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arXiv:cond-mat/0212248 · Physical Review Letters

Spectrum and Correlation Functions of a Quasi-One-Dimensional Quantum Ising Model

Sam T. Carr, Alexei M. Tsvelik

Published May 2, 2003• 25 citations
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Abstract

For a model of weakly coupled quantum Ising chains we outline the phase diagram and establish that well below the transition line the system has a remarkably one-dimensional spectrum. We study the dynamical magnetic susceptibility and find a very rich spectrum with several modes corresponding to a fundamental particle and its bound states. The approach is based on the Bethe ansatz and the random phase approximation applied to the interchain exchange.

DOI
10.1103/physrevlett.90.177206

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For a model of weakly coupled quantum Ising chains we outline the phase diagram and establish that well below the transition line the system has a remarkably one-dimensional spectrum. We study the dynamical magnetic susceptibility and find a very rich spectrum with several modes corresponding to a fundamental particle and its bound states. The approach is based on the Bethe ansatz and the random phase approximation applied to the interchain exchange.
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We study weakly coupled antiferromagnetic spin chains in their ordered phase by combining an exact solution of the single-chain problem with a random-phase-approximation analysis of the interchain interaction. A single chain is described by a quantum Sine-Gordon model and dynamical staggered susceptibilities are determined by employing the form-factor approach to quantum correlation functions. We consider both antiferromagnetic order encountered in quasi-one-dimensional materials such as KCuF3{\mathrm{KCuF}}_{3}KCuF3​ and spin-Peierls order as found in CuGeO3.{\mathrm{CuGeO}}_{3}.CuGeO3​.
Using large-scale determinant quantum Monte Carlo simulations in combination with the stochastic analytical continuation, we study two-particle dynamical correlation functions in the anisotropic square lattice of weakly coupled one-dimensional (1D) Hubbard chains at half filling and in the presence of weak frustration. The evolution of the static spin structure factor upon increasing the interchain coupling is suggestive of the transition from the power-law decay of spin-spin correlations in the 1D limit to long-range antiferromagnetic order in the quasi-1D regime and at T=0T=0T=0. In the numerically accessible regime of interchain couplings, the charge sector remains gapped. The low-energy momentum dependence of the spin excitations is well described by the linear spin-wave theory with the largest intensity located around the antiferromagnetic wave vector. This magnon mode corresponds to a bound state of two spinons. At higher energies the spinons deconfine and we observe signatures of the two-spinon continuum which progressively fade away as a function of interchain hopping.
We explore the phase diagram of Ising spins on one-dimensional chains that criss-cross in two perpendicular directions and that are connected by interchain couplings. This system is of interest as a simpler, classical analog of a quantum Hamiltonian that has been proposed as a model of magnetic behavior in Nb_{12}O_{29} and also, conceptually, as a geometry that is intermediate between one and two dimensions. Using mean-field theory as well as Metropolis Monte Carlo and Wang-Landau simulations, we locate quantitatively the boundaries of four ordered phases. Each becomes an effective Ising model with unique effective couplings at large interchain coupling. Away from this limit, we demonstrate nontrivial critical behavior, including tricritical points that separate first- and second-order phase transitions. Finally, we present evidence that this model belongs to the two-dimensional Ising universality class.
A simple kinetic model of the weakly coupled linear Ising chains is treated. The $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}$-dependent dynamical susceptibility is obtained, introducing the interchain molecular field into exact one-dimensional solutions. The usual molecular-field-type critical slowing down as $T\ensuremath{\rightarrow}{T}_{c}$ is accompanied by an additional noncritical, highly anisotropic slowing down due to high-intrachain short-range order. For the $\stackrel{\ensuremath{\rightarrow}}{\mathrm{q}}=0$ mode the factor of the one-dimensional slowing down goes with decreasing temperature as the square of the one-dimensional correlation length. In the case of a strong anisotropy (${J}_{\ensuremath{\parallel}}\ensuremath{\gg}{J}_{\ensuremath{\perp}}$), this factor surpasses the value of the ratio $\frac{{J}_{\ensuremath{\parallel}}}{{J}_{\ensuremath{\perp}}}$ already well above Tc{T}_{c}Tc​.
We consider a model of one-dimensional Mott insulators coupled by a weak interchain tunneling ${t}_{\ensuremath{\perp}}.$ We first determine the single-particle Green's function of a single chain by exact field-theoretical methods and then take the tunneling into account by means of a random-phase approximation. In order to embed this approximation into a well-defined expansion with a small parameter, the Fourier transform ${T}_{\ensuremath{\perp}}(k)$ of the interchain coupling is assumed to have a small support in momentum space such that every integration over transverse wave vector yields a small factor ${\ensuremath{\kappa}}_{0}^{2}\ensuremath{\ll}1.$ When ${T}_{\ensuremath{\perp}}(0)$ exceeds a critical value, a small Fermi surface develops in the form of electron and hole pockets. We demonstrate that Luttinger's theorem holds both in the insulating and in the metallic phases. We find that the quasiparticle residue Z increases very fast through the transition and quickly reaches a value of about 0.4--0.6. The metallic state close to the transition retains many features of the one-dimensional system in the form of strong incoherent continua.