Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1311.6959 · Symmetry Integrability and Geometry Methods and Applications

Functions Characterizing the Ground State of the XXZ Spin-1/2 Chain in the Thermodynamic Limit

Maxime Dugave

Published Apr 11, 2014• 11 citations
arXivHTMLPDF

Abstract

We establish several properties of the solutions to the linear integral equations describing the infinite volume properties of the XXZ spin-1/2 chain in the disordered regime.In particular, we obtain lower and upper bounds for the dressed energy, dressed charge and density of Bethe roots.Furthermore, we establish that given a fixed external magnetic field (or a fixed magnetization) there exists a unique value of the boundary of the Fermi zone.

DOI
10.3842/sigma.2014.043

Related papers

Dressed energy of the XXZ chain in the complex plane

Saskia Faulmann, G\"ohmann, Frank, Karol Kajetan Kozlowski

We consider the dressed energy $\varepsilon$ of the XXZ chain in the massless antiferromagnetic parameter regime at $0 < \Delta < 1$ 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 ${\rm Re}\, \varepsilon (\lambda) = 0$. We describe this curve and give lower bounds to the function ${\rm Re}\, \varepsilon$ in regions of the complex plane, where it is positive.

arXiv:2106.06732•Jun 12, 2021•0 citations

Spin transport on the integrable XXZ chain

Shinya Ae

Using the Bethe ansatz method and generalized hydrodynamics (GHD), properties of the spin transport is analyzed for the integrable XXZ chain with the anisotropy 0 ≤ ∆ < 1.Our approach starts from reconstructing the thermodynamic Bethe ansatz (TBA) equations for the integrable XXZ chain with arbitrary spin.We obtain the fusion hierarchy of the quantum transfer matrices (T -functions) in the arbitrary spin case, and we derive functional relations (T -and Y -systems) satisfied by the T -functions and certain combinations of them (Y -functions).Through analytical arguments, the Y -system is reduced to a set of non-linear integral equations, equivalent to the TBA equations.At first, we use the TBA equations for some higher-spin chains in order to examine the regular zero frequency contribution to the spin current correlation (spin dc conductivity) for the spin-1/2 XXZ chain.In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly obtain the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit.We find that the high temperature proportionality constant σ reg 0 of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π/ cos -1 ∆ ≥ 2 with the gap increasing larger than the second power of growing magnetization on one quasiparticle.The isotropic ∆ = 1 point is exceptional.Close to this point, σ reg 0 slowly increases in proportion to the first power of the one quasi-particle magnetization.On the other hand, σ reg 0 is proportional to the second power of the same when p 0 approaches irrational numbers.At second, we examine the finite-temperature spin Drude weight for the integrable XXZ chain with arbitrary spin, where GHD methods are employed.We calculate the spin Drude weight at zero magnetic field, and we find a characteristic fractal-like structure of the Drude weight at arbitrary spin across a broad range of temperatures, similar to the spin-1/2 case.In our approach, the solutions to the TBA equations (i.e., the Y -functions) can be explicitly written in terms of the T -functions, thus allowing for a systematic calculation of the hightemperature limit of the Drude weight, as well as the spin dc conductivity in the spin 1/2 case. B Proof of the equivalence between(2.43) and (2.73) C Delivation of the TBA equations for the integrable XXZ chain with arbitrary spin C.1 Free energy . . . . . . . . . .

•Mar 18, 2025•0 citations

Open XXZ chain and boundary modes at zero temperature

Sebastian Grijalva, Jacopo De Nardis, V. Terras

We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.

arXiv:1901.10932•SciPost Physics•Aug 20, 2019•28 citations

Bethe Ansatz and boundary energy of the open spin-1/2 XXZ chain

Rajan Murgan

We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{$\\alpha_-$,$\\alpha_+$,$\\beta_-$,$\\beta_+$}} are nonzero. A generalization of the Baxter $T-Q$ equation that involves more than one independent $Q$ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.

•Jan 1, 2006•0 citations

Dressed energy of the XXZ chain in the complex plane

Saskia Faulmann, Frank Göhmann, Karol Kajetan Kozlowski

Abstract We consider the dressed energy $$\varepsilon $$ ε of the XXZ chain in the massless antiferromagnetic parameter regime at $$0< \Delta < 1$$ 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 $$\mathrm{Re}\, \varepsilon (\lambda ) = 0$$ Re ε ( λ ) = 0 . We describe this curve and give lower bounds to the function $$\mathrm{Re}\, \varepsilon $$ Re ε in regions of the complex plane, where it is positive.

Letters in Mathematical Physics•Oct 1, 2021•1 citation

Surface free energy of the open XXZ spin-1/2 chain

Karol Kajetan Kozlowski, Balázs Pozsgay

We study the boundary free energy of the XXZ spin-1/2 chain subject to diagonal boundary fields. We first show that the representation for its finite Trotter number approximant obtained by Göhmann, Bortz and Frahm is related to the partition function of the six-vertex model with reflecting ends. Building on the Tsuchiya determinant representation for the latter quantity we are able to take the infinite Trotter number limit. This yields a representation for the surface free energy which involves the solution of the non-linear integral equation that governs the thermodynamics of the XXZ spin-1/2 chain subject to periodic boundary conditions. We show that this integral representation allows one to extract the low- T asymptotic behavior of the boundary magnetization at finite external magnetic field on the one hand and numerically plot this function on the other hand.

arXiv:1201.5884•Journal of Statistical Mechanics Theory and Experiment•May 22, 2012•64 citations
We establish several properties of the solutions to the linear integral equations describing the infinite volume properties of the XXZ spin-1/2 chain in the disordered regime.In particular, we obtain lower and upper bounds for the dressed energy, dressed charge and density of Bethe roots.Furthermore, we establish that given a fixed external magnetic field (or a fixed magnetization) there exists a unique value of the boundary of the Fermi zone.
We consider the dressed energy ε\varepsilonε of the XXZ chain in the massless antiferromagnetic parameter regime at 0<Δ<10 < \Delta < 10<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 Re ε(λ)=0{\rm Re}\, \varepsilon (\lambda) = 0Reε(λ)=0. We describe this curve and give lower bounds to the function Re ε{\rm Re}\, \varepsilonReε in regions of the complex plane, where it is positive.
Using the Bethe ansatz method and generalized hydrodynamics (GHD), properties of the spin transport is analyzed for the integrable XXZ chain with the anisotropy 0 ≤ ∆ < 1.Our approach starts from reconstructing the thermodynamic Bethe ansatz (TBA) equations for the integrable XXZ chain with arbitrary spin.We obtain the fusion hierarchy of the quantum transfer matrices (T -functions) in the arbitrary spin case, and we derive functional relations (T -and Y -systems) satisfied by the T -functions and certain combinations of them (Y -functions).Through analytical arguments, the Y -system is reduced to a set of non-linear integral equations, equivalent to the TBA equations.At first, we use the TBA equations for some higher-spin chains in order to examine the regular zero frequency contribution to the spin current correlation (spin dc conductivity) for the spin-1/2 XXZ chain.In the high temperature limit, we write down the dressed scattering kernels by one quasi-particle bare energies and exactly obtain the spin dc conductivity at zero magnetic field, which is proportional to the inverse temperature β at the β → 0 limit.We find that the high temperature proportionality constant σ reg 0 of the spin dc conductivity is discontinuous at all rational numbers of the anisotropy parameter p 0 = π/ cos -1 ∆ ≥ 2 with the gap increasing larger than the second power of growing magnetization on one quasiparticle.The isotropic ∆ = 1 point is exceptional.Close to this point, σ reg 0 slowly increases in proportion to the first power of the one quasi-particle magnetization.On the other hand, σ reg 0 is proportional to the second power of the same when p 0 approaches irrational numbers.At second, we examine the finite-temperature spin Drude weight for the integrable XXZ chain with arbitrary spin, where GHD methods are employed.We calculate the spin Drude weight at zero magnetic field, and we find a characteristic fractal-like structure of the Drude weight at arbitrary spin across a broad range of temperatures, similar to the spin-1/2 case.In our approach, the solutions to the TBA equations (i.e., the Y -functions) can be explicitly written in terms of the T -functions, thus allowing for a systematic calculation of the hightemperature limit of the Drude weight, as well as the spin dc conductivity in the spin 1/2 case. B Proof of the equivalence between(2.43) and (2.73) C Delivation of the TBA equations for the integrable XXZ chain with arbitrary spin C.1 Free energy . . . . . . . . . .
We study the open XXZ spin chain in the anti-ferromagnetic regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, for a chain of even length L and in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in L. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization. We finally discuss the case of chains of odd length.
We review recent results on the Bethe Ansatz solutions for the eigenvalues of the transfer matrix of an integrable open XXZ quantum spin chain using functional relations which the transfer matrix obeys at roots of unity. First, we consider a case where at most two of the boundary parameters {{alpha−\\alpha_-alpha−​,alpha+\\alpha_+alpha+​,beta−\\beta_-beta−​,beta+\\beta_+beta+​}} are nonzero. A generalization of the Baxter T−QT-QT−Q equation that involves more than one independent QQQ is described. We use this solution to compute the boundary energy of the chain in the thermodynamic limit. We conclude the paper with a review of some results for the general integrable boundary terms, where all six boundary parameters are arbitrary.
Abstract We consider the dressed energy ε\varepsilon ε ε of the XXZ chain in the massless antiferromagnetic parameter regime at 0<Δ<10< \Delta < 10<Δ<1 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 Re ε(λ)=0\mathrm{Re}\, \varepsilon (\lambda ) = 0Reε(λ)=0 Re ε ( λ ) = 0 . We describe this curve and give lower bounds to the function Re ε\mathrm{Re}\, \varepsilon Reε Re ε in regions of the complex plane, where it is positive.
We study the boundary free energy of the XXZ spin-1/2 chain subject to diagonal boundary fields. We first show that the representation for its finite Trotter number approximant obtained by Göhmann, Bortz and Frahm is related to the partition function of the six-vertex model with reflecting ends. Building on the Tsuchiya determinant representation for the latter quantity we are able to take the infinite Trotter number limit. This yields a representation for the surface free energy which involves the solution of the non-linear integral equation that governs the thermodynamics of the XXZ spin-1/2 chain subject to periodic boundary conditions. We show that this integral representation allows one to extract the low- T asymptotic behavior of the boundary magnetization at finite external magnetic field on the one hand and numerically plot this function on the other hand.