We consider the anisotropic one-dimensional $X\ensuremath{-}Y$ model of Lieb, Schultz, and Mattis and study the three zero-temperature correlation functions ${\ensuremath{\rho}}_{\ensuremath{\mu}\ensuremath{\nu}}(R,t)=〈{S}_{1}^{\ensuremath{\mu}}(0){S}_{R+1}^{\ensuremath{\nu}}(t)〉$, where $\ensuremath{\mu}\ensuremath{\nu}=x,y,z$ but $\ensuremath{\mu}\ensuremath{\ne}\ensuremath{\nu}$. We obtain the first-order term of ${\ensuremath{\rho}}_{\mathrm{xy}}$ for $R\ensuremath{\rightarrow}\ensuremath{\infty}$ or $t\ensuremath{\rightarrow}\ensuremath{\infty}$ and exact expressions for ${\ensuremath{\rho}}_{\mathrm{xz}}$ and ${\ensuremath{\rho}}_{\mathrm{yz}}$.
We consider the anisotropic one-dimensional X\ensuremath−Y model of Lieb, Schultz, and Mattis and study the three zero-temperature correlation functions \ensuremathρ\ensuremathμ\ensuremathν(R,t)=〈S1\ensuremathμ(0)SR+1\ensuremathν(t)〉, where \ensuremathμ\ensuremathν=x,y,z but \ensuremathμ\ensuremath=\ensuremathν. We obtain the first-order term of \ensuremathρxy for R\ensuremath→\ensuremath∞ or t\ensuremath→\ensuremath∞ and exact expressions for \ensuremathρxz and \ensuremathρyz.
We consider the one-dimensional $X\ensuremath{-}Y$ model of Lieb, Schulz, and Mattis and study the asymptotic behavior of each of the three correlation functions $〈{{\ensuremath{\sigma}}_{0}}^{i}{{\ensuremath{\sigma}}_{N}}^{i}〉={{\ensuremath{\rho}}_{N}}^{i}$, where i=x, y, or z. We study in detail the influence of $X\ensuremath{-}Y$ anisotropy by separately studying the correlation functions in both the isotropic and anisotropic cases at both nonzero and zero temperatures. For nonzero temperature we derive both low- and high-temperature expansions for all three correlations and show that these correlations go to zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. The behavior near T=0 is studied in the isotropic case by considering the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit with TN fixed, while in the anisotropic case we must hold T2N fixed as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. In this manner we obtain the T=0 result that if the interaction is stronger in the x direction, then ${{\ensuremath{\rho}}_{N}}^{x}$ approaches a constant exponentially while ${{\ensuremath{\rho}}_{N}}^{y}$ approaches zero exponentially as $N\ensuremath{\rightarrow}\ensuremath{\infty}$. We finally show that in the isotropic case at T=0 that ${{\ensuremath{\rho}}_{N}}^{x}={{\ensuremath{\rho}}_{N}}^{y}\ensuremath{\rightarrow}{N}^{\frac{\ensuremath{-}1}{2}}$. In all cases, at least the first two terms of the asymptotic series are explicitly given.
Carboni and Richards have performed exact numerical calculations of the time-dependent two-spin correlation function 〈S1z(t)S1z(0)〉 as $T\ensuremath{\rightarrow}\ensuremath{\infty}$ for a finite one-dimensional Heisenberg system. The non-Gaussian character of their result was characterized by a steep rise near zero frequency for the Fourier transform. We show here that these characteristics result from the inclusion of a Lorentzian form for $S(k, \ensuremath{\omega})$, the paramagnetic scattering function at small wave vectors. We also prove that ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}$, the time Fourier transform of 〈S1z(t)S1z(0)〉 in the $T\ensuremath{\rightarrow}\ensuremath{\infty}$ limit, obeys the inequality ${〈{{S}_{1}}^{z}(t){{S}_{1}}^{z}(0)〉}_{\ensuremath{\omega}}\ensuremath{\ge}\mathrm{const}\ifmmode\times\else\texttimes\fi{}\mathrm{ln}|\frac{1}{\ensuremath{\omega}}|$ as $\ensuremath{\omega}\ensuremath{\rightarrow}0$ for a one-dimensional system. We discuss the probable divergence of the same quantity in two dimensions.
Computer calculations of the time-dependent spin and energy correlation functions of the classical one-dimensional XY model at infinite temperature are reported. Plots of 〈SziSzi(t)〉, 〈SziSzi+1(t)〉, $〈{\ensuremath{\epsilon}}^{i}{\ensuremath{\epsilon}}^{i}(t)〉$, and (21)[〈SxiSxi(t)〉+〈SyiSyi(t)〉] out to times Jt=9 are given (${\ensuremath{\epsilon}}^{i}$ is the energy-density operator and $\ensuremath{\hbar}J$ is the exchange integral). Comparison is made with the exact results for the spin-1/2 XY model. After an appropriate scaling of the exchange integral and normalization to the classical value at t=0 the values of the spin-1/2 functions are close to the values of their classical counterparts for times up to $\mathrm{Jt}\ensuremath{\approx}4$.
A simple mathematical scheme is derived here. With this scheme, three problems of lattice-spin systems are solved exactly. The first one is the problem of solving, thermodynamically, a linear chain of the Lenz-Ising model in a zero magnetic field with nearest, as well as next-nearest, neighbor couplings. The problem turns out to be equivalent to the problem of a linear chain with only nearest-neighbor couplings but in a finite magnetic field. The second one is to solve an imperfect 1-dimensional Heisenberg-Dirac model, similar to the partially solved ``Ising-Heisenberg'' model of Lieb-Mattis-Schultz, in a zero magnetic field. The problem is solved completely in the sense that all the elementary excitations of this model are shown in terms of some pseudofermions and the spectra are given as ε1(q)=[(Aq+Bq)2+Eq2]12,ε2(q)=[(Aq−Bq)2+Eq2]12,where Aq, Bq, and Eq are three different functions of sin (q), cos (q), and coupling strengths involved. The third one, the XY model, is used to study the contribution of ``inhomogeneity'' in the coupling strengths to the system, as compared to the anisotropicity contribution to it.
Lieb-Schultz-Mattis (LSM) theorems impose non-perturbative constraints on the zero-temperature phase diagrams of quantum lattice Hamiltonians (always assumed to be local in this paper). LSM theorems have recently been interpreted as the lattice counterparts to mixed ’t Hooft anomalies in quantum field theories that arise from a combination of crystalline and global internal symmetry groups. Accordingly, LSM theorems have been reinterpreted as LSM anomalies. In this work, we provide a systematic diagnostic for LSM anomalies in one spatial dimension. We show that gauging subgroups of the global internal symmetry group of a quantum lattice model obeying an LSM anomaly delivers a dual quantum lattice Hamiltonian such that its internal and crystalline symmetries mix non-trivially through a group extension. This mixing of crystalline and internal symmetries after gauging is a direct consequence of the LSM anomaly, i.e., it can be used as a diagnostic of an LSM anomaly. We exemplify this procedure for a quantum spin-1/2 chain obeying an LSM anomaly resulting from combining a global internal \mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ 2 × ℤ 2 symmetry with translation or reflection symmetry. We establish a triality of models by gauging a \mathbb{Z}^{\,}_{2}\subset\mathbb{Z}^{\,}_{2}×\mathbb{Z}^{\,}_{2} ℤ 2 ⊂ ℤ 2 × ℤ 2 symmetry in two ways, one of which amounts to performing a Kramers-Wannier duality, while the other implements a Jordan-Wigner duality. We discuss the mapping of the phase diagram of the quantum spin-1/2 XYZ X Y Z chains under such a triality. We show that the deconfined quantum critical transitions between Neel and dimer orders are mapped to either topological or conventional Landau-Ginzburg transitions. Finally, we extend our results to \mathbb{Z}^{\,}_{n} ℤ n clock models with \mathbb{Z}^{\,}_{n}×\mathbb{Z}^{\,}_{n} ℤ n × ℤ n global internal symmetry, and provide a reinterpretation of the dual internal symmetries in terms of \mathbb{Z}^{\,}_{n} ℤ n charge and dipole symmetries.