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arXiv:1308.4395

Quantum-classical phase transition of the escape rate of two interacting large spins

S. A. Owerre, Manu B. Paranjape

Published Aug 20, 2013• 0 citations
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Abstract

The Hamiltonian of a two-sublattice antiferromagnetic spins, with single (hard-axis) and double ion anisotropies described by $H=J \bold{\hat{S}}_{1}\cdot\bold{\hat{S}}_{2} - 2J_{z}\hat{S}_{1z}\hat{S}_{2z}+K(\hat{S}_{1z}^2 +\hat{S}_{2z}^2)$ is investigated using the method of effective potential. The problem is mapped to a single particle quantum-mechanical Hamiltonian in terms of the relative coordinate and reduced mass. We study the quantum-classical phase transition of the escape rate of this model. We show that the first-order phase transition for this model sets in at the critical value $J_c=\frac{K+J_z}{2}$ while for the anisotropic Heisenberg coupling $H = J(S_{1x}S_{2x} +S_{1y}S_{2y}) + J_zS_{1z}S_{2z} + K(S_{1z}^2+ S_{2z}^2)$ we obtain $J_c=\frac{2K-J_z}{3}$ . The phase diagrams of the transition are also studied.

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The Hamiltonian of a two-sublattice antiferromagnetic spins, with single (hard-axis) and double ion anisotropies described by H=JS^1⋅S^2−2JzS^1zS^2z+K(S^1z2+S^2z2)H=J \bold{\hat{S}}_{1}\cdot\bold{\hat{S}}_{2} - 2J_{z}\hat{S}_{1z}\hat{S}_{2z}+K(\hat{S}_{1z}^2 +\hat{S}_{2z}^2)H=JS^1​⋅S^2​−2Jz​S^1z​S^2z​+K(S^1z2​+S^2z2​) is investigated using the method of effective potential. The problem is mapped to a single particle quantum-mechanical Hamiltonian in terms of the relative coordinate and reduced mass. We study the quantum-classical phase transition of the escape rate of this model. We show that the first-order phase transition for this model sets in at the critical value Jc=K+Jz2J_c=\frac{K+J_z}{2}Jc​=2K+Jz​​ while for the anisotropic Heisenberg coupling H=J(S1xS2x+S1yS2y)+JzS1zS2z+K(S1z2+S2z2)H = J(S_{1x}S_{2x} +S_{1y}S_{2y}) + J_zS_{1z}S_{2z} + K(S_{1z}^2+ S_{2z}^2)H=J(S1x​S2x​+S1y​S2y​)+Jz​S1z​S2z​+K(S1z2​+S2z2​) we obtain Jc=2K−Jz3J_c=\frac{2K-J_z}{3}Jc​=32K−Jz​​ . The phase diagrams of the transition are also studied.
The phase diagram of the anisotropic quantum spin-1/2 Heisenberg antiferromagnet in the presence of competitive nearest (J1)({J}_{1})(J1​)- and next-nearest $({J}_{2}=\ensuremath{\alpha}{J}_{1})$-neighbor interactions on a simple cubic lattice is studied within the framework of the differential operator technique. The Hamiltonian is solved by using an effective-field theory in a two-spin cluster. We propose a function for the free energy, which is similar to the Landau expansion, in order to find the ground state (T=0)(T=0)(T=0) and the finite-TTT phase diagram. At zero temperature, we find a first-order quantum phase transition in the point ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta})$ between the antiferromagnetic (AF) and superantiferromagnetic (SAF) phases. This is dependent on the anisotropy parameter $\ensuremath{\Delta}$ ($\ensuremath{\Delta}=0$ and 1 correspond to the isotropic Heisenberg and Ising models, respectively). The quantum critical values of ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}\ensuremath{\ne}1)={\ensuremath{\alpha}}_{c}^{q}(\ensuremath{\Delta})$ are smaller than those of the classical model ${\ensuremath{\alpha}}_{c}(\ensuremath{\Delta}=1)={\ensuremath{\alpha}}_{c}^{c}=1/4$ (for example, for $\ensuremath{\Delta}=0$, we find ${\ensuremath{\alpha}}_{c}^{q}(0)=0.21$). The former monotonically increases with the anisotropy parameter $\ensuremath{\Delta}$. The phase diagram in $T\text{\ensuremath{-}}\ensuremath{\alpha}$ plane presents a second-order transition at ${T}_{2}(\ensuremath{\alpha})$ between the AF and paramagnetic (P) phases. The phase transition between the SAF and P phases is of first-order ${T}_{1}(\ensuremath{\alpha})$ for all values of $\ensuremath{\Delta}∊[0,1]$. The phase transition lines ${T}_{1}(\ensuremath{\alpha})$ and ${T}_{2}(\ensuremath{\alpha})$ merge in a critical end point (CEP) that is dependent on $\ensuremath{\Delta}$ [i.e., ${T}_{\text{CEP}}(\ensuremath{\Delta})$ and ${\ensuremath{\alpha}}_{\text{CEP}}(\ensuremath{\Delta})$ are the temperature and frustration parameters at the CEP]. The ordered phases (AF and SAF) are separated by first-order transitions and exhibit a nontrivial re-entrant behavior at low temperatures.
The phase diagram at T = 0 of a family of spin-1 quantum Ising-Heisenberg models is studied using a mean-field approximation. The model Hamiltonian is characterized by single ion anisotropy terms D μ , (μ = x, y, z ) and by two positive parameters η x , η y measuring the degree of anisotropy of the interaction. They allow to describe a variety of models: among them the spin-1 Ising-Blume-Capel model (η x = η y = 0), and the isotropic Heisenberg model (η x = η y = 1). Both dipolar and quadrupolar order parameters are used to derive the phase diagram in the parameter space (η x , η y , D μ ). The resulting phase diagram is rather complex as it exhibits first and second order critical transitions as well as multicritical points.
We study the phase diagram of a spin-1 antiferromagnetic chain with isotropic antiferromagnetic interactions decaying with a power law $\ensuremath{\propto}{r}^{\ensuremath{-}\ensuremath{\alpha}}$ $(\ensuremath{\alpha}\ensuremath{\ge}1)$ accompanied by modulated single-ion anisotropy. Employing the techniques of the density-matrix renormalization group, the effects of long-range interactions and single-ion anisotropy on a variety of correlations are investigated. In order to check the consistency, the fidelity susceptibilities are evaluated across quantum phase transitions. The quantum critical points are faithfully detected and orders of phase transitions are determined. The correlation-length critical exponent is extracted from scaling functions of the fidelity susceptibility. The presence of long-range interactions leads to a quantitative change of the phase boundaries and reduces the order of the phase transition under certain conditions. A direct first-order transition between the periodic N\'eel phase and the large-DDD phase occurs for slowly decaying antiferromagnetic interactions.
The critical and hysteresis behaviors of the ferromagnetic mixed-spin system consisting of two sublattices A and B with two types of spins SA=32 and SB=12 described by anisotropic Classical-Heisenberg Model on a square lattice are examined by using Monte Carlo simulations based on metropolis algorithm. By exploiting the thermal variations of the order-parameters and the magnetic susceptibility, the phase diagrams are calculated on the (Dz/J, kBT/J), (Dx/J, kBT/J) and (Δ, kBT/J) planes and the effects of various interaction parameters in the Hamiltonian are explored. It is found that the model yields only second-order phase transitions. We have also estimated the critical exponents ν, β and γ of the model as functions of Dz and Dx by using the finite-size scaling arguments and found that they depend on the details of the model. Finally, the hysteresis behavior of the model is also investigated and it is revealed that the width of hysteresis loop decreases with temperature while it increases for positive Dz values.
We study classical and quantum Heisenberg antiferromagnets with exchange anisotropy of XXZ-type and crystal field single-ion terms of quadratic and quartic form in a field.The magnets display a variety of phases, including the spin-flop (or, in the quantum case, spin-liquid) and biconical (corresponding, in the quantum lattice gas description, to supersolid) phases.Applying ground-state considerations, Monte Carlo and density matrix renormalization group methods, the impact of quantum effects and lattice dimension is analysed.Interesting critical and multicritical behaviour may occur at quantum and thermal phase transitions.
We study quantum ferrimagnets in one, two, and three dimensions by using a variety of methods and approximations. These include: (i) a treatment based on the spin coherent state path-integral formulation of quantum ferrimagnets by taking into account the leading order quantum and thermal fluctuations (ii) a field-theoretical (non-linear σσσ-model type) formulation of the special case of one-dimensional quantum ferrimagnets at zero temperature (iii) an effective description in terms of dimers and quantum rotors, and (iv) a quantum renormalization group study of ferrimagnetic Heisenberg chains. Some of the formalism discussed here can be used for a unified treatment of both ferromagnets and antiferromagnets in the semiclassical limit. We show that the low (high) energy effective Hamiltonian of a (S_1, S_2) Heisenberg ferrimagnet is a ferromagnetic (antiferromagnetic) Heisenberg model. We also study the phase diagram of quantum ferrimagnets in the presence of an external magnetic field h (hc1<h<hc2h_{c1} < h < h_{c2}hc1​<h<hc2​) and show that the low- and the high-field phases correspond respectively to the classical Néel and the fully polarized ferromagnetic states. We also calculate the transition temperature for the Berezinskii-Kosterlitz-Thouless phase transition in the special case of two-dimensional quantum ferrimagnets.