arXiv subjects

Frank Göhmann

Publications and source records attributed to Frank Göhmann.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic analysis of a Fredholm determinant occurring in the description of the dynamical correlation functions of the Lieb--Liniger Bose gas

We perform a large-$x$ asymptotic analysis of the Fredholm determinant of an integrable integral operator with generalized sine kernel, where $x$ controls the strength of the oscillations along the integration contour of the operator and will play the role of the distance variable in applications to the correlation functions of integrable quantum systems. Our generalized sine kernel involves a number of functional parameters that will allow us to adapt it to the analysis of the dynamical correlation functions of the Lieb--Liniger model at finite temperature and for all positive values of the coupling constant. It will also allow us to consider a class of equilibrium correlation functions that are governed by generalized Gibbs ensembles. Our work is based on the analysis of a matrix Riemann--Hilbert problem that is canonically connected with our integrable integral operator.

math-ph↗

arXiv2026

Decay of spin helices in XXZ quantum spin chains with single-ion anisotropy

Long-lived spin-helix states facilitate the study of non-equilibrium dynamics in quantum magnets. We consider the decay of transverse spin-helices in antiferromagnetic spin-$S$ XXZ chains with single-ion anisostropy. The spin-helix decay is observable in the time evolution of the local magnetization that we calculate numerically for the system in the thermodynamic limit using infinite time-evolving block decimation simulations. Although the single-ion anisotropy prevents helix states from being eigenstates of the Hamiltonian, they still can be long-lived for appropriately chosen wave numbers. In case of an easy-axis exchange anisotropy the single-ion anisotropy may even stabilize the helices. Within a spin-wave approximation, we obtain a condition giving an estimate for the most stable wave number $Q$ that agrees qualitatively with our numerical results.

cond-mat.str-el↗

arXiv2025

Ballistic particle transport and Drude weight in gases

Owing to the fact that the particle current operator in non-relativistic gases is proportional to the total momentum operator, the particle transport in such systems is always ballistic and fully characterized by a Drude weight $Δ$. The Drude weight can be calculated within linear response theory. It is given by the formula $Δ= 2 πD$, where $D$ is the density of the gas. This holds in any dimension and for every equilibrium ensemble, in particular for generalized Gibbs ensembles that describe possible equilibrium states of isolated integrable quantum systems. In the canonical ensemble case, the Drude weight can be equivalently obtained from a generalized susceptibility related to the fluctuations of the conserved particle current. Such susceptibility can be rigorously calculated for the integrable Lieb-Liniger Bose gas in any generalized Gibbs ensemble using a generalized Yang-Yang thermodynamic formalism. The resulting expression agrees with a prediction made within the context of generalized hydrodynamics. It also allows us to see explicitly that, within truly generalized Gibbs ensembles, the conductivity related with the particle current is not determined by the corresponding current-current auto-correlation function.

cond-mat.quant-gas↗

arXiv2025

Spin conductivity of the XXZ chain in the antiferromagnetic massive regime

We present a series representation for the dynamical two-point function of the local spin current for the XXZ chain in the antiferromagnetic massive regime at zero temperature. From this series we can compute the correlation function with very high accuracy up to very long times and large distances. Each term in the series corresponds to the contribution of all scattering states of an even number of excitations. These excitations can be interpreted in terms of an equal number of particles and holes. The lowest term in the series comprises all scattering states of one hole and one particle. This term determines the long-time large-distance asymptotic behaviour which can be obtained explicitly from a saddle-point analysis. The space-time Fourier transform of the two-point function of currents at zero momentum gives the optical spin conductivity of the model. We obtain highly accurate numerical estimates for this quantity by numerically Fourier transforming our data. For the one-particle, one-hole contribution, equivalently interpreted as a two-spinon contribution, we obtain an exact and explicit expression in terms of known special functions. For large enough anisotropy, the two-spinon contribution carries most of the spectral weight, as can be seen by calculating the f-sum rule.

cond-mat.stat-mech↗

arXiv2025

Chiral eigenbases of the XX and XY quantum spin chains

We calculate the values of observables in chiral eigenstates of the XX quantum spin chain that were introduced in previous work and compare the form of the result with the respective expressions obtained in the more familiar eigenbasis of states with fixed magnetization in the $z$ direction. We carry out the diagonalization of the XY spin chain in the chiral basis. We calculate the norm of the chiral XY eigenstates, and the values of the one-point functions and some neighbor two-point correlation functions. We interpret the spectrum and the particle content of the XY chain in terms of scattering states of an even number of kink and antikink excitations that are created over a reduced Brillouin zone.

cond-mat.stat-mech↗

arXiv2025

Low-temperature spectrum of the quantum transfer matrix of the XXZ chain in the massless regime

The free energy per lattice site of a quantum spin chain in the thermodynamic limit is determined by a single `dominant' Eigenvalue of an associated quantum transfer matrix in the infinite Trotter number limit. For integrable quantum spin chains, related with solutions of the Yang-Baxter equation, an appropriate choice of the quantum transfer matrix enables to study its spectrum, e.g.\ by means of the algebraic Bethe Ansatz. In its turn, the knowledge of the full spectrum allows one to study its universality properties such as the appearance of a conformal spectrum in the low-temperature regime. More generally, accessing the full spectrum is a necessary step for deriving thermal form factor series representations of the correlation functions of local operators for the spin chain under consideration. These are statements that have been established by physicists on a heuristic level and that are calling for a rigorous mathematical justification. In this work we implement certain aspects of this programme with the example of the XXZ quantum spin chain in the antiferromagnetic massless regime and in the low-temperature limit. We rigorously establish the existence, uniqueness and characterise the form of the solutions to the non-linear integral equations that are equivalent to the Bethe Ansatz equations for the quantum transfer matrix of this model. This allows us to describe that part of the quantum transfer matrix spectrum that is related to the Bethe Ansatz and that does not collapse to zero in the infinite Trotter number limit. Within the considered part of the spectrum we rigorously identify the dominant Eigenvalue and show that those correlations lengths that diverge in the low-temperature limit are given, to the leading order, by the spectrum of the free Boson $c=1$ conformal field theory. This rigorously establishes a long-standing conjecture present in the physics literature.

math-ph↗

arXiv2024

Bethe Ansatz

The term Bethe Ansatz stands for a multitude of methods in the theory of integrable models in statistical mechanics and quantum field theory that were designed to study the spectra, the thermodynamic properties and the correlation functions of these models non-perturbatively. This essay attempts to a give a brief overview of some of these methods and their development, mostly based on the example of the Heisenberg model and the corresponding six-vertex model.

math-ph↗

arXiv2024

Chiral basis for qubits and spin-helix decay

We propose a qubit basis composed of transverse spin helices with kinks. Unlike the usual computational basis, this chiral basis is well suited for describing quantum states with nontrivial topology. Choosing appropriate parameters the operators of the transverse spin components, $σ_n^x$ and $σ_n^y$, become diagonal in the chiral basis, which facilitates the study of problems focused on transverse spin components. As an application, we study the temporal decay of the transverse polarization of a spin helix in the XX model that has been measured in recent cold atom experiments. We obtain an explicit universal function describing the relaxation of helices of arbitrary wavelength.

quant-ph↗

arXiv2023

Thermal form-factor expansion of the dynamical two-point functions of local operators in integrable quantum chains

Evaluating a lattice path integral in terms of spectral data and matrix elements pertaining to a suitably defined quantum transfer matrix, we derive form-factor series expansions for the dynamical two-point functions of arbitrary local operators in fundamental Yang-Baxter integrable lattice models at finite temperature. The summands in the series are parameterised by solutions of the Bethe Ansatz equations associated with the eigenvalue problem of the quantum transfer matrix. We elaborate on the example of the XXZ chain for which the solutions of the Bethe Ansatz equations are sufficiently well understood in certain limiting cases. We work out in detail the case of the spin-zero operators in the antiferromagnetic massive regime at zero temperature. In this case the thermal form-factor series turn into series of multiple integrals with fully explicit integrands. These integrands factorize into an operator-dependent part, determined by the so-called Fermionic basis, and a part which we call the universal weight as it is the same for all spin-zero operators. The universal weight can be inferred from our previous work. The operator-dependent part is rather simple for the most interesting short-range operators. It is determined by two functions $ρ$ and $ω$ for which we obtain explicit expressions in the considered case. As an application we rederive the known explicit form-factor series for the two-point function of the magnetization operator and obtain analogous expressions for the magnetic current and the energy operators.

cond-mat.stat-mech↗

arXiv2023

Properties of linear integral equations related to the six-vertex model with disorder parameter

One of the key steps in recent work on the correlation functions of the XXZ chain was to regularize the underlying six-vertex model by a disorder parameter $α$. For the regularized model it was shown that all static correlation functions are polynomials in only two functions. It was further shown that these two functions can be written as contour integrals involving the solutions of a certain type of linear and non-linear integral equations. The linear integral equations depend parametrically on $α$ and generalize linear integral equations known from the study of the bulk thermodynamic properties of the model. In this note we consider the generalized dressed charge and a generalized magnetization density. We express the generalized dressed charge as a linear combination of two quotients of $Q$-functions, the solutions of Baxter's $t$-$Q$-equation. With this result we give a new proof of a lemma on the asymptotics of the generalized magnetization density as a function of the spectral parameter.

math-ph↗

arXiv2021

Short-distance thermal correlations in the XXZ chain

Recent studies have revealed much of the mathematical structure of the static correlation functions of the XXZ chain. Here we use the results of those studies in order to work out explicit examples of short-distance correlation functions in the infinite chain. We compute two-point functions ranging over 2, 3 and 4 lattice sites as functions of the temperature and the magnetic field for various anisotropies in the massless regime $- 1 < Δ< 1$. It turns out that the new formulae are numerically efficient and allow us to obtain the correlations functions over the full parameter range with arbitrary precision.

cond-mat.str-el↗

arXiv2021

A thermal form factor series for the longitudinal two-point function of the Heisenberg-Ising chain in the antiferromagnetic massive regime

We consider the longitudinal dynamical two-point function of the XXZ quantum spin chain in the antiferromagnetic massive regime. It has a series representation based on the form factors of the quantum transfer matrix of the model. The $n$th summand of the series is a multiple integral accounting for all $n$-particle $n$-hole excitations of the quantum transfer matrix. In previous works the expressions for the form factor amplitudes appearing under the integrals were either again represented as multiple integrals or in terms of Fredholm determinants. Here we obtain a representation which reduces, in the zero-temperature limit, essentially to a product of two determinants of finite matrices whose entries are known special functions. This will facilitate the further analysis of the correlation function.

cond-mat.stat-mech↗

arXiv2021

Dressed energy of the XXZ chain in the complex plane

We consider the dressed energy $\varepsilon$ of the XXZ chain in the massless antiferromagnetic parameter regime at $0 < Δ< 1$ and at finite magnetic field. This function is defined as a solution of a Fredholm integral equation of the second kind. Conceived as a real function over the real numbers it describes the energy of particle-hole excitations over the ground state at fixed magnetic field. The extension of the dressed energy to the complex plane determines the solutions to the Bethe Ansatz equations for the eigenvalue problem of the quantum transfer matrix of the model in the low-temperature limit. At low temperatures the Bethe roots that parametrize the dominant eigenvalue of the quantum transfer matrix come close to the curve ${\rm Re}\, \varepsilon (λ) = 0$. We describe this curve and give lower bounds to the function ${\rm Re}\, \varepsilon$ in regions of the complex plane, where it is positive.

math-ph↗

arXiv2021

Exact real-time longitudinal correlation functions of the massive XXZ chain

We apply the recently developed thermal form factor expansion method to evaluate the real-time longitudinal spin-spin correlation functions of the spin-$\frac{1}{2}$ XXZ chain in the antiferromagnetically ordered regime at temperature $T=0$. An analytical result containing all types of excitations in the model is obtained, without any approximations. This allows for the accurate calculation of the real-time correlation functions in this strongly interacting quantum system for arbitrary distances and times.

cond-mat.stat-mech↗

arXiv2021

Fourth-neighbour two-point functions of the XXZ chain and the Fermionic basis approach

We give a descriptive review of the Fermionic basis approach to the theory of correlation functions of the XXZ quantum spin chain. The emphasis is on explicit formulae for short-range correlation functions which will be presented in a way that allows for their direct implementation on a computer. Within the Fermionic basis approach a huge class of stationary reduced density matrices, compatible with the integrable structure of the model, assumes a factorized form. This means that all expectation values of local operators and all two-point functions, in particular, can be represented as multivariate polynomials in only two functions $ρ$ and $ω$ and their derivatives with coefficients that are rational in the deformation parameter $q$ of the model. These coefficients are of `algebraic origin'. They do not depend on the choice of the density matrix, which only impacts the form of $ρ$ and $ω$. As an example we work out in detail the case of the grand canonical ensemble at temperature $T$ and magnetic field $h$ for $q$ in the critical regime. We compare our exact results for the fourth-neighbour two-point functions with asymptotic formulae for $h, T = 0$ and for finite $h$ and $T$.

cond-mat.stat-mech↗

arXiv2021

Introduction to Solid State Physics

Lecture Goals: (i) Introduction to the basic concepts, meaning that the emphasis is, in the first instance, on the single-particle aspects. (ii) Service for Experimental Solid State Physics. (iii) Emphasis on the explanation of concepts and basic ideas, not always quantitative, justification of the use of simplified `model Hamiltonians'. (iv) Raise some understanding why many-body physics is mostly phenomenology. (v) Convey the following main idea: (Collective) elementary excitations are `quasi-particles' characterized by their dispersion relation $\mathbf{p} \mapsto \varepsilon (\mathbf{p})$ and by certain quantum numbers like spin and charge. The most important two are `the phonon' (= quantized lattice vibration) and `the electron' (= quantized charge excitation of the solid, which has as much to do with the electron of elementary particle physics as water waves have to do with water).

cond-mat.str-el↗

arXiv2020

Thermal form-factor approach to dynamical correlation functions of integrable lattice models

We propose a method for calculating dynamical correlation functions at finite temperature in integrable lattice models of Yang-Baxter type. The method is based on an expansion of the correlation functions as a series over matrix elements of a time-dependent quantum transfer matrix rather than the Hamiltonian. In the infinite Trotter-number limit the matrix elements become time independent and turn into the thermal form factors studied previously in the context of static correlation functions. We make this explicit with the example of the XXZ model. We show how the form factors can be summed utilizing certain auxiliary functions solving finite sets of nonlinear integral equations. The case of the XX model is worked out in more detail leading to a novel form-factor series representation of the dynamical transverse two-point function.

cond-mat.stat-mech↗

arXiv2020

Statistical mechanics of integrable quantum spin systems

This script is based on the notes the author prepared to give a set of six lectures at the Les Houches School "Integrability in Atomic and Condensed Matter Physics" in the summer of 2018. The school had its focus on the application of integrability based methods to problems in non-equilibrium statistical mechanics. The lectures were meant to complement this subject with background material on the equilibrium statistical mechanics of quantum spin chains from a vertex model perspective. The author was asked to provide a minimal introduction to quantum spin systems including notions like the reduced density matrix and correlation functions of local observables. He was further asked to explain the graphical language of vertex models and to introduce the concepts of the Trotter decomposition and the quantum transfer matrix. This was basically the contents of the first four lectures presented at the school. In the remaining two lectures these notions were filled with life by deriving an integral representation of the free energy per lattice site for the Heisenberg-Ising chain (alias XXZ model) using techniques based on non-linear integral equations.

cond-mat.stat-mech↗