A triangular, half-infinite, nearest-neighbor Ising system is studied whose coupling strengths close to the boundary are weaker than those in the bulk. The deviations decay as $\ensuremath{-}\frac{A}{{m}^{y}}$ with the distance $m$ from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation $g(\mathcal{r})$ is studied for critical bulk coupling. For $y=1$ the behavior is $g(\mathcal{r})\ensuremath{\sim}\frac{1}{{\mathcal{r}}^{\ensuremath{\eta}}}$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$, with a nonuniversal exponent $\ensuremath{\eta}$ depending on $A$ . For $0<y<1$ anomalous decay $g(\mathcal{r})\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(\frac{r}{\ensuremath{\xi}})}^{1\ensuremath{-}y}]$ is found. Expressions for $\ensuremath{\eta}$ and $\ensuremath{\xi}$ in terms of $A$ and $y$ are given.A triangular, half-infinite, nearest-neighbor Ising system is studied whose coupling strengths close to the boundary are weaker than those in the bulk. The deviations decay as $\ensuremath{-}\frac{A}{{m}^{y}}$ with the distance $m$ from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation $g(\mathcal{r})$ is studied for critical bulk coupling. For $y=1$ the behavior is $g(\mathcal{r})\ensuremath{\sim}\frac{1}{{\mathcal{r}}^{\ensuremath{\eta}}}$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$, with a nonuniversal exponent $\ensuremath{\eta}$ depending on $A$. For $0<y<1$ anomalous decay $g(\mathcal{r})\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(\frac{r}{\ensuremath{\xi}})}^{1\ensuremath{-}y}]$ is found. Expressions for $\ensuremath{\eta}$ and $\ensuremath{\xi}$ in terms of $A$ and $y$ are given.

Physical Review LettersOct 26, 198180 citations

Quantum correlations in spin models

Guo-Feng Zhang, Heng Fan, Ai-Ling Ji +3 more

arXiv:1105.2866Annals of PhysicsMay 22, 201142 citations

From Decay of Correlations to Locality and Stability of the Gibbs State

Ángela Capel, Massimo Moscolari, Stefan Teufel +1 more

Abstract We show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb–Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results.Abstract We show that whenever the Gibbs state of a quantum spin system satisfies decay of correlations, then it is stable, in the sense that local perturbations affect the Gibbs state only locally, and it satisfies local indistinguishability, i.e. it exhibits local insensitivity to system size. These implications hold in any dimension, require only locality of the Hamiltonian, and are based on Lieb–Robinson bounds and on a detailed analysis of the locality properties of the quantum belief propagation for Gibbs states. To demonstrate the versatility of our approach, we explicitly apply our results to several physically relevant models in which the decay of correlations is either known to hold or is proved by us. These include Gibbs states of one-dimensional spin chains with polynomially decaying interactions at any temperature, and high-temperature Gibbs states of quantum spin systems with finite-range interactions in any dimension. We also prove exponential decay of correlations above a threshold temperature for Gibbs states of one-dimensional finite spin chains with translation-invariant and exponentially decaying interactions, and then apply our general results.

arXiv:2310.09182Communications in Mathematical PhysicsJan 18, 20258 citations

Anomalous decay of pair correlations for two-dimensional critical wetting

Andrew O. Parry

Develops a scaling theory for the decay of the spin-spin correlation function at two-dimensional (d=2) wetting transitions in systems with short-ranged forces. For the critical wetting transition effective Hamiltonian results show that the scaling theory is obeyed but the decay of correlations is anomalous. In contrast, for the complete wetting transition, scaling and the Ornstein-Zernike theory are both valid. The author argues that the anomalous decay is specific to zero bulk field critical wetting transitions ad d=2.Develops a scaling theory for the decay of the spin-spin correlation function at two-dimensional (d=2) wetting transitions in systems with short-ranged forces. For the critical wetting transition effective Hamiltonian results show that the scaling theory is obeyed but the decay of correlations is anomalous. In contrast, for the complete wetting transition, scaling and the Ornstein-Zernike theory are both valid. The author argues that the anomalous decay is specific to zero bulk field critical wetting transitions ad d=2.

Journal of Physics A Mathematical and GeneralApr 21, 19924 citations

Weak Correlation Decay in Classical and Quantum Systems

Roberto Artuso

Jan 1, 19980 citations

m

from the boundary. By means of a repeated star-triangle transformation the boundary spin-spin correlation

g(\mathcal{r})

is studied for critical bulk coupling. For

y=1

the behavior is $g(\mathcal{r})\ensuremath{\sim}\frac{1}{{\mathcal{r}}^{\ensuremath{\eta}}}$ as $r\ensuremath{\rightarrow}\ensuremath{\infty}$, with a nonuniversal exponent $\ensuremath{\eta}$ depending on

A

. For

0<y<1

anomalous decay $g(\mathcal{r})\ensuremath{\sim}\mathrm{exp}[\ensuremath{-}{(\frac{r}{\ensuremath{\xi}})}^{1\ensuremath{-}y}]$ is found. Expressions for $\ensuremath{\eta}$ and $\ensuremath{\xi}$ in terms of

A

and

y

are given.