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:cond-mat/0309593

Combinatorial Solution of One-Dimensional Quantum Systems

Domingos H. U. Marchetti, Claudio Fernandes de Souza Rodrigues

Published Sep 25, 2003• 0 citations
arXivHTMLPDF

Abstract

We give a self-contained exposition of the combinatorial solution of quantum mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter formula, we write the partition function as a generating function of a spanning subgraph of a two-dimensional lattice and solve the combinatorial problem by the method of Pfaffians provided the weights satisfy the so-called free fermion condition. The free energy and the ground state energy as a function of the inverse temperature, couplings J and magnetic fields h, for the XY model in a transverse field with period p=1 and 2, is then obtained.

DOI
10.48550/arxiv.cond-mat/0309593

Related papers

The ST-Transformation Approach to Analytic Solutions of Quantum Systems. II: Transfer-Matrix and Pfaffian Methods

M. Inoue, Masuo Suzuki

The general formalism to map d-dimensional quantum systems to (d + 1)-dimensional classical systems by the Trotter-like formula and to solve exactly the mapped systems by some classical methods such as the transfer-matrix (including Bethe Ansatz) or Pfaffian methods is applied to find the free energies, ground state energies, excitation gaps and correlations of the transverse Ising chain and the one-dimensional XY-model.

Progress of Theoretical Physics•Mar 1, 1988•47 citations

Relationship between d-Dimensional Quantal Spin Systems and (d+1)-Dimensional Ising Systems: Equivalence, Critical Exponents and Systematic Approximants of the Partition Function and Spin Correlations

Masuo Suzuki

The partition function of a quantal spin system is expressed by that of the Ising model, on the basis of the generalized Trotter formula. Thereby the ground state of the d-dimensional Ising model with a transverse field is proven to be equivalent to the (d+1)-dimensional Ising model at finite temperatures. A general relationship is established between the two partition functions of a general quantal spin system and the corresponding Ising model with many-spin interactions, which yields some rigorous results on quantum systems. Some applications are given.

Progress of Theoretical Physics•Nov 1, 1976•1,135 citations

Spin-Boson Systems: One-Dimensional Equivalents and the Kondo Problem

Martin Blume, Victor J. Emery, Alan H. Luther

The partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is shown to be exactly equal to that of a one-dimensional gas and also, in a delicate limit, to that of a one-dimensional Ising model. Both of these are peculiar in that they must provide a free energy which is an intensive quantity. For special values of the parameters in the Hamiltonian, the partition function is equal to that first derived by Anderson and Yuval, and by Hamann, for the Kondo problem.

Physical Review Letters•Aug 17, 1970•68 citations

Application of many-fermion techniques to lattice statistical problems

R. W. Gibberd

The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.

Canadian Journal of Physics•Apr 1, 1969•2 citations

Exact thermal properties of free-fermionic spin chains

Michał Białończyk, Fernando Gómez-Ruiz, Adolfo del Campo

An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.

arXiv:2103.16322•SciPost Physics•Jul 15, 2021•11 citations

Exact Thermal Properties of Integrable Spin Chains

Michał Białończyk, Fernando J. Gómez-Ruiz, Adolfo del Campo

An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of an arbitrary observable at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.

•Apr 9, 2021•0 citations
We give a self-contained exposition of the combinatorial solution of quantum mechanical systems of coupled spins on a one-dimensional lattice. Using Trotter formula, we write the partition function as a generating function of a spanning subgraph of a two-dimensional lattice and solve the combinatorial problem by the method of Pfaffians provided the weights satisfy the so-called free fermion condition. The free energy and the ground state energy as a function of the inverse temperature, couplings J and magnetic fields h, for the XY model in a transverse field with period p=1 and 2, is then obtained.
The partition function of a quantal spin system is expressed by that of the Ising model, on the basis of the generalized Trotter formula. Thereby the ground state of the d-dimensional Ising model with a transverse field is proven to be equivalent to the (d+1)-dimensional Ising model at finite temperatures. A general relationship is established between the two partition functions of a general quantal spin system and the corresponding Ising model with many-spin interactions, which yields some rigorous results on quantum systems. Some applications are given.
The partition function for a spin \textonehalf{} in a magnetic field coupled to a set of bosons is shown to be exactly equal to that of a one-dimensional gas and also, in a delicate limit, to that of a one-dimensional Ising model. Both of these are peculiar in that they must provide a free energy which is an intensive quantity. For special values of the parameters in the Hamiltonian, the partition function is equal to that first derived by Anderson and Yuval, and by Hamann, for the Kondo problem.
The partition functions of many lattice statistical problems, such as the Ising models, the dimer model, and the ferroelectric model, have been written in the following general form:[Formula: see text]For models that can be solved by the pfaffian technique, H(j) is a quadratic function of fermion operators, but otherwise H(j) also contains quartic functions of fermion operators, making the above expression intractable to exact evaluation. General field theoretical perturbation methods, such as the Green's function techniques and the diagonalization (by a Fourier transformation) of the Hamiltonian, Ho(j), are developed in relation to the above expression. It is shown that the partition function for unsolved problems can be expressed in terms of an "irreducible vertex part" for which low-order approximations can be obtained. This general formalism is applied to the next-nearest neighbor two-dimensional Ising lattice. A first-order approximation is obtained for the irreducible vertex part, and the critical properties are obtained. For the equivalent-neighbor model, the critical temperature obtained is in excellent agreement with the series expansion results, being only 0.4 % higher.
An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of a wide class of observables at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.
An exact description of integrable spin chains at finite temperature is provided using an elementary algebraic approach in the complete Hilbert space of the system. We focus on spin chain models that admit a description in terms of free fermions, including paradigmatic examples such as the one-dimensional transverse-field quantum Ising and XY models. The exact partition function is derived and compared with the ubiquitous approximation in which only the positive parity sector of the energy spectrum is considered. Errors stemming from this approximation are identified in the neighborhood of the critical point at low temperatures. We further provide the full counting statistics of an arbitrary observable at thermal equilibrium and characterize in detail the thermal distribution of the kink number and transverse magnetization in the transverse-field quantum Ising chain.