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Physical review. B, Condensed matter

Discontinuous spin-glass transition in a random quantum Heisenberg magnet

T. K. Kopeć

Published Oct 1, 1995• 10 citations
Publisher

Abstract

We study the Schwinger bosons SU(M) generalization of the spin-S (S\ensuremath{\equiv}\ensuremath{\kappa}M) quantum Heisenberg magnet with infinite-range, Gaussian-random exchange interactions. The spin-glass transition was accessed in the spherical limit M=\ensuremath{\infty} with fixed \ensuremath{\kappa}. The model is solved exactly in the spin-glass phase wthin the Parisi ansatz with a single step of the replica symmetry breaking. The physical consequences of this quantum spin-glass transition scenario are discussed.

DOI
10.1103/physrevb.52.9590

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We study the Schwinger bosons SU(M) generalization of the spin-S (S\ensuremath{\equiv}\ensuremath{\kappa}M) quantum Heisenberg magnet with infinite-range, Gaussian-random exchange interactions. The spin-glass transition was accessed in the spherical limit M=\ensuremath{\infty} with fixed \ensuremath{\kappa}. The model is solved exactly in the spin-glass phase wthin the Parisi ansatz with a single step of the replica symmetry breaking. The physical consequences of this quantum spin-glass transition scenario are discussed.
We examine the spin-S quantum Heisenberg magnet with Gaussian-random, infinite-range exchange interactions. The quantum-disordered phase is accessed by generalizing to SU(M) symmetry and studying the large M limit. For large S the ground state is a spin glass, while quantum fluctuations produce a spin-fluid state for small S. The spin-fluid phase is found to be generically gapless---the average, zero temperature, local dynamic spin susceptibility obeys \ensuremath{\chi}\ifmmode\bar\else\textasciimacron\fi{}(\ensuremath{\omega})\ensuremath{\sim}ln(1/\ensuremath{\Vert}\ensuremath{\omega}\ensuremath{\Vert})+i(\ensuremath{\pi}/2)sgn(\ensuremath{\omega}) at low frequencies.
We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ( h ) fields. We show that, for all h , the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p -rotor model at nonzero h : here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024
We study with exact diagonalization techniques the Heisenberg model for a system of SU(2) spins with S=1/2S=1/2S=1/2 and random infinite-range exchange interactions. We calculate the critical temperature Tg{T}_{g}Tg​ for the spin-glass to paramagnetic transition. We obtain ${T}_{g}\ensuremath{\approx}0.13,$ in good agreement with previous quantum Monte Carlo and analytical estimates. We provide a detailed picture for the different kind of excitations which intervene in the dynamical response ${\ensuremath{\chi}}^{\ensuremath{''}}(\ensuremath{\omega},T)$ at T=0T=0T=0 and analyze their evolution as T increases. We also calculate the specific heat Cv(T).{C}_{v}(T).Cv​(T). We find that it displays a smooth maximum at ${T}_{M}\ensuremath{\approx}0.25,$ in good qualitative agreement with experiments. We argue that the fact that TM>Tg{T}_{M}>{T}_{g}TM​>Tg​ is due to a quantum disorder effect.
We studied the phase transition of the $\ifmmode\pm\else\textpm\fi{}J$ Heisenberg model with and without a random anisotropy on four-dimensional lattice $L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}L\ifmmode\times\else\texttimes\fi{}(L+1)(L<~9).$ We showed that the Binder parameters g(L,T)g(L,T)g(L,T)'s for different sizes do not cross even when the anisotropy is present. On the contrary, when a strong anisotropy exists, g(L,T)g(L,T)g(L,T) exhibits a steep negative dip near the spin-glass phase-transition temperature TSG{T}_{\mathrm{SG}}TSG​ similarly to the p-state infinite-range Potts glass model with p> 3,p>~3,p> 3, in which the one-step replica-symmetry breaking (RSB) occurs. We speculated that a one-step RSB-like state occurs below TSG,{T}_{\mathrm{SG}},TSG​, which breaks the usual crossing behavior of g(L,T).g(L,T).g(L,T).
The infinite-range M-component models of quantum rotors and anharmonic oscillators coupled by random exchange interactions with Gaussian distributions are considered. Both models exhibit a quantum phase transition separating the spin glass and disordered phases where the quantum effects are regulated by a parameter \ensuremath{\Delta} (inverse momentum of inertia for rotor model and inverse mass for oscillator model) parametrizing the kinetic energy. The models allow, in the limit M\ensuremath{\rightarrow}\ensuremath{\infty}, for an exact solution in both the spin glass and disordered phases. Employing the quantum finite-temperature real-time description, we examine the static and dynamic properties of the systems without making use of the n-replica method. We find that the dynamical solution is marginally stable in the spin-glass phase with a constant static susceptibility below the spin-glass freezing temperature. A closed-form solution for the averaged free energy has been obtained and various thermodynamic functions are derived. In particular, the entropy is always positive in the spin-glass plane and goes to zero with a T3{\mathit{T}}^{3}T3 dependency in the low-temperature region. This is in contrast to the classical M-component Heisenberg model in the M\ensuremath{\rightarrow}\ensuremath{\infty} limit.
Using exact diagonalization techniques we study the dynamical response of the anisotropic disordered Heisenberg model for systems of S=1/2 spins with infinite range random exchange interactions at temperature T=0. The model can be considered as a generalization, to the quantum case, of the well known Sherrington-Kirkpatrick classical spin-glass model. We also compute and study the behavior of the Edwards Anderson order parameter and energy per spin as the anisotropy evolves from the Ising to the Heisenberg limits.