorous Results

We investigate, in the Luttinger model with fixed box potential, the time evolution of an inhomogeneous state prepared as a localized fermion added to the noninteracting ground state. We proved that, if the state is evolved with the interacting Hamiltonian, the averaged density has two peaks moving in opposite directions, with a constant but renormalized velocity. We also proved that a dynamical `Landau quasi-particle weight' appears in the oscillating part of the averaged density, asymptotically vanishing with large time. The results are proved with the Mattis-Lieb diagonalization method. A simpler proof with the exact Bosonization formulas is also provided.

math-ph↗
arXiv2015

Corrected Loop Vertex Expansion for Phi42 Theory

This paper is an extended erratum to J. Math. Phys.53, 042302 (2012) and arXiv:1104.3443, in which the classic construction and Borel summability of the phi^4_2 Euclidean quantum field theory was revisited combining a multi-scale analysis with the constructive method called Loop Vertex Expansion (LVE). Unfortunately we discovered an important error in the method of J. Math. Phys.53, 042302 (2012). We explain the mistake, and provide a new, correct construction of the phi^4_2 theory according to the LVE.

math-ph↗
arXiv2014

Quantum Quench for inhomogeneous states in the non-local Luttinger model

In the Luttinger model with non-local interaction we investigate, by exact analytical methods, the time evolution of an inhomogeneous state with a localized fermion added to the non interacting ground state. In absence of interaction the averaged density has two peaks moving in opposite directions with constant velocities. If the state is evolved with the interacting Hamiltonian two main effects appear. The first is that the peaks have velocities which are not constant but vary between a minimal and maximal value. The second is that a dynamical `Landau quasi-particle weight' appears in the oscillating part of the averaged density, asymptotically vanishing with time, as consequence of the fact that fermions are not excitations of the interacting Hamiltonian.

cond-mat.stat-mech↗
arXiv2014

Construction of the Noncommutative Complex Ball

We describe the construction of the noncommutative complex ball whose commutative analog is the Hermitian symmetric space $D=SU(m,1)/U(m)$, with the method of coherent state quantization. In the commutative limit we obtain the standard manifold. We consider also a quantum field theory model on the noncommutative manifold.

math-ph↗
arXiv2014

Constructive Renormalization for $Φ^{4}_2$ Theory with Loop Vertex Expansion

In this paper we construct the 2 dimensional Euclidean $ϕ^4$ quantum field theory using the method of loop vertex expansion. We reproduce the results of standard constructive theory, for example the Borel summability of the Schwinger functions in the coupling constant. Our method should be also suitable for the future construction of Grosse-Wulkenhaar models on non-commutative space-time.

math-ph↗
arXiv2013

How to Resum Feynman Graphs

In this paper we reformulate in a simpler way the combinatoric core of constructive quantum field theory We define universal rational combinatoric weights for pairs made of a graph and one of its spanning trees. These weights are nothing but the percentage of Hepp's sectors in which the tree is leading the ultraviolet analysis. We explain how they allow to reshuffle the divergent series formulated in terms of Feynman graphs into convergent series indexed by the trees that these graphs contain. The Feynman graphs to be used are not the ordinary ones but those of the intermediate field representation, and the result of the reshuffling is called the Loop Vertex Expansion.

math-ph↗
arXiv2012

Constructive Renormalization of 2-dimensional Grosse-Wulkenhaar Model

In this talk we briefly report the recent work on the construction of the 2-dimensional Grosse-Wulkenhaar model with the method of loop vertex expansion. We treat renormalization with this new tool, adapt Nelson's argument and prove Borel summability of the perturbation series. This is the first non-commutative quantum field theory model to be built in a non-perturbative sense.

hep-th↗
arXiv2012

Construction of the noncommutative rank I Bergman domain

In this paper we present a harmonic oscillator realization of the most degenerate discrete series representations of the SU(2,1) group and the deformation quantization of the coset space $D=SU(2,1)/U(2)$ with the method of coherent state quantization.

math-ph↗
arXiv2010

Quantum Field Theory on quantized Bergman domain

We present an oscillator realization of discrete series representations of group SU(2,2). We give formulas for the coherent state star-product quantization of a Bergman domain $D$. A formulation of a (regularized) non-commutative scalar field on a quantized $D$ is given.

math-ph↗
arXiv2010

Loop Vertex Expansion for Phi^2k Theory in Zero Dimension

In this paper we extend the method of loop vertex expansion to interactions with degree higher than 4. As an example we provide through this expansion an explicit proof that the free energy of Phi^2k scalar theory in zero dimension is Borel-Le Roy summable of order k-1. We detail the computations in the case of a Phi^6 interaction.

math-ph↗