Physics Faculty PublicationsCopyright (c) 2015 University of Rhode Island All rights reserved.
http://digitalcommons.uri.edu/phys_facpubs
Recent documents in Physics Faculty Publicationsen-usSat, 24 Jan 2015 01:47:22 PST3600Implications of Direct-Product Ground States in the One-Dimensional Quantum XYZ and XY Spin Chains
http://digitalcommons.uri.edu/phys_facpubs/92
http://digitalcommons.uri.edu/phys_facpubs/92Thu, 22 Jan 2015 07:26:14 PST
We state the conditions under which the general spin-s quantum XYZ ferromagnet (H−) and antiferromagnet (H+) with an external magnetic field along one axis, specified by the Hamiltonian H±=± 𝒥Nl=1 (JxSxlSxl+1+JySylS l+1y+JzSzlSzl+1)-h 𝒥Nl=1Szl exhibits a fully ordered ground state described by a wave function which is a direct product of single-site wave functions. We present a detailed analysis of the implications for the zero-temperature dynamical properties of this model. In particular, we derive a rigorous relation between the three dynamic structure factors Sμμ(q,ω), μ=x,y,z at T=0. For the special case of the s=(1/2) anisotropic XY model (Jz=0), these relations are used to determine the dynamic structure factors Sxx(q,ω) and Syy(q,ω) at T=0 and h=(JxJy)1/2 in terms of the known dynamic structure factor Szz(q,ω).
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Gerhard Müller et al.Wave-Number Dependent Susceptibilities of One-Dimensional Quantum Spin Models
http://digitalcommons.uri.edu/phys_facpubs/91
http://digitalcommons.uri.edu/phys_facpubs/91Thu, 22 Jan 2015 07:20:20 PST
We calculate the zero-temperature q-dependent susceptibilities of the one-dimensional, S=1/2, transverse Ising model at the critical magnetic field and of the isotropic XY model in zero field which have not been previously determined. Our method, which is based on a rigorous method of calculating dynamic correlation functions for these models, provides precise numerical values for the susceptibilities at wave numbers q=kπ/M for integral M and odd integral k, as well as exact analytic results for the dominant singularities at q=0 and q=π.
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Gerhard Müller et al.Susceptibilities of One-Dimensional Quantum Spin Models at Zero Temperature
http://digitalcommons.uri.edu/phys_facpubs/90
http://digitalcommons.uri.edu/phys_facpubs/90Thu, 22 Jan 2015 07:15:10 PST
We calculate precise numerical values for the nondivergent direct or staggered zero-temperature susceptibilities of the one-dimensional, S=1/2, transverse Ising model at the critical field and for the isotropic XY model in zero field which have not been previously determined analytically. Our method is based on a rigorous approach to calculate dynamic correlation functions for these models. We also investigate the exact nature of the divergenices in the q-dependent susceptibilities. Our results are compared with existing predictions of approximate analytic approaches and numerical finite-chain calculations. Our result for the XY case is directly relevant for the interpretation of recent susceptibility measurements on the quasi-one-dimensional magnetic compound Cs2CoCl4.
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Gerhard Müller et al.Dynamic Correlation Functions for One-Dimensional Quantum Spin Systems: New Results Based on a Rigorous Approach
http://digitalcommons.uri.edu/phys_facpubs/89
http://digitalcommons.uri.edu/phys_facpubs/89Thu, 22 Jan 2015 07:08:23 PST
We present new results on the time-dependent correlation functions Ξn(t)=4⟨Sξ0(t)Sξn⟩, ξ=x,y, at zero temperature of the one-dimensional S=12 isotropic XY model (h=γ=0) and of the transverse Ising (TI) model at the critical magnetic field (h=γ=1). Both models are characterized by special cases of the Hamiltonian H=−JΣl[(1+γ)SxlSxl+1+(1−γ)SylSyl+1+hSz1]. We have derived exact results on the long-time asymptotic expansions of the autocorrelation functions Ξ0(0) and on the singularities of their frequency-dependent Fourier transforms Φξξ0(ω). We have also determined the latter functions by high-precision numerical calculations. The functions Φξξ0(ω), ξ=x,y, have singularities at the infinite sequence of frequencies ω=mω0, m=0,1,2,3,…, where ω0=J for the XY model and ω0=2J for the TI model. In the TI case, the leading singularities in φxx0(ω) are alternately one-sided and two-sided power-law singularities, the first two of which (at ω=0,2J) are divergent. The dominant singularities in the XY case are alternate one-sided power laws and two-sided power laws with logarithmic corrections, the first two of which (at ω=0,J) are divergent. The singularities at higher frequencies in both models are finite and become increasingly weaker. We point out that the nonanalyticities at ω≠0 are intrinsic features of the discrete quantum chain and have therefore not been found in the context of a continuum analysis (Luttinger model). At least the most prominent features of our new results should be observable in low-temperature dynamical experiments on quasi-one-dimensional compounds such as the XY-like substances Cs2CoCl4 and PrCl3 and the S=12 Ising-like substance CsCoCl3·2H2O.
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Gerhard Müller et al.Impact of Criticality and Phase Separation on the Spin Dynamics of the One-Dimensional t–J Model
http://digitalcommons.uri.edu/phys_facpubs/88
http://digitalcommons.uri.edu/phys_facpubs/88Mon, 08 Dec 2014 08:24:16 PST
The recursion method is used to determine the T=0 spin dynamic structure factor S_{ zz }(q,ω) in the Luttinger liquid state and in the phase‐separated state of the one‐dimensional t–Jmodel. As the exchange coupling increases from zero, the dispersions and line shapes of the dominant spin excitations are observed to undergo a major metamorphosis between the free‐fermion limit and the onset of phase separation. The familiar two‐spinon spectrum of the Heisenberg antiferromagnetic chain emerges gradually in the strongly phase‐separated state.
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Shu Zhang et al.Spin Correlation Functions in Randomexchange s=1/2 XXZ Chains
http://digitalcommons.uri.edu/phys_facpubs/87
http://digitalcommons.uri.edu/phys_facpubs/87Mon, 08 Dec 2014 08:17:14 PST
The decay of (disorder‐averaged) static spin correlation functions at T=0 for the one‐dimensional spin‐1/2 XXZantiferromagnet with uniform longitudinal coupling JΔ and random transverse coupling Jλ_{ i } is investigated by numerical calculations for ensembles of finite chains. At Δ=0 (XXmodel) the calculation is based on the Jordan‐Wigner mapping to free lattice fermions for chains with up to N=100 sites. At Δ≠0 Lanczos diagonalizations are carried out for chains with up to N=22 sites. The longitudinal correlation function 〈S^{ z }_{0}S^{ z }_{ r }〉 is found to exhibit a power‐law decay with an exponent that varies with Δ and, for nonzero Δ, also with the width of the λ_{ i }‐distribution. The results for the transverse correlation function 〈S^{ x }_{0}S^{ x }_{ r }〉 show a crossover from power‐law decay to exponential decay as the exchange disorder is turned on.
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Heinrich Röder et al.Excitation Spectra of the Linear Alternating Antiferromagnet
http://digitalcommons.uri.edu/phys_facpubs/86
http://digitalcommons.uri.edu/phys_facpubs/86Mon, 08 Dec 2014 08:09:12 PST
The linear, spin-1/2, alternating Heisenberg chain has attracted theoretical and experimental attention from physical chemists for about two decades, particularly in relation to spin exciton theory and the properties of linear, exchange-coupled free radicals. The model is somewhat unfamiliar to physicists but has become of increasing interest recently, primarily because of its relation to spin-Peierls transition systems. A striking feature of this model is that it has so far proved resistant to any form of analytic attack. Existing theories are therefore all approximate, and are not in agreement with one another. In particular, there is disagreement about the existence of an energy gap in the excitation spectrum for nonzero alternation, such a gap being crucial to spin-Peierls theory and spin exciton theory. In this paper we employ the method which has so far proved more reliable than any other approximate technique, namely the method of extrapolating exact finite-chain calculations to the thermodynamic limit. Our study is an extension of earlier work in this direction, and focuses on the nature of the ground state and on low-lying excitations in general, and the existence and properties of the gap in particular. We introduce the features of the linear alternating antiferromagnet through an initial description of the spin-Peierls transition and with brief reference to organic free radicals and spin exciton theory. This is followed by a survey of existing approximate theories. Features of the excitation spectrum are discussed and finite-chain extrapolations for the ground-state energy and energy gap as a function of alternation are presented. Comparisons are made with similar procedures performed on exactly solvable models, as a test of the expected accuracy of the extrapolations. Excitation spectra for a variety of other alternating models, classical and quantum, are calculated and surveyed comparatively. An unusual variety of behavior is observed, with striking differences between quantum and classical systems. Finally, a detailed comparison is made between our results and those of other approximate methods, including the new quantum renormalization-group approach. Particular attention is paid to values for the T=0 spin-Peierls critical exponents.
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Jill C. Bonner et al.Generalized Heisenberg Quantum Spin Chains (Invited)
http://digitalcommons.uri.edu/phys_facpubs/85
http://digitalcommons.uri.edu/phys_facpubs/85Mon, 08 Dec 2014 08:01:58 PST
Since the Heisenberg spin chain can be considered the simplest realistic model of magnetism, surprise and some degree of controversy have resulted from recent work of Haldane. The prediction is that quantum spin chains with half‐integer spin should all display T=0 phase behavior equivalent to that of the Bethe Ansatz integrable (solvable) spin‐1/2 quantum chain. More remarkably, the class of integerspin chains is predicted to show very different phase behavior. In particular, a gap should be present in the spectrum of a Heisenberg antiferromagnetic chain. This remarkable feature is counterintuitive in terms of accepted wisdom in magnetism (spin‐wave theory, spin‐Peierls theory) and critical phenomena. Consequently the vertification of the prediction is of great interest. A considerable amount of numerical work has been done, involving finite‐chain, finite‐size scaling, variational, Monte Carlo and other calculations, which will be reviewed here. The present consensus is that the weight of numerical evidence supports the prediction, although puzzling features still remain. Adding additional interactions to the basic Heisenberg Hamiltonian such as spin (XXZ) anisotropy, single‐ion anisotropy, biquadratic exchange, and an applied magnetic field, generates a rich and complicated phase diagram for chains with spin >1/2, particularly for the case of integer spin. The s=1 phase diagram seems to display critical behavior of a type not previously encountered. A theoretical appraisal of the Haldane phenomenon will include a discussion of the possible role of nonintegrability. Mention will also be made of current progress in experimental investigation of the phenomenon, including problems that might be encountered. More recent work of Affleck has greatly generalized the field‐theoretic mappings which underlay the original work of Haldane. A number of interesting problems have been mapped into quantum spin chains of various types, including field theoretic phenomena and the localization problem of the quantum Hall effect.
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Jill C. BonnerQuantum Spin Chains and the Conformal Anomaly
http://digitalcommons.uri.edu/phys_facpubs/84
http://digitalcommons.uri.edu/phys_facpubs/84Mon, 08 Dec 2014 08:01:56 PST
The conformal anomaly c determines the universality class of a model system in statistical mechanics. The value of c characterizes both 2D classical models and their 1D quantum counterparts. The conformal anomaly may therefore be determined numerically for quantum spin chains using the relation: E_{0}(N)≂E_{0}(∞)−(NΔE/12)c(1/N^{2}), where E_{0} (N) is the ground‐state energy of an N‐spin finite system, E_{0} (∞) is the ground‐state energy in the thermodynamic limit, and ΔE is the energy gap between the ground state at k=0 and the first excited state of the dispersion curve at k=2π/N. The numerical approach is highly successful when tested on the integrable s= 1/2 Heisenberg antiferromagnetic XXZ chain and the integrable s=1 SU(2) model. The method gives c=1 to within 2% accuracy for the s=1 and (3)/(2) XY chains, placing them in the universality class of the 2D XY model. The result c=1 (2% accuracy) is obtained for the s= (3)/(2) Heisenberg antiferromagnetic chain, in agreement with the Haldane prediction. The s=1 pure antiferromagnetic biquadratic chain and the s=1 XXZ model with uniaxial anisotropy in the vicinity of the critical point Δ=Δ_{2} ∼1.15 −1.18 have also been studied.
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Jill C. Bonner et al.Quantum Spin Dynamics of the Antiferromagnetic Linear Chain in Zero and Nonzero Magnetic Field
http://digitalcommons.uri.edu/phys_facpubs/83
http://digitalcommons.uri.edu/phys_facpubs/83Thu, 04 Dec 2014 07:47:01 PST
Spin-dynamical calculations on one-dimensional systems have relied heavily on classical (s=∞) theories, despite abundant evidence that quantum effects can be extremely important at low temperatures. We present a new approach to the spin dynamics of the one-dimensional isotropic s=1/2 Heisenberg antiferromagnetic (HB AF) which does not involve the many-body techniques usually employed. It is based on analytic Bethe ansatz calculations of excitation energies and densities of states combined with finite-chain calculations of matrix elements. An important feature of our method is the use of rigorous selection rules and the introduction of new selection rules, which are valid for macroscopic systems in a magnetic field. We show that in zero field the dynamical two-spin correlation function Sμμ(q,ω) at T=0 is governed by a two-parameter continuum of spin-wave-type excitations. In nonzero field, the longitudinal component Szz(q,ω) and the transverse components Sxx(q,ω)≡Syy(q,ω) behave quite differently because they are dominated by different continua of excitations. The former is characterized by a lowest excitation branch with a zero-frequency mode moving from the zone boundary (q=π) towards the zone center (q=0) as the field increases, whereas the latter is characterized by a lowest branch with a zero frequency mode moving from q=0 to π with increasing field. The first part of our work features an approximate analytic expression for Sμμ(q,ω) at zero temperature and in zero field. Although our expression is not rigorous, exact sum rules are violated only by a small amount, and good agreement exists with the few known exact results. Our studies are extended to nonzero temperatures by placing major reliance on exact finite-chain calculations. Our work was stimulated by recent neutron scattering experiments and is oriented towards experimental comparisons. Our result for the s=1/2 integrated intensity is in much better agreement with neutron scattering data on CuCl2·2N(C5D5) (CPC) than the corresponding semiclassical result. Moreover, the spectral-weight distribution in Sμμ(q,ω) shows increasing asymmetry as q→π, a quantum effect, again in agreement with more recent neutron scattering data. The second part of our work deals with the effects of an applied magnetic field. We extend the analytic work of Ishimura and Shiba to obtain expressions for the energies and densities of states of the various excitation continua. It is shown that these continua are expected to give rise to multiple structures in the scattering intensity. Our results appear to be in quantitative agreement with preliminary results of a neutron study in CPC in a field of 70 kOe, revealing anomalous scattering intensity peaks. Our results repeatedly demonstrate the inadequacy of classical spin-wave theory for this problem. They call for additional experimental studies on quasi-one-dimensional antiferromagnets to examine other unusual and interesting phenomena predicted by our approach.
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Gerhard Müller et al.Quantum Effects in the Spin Dynamics of the Linear Heisenberg Antiferromagnet
http://digitalcommons.uri.edu/phys_facpubs/82
http://digitalcommons.uri.edu/phys_facpubs/82Thu, 04 Dec 2014 07:38:17 PST
We present an approximate analytic expression for the dynamical spin correlation function of the S=1/2 linear Heisenberg antiferromagnet at T=0. The basis for our approach is that in zero field the spectrum is dominated by a double continuum [in (q,ω) ‐space] of triplet spin waveexcitations. The S=1/2 integrated intensity agrees very well with recent neutron scattering results on CPC, unlike the corresponding classical intensity. Moreover, the S=1/2 spectral weight function shows increasing asymmetry as q→π, a quantum effect, observable in more recent neutron scattering data. In non‐zero magnetic field, there exist two, partly overlapping, double continua, each giving rise to a peak situated at the lower boundary. The (zz component of) spectral weight function therefore has a double‐peaked structure, as observed experimentally. Theory and experiment are in apparent agreement concerning the energy difference between the peaks.
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Gerhard Müller et al.Zero-Temperature Dynamics of the S=1/2 Linear Heisenberg Antiferromagnet
http://digitalcommons.uri.edu/phys_facpubs/81
http://digitalcommons.uri.edu/phys_facpubs/81Thu, 04 Dec 2014 07:30:47 PST
We present an analytical expression for the dynamical spin-correlation function of the S=1/2 linear Heisenberg antiferromagnet based on the fact that the spectrum is dominated by a double continuum of spin-wave excitations. Our expression is not exact, but exact sum rules show that the degree of approximation is small. We predict that in a uniform magnetic field the spectral weight function will display a double-peaked structure. Such a feature is observed in recent neutron-scattering experiments.
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Gerhard Müller et al.Linear Magnetic Chains with Anisotropic Coupling
http://digitalcommons.uri.edu/phys_facpubs/80
http://digitalcommons.uri.edu/phys_facpubs/80Tue, 02 Dec 2014 13:33:19 PST
inear chains (and rings) of S=12 spins with the anisotropic (Ising-Heisenberg) Hamiltonian ℋ=−2JΣi=1N{SizSi+1z+γ(SixSi+1x+SiySi+1y)}−gβΣi=1NH·Si have been studied by exact machine calculations for N=2 to 11, γ=0 to 1 and for ferro- and antiferro-magnetic coupling. The results reveal the dependence on finite size and anisotropy of the spectrum and dispersion laws, of the energy, entropy, and specific heat, of the magnetization and susceptibilities, and of the pair correlations. The limiting N→∞ behavior is accurately indicated, for all γ, in the region kT∣∣J∣∣>~0.5 which includes the maxima in the specific heat and susceptibility. The behavior of thermal and magnetic properties of infinite chains at lower temperatures is estimated by extrapolation. For infinite antiferromagnetic chains the ground-state degeneracy, the anisotropy gap, and the magnetization, perpendicular susceptibility, and pair correlations at T=0 are similarly studied. Estimates of the long-range order suggest that it vanishes only at the Heisenberg limit γ=1 and confirm the accuracy of Walker's perturbation series in γ.
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Jill C. Bonner et al.Integrable and Nonintegrable Classical Spin Clusters: Trajectories and Geometric Structure of Invariants
http://digitalcommons.uri.edu/phys_facpubs/79
http://digitalcommons.uri.edu/phys_facpubs/79Mon, 20 Oct 2014 10:34:53 PDT
This study investigates the nonlinear dynamics of a pair of exchange-coupled spins with biaxial exchange and single-site anisotropy. It represents a Hamiltonian system with 2 degrees of freedom for which we have already established the (nontrivial) integrability criteria and constructed the integrals of the motion provided they exist. Here we present a comparative study of the phase-space trajectories for two specific models with the same symmetry properties, one of which (the XY model with exchange anisotropy) is integrable, and the other (the XY model with single-site anisotropy) nonintegrable. In the integrable model, the integrals of the motion (analytic invariants) can be reconstructed numerically by means of time averages of dynamical variables over all trajectories. In the nonintegrable model, such time averages over trajectories define nonanalytic invariants, where the nonanalyticities are associated with the presence of chaotic trajectories. A prominent feature in the nonintegrable model is the occurrence of very long time scales caused by the presence of low- ux cantori, which form "sticky" coats on the boundary between chaotic regions and regular islands or "leaky" walls between dierent chaotic regions. These cantori dominate the convergence properties of time averages and presumably determine the long-time asymptotic properties of dynamic correlation functions. Finally, we present a special class of integrable systems containing arbitrarily many spins coupled by general biaxial exchange anisotropy.
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Niraj Srivastava et al.Quantum and Classical Spin Clusters: Disappearance of Quantum Numbers and Hamiltonian Chaos
http://digitalcommons.uri.edu/phys_facpubs/78
http://digitalcommons.uri.edu/phys_facpubs/78Wed, 01 Oct 2014 13:10:11 PDT
We present a direct link between manifestations of classical Hamiltonian chaos and quantum nonintegrability effects as they occur in quantum invariants. In integrable classical Hamiltonian systems, analytic invariants (integrals of the motion) can be constructed numerically by means of time averages of dynamical variables over phase-space trajectories, whereas in near-integrable models such time averages yield nonanalytic invariants with qualitatively different properties. Translated into quantum mechanics, the invariants obtained from time averages of dynamical variables in energy eigenstates provide a topographical map of the plane of quantized actions (quantum numbers) with properties which again depend sensitively on whether or not the classical integrability condition is satisfied. The most conspicuous indicator of quantum chaos is the disappearance of quantum numbers, a phenomenon directly related to the breakdown of invariant tori in the classical phase flow. All results are for a system consisting of two exchange-coupled spins with biaxial exchange and single-site anisotropy, a system with a nontrivial integrability condition.
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Niraj Srivastava et al.Dynamics of Semi-infinite Quantam Spin Chains at T = ∞
http://digitalcommons.uri.edu/phys_facpubs/77
http://digitalcommons.uri.edu/phys_facpubs/77Tue, 16 Sep 2014 06:24:58 PDT
Time-dependent spin autocorrelation functions and their spectral densities for the semi-infinite one-dimensionals=1/2 XY and XXZ models atT=∞ are determined in part by rigorous calculations in the fermion representation and in part by the recursion method in the spin representation. Boundary effects yield valuable new insight into the different dynamical processes which govern the transport of spin fluctuations in the two models. The results obtained for theXXX model bear the unmistakable signature of spin diffusion in the form of a squareroot infrared divergence in the spectral density.
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Joachim Stolze et al.InfInfinite-temperature Dynamics of the Equivalent-neighbor XYZ Model
http://digitalcommons.uri.edu/phys_facpubs/76
http://digitalcommons.uri.edu/phys_facpubs/76Tue, 09 Sep 2014 06:48:14 PDT
The dynamics of the classical XYZ model with uniform interaction is investigated by the recursion method and, in part, by exact analysis. The time evolution is anharmonic for arbitrary N (number of spins); only the cases N=2 and ∞ are completely integrable. For the special (uniaxially symmetric) equivalent-neighbor XXZ model, the nonlinearities in the equations of motion disappear in the limit N-->∞, and the spin autocorrelation functions are determined exactly for infinite temperature: The function exhibits a Gaussian decay to a nonzero constant, and the function decays to zero, algebraically or like a Gaussian, depending on the amount of uniaxial anisotropy. For the general XYZ case, the T=∞ dynamical behavior includes four different universality classes, categorized according to the decay law of the spectral densities at high frequencies. That decay law governs the growth rate of the sequence of recurrents that determine the relaxation function in the continued-fraction representation. The four universality classes may serve as prototypes for a classification of the dynamics of classical and quantum many-body systems in general.
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Jian-Min Liu et al.Dynamics of an Integrable Two-sublattice Spin Model with Long-range Interaction
http://digitalcommons.uri.edu/phys_facpubs/75
http://digitalcommons.uri.edu/phys_facpubs/75Tue, 09 Sep 2014 06:43:24 PDT
The dynamics of the classical two-sublattice XYZ model with uniform intersublattice interaction and zero intrasublattice interaction is completely integrable for arbitrary system sizes. This makes the system amenable to an exact analysis of dynamic correlation functions. Here we present some exact results for the case with isotropic interaction (XXX model). The dynamical properties of the two-sublattice XYZ model are compared with those of the equivalent-neighbor XYZ model and categorized into universality classes of dynamical behavior.
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Jian-Min Liu et al.Computation of Dominant Eigenvalues and Eigenvectors: A Comparative Study of Algorithms
http://digitalcommons.uri.edu/phys_facpubs/74
http://digitalcommons.uri.edu/phys_facpubs/74Tue, 09 Sep 2014 06:34:39 PDT
We investigate two widely used recursive algorithms for the computation of eigenvectors with extreme eigenvalues of large symmetric matrices -- the modified Lanczös method and the conjugate-gradient method. The goal is to establish a connection between their underlying principles and to evaluate their performance in applications to Hamiltonian and transfer matrices of selected model systems of interest in condensed matter physics and statistical mechanics. The conjugate-gradient method is found to converge more rapidly for understandable reasons, while storage requirements are the same for both methods.
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M. P. Nightingale et al.Spin diffusion in the one-dimensional s = 1/2 XXZ model at infinite temperature
http://digitalcommons.uri.edu/phys_facpubs/73
http://digitalcommons.uri.edu/phys_facpubs/73Thu, 24 Jul 2014 07:49:44 PDT
Time-dependent spin-autocorrelation functions at T = ∞ and (in particular) their spectral densities for the bulk spin and the boundary spin of the semi-infinite spin-1/2 XXZ model (with exchange parameters Jx = Jy = J, Jz) are investigated on the basis of (i) rigorous bounds in the time domain and (ii) a continued-fraction analysis in the frequency domain. We have found strong numerical evidence for spin diffusion in quantum spin models. For Jz/J increasing from zero, the results of the short-time expansion indicate a change of the bulk-spin xx-autocorrelation function from Gaussian decay to exponential decay. The continued-fraction analysis of the same dynamic quantity signals a change from exponential decay to power-law decay as Jz/J pproaches unity and back to a more rapid decay upon further increase of that parameter. By contrast, the change in symmetry at Jz/J = 1 has virtually no impact on the bulk-spin zz-autocorrelation function (as expected). Similar contrasting properties are observable in the boundary-spin autocorrelation functions.
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Markus Böhm et al.