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Journal of Physics Condensed Matter

Phase diagrams of quantum spin glass models

Ferenc Pázmándi, Zbigniew Domański

Published Mar 1, 1993• 6 citations
Publisher

Abstract

Infinite-range quantum spin glass models with arbitrary spin are studied within the Matsubara time formalism. As an example the dynamic spin self-interaction R(t, t') and the resulting phase diagrams are presented for the quantum Ising model in a transverse field. Some results, such as the phase transition temperatures, are also reported for the XY and the Heisenberg models.

DOI
10.1088/0953-8984/5/9/001

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•Jan 1, 2004•0 citations
Infinite-range quantum spin glass models with arbitrary spin are studied within the Matsubara time formalism. As an example the dynamic spin self-interaction R(t, t') and the resulting phase diagrams are presented for the quantum Ising model in a transverse field. Some results, such as the phase transition temperatures, are also reported for the XY and the Heisenberg models.
We use the static approximation to investigate the phase diagram of the infinite-ranged quantum Heisenberg spin glass (QHSG) in an external field, as well as the phase diagram of a two-component quantum spin glass (QXYSG) in a transverse or a longitudinal field. For the QHSG we find that, for strong field (H) and low temperature (T), the phase transition into the spin-glass phase characterized by nonzero transverse spin-glass ordering is given by Tc{\mathit{T}}_{\mathit{c}}Tc​\ensuremath{\sim}exp[-H2{\mathit{H}}^{2}H2/(2J2{\mathit{J}}^{2}J2S2{\mathit{S}}^{2}S2)], as compared with Tc{\mathit{T}}_{\mathit{c}}Tc​\ensuremath{\sim}exp{-H2{\mathit{H}}^{2}H2/[2J2{\mathit{J}}^{2}J2S(S+1)]} for classical spins of magnitude \ensuremath{\surd}S(S+1) . In the case of QXYSG in a transverse field (\ensuremath{\Gamma}), the phase boundary terminates at \ensuremath{\Gamma}/J=2 for T=0.
Several publications dealing with the infinite-range quantum XY and Heisenberg spin glasses in an external magnetic field K in the z direction share a common supposition: the correlation between the x and y components of the spins is zero. Using the Matsubara imaginary-time formalism we show that this assumption holds only for those models where K=0. With the help of the Trotter-Suzuki formulation we investigate the phase diagram of the XY model for spin S=1/2, and find that it is different from those previously reported because of the nonzero xy correlation. It is also argued that for a general spin S in the XY model at zero temperature the spin-glass phase persists up to Kc{\mathit{K}}_{\mathit{c}}Kc​=2S.
We study the three-dimensional quantum Ising spin glass in a transverse magnetic field following the evolution of the bond probability distribution under renormalization-group transformations. The phase diagram (critical temperature Tc{\mathit{T}}_{\mathit{c}}Tc​ vs transverse field \ensuremath{\Gamma}) we obtain shows a finite slope near T=0, in contrast with the infinite slope for the pure case. Our results compare very well with the experimental data recently obtained for the dipolar Ising spin glass LiHo0.167{\mathrm{LiHo}}_{0.167}LiHo0.167​Y0.833{\mathrm{Y}}_{0.833}Y0.833​F4{\mathrm{F}}_{4}F4​, in a transverse field. This indicates that this system is more appropriately described by a model with short-range interactions than by an equivalent Sherrington-Kirkpatrick model in a transverse field.
The infinite-range Ising spin-glass in the presence of a transverse field is studied by Monte Carlo simulations. The overlap probability distribution P ( q ) is defined and calculated for the present quantum case. The P ( q ) in the spin-glass phase has a long tail extended down to q = 0 qualitatively similar to the classical Sherrington-Kirkpatrick model. The phase boundary of the glass transition temperature vs. the transverse field is obtained. Our results for the phase diagram agree with our recent replica symmetry-breaking solution but disagree with the results of Yokata and of Ray et al.
This thesis aims at a description of the equilibrium dynamics of quantum spin glass systems. To this end a generic fermionic SU(2), spin 1/2 spin glass model with infinite-range interactions is defined in the first part. The model is treated in the framework of imaginary-time Grassmann field theory along with the replica formalism. A dynamical two-step decoupling procedure, which retains the full time dependence of the (replica-symmetric) saddle point, is presented. As a main result, a set of highly coupled self-consistency equations for the spin-spin correlations can be formulated. Beyond the so-called spin-static approximation two complementary systematic approximation schemes are developed in order to render the occurring integration problem feasible. One of these methods restricts the quantum-spin dynamics to a manageable number of bosonic Matsubara frequencies. A sequence of improved approximants to some quantity can be obtained by gradually extending the set of employed discrete frequencies. Extrapolation of such a sequence yields an estimate of the full dynamical solution. The other method is based on a perturbative expansion of the self-consistency equations in terms of the dynamical correlations. In the second part these techniques are applied to the isotropic Heisenberg spin glass both on the Fock space (HSGF) and, exploiting the Popov-Fedotov trick, on the spin space (HSGS). The critical temperatures of the paramagnet to spin glass phase transitions are determined accurately. Compared to the spin-static results, the dynamics causes slight increases of T_c by about 3% and 2%, respectively. For the HSGS the specific heat C(T) is investigated in the paramagnetic phase and, by way of a perturbative method, below but close to T_c. The exact C(T)-curve is shown to exhibit a pronounced non-analyticity at T_c and, contradictory to recent reports by other authors, there is no indication of maximum above T_c. In the last part of this thesis the spin glass model is augmented with a nearest-neighbor hopping term on an infinite-dimensional cubic lattice. An extended self-consistency structure can be derived by combining the decoupling procedure with the dynamical CPA method. For the itinerant Ising spin glass numerous solutions within the spin-static approximation are presented both at finite and zero temperature. Systematic dynamical corrections to the spin-static phase diagram in the plane of temperature and hopping strength are calculated, and the location of the quantum critical point is determined.