Quantum spin chains: Simple models with complex dynamics

The present study highlights some of the complexities observed in the dynamical properties of one-dimensional quantum spin systems. Exact results for zero-temperature dynamic correlation functions are presented for two contrasting situations:

1. a system with a fully ordered ferromagnetic ground state;
2. a system at aT _c=0 critical point.

For both situations it is found that the exact results are considerably more complex than has been anticipated on the basis of approximate approaches which are considered to be appropriate and reliable for such situations. A still higher degree of complexity is expected for the dynamics of quantum spin systems which are nonintegrable. The paper concludes with some observations concerning nonintegrability effects and quantum chaos in spin systems.     

Reduced models for the medium-frequency dynamics of stochastic systems

In this paper, a frequency domain vibration analysis procedure of a randomly parametered structural system is described for the medium-frequency range. In this frequency range, both traditional modal analysis and statistical energy analysis (SEA) procedures well-suited for low- and high-frequency vibration analysis respectively, lead to computational and conceptual difficulties. The uncertainty in the structural system can be attributed to various reasons such as the coupling of the primary structure with a variety of secondary systems for which conventional modeling is not practical. The methodology presented in the paper consists of coupling probabilistic reduction methods with dynamical reduction methods. In particular, the Karhunen-Loeve and Polynomial Chaos decompositions of stochastic processes are coupled with an operator decomposition scheme based on the spectrum of an energy operator adapted to the frequency band of interest.     

Finite-size scaling for correlations of quantum spin chains at criticality

We study the finite-size scaling behavior of two-point correlation functions of translationally invariant many-body systems at criticality. We propose an efficient method for calculating the two-point correlation functions in the thermodynamic limit from numerical data of finite systems. Our method is most effective when applied to a two-dimensional (classical) system which possesses a conformal invariance. By using this method with numerical data obtained from exact diagonalizations and Monte Carlo simulations, we study the spin-spin correlations of the quantum spin-1/2 and-3/2 antifierromagnetic chains. In particular, the logarithmic corrections to power-law decay of the correlation of the spin-1/2 isotropic Heisenberg antiferromagnetic chain are studied thoroughly. We clarify the cause of the discrepancy in previous calculations for the logarithmic corrections. Our result strongly supports the field-theoretic prediction based on the mappings to the Wess-Zumino-Witten nonlinear -model or the sine-Gordon model. We also treat logarithmic corrections and crossover phenomena in the spin-spin correlation of the spin-3/2 isotropic Heisenberg antiferromagnetic chain. Our results are consistent with the Affleck-Haldane prediction that the correlation of the spin-3/2 chain exhibits a crossover to the same asymptotic behavior as in the spin-1/2 chain.     

Markov chains and discrete chaos

The connection between the statistics of a one-dimensional system exhibiting discrete chaos and its underlying deterministic law is formulated in terms of Markov chains. The correlation functions of piecewise linear Markovian maps are computed exactly using linear recursion formulae. An example of a non-Markovian map is given which can be approximated by Markovian maps.     

Monte-Carlo study of correlations in quantum spin chains at non-zero temperature

Antiferromagnetic Heisenberg spin chains with various spin values (S=1/2,1,3/2,2,5/2) are studied numerically with the quantum Monte-Carlo method. Effective spin S chains are realized by ferromagnetically coupling n=2S antiferromagnetic spin chains with S=1/2. The temperature dependence of the uniform susceptibility, the staggered susceptibility, and the static structure factor peak intensity are computed down to very low temperatures, . The correlation length at each temperature is deduced from numerical measurements of the instantaneous spin-spin correlation function. At high temperatures, very good agreement with exact results for the classical spin chain is obtained independent of the value of S. For the S=2 chain which has a gap , the correlation length and the uniform susceptibility in the temperature range are well predicted by the semi-classical theory of Damle and Sachdev. Received: 23 December 1997 / Revised and Accepted: 11 March 1998     

Quantum crystals and spin chains

In this article, we discuss the quantum version of the melting crystal corner in one, two, and three dimensions, generalizing the treatment for the quantum dimer model. Using a mapping to spin chains we find that the two-dimensional case (growth of random partitions) is integrable and leads directly to the Hamiltonian of the Heisenberg XXZ ferromagnet. The three-dimensional case of the melting crystal corner is described in terms of a system of coupled XXZ spin chains. We give a conjecture for its mass gap and analyze the system numerically.     

Multiplicity in supersymmetric spin chains

We discuss a simple procedure for obtaining new integrable spin chains from old by replacing each single state of the original model by some collection of states. This works whenever the Lax matrix of the chain has a certain form. The simplest example is the su(n)$\mathfrak{su}(n)$ XX model. We apply the techniques of the nested algebraic Bethe ansatz to solve such systems, in the bosonic and supersymmetric cases.     

Solitons in stochastic molecular chains

The dynamics of solitary collective excitations in one-dimensional molecular crystals under the action of stochastic perturbations is studied. An equation is derived for the mean field of nonlinear collective excitations a(x, t) and deformation of the lattice (x, t). The law describing the decay of collective excitations is obtained.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 21–25, December, 1981.     

Algebraic decay in self-similar Markov chains

A continuous-time Markov chain is used to model motion in the neighborhood of a critical invariant circle for a Hamiltonian map. States in the infinite chain represent successive rational approximants to the frequency of the invariant circle. For the case of a noble frequency, the chain is self-similar and the nonlinear integral equation for the first passage time distribution is solved exactly. The asymptotic distribution is a power law times a function periodic in the logarithm of the time. For parameters relevant to the critical noble circle, the decay proceeds ast ^–4.05.     

Magnetic resonance in quantum spin chains

The present understanding of quantum spin chains is reviewed from the magnetic resonance point of view. This includes both the ideal one-dimensional properties in the spin sector as well as the complex interplay with orbital, charge, and lattice degrees of freedom which govern the ground state. In copper-phosphates we observe an extremely extended paramagnetic regime governed by strong antiferromagnetic correlations with record values of the ratio k_BT_N/J < 6×10⁻⁴, which compares the ordering temperature of a Néel state to the magnitude of the exchange J between neighbouring spins. A detailed quantitative discussion of NMR and ESR relaxation within this paramagnetic regime elucidates the relevant exchange interactions in typical bonding geometries of most common quantum-spin-chain systems like KCuF₃, CuGeO₃, Na_xV₂O₅, and LiCuVO₄. Concerning the ground state, paramount topics of modern solid-state physics arise among these examples as there are multiferroicity, charge order, metal-insulator transition, and spin dimerization as well as phase separation.     

Measurement-based teleportation along quantum spin chains

We examine the teleportation of an unknown spin-1/2 quantum state along a quantum spin chain with an even number of sites. Our protocol, using a sequence of Bell measurements, may be viewed as an iterated version of the 2-qubit protocol of C. H. Bennett et al. [Phys. Rev. Lett. 70, 1895 (1993)]. A decomposition of the Hilbert space of the spin chain into 4 vector spaces, called Bell subspaces, is given. It is established that any state from a Bell subspace may be used as a channel to perform unit fidelity teleportation. The space of all spin-0 many-body states, which includes the ground states of many known antiferromagnetic systems, belongs to a common Bell subspace. A channel-dependent teleportation parameter [symbol: see text] is introduced, and a bound on the teleportation fidelity is given in terms of [symbol: see text].     

Stochastic resonance phenomena in spin chains

We discuss stochastic resonance-like effects in the context of coupled quantum spin systems. We focus here on an information-theoretic approach and analyze the steady state quantum correlations (entanglement) as well as the global correlations in the system when subject to different forms of local decoherence. In the presence of decay, it has been shown that the system displays quantum correlations only when the noise strength is above a certain threshold. We extend this result to the case of a Heisenberg XYZ exchange interaction and revise and clarify the mechanisms underlying this behaviour. In the presence of pure dephasing, we show that the system always remains separable in the steady state. When both types of noise are present, we show that the system can still exhibit entanglement for long times, provided that the pure dephasing rate is not too large.     

Quantum spin chains with fractional revival

A systematic study of fractional revival at two sites in XX$XX$ quantum spin chains is presented. Analytic models with this phenomenon are obtained by combining two basic ways of realizing fractional revival in a spin chain. The first proceeds through isospectral deformations of spin chains with perfect state transfer. The second makes use of couplings provided by the recurrence coefficients of polynomials with a bi-lattice orthogonality grid. The latter method leads to analytic models previously identified that can exhibit perfect state transfer in addition to fractional revival.     

Phase transition in aperiodic spin chains

The quantum-mechanical transition in the ground state of some aperiodic spin chains is considered. Exact expressions for the energy gap and the dispersion relation of low-energy excitations close to the critical point are derived. Applications of these results to the study of particular quasiperiodic and random models are discussed.     

Entangled finitely correlated quantum spin chains

We study the multipartite entanglement properties of translation-invariant states of infinite quantum spin chains with a valence-bond structure.     

Thermodynamics of antiferromagnetic alternating spin chains

We consider integrable quantum spin chains with alternating spins (S₁,S₂)$(S_1,S_2)$. We derive a finite set of non-linear integral equations for the thermodynamics of these models by use of the quantum transfer matrix approach. Numerical solutions of the integral equations are provided for quantities like specific heat, magnetic susceptibility and in the case S₁=S₂$S_1=S_2$ for the thermal Drude weight. At low temperatures one class of models shows finite magnetization and the other class presents antiferromagnetic behaviour. The thermal Drude weight behaves linearly on T at low temperatures and is proportional to the central charge c   of the system. Quite generally, we observe residual entropy for S₁≠S₂$S_1\ne S_2$.