We solve the
S=1/2 infinite-range random Heisenberg Hamiltonian in the paramagnetic phase using quantum Monte Carlo and analytical techniques. We find that the spin-glass susceptibility diverges at a finite temperature
Tg which demonstrates the existence of a low-temperature ordered phase. Quantum fluctuations reduce the critical temperature and the effective Curie constant with respect to their classical values. They also give rise to a redistribution of spectral weight in the dynamic structure factor in the paramagnetic phase. As the temperature decreases the spectrum of magnetic excitations gradually splits into quasielastic and inelastic contributions whose weights scale as
S2 and
S at low temperature.
We study with exact diagonalization techniques the Heisenberg model for a system of SU(2) spins with
S=1/2 and random infinite-range exchange interactions. We calculate the critical temperature
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=0 and analyze their evolution as T increases. We also calculate the specific heat
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 is due to a quantum disorder effect.
A central question in frustrated quantum magnets is how quantum fluctuations destabilize spin-glass order and promote quantum spin-liquid behavior. We study the equilibrium dynamics of an infinite-range quantum Heisenberg model with random couplings, in which local magnetic moments arise from N_{f} flavors of spinful fermions. We employ an expansion in N_{f}, which controls the strength of quantum fluctuations, and self-consistently include 1/N_{f} corrections to the Luttinger-Ward functional. In the large-N_{f} limit, where quantum fluctuations are weak, the high- and low-temperature phases are respectively paramagnetic and spin glass ordered, with a transition temperature independent of N_{f}. For small numbers of fermionic flavors, however, quantum fluctuations substantially suppress the ordering temperature. We show that this behavior reflects the proximity of the system to a Sachdev-Ye-Kitaev (SYK) regime, realizing quantum spin-liquid dynamics over a broad range of finite frequencies, where both fermionic and spin spectral densities display critical behavior over a broad range of finite frequencies, with the latter exhibiting the scale-invariant form χ^{''}(ω)∼sgn(ω). At the lowest energies and temperatures, spin glass dynamics ultimately take over, producing a universal sub-Ohmic dynamical spin susceptibility χ^{''}(ω)∼sgn(ω)sqrt[|ω|]. Our results highlight a direct connection between the suppression of spin-glass order and the emergence of SYK criticality in a frustrated quantum magnet.
A mean-field spherical model with random couplings between pairs, quartets, and possibly higher multiplets of spins is considered. It has the same critical behavior as the Sherrington-Kirkpatrick model. It thus exhibits replica symmetry breaking. The order parameter function is solved exactly in the whole low temperature phase. The zero-field-cooled susceptibility remains finite at low
T. Next a quantum version of the system is considered. Whereas the magnetic properties are not altered qualitatively, the thermodynamics is now regular at small temperatures.
We explore the behavior of the order parameter distribution of the quantum Sherrington-Kirkpatrick model in the spin glass phase using Monte Carlo technique for the effective Suzuki-Trotter Hamiltonian at finite temperatures and that at zero temperature obtained using the exact diagonalization method. Our numerical results indicate the existence of a low- but finite-temperature quantum-fluctuation-dominated ergodic region along with the classical fluctuation-dominated high-temperature nonergodic region in the spin glass phase of the model. In the ergodic region, the order parameter distribution gets narrower around the most probable value of the order parameter as the system size increases. In the other region, the Parisi order distribution function has nonvanishing value everywhere in the thermodynamic limit, indicating nonergodicity. We also show that the average annealing time for convergence (to a low-energy level of the model, within a small error range) becomes system size independent for annealing down through the (quantum-fluctuation-dominated) ergodic region. It becomes strongly system size dependent for annealing through the nonergodic region. Possible finite-size scaling-type behavior for the extent of the ergodic region is also addressed.
The behavior of the nonlinear susceptibility ${\ensuremath{\chi}}_{3}$ and its relation to the spin-glass transition temperature
Tf in the presence of random fields are investigated. To accomplish this task, the Sherrington-Kirkpatrick model is studied through the replica formalism, within a one-step replica-symmetry-breaking procedure. In addition, the dependence of the Almeida-Thouless eigenvalue ${\ensuremath{\lambda}}_{\mathrm{AT}}$ (replicon) on the random fields is analyzed. Particularly, in the absence of random fields, the temperature
Tf can be traced by a divergence in the spin-glass susceptibility ${\ensuremath{\chi}}_{\mathrm{SG}}$, which presents a term inversely proportional to the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$. As a result of a relation between ${\ensuremath{\chi}}_{\mathrm{SG}}$ and ${\ensuremath{\chi}}_{3}$, the latter also presents a divergence at
Tf, which comes as a direct consequence of ${\ensuremath{\lambda}}_{\mathrm{AT}}=0$ at
Tf. However, our results show that, in the presence of random fields, ${\ensuremath{\chi}}_{3}$ presents a rounded maximum at a temperature
T∗ which does not coincide with the spin-glass transition temperature
Tf (i.e.,
T∗>Tf for a given applied random field). Thus, the maximum value of ${\ensuremath{\chi}}_{3}$ at
T∗ reflects the effects of the random fields in the paramagnetic phase instead of the nontrivial ergodicity breaking associated with the spin-glass phase transition. It is also shown that ${\ensuremath{\chi}}_{3}$ still maintains a dependence on the replicon ${\ensuremath{\lambda}}_{\mathrm{AT}}$, although in a more complicated way as compared with the case without random fields. These results are discussed in view of recent observations in the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound.