We prove that for the spin-1/2 Heisenberg XXZ chain in the range $Δ\in[-1,1/2]$ , the ground state on the torus of length $L$ converges to an infinite volume ground state $\langle\cdot\rangle$ as $L\to\infty$ , and that the spin-spin correlation $\langle S_0^{(1)}S_x^{(1)}\rangle$ decays polynomially fast in $x$ . In the range $Δ\in[-1,0]$ we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that $L$ and $β=1/T$ can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that $\langle S_0^{(1)}S_x^{(1)}(t)\rangle$ decays to zero in $|(x,t)|$ , and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points $Δ=0,-1$ ), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.We prove that for the spin-1/2 Heisenberg XXZ chain in the range $Δ\in[-1,1/2]$, the ground state on the torus of length $L$ converges to an infinite volume ground state $\langle\cdot\rangle$ as $L\to\infty$, and that the spin-spin correlation $\langle S_0^{(1)}S_x^{(1)}\rangle$ decays polynomially fast in $x$. In the range $Δ\in[-1,0]$ we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that $L$ and $β=1/T$ can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that $\langle S_0^{(1)}S_x^{(1)}(t)\rangle$ decays to zero in $|(x,t)|$, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points $Δ=0,-1$), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.

arXiv:2608.21306Aug 21, 20260 citations

Accelerated convergence in exact-diagonalization studies

M. A. Novotny, José A. Riera, P. W. Leung +1 more

A simple method is proposed and tested for obtaining accelerated convergence of quantum systems on small lattices with N sites. The main idea is to perform exact diagonalizations with some added irrelevant parameter, and use this parameter to accelerate the convergence to the infinite-lattice limit. In this paper different boundary conditions are used to improve the convergence for the Heisenberg model. In particular, we find that the application of the method to the d=1 antiferromagnetic Heisenberg model changes the rate of convergence of the ground-state energy per site, \ensuremath{\Vert} ${\mathit{E}}_{0}$ (N)- ${\mathit{E}}_{0}$ (\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.A simple method is proposed and tested for obtaining accelerated convergence of quantum systems on small lattices with N sites. The main idea is to perform exact diagonalizations with some added irrelevant parameter, and use this parameter to accelerate the convergence to the infinite-lattice limit. In this paper different boundary conditions are used to improve the convergence for the Heisenberg model. In particular, we find that the application of the method to the d=1 antiferromagnetic Heisenberg model changes the rate of convergence of the ground-state energy per site, \ensuremath{\Vert}${\mathit{E}}_{0}$(N)-${\mathit{E}}_{0}$(\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.

Physical review. B, Condensed matterSep 1, 19930 citations

On the existence of a heisenberg dynamics

I. M. Burban

Ukrainian Mathematical JournalJan 1, 19740 citations

Dynamics of the Classical Heisenberg Chain

Z. Slanič, Harvey Gould, Jan Tobochnik

Computers in PhysicsNov 1, 19913 citations

We prove that for the spin-1/2 Heisenberg XXZ chain in the range

Δ\in[-1,1/2]

, the ground state on the torus of length

L

converges to an infinite volume ground state

\langle\cdot\rangle

L\to\infty

, and that the spin-spin correlation

\langle S_0^{(1)}S_x^{(1)}\rangle

decays polynomially fast in

x

. In the range

Δ\in[-1,0]

we have the stronger results: that the convergence holds for several finite-volume, finite temperature states, that

L

and

β=1/T

can be taken to infinity in any order, that the convergence holds on (Euclidean) dynamic correlators, and that

\langle S_0^{(1)}S_x^{(1)}(t)\rangle

decays to zero in

|(x,t)|

, and cannot decay exponentially fast. The convergence of the ground state in infinite volume is known rigorously by Bethe Ansatz methods, while our correlation decay results (apart from the points

Δ=0,-1

), the convergence on dynamic correlators and the interchangeability of limits are new at the rigorous level. Moreover our methods are new and do not use any Bethe Ansatz techniques, using only the following related probabilistic models. In the Lorentz mirror model with loop weight 2, a model of random loops on the square lattice, in a large range of parameters we prove that connection probabilities tend to 0, and do so polynomially fast in the symmetric case of the model. Our proof uses two couplings of this mirror model with the six-vertex model, one of which is new. The main input is the delocalisation of the height function of the six-vertex model proved by several authors. We further prove that the height function of a certain space-time version of the six-vertex model delocalises and then prove through analogous couplings to those mentioned above that connection probabilities converge to 0 in the loop representation of the XXZ model introduced by Ueltschi.

{\mathit{E}}_{0}

(N)-

{\mathit{E}}_{0}

(\ensuremath{\infty})\ensuremath{\Vert}\ensuremath{\sim}${\mathit{N}}^{\mathrm{\ensuremath{-}}\mathit{x}}$, to x\ensuremath{\approxeq}4 from the value x=2, which is found using only periodic boundary conditions.