arXiv subjects

Giuseppe Mussardo

Publications and source records attributed to Giuseppe Mussardo.

At least 37 records · Page 2Linked to original sources

arXiv2020

Finite Temperature Off-Diagonal Long-Range Order for Interacting Bosons

Characterizing the scaling with the total particle number ($N$) of the largest eigenvalue of the one--body density matrix ($λ_0$), provides informations on the occurrence of the off-diagonal long-range order (ODLRO) according to the Penrose-Onsager criterion. Setting $λ_0\sim N^{\mathcal{C}_0}$, then $\mathcal{C}_0=1$ corresponds to ODLRO. The intermediate case, $0<\mathcal{C}_0<1$, corresponds for translational invariant systems to the power-law decaying of (non-connected) correlation functions and it can be seen as identifying quasi-long-range order. The goal of the present paper is to characterize the ODLRO properties encoded in $\mathcal{C}_0$ [and in the corresponding quantities $\mathcal{C}_{k \neq 0}$ for excited natural orbitals] exhibited by homogeneous interacting bosonic systems at finite temperature for different dimensions. We show that $\mathcal{C}_{k \neq 0}=0$ in the thermodynamic limit. In $1D$ it is $\mathcal{C}_0=0$ for non-vanishing temperature, while in $3D$ $\mathcal{C}_0=1$ ($\mathcal{C}_0=0$) for temperatures smaller (larger) than the Bose-Einstein critical temperature. We then focus our attention to $D=2$, studying the $XY$ and the Villain models, and the weakly interacting Bose gas. The universal value of $\mathcal{C}_0$ near the Berezinskii--Kosterlitz--Thouless temperature $T_{BKT}$ is $7/8$. The dependence of $\mathcal{C}_0$ on temperatures between $T=0$ (at which $\mathcal{C}_0=1$) and $T_{BKT}$ is studied in the different models. An estimate for the (non-perturbative) parameter $ξ$ entering the equation of state of the $2D$ Bose gases, is obtained using low temperature expansions and compared with the Monte Carlo result. We finally discuss a double jump behaviour for $\mathcal{C}_0$, and correspondingly of the anomalous dimension $η$, right below $T_{BKT}$ in the limit of vanishing interactions.

cond-mat.stat-mech↗

arXiv2020

Approaching the Self-Dual Point of the Sinh-Gordon model

One of the most striking but mysterious properties of the sinh-Gordon model (ShG) is the $b \rightarrow 1/b$ self-duality of its $S$-matrix, of which there is no trace in its Lagrangian formulation. Here $b$ is the coupling appearing in the model's eponymous hyperbolic cosine present in its Lagrangian, $\cosh(bϕ)$. In this paper we develop truncated spectrum methods (TSMs) for studying the sinh-Gordon model at a finite volume as we vary the coupling constant. We obtain the expected results for $b \ll 1$ and intermediate values of $b$, but as the self-dual point $b=1$ is approached, the basic application of the TSM to the ShG breaks down. We find that the TSM gives results with a strong cutoff $E_c$ dependence, which disappears according only to a very slow power law in $E_c$. Standard renormalization group strategies -- whether they be numerical or analytic -- also fail to improve upon matters here. We thus explore three strategies to address the basic limitations of the TSM in the vicinity of $b=1$. In the first, we focus on the small-volume spectrum. We attempt to understand how much of the physics of the ShG is encoded in the zero mode part of its Hamiltonian, in essence how `quantum mechanical' vs `quantum field theoretic' the problem is. In the second, we identify the divergencies present in perturbation theory and perform their resummation using a supra-Borel approximate. In the third approach, we use the exact form factors of the model to treat the ShG at one value of $b$ as a perturbation of a ShG at a different coupling. In the light of this work, we argue that the strong coupling phase $b > 1$ of the Lagrangian formulation of model may be different from what is naïvely inferred from its $S$-matrix. In particular, we present an argument that the theory is massless for $b>1$.

hep-th↗

arXiv2020

Dynamics of one-dimensional quantum many-body systems in time-periodic linear potentials

We study a system of one-dimensional interacting quantum particles subjected to a time-periodic potential linear in space. After discussing the cases of driven one- and two-particles systems, we derive the analogous results for the many-particles case in presence of a general interaction two-body potential and the corresponding Floquet Hamiltonian. When the undriven model is integrable, the Floquet Hamitlonian is shown to be integrable too. We determine the micro-motion operator and the expression for a generic time evolved state of the system. We discuss various aspects of the dynamics of the system both at stroboscopic and intermediate times, in particular the motion of the center of mass of a generic wavepacket and its spreading over time. We also discuss the case of accelerated motion of the center of mass, obtained when the integral of the coeffcient strenght of the linear potential on a time period is non-vanishing, and we show that the Floquet Hamiltonian gets in this case an additional static linear potential. We also discuss the application of the obtained results to the Lieb-Liniger model.

cond-mat.stat-mech↗

arXiv2020

Exact out-of-equilibrium steady states in the semiclassical limit of the interacting Bose gas

We study the out-of-equilibrium properties of a classical integrable non-relativistic theory, with a time evolution initially prepared with a finite energy density in the thermodynamic limit. The theory considered here is the Non-Linear Schrodinger equation which describes the dynamics of the one-dimensional interacting Bose gas in the regime of high occupation numbers. The main emphasis is on the determination of the late-time Generalised Gibbs Ensemble (GGE), which can be efficiently semi-numerically computed on arbitrary initial states, completely solving the famous quench problem in the classical regime. We take advantage of known results in the quantum model and the semiclassical limit to achieve new exact results for the momenta of the density operator on arbitrary GGEs, which we successfully compare with ab-initio numerical simulations. Furthermore, we determine the whole probability distribution of the density operator (full counting statistics), whose exact expression is still out of reach in the quantum model.

cond-mat.stat-mech↗

arXiv2019

Integrable Floquet Hamiltonian for a Periodically Tilted 1D Gas

An integrable model subjected to a periodic driving gives rise generally to a non-integrable Floquet Hamiltonian. Here we show that the Floquet Hamiltonian of the integrable Lieb--Liniger model in presence of a linear potential with a periodic time--dependent strength is instead integrable and its quasi-energies can be determined using the Bethe ansatz approach. We discuss various aspects of the dynamics of the system at stroboscopic times and we also propose a possible experimental realisation of the periodically driven tilting in terms of a shaken rotated ring potential.

cond-mat.stat-mech↗

arXiv2018

Generalized Riemann Hypothesis, Time Series and Normal Distributions

$L$ functions based on Dirichlet characters are natural generalizations of the Riemann $ζ(s)$ function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. In this paper we address the Generalized Riemann Hypothesis relative to the non-trivial complex zeros of the Dirichlet $L$ functions by studying the possibility to enlarge the original domain of convergence of their Euler product. The feasibility of this analytic continuation is ruled by the asymptotic behavior in $N$ of the series $B_N = \sum_{n=1}^N \cos \bigl( t \log p_n - \arg χ(p_n) \bigr)$ involving Dirichlet characters $χ$ modulo $q$ on primes $p_n$. Although deterministic, these series have pronounced stochastic features which make them analogous to random time series. We show that the $B_N$'s satisfy various normal law probability distributions. The study of their large asymptotic behavior poses an interesting problem of statistical physics equivalent to the Single Brownian Trajectory Problem, here addressed by defining an appropriate ensemble $\mathcal{E}$ involving intervals of primes. For non-principal characters, we show that the series $B_N$ present a universal diffusive random walk behavior $B_N = O(\sqrt{N})$ in view of the Dirichlet theorem on the equidistribution of reduced residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes. This purely diffusive behavior of $B_N$ implies that the domain of convergence of the infinite product representation of the Dirichlet $L$-functions for non-principal characters can be extended from $\Re(s) > 1$ down to $\Re (s) = \frac{1}{2}$, without encountering any zeros before reaching this critical line.

math.NT↗

arXiv2018

Universal off-diagonal long-range order behaviour for a trapped Tonks-Girardeau gas

The scaling of the largest eigenvalue $λ_0$ of the one-body density matrix of a system with respect to its particle number $N$ defines an exponent $\mathcal{C}$ and a coefficient $\mathcal{B}$ via the asymptotic relation $λ_0 \sim \mathcal{B}\,N^{\mathcal{C}}$. The case $\mathcal{C}=1$ corresponds to off-diagonal long-range order. For a one-dimensional homogeneous Tonks-Girardeau gas, a well known result also confirmed by bosonization gives instead $\mathcal{C}=1/2$. Here we investigate the inhomogeneous case, initially addressing the behaviour of $\mathcal{C}$ in presence of a general external trapping potential $V$. We argue that the value $\mathcal{C}= 1/2$ characterises the hard-core system independently of the nature of the potential $V$. We then define the exponents $γ$ and $β$ which describe the scaling with $N$ of the peak of the momentum distribution and the natural orbital corresponding to $λ_0$ respectively, and we derive the scaling relation $γ+ 2β= \mathcal{C}$. Taking as a specific case the power-law potential $V(x)\propto x^{2n}$, we give analytical formulas for $γ$ and $β$ as functions of $n$. Analytical predictions for the coefficient $\mathcal{B}$ are also obtained. These formulas are derived exploiting a recent field theoretical formulation and checked against numerical results. The agreement is excellent.

cond-mat.stat-mech↗

arXiv2018

Generalized Riemann Hypothesis and Stochastic Time Series

Using the Dirichlet theorem on the equidistribution of residue classes modulo $q$ and the Lemke Oliver-Soundararajan conjecture on the distribution of pairs of residues on consecutive primes, we show that the domain of convergence of the infinite product of Dirichlet $L$-functions of non-principal characters can be extended from $\Re(s) > 1$ down to $\Re(s) > \half$, without encountering any zeros before reaching this critical line. The possibility of doing so can be traced back to a universal diffusive random walk behavior $C_N = {\cal O}(N^{1/2})$ of the series $C_N = \sum_{n=1}^N χ(p_n)$ over the primes $p_n$ where $χ$ is a Dirichlet character, which underlies the convergence of the infinite product of the Dirichlet functions. The series $C_N$ presents several aspects in common with stochastic time series and its control requires to address a problem similar to the Single Brownian Trajectory Problem in statistical mechanics. In the case of the Dirichlet functions of non principal characters, we show that this problem can be solved in terms of a self-averaging procedure based on an ensemble $\CE$ of block variables computed on extended intervals of primes. Those intervals, called {\em inertial intervals}, ensure the ergodicity and stationarity of the time series underlying the quantity $C_N$. The infinity of primes also ensures the absence of rare events which would have been responsible for a different scaling behavior than the universal law $C_N = {\cal O}(N^{1/2})$ of the random walks.

math.NT↗

arXiv2018

Deviations from Off-Diagonal Long-Range Order in One-Dimensional Quantum Systems

A quantum system exhibits off-diagonal long-range order (ODLRO) when the largest eigenvalue $λ_0$ of the one-body-density matrix scales as $λ_0 \sim N$, where $N$ is the total number of particles. Putting $λ_0 \sim N^{\cal C}$ to define the scaling exponent ${\cal C}$, then ${\cal C}=1$ corresponds to ODLRO and ${\cal C}=0$ to the single-particle occupation of the density matrix orbitals. When $0<{\cal C}<1$, ${\cal C}$ can be used to quantify deviations from ODLRO. In this paper we study the exponent ${\cal C}$ in a variety of one-dimensional bosonic and anyonic quantum systems. For the $1D$ Lieb-Liniger Bose gas we find that for small interactions ${\cal C}$ is close to $1$, implying a mesoscopic condensation, i.e. a value of the "condensate" fraction $λ_0/N$ appreciable at finite values of $N$ (as the ones in experiments with $1D$ ultracold atoms). $1D$ anyons provide the possibility to fully interpolate between ${\cal C}=1$ and $0$. The behaviour of ${\cal C}$ for these systems is found to be non-monotonic both with respect to the coupling constant and the statistical parameter.

cond-mat.stat-mech↗

arXiv2017

Yang-Lee Zeros of the Yang-Lee Model

To understand the distribution of the Yang-Lee zeros in quantum integrable field theories we analyse the simplest of these systems given by the two-dimensional Yang-Lee model. The grand-canonical partition function of this quantum field theory, as a function of the fugacity z and the inverse temperature beta, can be computed in terms of the Thermodynamics Bethe Ansatz based on its exact S-matrix. We extract the Yang-Lee zeros in the complex plane by using a sequence of polynomials of increasing order N in z which converges to the grand-canonical partition function. We show that these zeros are distributed along curves which are approximate circles as it is also the case of the zeros for purely free theories. There is though an important difference between the interactive theory and the free theories, for the radius of the zeros in the interactive theory goes continuously to zero in the high-temperature limit beta ->0 while in the free theories it remains close to 1 even for small values of beta, jumping to 0 only at beta = 0.

cond-mat.stat-mech↗

arXiv2017

Non Relativistic Limit of Integrable QFT with fermionic excitations

The aim of this paper is to investigate the non-relativistic limit of integrable quantum field theories with fermionic fields, such as the O(N) Gross-Neveu model, the supersymmetric Sinh-Gordon and non-linear sigma models. The non-relativistic limit of these theories is implemented by a double scaling limit which consists of sending the speed of light c to infinity and rescaling at the same time the relevant coupling constant of the model in such a way to have finite energy excitations. For the general purpose of mapping the space of continuous non-relativistic integrable models, this paper completes and integrates the analysis done in Ref.[1] on the non-relativistic limit of purely bosonic theories.

hep-th↗

arXiv2017

The Coprime Quantum Chain

In this paper we introduce and study the coprime quantum chain, i.e. a strongly correlated quantum system defined in terms of the integer eigenvalues $n_i$ of the occupation number operators at each site of a chain of length $M$. The $n_i$'s take value in the interval $[2,q]$ and may be regarded as $S_z$ eigenvalues in the spin representation $j = (q-2)/2$. The distinctive interaction of the model is based on the coprimality matrix $\bf Φ$: for the ferromagnetic case, this matrix assigns lower energy to configurations where occupation numbers $n_i$ and $n_{i+1}$ of neighbouring sites share a common divisor, while for the anti-ferromagnetic case it assigns lower energy to configurations where $n_i$ and $n_{i+1}$ are coprime. The coprime chain, both in the ferro and anti-ferromagnetic cases, may present an exponential number of ground states whose values can be exactly computed by means of graph theoretical tools. In the ferromagnetic case there are generally also frustration phenomena. A fine tuning of local operators may lift the exponential ground state degeneracy and, according to which operators are switched on, the system may be driven into different classes of universality, among which the Ising or Potts universality class. The paper also contains an appendix by Don Zagier on the exact eigenvalues and eigenvectors of the coprimality matrix in the limit $q \rightarrow \infty$.

cond-mat.str-el↗

arXiv2017

Non Relativistic Limit of Integrable QFT and Lieb-Liniger Models

In this paper we study a suitable limit of integrable QFT with the aim to identify continuous non-relativistic integrable models with local interactions. This limit amounts to sending to infinity the speed of light c but simultaneously adjusting the coupling constant g of the quantum field theories in such a way to keep finite the energies of the various excitations. The QFT considered here are Toda Field Theories and the O(N) non-linear sigma model. In both cases the resulting non-relativistic integrable models consist only of Lieb-Liniger models, which are fully decoupled for the Toda theories while symmetrically coupled for the O(N) model. These examples provide explicit evidence of the universality and ubiquity of the Lieb-Liniger models and, at the same time, suggest that these models may exhaust the list of possible non-relativistic integrable theories of bosonic particles with local interactions.

cond-mat.stat-mech↗

arXiv2016

Equilibration Properties of Classical Integrable Field Theories

We study the equilibration properties of classical integrable field theories at a finite energy density, with a time evolution that starts from initial conditions far from equilibrium. These classical field theories may be regarded as quantum field theories in the regime of high occupation numbers. This observation permits to recover the classical quantities from the quantum ones by taking a proper $\hbar \rightarrow 0$ limit. In particular, the time averages of the classical theories can be expressed in terms of a suitable version of the LeClair-Mussardo formula relative to the Generalized Gibbs Ensemble. For the purposes of handling time averages, our approach provides a solution of the problem of the {\em infinite gap solutions} of the Inverse Scattering Method.

cond-mat.stat-mech↗

arXiv2015

Quench Dynamics in Two-Dimensional Integrable SUSY Models

We analyse quench processes in two dimensional quantum field theories with infinite number of conservation laws which also include fermionic charges that close a $N=1$ supersymmetric algebra. While in general the quench protocol induces a breaking of supersymmetry, we show that there are particular initial states which ensure the persistence of supersymmetry also for the dynamics out of equilibrium. We discuss the conditions that identify such states and, as application, we present the significant cases of the Tricritical Ising Model and the Sine-Gordon model at its supersymmetric point. We also address the issue of the Generalized Gibbs Ensemble in the presence of fermionic conserved charges.

cond-mat.stat-mech↗

arXiv2015

Bound States of Majorana Fermions in Semi-classical Approximation

We derive a semi-classical formula for computing the spectrum of bound states made of Majorana fermions in a generic non-integrable 2d quantum field theory with a set of degenerate vacua. We illustrate the application of the formula in a series of cases, including an asymmetric well potential where the spectra of bosons and fermions may have some curious features. We also discuss the merging of fermionic and bosonic spectra in the presence of supersymmetry. Finally, we use the semi-classical formula to analyse the evolution of the particle spectra in a class of non-integrable supersymmetry models.

cond-mat.stat-mech↗

arXiv2014

Truncated Conformal Space Approach for 2D Landau-Ginzburg Theories

We study the spectrum of Landau-Ginzburg theories in 1+1 dimensions using the truncated conformal space approach employing a compactified boson. We study these theories both in their broken and unbroken phases. We first demonstrate that we can reproduce the expected spectrum of a $Φ^2$ theory (i.e. a free massive boson) in this framework. We then turn to $Φ^4$ in its unbroken phase and compare our numerical results with the predictions of two-loop perturbation theory, finding excellent agreement. We then analyze the broken phase of $Φ^4$ where kink excitations together with their bound states are present. We confirm the semiclassical predictions for this model on the number of stable kink-antikink bound states. We also test the semiclassics in the double well phase of $Φ^6$ Landau-Ginzburg theory, again finding agreement.

hep-th↗

arXiv2014

Energy-pressure relation for low-dimensional gases

A particularly simple relation of proportionality between internal energy and pressure holds for scale invariant thermodynamic systems, including classical and quantum Bose and Fermi ideal gases. One can quantify the deviation from such a relation by introducing the internal energy shift as the difference between the internal energy of the system and the corresponding value for scale invariant gases. We discuss general thermodynamic properties associated to the scale invariance, provide criteria for which the internal energy shift density is a bounded function of temperature. We then study the internal energy shift and deviations from the energy-pressure proportionality in low dimensional models of gases interpolating between the ideal Bose and the ideal Fermi gases, focusing on the Lieb-Liniger model in 1d and on the anyonic gas in 2d. In 1d the internal energy shift is determined from the thermodynamic Bethe ansatz integral equations and an explicit relation for it is given at high temperature. Our results show that the internal energy shift is positive, it vanishes in the two limits of zero and infinite coupling (respectively the ideal Bose and the Tonks Girardeau gas) and it has a maximum at a finite, temperature depending, value of the coupling. Remarkably, at fixed coupling the energy shift density saturates to a finite value for infinite temperature. In 2d we consider systems of Abelian anyons and non Abelian Chern-Simons particles: as it can be seen also directly from a study of the virial coefficients, in the usually considered hard-core limit the internal energy shift vanishes and the energy is just proportional to the pressure, with the proportionality constant being simply the area of the system. Soft-core boundary conditions at coincident points for the two-body wavefunction introduce a length scale, and induce a non-vanishing internal energy shift.

quant-ph↗