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arXiv:0806.3192 · Physical Review A

Semiclassical analysis of spin systems near critical energies

Pedro Ribeiro, Thierry Paul

Published Mar 16, 2009• 9 citations
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Abstract

The spectral properties of su(2) Hamiltonians are studied for energies near the critical classical energy ${\ensuremath{\epsilon}}_{c}$ for which the corresponding classical dynamics presents hyperbolic points. A general method leading to an algebraic relation for eigenvalues in the vicinity of ${\ensuremath{\epsilon}}_{c}$ is obtained in the thermodynamic limit, when the semiclassical parameter ${n}^{\ensuremath{-}1}={(2s)}^{\ensuremath{-}1}$ goes to zero (where $s$ is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near ${\ensuremath{\epsilon}}_{c}$, are also computed and shown to be in agreement with the numerical results.

DOI
10.1103/physreva.79.032107

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The spectral properties of su(2) Hamiltonians are studied for energies near the critical classical energy \ensuremathϵc{\ensuremath{\epsilon}}_{c}\ensuremathϵc​ for which the corresponding classical dynamics presents hyperbolic points. A general method leading to an algebraic relation for eigenvalues in the vicinity of \ensuremathϵc{\ensuremath{\epsilon}}_{c}\ensuremathϵc​ is obtained in the thermodynamic limit, when the semiclassical parameter n\ensuremath−1=(2s)\ensuremath−1{n}^{\ensuremath{-}1}={(2s)}^{\ensuremath{-}1}n\ensuremath−1=(2s)\ensuremath−1 goes to zero (where sss is the total spin of the system). Two applications of this method are given and compared with numerics. Matrix elements of observables, computed between states with energy near \ensuremathϵc{\ensuremath{\epsilon}}_{c}\ensuremathϵc​, are also computed and shown to be in agreement with the numerical results.
Abstract Part I: Quantum Collective Spin Systems We use a coherent spin state formalism in order to study collective spin models, which have many applications in physics. In particular, the Lipkin-Meshkov-Glick (LMG) has been analyzed in the thermodynamic limit. The method developed during this work can, in principle, be used for more general Hamiltonians written as a function of the su(2) algebra generators. We have derived exact expressions for the integrated density of states for this model. The nature of the density of states singularities has been detailed. The first order finite size corrections as well as observables expectation values were also computed. The standard Bohr-Sommerfeld quantization approach, adapted to the spin case, is no longer valid near the spectral singularities. In order to treat these cases, we have developed a new approach that permits to describe the spectrum in the neighbourhood of the critical points. Part II: Adiabatic Quantum Computation We have proposed a simple model which highlights the relations between quantum phase transitions and adiabatic (quantum) computation. This model puts in evidence the importance of the choice of the initial Hamiltonian and of the adiabatic path in the parameter space; it should be helpful as a toy model for more realistic cases. We have also studied the state populations dynamics when a quantum phase transition point is crossed, for the LMG model studied in the first part. Numerical simulations show that the dynamics of the populations is very sensitive to the presence of exceptional points in the spectrum, which a simplified model for the quantum evolution already suggests.
An approximation for calculating the thermodynamic properties of spin-\textonehalf{} Ising models is described. The method is based on the formulation of Kramers and Wannier for the square lattice, and involves expressing the free energy in terms of the limiting maximum eigenvalue of a matrix sequence which represents the contribution from a single spin of the lattice. The application of this formulation to various two- and three-dimensional lattices shows that many of these are characterized by an identically structured matrix sequence with exactly two nonvanishing elements in each row and column. To explore the potential of using this similarity and its simplicity for analyzing Ising models we consider the two-dimensional square lattice in an external field and approximate the doubly infinite lattice by a semi-infinite one consisting of $n (\ensuremath{\le}10)$ infinitely long rows. Calculations are carried out and compared with (1) Onsager's formula for the free energy, (2) the Onsanger-Yang formula for the spontaneous magnetization, and (3) values calculated from series. Except for the region around the critical point where the correlation length exceeds nnn lattice spacings, substantial agreement---to better than 12 places of decimals in some cases---is obtained. As an additional measure of the potential of this method, a value for the critical exponent $\ensuremath{\delta}$ is computed by the extrapolation of the magnetization along the critical isotherm. The result is found to be in excellent agreement with the known value for $\ensuremath{\delta}$.
Our study emphasizes the efficacy of employing matrices resembling Wishart matrices, derived from magnetization time series data within specific dynamics, to elucidate phase transitions and critical phenomena in the Q-state Potts model. Through the application of appropriate statistical methods, we not only identify second-order transitions but also distinguish weaker first-order transitions by carefully analyzing the density of eigenvalues and their fluctuations. Additionally, we investigate the method’s sensitivity to stronger first-order transition points. Notably, we establish a robust correlation between the system’s actual thermodynamics and the spectral thermodynamics encapsulated within the eigenvalues. Our findings are further supported by correlation histograms of the time series data, revealing insightful patterns. Building upon our core results, we provide a didactic analysis that draws parallels between the spectral properties of criticality in a spin system and matrices intentionally imbued with correlations (a toy model). Within this framework, we observe a universal behavior characterized by the distribution of eigenvalues into two distinct groups, separated by a gap dependent on the level of correlation, influenced by temperature-induced changes in the spin system.
We provide an extensive study of the subohmic spin-boson model with power-law density of states $J(\ensuremath{\omega})\ensuremath{\propto}{\ensuremath{\omega}}^{s}$ (with $0<s\ensuremath{\le}1$), focusing on the equilibrium dynamics of the three possible spin components, from very weak dissipation to the quantum critical regime. Two complementary methods, the bosonic numerical renormalization group and Majorana diagrammatics, are used to explore the physical properties in a wide range of parameters. We show that the bosonic self-energy is the crucial ingredient for the description of critical fluctuations, but that many-body vertex corrections need to be incorporated as well in order to obtain quantitative agreement of the diagrammatics with the numerical simulations. Our results suggest that the out-of-equilibrium dynamics in dissipative models beyond the Bloch-Redfield regime should be reconsidered in the long-time limit. Regarding also the spin-boson Hamiltonian as a toy model of quantum criticality, some of the insights gained here may be relevant for field theories of electrons coupled to bosons in higher dimensions.
Starting with path integrals in the SU(2) coherent state representation, the semiclassical approximation of the propagator for the spin system is investigated. By extending the idea of the semiclassical expansion method, which was developed in the usual phase-space path integrals, to the path integrals in the curved phase space, which is characteristic of the SU(2) coherent states, we obtain a closed form for the semiclassical propagator. As an application, we discuss the semiclassical quantization condition for the spin system.