Six-Vertex Model

2019 ◽  
pp. 454-473
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Bethe Ansatz ◽  
Transfer Matrix ◽  
Spin Chain ◽  
Quantum Spin ◽  
Square Lattice ◽  
Vertex Model ◽  
Quantum Spin Chain ◽  
Special Value ◽  
Quantum Operators ◽  
The One

This chapter considers the special case of the six-vertex model on a square lattice using a trigonometric parameterization of the vertex weights. It demonstrates how, by exploiting the Yang-Baxter relations, the six-vertex model is diagonalized and the Bethe ansatz equations are derived. The Hamiltonian of the Heisenberg quantum spin chain is obtained from the transfer matrix for a special value of the spectral parameter together with an infinite set of further conserved quantum operators. By the diagonalization of the transfer matrix the exact solution of the one-dimensional quantum spin chain Hamiltonian has automatically also been obtained, which is given by the same Bethe ansatz equations.

scholarly journals Spin chains with boundary inhomogeneities

2021 ◽  
Vol 2021 (8) ◽  
Author(s):  
Rafael I. Nepomechie ◽  
Ana L. Retore
Keyword(s):  
Quantum Group ◽  
Bethe Ansatz ◽  
Transfer Matrix ◽  
Spin Chain ◽  
Quantum Spin ◽  
Spin Chains ◽  
Group Symmetry ◽  
Quantum Spin Chain ◽  
Open Quantum ◽  
Bethe Ansatz Solution

Abstract We investigate the effect of introducing a boundary inhomogeneity in the transfer matrix of an integrable open quantum spin chain. We find that it is possible to construct a local Hamiltonian, and to have quantum group symmetry. The boundary inhomogeneity has a profound effect on the Bethe ansatz solution.

Bethe ansatz solution of the Fateev-Zamolodchikov quantum spin chain with boundary terms

1990 ◽  
Vol 147 (1) ◽  
pp. 70-78 ◽  
Author(s):  
Luca Mezincescu ◽  
Rafael I. Nepomechie ◽  
V. Rittenberg
Keyword(s):  
Bethe Ansatz ◽  
Spin Chain ◽  
Quantum Spin ◽  
Quantum Spin Chain ◽  
Boundary Terms ◽  
Bethe Ansatz Solution

scholarly journals Bethe ansatz on a quantum computer?

2021 ◽  
Vol 21 (3&4) ◽  
pp. 255-265
Author(s):  
Rafael I. Nepomechie
Keyword(s):  
Bethe Ansatz ◽  
Spin Chain ◽  
Quantum Computer ◽  
Quantum Spin ◽  
Variational Parameter ◽  
Quantum Spin Chain

We consider the feasibility of studying the anisotropic Heisenberg quantum spin chain with the Variational Quantum Eigensolver (VQE) algorithm, by treating Bethe states as variational states, and Bethe roots as variational parameters. For short chains, we construct exact one-magnon trial states that are functions of the variational parameter, and implement the VQE calculations in Qiskit. However, exact multi-magnon trial states appear to be out out of reach.

scholarly journals A SPIN CHAIN PRIMER

1999 ◽  
Vol 13 (24n25) ◽  
pp. 2973-2985 ◽  
Author(s):  
RAFAEL I. NEPOMECHIE
Keyword(s):  
Bethe Ansatz ◽  
Spin Chain ◽  
Quantum Spin ◽  
Baxter Equation ◽  
Quantum Spin Chain ◽  
Algebraic Bethe Ansatz ◽  
Chain Primer

This is a very elementary introduction to the Heisenberg (XXX) quantum spin chain, the Yang–Baxter equation, and the algebraic Bethe Ansatz.

Bethe Ansatz for the Anisotropic Heisenberg Quantum Spin Chain

2019 ◽  
pp. 502-544
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Wave Function ◽  
Boundary Conditions ◽  
Bethe Ansatz ◽  
Ground State ◽  
Spin Chain ◽  
Energy Eigenvalue ◽  
Quantum Spin ◽  
Quantum Spin Chain ◽  
Complex Solutions ◽  
New Variables

This chapter verifies the conjecture for the wave function, the Bethe ansatz wave function, of the anisotropic Heisenberg quantum spin chain by examining first the cases for one, two, and three spin deviations. The equations determining the quasi- momenta are the Bethe ansatz equations, now obtained from the coordinate Bethe ansatz. The Bethe ansatz equations derive from the eigenvalue equation in combination with boundary conditions, here periodic boundary conditions. These quasi-momenta also determine the energy eigenvalue. However, solving the Bethe ansatz equations to obtain a particular state requires more considerations. New variables, called rapidities, are useful. The consideration of the thermodynamic limit then allows to extract information about the ground state and low-lying excitations of the anisotropic quantum spin chain from the Bethe ansatz equations. Furthermore, complex solutions of the Bethe ansatz equations, called strings, are considered.

Thermodynamics of the Isotropic Heisenberg Quantum Spin Chain

2019 ◽  
pp. 641-654
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Free Energy ◽  
Bethe Ansatz ◽  
Finite Temperature ◽  
Spin Chain ◽  
Conformal Symmetry ◽  
Quantum Spin ◽  
Quantum Spin Chain ◽  
Explicit Formulas ◽  
Thermodynamic Bethe Ansatz ◽  
Physical Quantities

This chapter presents the extension of the Bethe ansatz to finite temperature, the thermodynamic Bethe ansatz, for the antiferromagnetic isotropic Heisenberg quantum spin chain, the XXX quantum spin chain. It discusses how the added complications of this model arise from the more complicated structure of excitations of the quantum spin chain, the complex string excitations, which have to be included in the Bethe ansatz thermodynamics. It derives the integral equations of the thermodynamic Bethe ansatz for the XXX quantum spin chain and mentions explicit formulas for the free energy of the quantum spin chain and some interesting physical quantities, especially making contact with predictions of conformal symmetry.

The Anisotropic Heisenberg Quantum Spin Chain

2019 ◽  
pp. 491-501
Author(s):  
Hans-Peter Eckle
Keyword(s):  
Magnetic Materials ◽  
Spin Chain ◽  
Heisenberg Model ◽  
Magnetic Moments ◽  
Quantum Spin ◽  
Quantum Spin Chain ◽  
Good Picture ◽  
One Dimensional ◽  
Dimensional Version ◽  
The One

This chapter introduces the Heisenberg model, a fully quantum mechanical model that describes the magnetism of localized magnetic moments. The one-dimensional version of the Heisenberg model, the Heisenberg quantum spin chain, provides a good picture of magnetic materials that belong to a class of insulating magnetic materials where the interaction of the magnetic moments in one particular direction is much larger than in the perpendicular directions, and which can be described with high accuracy as quasi- one-dimensional magnets. A detailed description of the Heisenberg quantum spin chain is followed by a discussion of its various special cases, in particular the special case of the anisotropic Heisenberg quantum spin chain, the so-called XXZ quantum spin chain. It considers the solution of eigenvalue problem of this quantum spin and leads to Bethe’s conjecture for the wave function.

scholarly journals FATEEV-ZAMOLODCHIKOV SPIN CHAIN: EXCITATION SPECTRUM, COMPLETENESS AND THERMODYNAMICS

1994 ◽  
Vol 09 (28) ◽  
pp. 4921-4947 ◽  
Author(s):  
GIUSEPPE ALBERTINI
Keyword(s):  
Specific Heat ◽  
Bethe Ansatz ◽  
Excitation Spectrum ◽  
Spin Chain ◽  
Central Charge ◽  
Quantum Spin ◽  
Quantum Spin Chain

The sector of zero ZN charge is studied for the ferromagnetic (FM) and antiferromagnetic (AFM) version of the (ZN×Z2)-invariant Fateev-Zamolodchikov quantum spin chain. We conjecture that the relevant Bethe ansatz equations should admit, besides the usual stringlike solutions, exceptional multiplets, and a number of nonphysical solutions. Once the physical ones are identified, we show how completeness and the gapless excitation spectrum can be obtained. The central charge is computed from the specific heat and found to be [Formula: see text] (FM) and c=1 (AFM).

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