We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spins. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface does not fluctuate, i.e. has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction formula for the partition functions in two dimensions in terms of the partition function of the one-dimensional model.We develop a geometric representation for the ground state of the spin-1/2 quantum XXZ ferromagnetic chain in terms of suitably weighted random walks in a two-dimensional lattice. The path integral model so obtained admits a genuine classical statistical mechanics interpretation with a translation invariant Hamiltonian. This new representation is used to study the interface ground states of the XXZ model. We prove that the probability of having a number of down spins in the up phase decays exponentially with the sum of their distances to the interface plus the square of the number of down spins. As an application of this bound, we prove that the total third component of the spin in a large interval of even length centered on the interface does not fluctuate, i.e. has zero variance. We also show how to construct a path integral representation in higher dimensions and obtain a reduction formula for the partition functions in two dimensions in terms of the partition function of the one-dimensional model.

arXiv:math-ph/9908004Reviews in Mathematical PhysicsOct 1, 200017 citations

Study of One-Dimensional XY Model by the Transfer-Matrix Method

Hiroshi Betsuyaku

The Suzuki-Trotter formula has been used to get the $m$ th approximant to the classical representation of the partition function of the $N$ -spin $S=\frac{1}{2}$ $\mathrm{XY}$ chain. The equivalent two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) Ising model with four-spin interactions has been studied in detail by use of the numerically exact transfer-matrix method for $T>~0.05$ and $m<~8$ . The limiting $m\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior has been estimated from the extrapolation formula of the form $E(m)=E(\ensuremath{\infty})+\frac{a}{{m}^{2}}$. The limiting value of the energy agrees with the exact solution of Katsura.The Suzuki-Trotter formula has been used to get the $m$ th approximant to the classical representation of the partition function of the $N$-spin $S=\frac{1}{2}$ $\mathrm{XY}$ chain. The equivalent two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) Ising model with four-spin interactions has been studied in detail by use of the numerically exact transfer-matrix method for $T>~0.05$ and $m<~8$. The limiting $m\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior has been estimated from the extrapolation formula of the form $E(m)=E(\ensuremath{\infty})+\frac{a}{{m}^{2}}$. The limiting value of the energy agrees with the exact solution of Katsura.

Physical Review LettersAug 13, 198468 citations

The classical limit of quantum partition functions

Barry Simon A

Communications in Mathematical PhysicsOct 1, 1980162 citations

Algebraic Representation of Correlation Functions in Integrable Spin Chains

Herman E. Boos, M. Jimbo, T. Miwa +2 more

arXiv:hep-th/0601132Annales Henri PoincaréNov 24, 200666 citations

Computation of disordered system from the first principles of classical mechanics and ℕℙ hard problem

Ashot S. Gevorkyan, Vladimir Sahakyan

We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ℕℙ hard problem, that however in the limit of the statistical equilibrium can be calculated by ℙ algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.We study the classical 1D Heisenberg spin glasses in the framework of nearest-neighboring model. Based on the Hamilton equations we obtained the system of recurrence equations which allows to perform node-by-node calculations of a spin-chain. It is shown that calculations from the first principles of classical mechanics lead to ℕℙ hard problem, that however in the limit of the statistical equilibrium can be calculated by ℙ algorithm. For the partition function of the ensemble a new representation is offered in the form of one-dimensional integral of spin-chains’ energy distribution.

Physics of Atomic NucleiMar 1, 20171 citation

The Suzuki-Trotter formula has been used to get the

m

th approximant to the classical representation of the partition function of the

N

-spin

S=\frac{1}{2}

\mathrm{XY}

chain. The equivalent two-dimensional ($N\ifmmode\times\else\texttimes\fi{}2m$) Ising model with four-spin interactions has been studied in detail by use of the numerically exact transfer-matrix method for

T>~0.05

and

m<~8

. The limiting $m\ensuremath{\rightarrow}\ensuremath{\infty}$ behavior has been estimated from the extrapolation formula of the form $E(m)=E(\ensuremath{\infty})+\frac{a}{{m}^{2}}$. The limiting value of the energy agrees with the exact solution of Katsura.