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Physical review. B, Condensed matter

Spreading of damage and susceptibility in the quantum S=1/2 Heisenberg ferromagnet

U.M.S. Costa, I. V. Rojdestvenski, M. L. Lyra

Published Aug 1, 1998• 6 citations
Publisher

Abstract

In this work we extend a recently proposed prescription for studying the dynamical properties of quantum Heisenberg ferromagnets which is based on the spreading of the damage technique. Within the Handscomb Monte Carlo method, we introduce a Hamming distance defined as a difference in the cyclic structure of Mayers diagrams in such a way that the quantum nature of the Hamiltonian is accounted for. We show that the cyclic Hamming distance is closely related to the static spin susceptibility and illustrate our results by comparing them with previous two-loop renormalization group calculations of the susceptibility low-temperature behavior in two-dimensional quantum Heisenberg models.

DOI
10.1103/physrevb.58.2403

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In this work we extend a recently proposed prescription for studying the dynamical properties of quantum Heisenberg ferromagnets which is based on the spreading of the damage technique. Within the Handscomb Monte Carlo method, we introduce a Hamming distance defined as a difference in the cyclic structure of Mayers diagrams in such a way that the quantum nature of the Hamiltonian is accounted for. We show that the cyclic Hamming distance is closely related to the static spin susceptibility and illustrate our results by comparing them with previous two-loop renormalization group calculations of the susceptibility low-temperature behavior in two-dimensional quantum Heisenberg models.
We suggest a tool to investigate the processes of the ferromagnetic ordering in the quantum Heisenberg model. We introduce the spreading of a damage technique based on the Handscomb Monte Carlo method and apply it to the three-dimensional simple cubic Heisenberg ferromagnet. Particularly, we study the thermal dependence of the graph Hamming distance and employ a finite-size scaling analysis of its size dependence. It is obtained that a properly defined Hamming distance displays a maximum at a characteristic temperature Ts,{T}_{s},Ts​, below which it becomes roughly size independent. We discuss the properties of these dependences in relevance to other thermodynamic functions of the model.
The low-temperature behavior of quantum Heisenberg models can be obtained by combining a renormalization-group analysis of the quantum models with a Monte Carlo simulation of the corresponding classical model, where the ``spins'' are represented by unit vectors defined on a lattice. In this approach renormalization-group equations for the quantum models lead to an effective classical O(3) nonlinear \ensuremath{\sigma} model in appropriate regimes of the parameter space. In the present work we apply this technique, which was recently used by Chakravarty, Halperin, and Nelson to discuss the low-temperature behavior of quantum antiferromagnets, to ferromagnets and calculate the correlation length and the susceptibility of a two-dimensional quantum ferromagnet at low temperatures. The results obtained with this method are then compared with those obtained from a direct simulation of the spin S=1/2 quantum ferromagnet. Excellent agreement between the two approaches leads us to believe that the analytic expressions obtained in the renormalization-group method are accurate at low temperatures. This strengthens the validity of the procedure adopted for antiferromagnets as well, where no such simulation results are yet known. The present paper also further substantiates recent work on the properties of solid 3He^{3}\mathrm{He}3He in two dimensions.
We apply the spreading of the damage technique to a S=1/2 Ising ferromagnetic thin film. We use Metropolis dynamics to calculate the temperature dependence of the Hamming distance as well as the Hamming distance distributions along the Markov chain. We suggest a modification of the algorithm for calculating the Hamming distance in the case of Metropolis dynamics and we discuss the results for the two-dimensional model obtained by the modified approach. For a film with three layers we present the thermal dependences of the above quantities. We also suggest a new way to introduce damage to the system as the difference in the boundary conditions. We discuss the results of this new approach for the two-dimensional model.
We investigate the critical relaxational dynamics of the S=1/2S=1/2S=1/2 Heisenberg ferromagnet on a simple cubic lattice within the Handscomb prescription, on which a diagrammatic series expansion of the partition function is computed by means of a Monte Carlo procedure. Using a phenomenological renormalization-group analysis of graph quantities related to the spin susceptibility and order parameter, we obtain precise estimates of the critical exponents relations $\ensuremath{\gamma}/\ensuremath{\nu}=1.98\ifmmode\pm\else\textpm\fi{}0.01$ and $\ensuremath{\beta}/\ensuremath{\nu}=0.512\ifmmode\pm\else\textpm\fi{}0.002$ and of the Curie temperature ${k}_{B}{T}_{c}/J=1.6778\ifmmode\pm\else\textpm\fi{}0.0002.$ The critical correlation time ${\ensuremath{\tau}}_{\mathrm{int}}$ of both energy and susceptibility is also computed. We found the number of Monte Carlo steps needed to generate uncorrelated diagram configurations scales with the system's volume. We estimate the efficiency of the Handscomb method, comparing its ability in dealing with the critical slowing down to that of other quantum and classical Monte Carlo prescriptions.
We investigate the damage or Hamming distance between two configurations of Ising spins. We find an exact relation between the difference of the two possible types of damage and the spin-spin correlation function, which is generally valid. For the specific case of ferromagnetic interactions, heat bath dynamics and same sequence of random numbers, this relation involves only one type of damage. The susceptibility and the magnetization can also be expressed in terms of the damage. Numerical determination of the damage for the 2 d Ising model is not only an efficient way to calculate correlation functions but also gives access to spin fluctuations visualized as clusters of damaged sites which have a fractal dimension d −β/ν at T c and whose size distribution is also related to static exponents.
The thermodynamic properties and critical behavior of a Heisenberg ferromagnet in two dimensions are studied using the Handscomb quantum Monte Carlo method. The spins (with quantum number S=12) are arranged on a square lattice and have a long-range interaction with a power-law decrease, J(r)∝r−p, with distance r. The parameter p is taken to be in the range 2<p⩽6. The system has an ordering transition at a nonzero temperature Tc when 2<p<4. The dependence of Tc on p and the critical behavior of the susceptibility are investigated. Properties of spin correlation functions in different regimes (of p and T) are discussed. The results are compared with previous approximate analytic quantum theories.