On the Dynamics of Free-Fermionic Tau-Functions at Finite Temperature

Abstract

In this work we explore an instance of the $\tau$-function of vertex type operators, specified in terms of a constant phase shift in a free-fermionic basis. From the physical point of view this $\tau$-function has multiple interpretations: as a correlator of Jordan-Wigner strings, a Loschmidt Echo in the Aharonov-Bohm effect, or the generating function of the local densities in the Tonks-Girardeau gas. We present the $\tau$-function as a form-factors series and tackle it from four vantage points: (i) we perform an exact summation and express it in terms of a Fredholm determinant in the thermodynamic limit, (ii) we use bosonization techniques to perform partial summations of soft modes around the Fermi surface to acquire the scaling at zero temperature, (iii) we derive large space and time asymptotic behavior for the thermal Fredholm determinant by relating it to effective form-factors with an asymptotically similar kernel, and (iv) we identify and sum the important basis elements directly through a tailor-made numerical algorithm for finite-entropy states in a free-fermionic Hilbert space. All methods confirm each other. We find that, in addition to the exponential decay in the finite-temperature case the dynamic correlation functions exhibit an extra power law in time, universal over any distribution and time scale.

arXiv comment83 pages, 21 figures. Revised based on reviews on SciPost Physics Core

References in corpus (83)

• Nonequilibrium dynamics of closed interacting quantum systems
• One dimensional Bosons: From Condensed Matter Systems to Ultracold Gases
• Quantum Inverse Scattering Method and Correlation Functions
• Transport in Out-of-Equilibrium XXZ Chains: Exact Profiles of Charges and Currents
• Non-equilibrium coherence dynamics in one-dimensional Bose gases
• Emergent hydrodynamics in integrable quantum systems out of equilibrium
• Time evolution of local observables after quenching to an integrable model
• One-Dimensional Quantum Liquids: Beyond the Luttinger Liquid Paradigm
• Fractional spinon excitations in the quantum Heisenberg antiferromagnetic chain
• On the Numerical Evaluation of Fredholm Determinants
• Quantum rings for beginners: Energy spectra and persistent currents
• Universal theory of nonlinear Luttinger liquids
• The Quench Action
• Decoherence and the Loschmidt echo
• Dynamical density-density correlations in the one-dimensional Bose gas
• Multispinon continua at zero and finite temperature in a near-ideal Heisenberg chain
• Probing the Excitations of a Lieb-Liniger Gas from Weak to Strong Coupling
• Finite Temperature Correlation Functions in Integrable QFT
• Form factor approach to dynamical correlation functions in critical models
• Dynamical structure factor of one-dimensional Bose gases: experimental signatures of beyond-Luttinger liquid physics
• Temperature Correlations of Quantum Spins
• Correlation functions and transport coefficients in generalised hydrodynamics
• Finite temperature correlations in the Lieb-Liniger 1D Bose gas
• Non-Unitary Spin-Charge Separation in One-Dimensional Fermion Gas
• A comment on finite temperature correlations in integrable QFT
• Correlation functions of integrable models: a description of the ABACUS algorithm
• Free fermions and tau-functions
• Form factor expansion for thermal correlators
• Form factor approach to the asymptotic behavior of correlation functions in critical models
• Green's function for magnetically incoherent interacting electrons in one dimension
• Exact prefactors in static and dynamic correlation functions of 1D quantum integrable models: applications to the Calogero-Sutherland, Lieb-Liniger and XXZ models
• Nonuniversal prefactors in correlation functions of 1D quantum liquids
• Low temperature correlation functions in integrable models: Derivation of the large distance and time asymptotics from the form factor expansion
• Riemann-Hilbert approach to a generalized sine kernel and applications
• Finite-temperature form factors in the free Majorana theory
• Spin decoherence due to a randomly fluctuating spin bath
• Correlation Functions of One-Dimensional Lieb-Liniger Anyons
• Finite Temperature Dynamical Correlations in Massive Integrable Quantum Field Theories
• Zero temperature correlation functions for the impenetrable fermion gas
• Thermal form-factor approach to dynamical correlation functions of integrable lattice models
• Thermal form factors of the XXZ chain and the large-distance asymptotics of its temperature dependent correlation functions
• Dynamics of the attractive 1D Bose gas: analytical treatment from integrability
• On the structure of typical states of a disordered Richardson model and many-body localization
• Generalized hydrodynamics regime from the thermodynamic bootstrap program
• Finite temperature correlation functions from form factors
• The impact of the injection protocol on an impurity's stationary state
• Impurity Green's function of a one-dimensional Fermi-gas
• Exact correlations in the Lieb-Liniger model and detailed balance out-of-equilibrium
• Temperature correlators in the two-component one-dimensional gas
• Particle-hole pairs and density-density correlations in the Lieb-Liniger model
• Low-temperature large-distance asymptotics of the transversal two-point functions of the XXZ chain
• Finite-Temperature Form Factors: a Review
• Density form factors of the 1D Bose gas for finite entropy states
• One-Dimensional Impenetrable Anyons in Thermal Equilibrium. I. Anyonic Generalization of Lenard's Formula
• One-Dimensional Impenetrable Anyons in Thermal Equilibrium. II. Determinant Representation for the Dynamic Correlation Functions
• Thermodynamic bootstrap program for integrable QFT's: Form factors and correlation functions at finite energy density
• Finite temperature and quench dynamics in the Transverse Field Ising Model from form factor expansions
• Strongly interacting mesoscopic systems of anyons in one dimension
• Equilibrium dynamics of the XX chain
• On the Finite Temperature Formalism in Integrable Quantum Field Theories
• A systematic $1/c$-expansion of form factor sums for dynamical correlations in the Lieb-Liniger model
• Long-time and large-distance asymptotic behavior of the current-current correlators in the non-linear Schrödinger model
• Time and temperature-dependent correlation function of an impurity in a one-dimensional Fermi gas as a Fredholm determinant
• Time and temperature dependent correlation functions of the 1D impenetrable electron gas
• Temperature Correlation of Quantum Spins
• Zero temperature momentum distribution of an impurity in a polaron state of one-dimensional Fermi and Tonks-Girardeau gases
• Microscopic approach to a class of 1D quantum critical models
• Correlation functions of one-dimensional strongly interacting two-component gases
• Euler-scale dynamical fluctuations in non-equilibrium interacting integrable systems
• One-Dimensional Impenetrable Anyons in Thermal Equilibrium. III. Large distance asymptotics of the space correlations
• How generalized hydrodynamics time evolution arises from a form factor expansion
• Effective free-fermionic form factors and the XY spin chain
• Integral operators with the generalized sine-kernel on the real axis
• Correlations in the impenetrable electron gas
• One-Dimensional Impenetrable Anyons in Thermal Equilibrium. IV. Large Time and Distance Asymptotic Behavior of the Correlation Functions
• High-temperature analysis of the transverse dynamical two-point correlation function of the XX quantum-spin chain
• Anderson's orthogonality catastrophe
• Correlation functions and momentum distribution of one-dimensional hard-core anyons in optical lattices
• Large-Distance Asymptotic Behavior of the Correlation Functions of 1D Impenetrable Anyons at Finite Temperatures
• The two particle-hole pairs contribution to the dynamic correlation functions of quantum integrable models
• Statistical mechanics of integrable quantum spin systems
• Late-time large-distance asymptotics of the transverse correlation functions of the XX chain in the space-like regime
• The relevant excitations for the one-body function in the Lieb-Liniger model

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• On Landauer--Büttiker formalism from a quantum quench
• Kubo-Martin-Schwinger relation for an interacting mobile impurity
• Out-of-equilibrium dynamics of the Kitaev model on the Bethe lattice via coupled Heisenberg equations
• One-dimensional Fermi polaron after a kick: two-sided singularity of the momentum distribution, Bragg reflection and other exact results
• Universal properties and dynamical bosonization of strongly interacting one-dimensional anyons
• Dynamical correlation functions in the Ising field theory
• Two-spinon effects on the thermal Tonks-Girardeau gas