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Physical review. B, Condensed matter

Correlations in vector-spin-glass models in a transverse field

Ferenc Pázmándi, Zbigniew Domański

Published Mar 1, 1994• 2 citations
Publisher

Abstract

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 ${\mathit{K}}_{\mathit{c}}$=2S.

DOI
10.1103/physrevb.49.6794

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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 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.
In this paper we investigate for the infinite-range Ising spin-glass model [i.e., the Sherrington-Kirkpatrick (SK) model] with a transverse field under the static approximation by using the imaginary-time replica formalism. From the investigations we show three important results: First, we show that a replica-symmetric quantum spin-glass phase is stable in most of the area of the spin-glass phase in the temperature-transverse field phase diagram. This confirms the existence of a stable replica-symmetric spin glass phase under the static approximation, which is contrary to some previous results derived without the static approximation where the replica-symmetric solution is always unstable in the whole spin-glass phase. Second, we show our theoretical result for the nonlinear susceptibility ${\ensuremath{\chi}}_{\mathrm{nl}}$ which conforms to the experimental result of nonlinear susceptibility measurement by Wu et al. [Phys. Rev. Lett. 71, 1919 (1993)] in a quantum spin glass ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}.$ Third, in a classical (SK) spin-glass system, we confirm the anomaly in the second temperature derivative of CH/T{C}_{H}/TCH​/T near the glass transition temperature Tg,{T}_{g},Tg​, associated possibly with the field-dependent variation of entropy in the spin glass transition, which agrees with the previous experimental observation in a classical spin glass system CuMn by Fogle et al. [Phys. Rev. Lett. 50, 1815 (1983)]. We also show that this anomaly is suppressed by the nonzero transverse field of the quantum spin glass system, by which we can check for the quantum tunneling competing against spin freezing.
The infinite-range quantum Heisenberg spin glass with uniaxial anisotropy (D) and coaxial external magnetic field (h) is studied by means of the thermo-field-dynamics as a substitute for the n-replica trick. Within the mean-field theory a multiplicity of spin-glass phases has been found for the spin values S=1 and S=3/2, respectively, including: longitudinal, transverse, and mixed phases. The stability of the mean-field-type solution against the action of fluctuations has been investigated and the existence of a crossover from the longitudinal-to-transverse freezing is demonstrated explicitly in the temperature-field (T-h) phase diagram for various values of the constant D.
We study the zero-temperature behaviour of the infinite-ranged Ising spin glass in a transverse field. Using spin summation and Monte Carlo methods we characterize the zero-temperature quantum transition. Our results are well compatible with a value for the correlation length exponent, z=4 for the dynamical exponent and an algebraic decay for the imaginary-time correlation function. The zero-temperature Monte Carlo relaxation of the energy in the presence of the transverse field shows that the system monotonically reaches the ground-state energy due to quantum fluctuations and displays glassy effects due to the strong anisotropy in the effective Hamiltonian.
The Ising infinite-range spin-glass model with p-spin interactions in the presence of a transverse field is studied for large but finite p using the Matsubara time representation and Parisi's scheme of replica-symmetry breaking. In the spin-glass phase, the corrections to the limit appear much more essential than in the classical counterpart. It is shown that the quantum fluctuations have the effect of destroying the spin-glass order and a lower temperature is required to stabilize the spin-glass phase. The spin autocorrelation function in the spin-glass phase is explicitly calculated as a function of p and the Matsubara time. The result is just complementary and consistent with that previously obtained for the paramagnetic state without using the replica method.