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

Physical Review Letters

Bethe-Ansatzresults for a solvable O(n) model on the square lattice

Murray T. Batchelor, Bernard Nienhuis, S. Ole Warnaar

Published May 22, 1989• 54 citations
Publisher

Abstract

The Bethe-Ansatz approach is used to solve a nineteen-vertex model equivalent to a spin model with O(n) symmetry on the square lattice. The bulk free energy is determined exactly on four branches of critical points. On two of these branches, the central charge is derived from finite-size corrections. On the two other branches, the nature of the Bethe Ansatz solutions prevents us from computing the central charge analytically. In this case, our approach still allows the exact numerical determination of the free energy on strips of hundreds of sites wide.

DOI
10.1103/physrevlett.62.2425

Related papers

Finite lattice methods in statistical mechanics

Murray T. Batchelor

This thesis is concerned with the finite lattice study of spin models. The underlying theme in Parts I and II is the exploitation of Bethe ansatz equations to provide finite lattice data well beyond the range normally available. In Part I the Bethe ansatz equations for eigenvalues of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to agree with recent studies of the XXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data. In Part II, the numerical solutions of the Bethe ansatz equations for the eigen-energies of the XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkm-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. Eigenstates of the Ashkin-Teller and Potts models with free boundaries are also obtained, allowing an examination of their critical surface properties. In Part HI the critical behaviour of an Ising model with competing firstand third- nearest neighbour interactions on the square lattice is investigated using the finite lattice method. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in the finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.

•Jan 1, 1987•2 citations

New Approach to Thermal Bethe Ansatz

C. Destri, Hector J. de Vega

We present a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisemberg chain we derive, for arbitrary values of the anysotropy, a {\bf single} non--linear integral equation from which the free energy can be exactly calculated. The high--temperature expansion follows in a sistematic and relatively simple way. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around $T=0$. Furthermore, we derive a single non-linear integral equation describing the finite--size ground--state energy of the Sine--Gordon quantum field theory. PACS: 05.30, 03.70. 75.10.5

arXiv:hep-th/9203064•Mar 24, 1992•8 citations

Critical properties of the Izergin-Korepin and solvable O(n) models and their related quantum spin chains

S. Ole Warnaar, Murray T. Batchelor, Bernard Nienhuis

The central charge of the Izergin-Korepin model the corresponding quantum spin chain, and the O (n) model is calculated analytically via the Bethe ansatz. The calculation extends a technique recently developed for the Zamolodchikov-Fateev model. In addition critical exponents and the central charge for these models are obtained from numerical solutions of Bethe ansatz equations for finite systems. As a physical application the authors find the exponents v=12/23 and gamma =53/46 for the Theta -transition of polymers in two dimensions.

Journal of Physics A Mathematical and General•Jun 7, 1992•85 citations

Exactly solvable interacting spin-ice vertex model

Anderson A. Ferreira, Francisco Castilho Alcaraz

A special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.

arXiv:cond-mat/0608504•Physical Review E•Jul 24, 2006•3 citations

Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice

Murray T. Batchelor, Junji Suzuki, C M Yung

We derive the nested Bethe ansatz solution of the fully packed $\mathrm{O}(n)$ loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behavior. In the $n=0$ limit we obtain the exact compact exponents $\ensuremath{\gamma}=1$ and $\ensuremath{\nu}=\frac{1}{2}$ for Hamiltonian walks, along with the exact value ${\ensuremath{\kappa}}^{2}=3\sqrt{\frac{3}{4}}$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.

arXiv:cond-mat/9408083•Physical Review Letters•Nov 14, 1994•53 citations

Finite-size corrections for nested Bethe ansatz models and conformal invariance

Hector J. de Vega

The method presented permits us to systematically compute physical magnitudes for large but finite size in theories solvable by the nested Bethe ansatz. The value of the central charge c and the surface tension are explicitly calculable in this way. For the fundamental vertex and spin models associated to simply laced Lie algebras, c turns out to be equal to the rank of the algebra.

Journal of Physics A Mathematical and General•Dec 1, 1987•33 citations
The Bethe-Ansatz approach is used to solve a nineteen-vertex model equivalent to a spin model with O(n) symmetry on the square lattice. The bulk free energy is determined exactly on four branches of critical points. On two of these branches, the central charge is derived from finite-size corrections. On the two other branches, the nature of the Bethe Ansatz solutions prevents us from computing the central charge analytically. In this case, our approach still allows the exact numerical determination of the free energy on strips of hundreds of sites wide.
This thesis is concerned with the finite lattice study of spin models. The underlying theme in Parts I and II is the exploitation of Bethe ansatz equations to provide finite lattice data well beyond the range normally available. In Part I the Bethe ansatz equations for eigenvalues of the eight-vertex model are solved numerically to yield mass gap data on infinitely long strips of up to 512 sites in width. The finite-size corrections, at criticality, to the free energy per site and polarization gap are found to agree with recent studies of the XXZ spin chain. The leading corrections to the finite-size scaling estimates of the critical line and thermal exponent are also found, providing an explanation of the poor convergence seen in earlier studies. Away from criticality, the linear scaling fields are derived exactly in the full parameter space of the spin system, allowing a thorough test of a recently proposed method of extracting linear scaling fields and related exponents from finite lattice data. In Part II, the numerical solutions of the Bethe ansatz equations for the eigen-energies of the XXZ Hamiltonian on very large chains are used to identify, via conformal invariance, the scaling dimensions of various two-dimensional models. With periodic boundary conditions, eight-vertex and Gaussian model operators are found. The scaling dimensions of the Ashkm-Teller and Potts models are obtained by the exact relating of eigenstates of their quantum Hamiltonians to those of the XXZ chain with modified boundary conditions. Eigenstates of the Ashkin-Teller and Potts models with free boundaries are also obtained, allowing an examination of their critical surface properties. In Part HI the critical behaviour of an Ising model with competing firstand third- nearest neighbour interactions on the square lattice is investigated using the finite lattice method. In the ferromagnetic region, the phase boundary is located with an accuracy at least equal to that of alternative methods. In the antiphase region, distinctive structure in the finite lattice estimators is found over an extended temperature range. However, the nature of the transition remains unclear.
We present a new approach to the calculation of thermodynamic functions for crossing-invariant models solvable by Bethe Ansatz. In the case of the XXZ Heisemberg chain we derive, for arbitrary values of the anysotropy, a {\bf single} non--linear integral equation from which the free energy can be exactly calculated. The high--temperature expansion follows in a sistematic and relatively simple way. For low temperatures we obtain the correct central charge and predict the analytic structure of the full expansion around T=0T=0T=0. Furthermore, we derive a single non-linear integral equation describing the finite--size ground--state energy of the Sine--Gordon quantum field theory. PACS: 05.30, 03.70. 75.10.5
The central charge of the Izergin-Korepin model the corresponding quantum spin chain, and the O (n) model is calculated analytically via the Bethe ansatz. The calculation extends a technique recently developed for the Zamolodchikov-Fateev model. In addition critical exponents and the central charge for these models are obtained from numerical solutions of Bethe ansatz equations for finite systems. As a physical application the authors find the exponents v=12/23 and gamma =53/46 for the Theta -transition of polymers in two dimensions.
A special family of solvable five-vertex model is introduced on a square lattice. In addition to the usual nearest-neighbor interactions, the vertices defining the model also interact along one of the diagonals of the lattice. This family of models includes in a special limit the standard six-vertex model. The exact solution of these models is an application of the matrix product ansatz introduced recently and applied successfully in the solution of quantum chains. The phase diagram and the free energy of the models are calculated in the thermodynamic limit. The models exhibit massless phases, and our analytical and numerical analyses indicate that such phases are governed by a conformal field theory with central charge c=1 and continuously varying critical exponents.
We derive the nested Bethe ansatz solution of the fully packed O(n)\mathrm{O}(n)O(n) loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behavior. In the n=0n=0n=0 limit we obtain the exact compact exponents $\ensuremath{\gamma}=1$ and $\ensuremath{\nu}=\frac{1}{2}$ for Hamiltonian walks, along with the exact value ${\ensuremath{\kappa}}^{2}=3\sqrt{\frac{3}{4}}$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.