We present the mean field solution of the quantum and classical Heisenberg spin glasses, using the combination of a high precision numerical solution of the Parisi full replica symmetry breaking equations and a continuous time quantum Monte Carlo algorithm. We characterize the spin glass order and its low-energy excitations down to zero temperature. The Heisenberg spin glass has a rougher energy landscape than its Ising analog, and exhibits a very slow temperature evolution of its dynamical properties. We extend our analysis to the doped, metallic Heisenberg spin glass, which displays unexpectedly slow spin dynamics, reflecting the proximity to the melting quantum critical point and its associated Sachdev-Ye-Kitaev Planckian dynamics.
We present the mean field solution of the quantum and classical Heisenberg spin glasses, using the combination of a high precision numerical solution of the Parisi full replica symmetry breaking equations and a continuous time quantum Monte Carlo algorithm. We characterize the spin glass order and its low-energy excitations down to zero temperature. The Heisenberg spin glass has a rougher energy landscape than its Ising analog, and exhibits a very slow temperature evolution of its dynamical properties. We extend our analysis to the doped, metallic Heisenberg spin glass, which displays unexpectedly slow spin dynamics, reflecting the proximity to the melting quantum critical point and its associated Sachdev-Ye-Kitaev Planckian dynamics.
We construct a complete numerically exact solution of a mean-field quantum spin glass model---the transverse field Sherrington-Kirkpatrick model---by implementing a continuous-time quantum Monte Carlo method in the presence of full replica symmetry breaking. We extract the full numerically exact phase diagram, displaying a glassy phase with continuous replica symmetry breaking at small transverse fields and low temperatures. A paramagnetic phase emerges once thermal and quantum fluctuations melt the spin glass. We characterize both phases by extracting the order parameter as well as the static and dynamical local spin susceptibilities. The static susceptibility shows a plateau in the glassy phase, but it remains smooth across the phase boundary. For the imaginary part of the dynamical susceptibility, we find an Ohmic, i.e., linear in $\ensuremath{\omega}$, scaling for small frequencies $\ensuremath{\omega}$, with a slope independent of the transverse field. These results compare qualitatively well with ac susceptibility measurements on a dipole-coupled three-dimensional Ising magnet---the ${\mathrm{LiHo}}_{x}{\mathrm{Y}}_{1\ensuremath{-}x}{\mathrm{F}}_{4}$ compound---in a transverse magnetic field. Our work provides a general framework for the exact numerical solution of mean-field quantum glass models, and it constitutes an important step towards understanding glassiness in realistic systems.
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
A full mean-field solution of a quantum Heisenberg spin-glass model is presented in a large- N limit. A spin-glass transition is found for all values of the spin S. The quantum critical regime associated with the quantum transition at S = 0 and the various regimes in the spin-glass phase at high spin are analyzed. The specific heat is shown to vanish linearly with temperature. In the spin-glass phase, intriguing connections between the equilibrium properties of the quantum problem and the out-of-equilibrium dynamics of classical models are pointed out.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
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.
A class of infinite-ranged random model Hamiltonians is defined as a limiting case in which the appropriate form of mean-field theory, order parameters and phase diagram to describe spin-glasses may be established. It is believed that these Hamiltonians may be exactly soluble, although a complete solution is not yet available. Thermodynamic properties of the model for Ising and XY spins are evaluated using a many-replica procedure. Results of the replica theory reproduce properties at and above the ordering temperature which are also predicted by high-temperature expansions, but are in error at low temperatures. Extensive computer simulations of infinite-ranged Ising spin-glasses are presented. They confirm the general details of the predicted phase diagram. The errors in the replica solution are found to be small, and confined to low temperatures. For this model, the extended mean-field theory of Thouless, Anderson, and Palmer gives physically sensible low-temperature predictions. These are in quantitative agreement with the Monte Carlo statics. The dynamics of the infinite-ranged Ising spin-glass are studied in a linearized mean-field theory. Critical slowing down is predicted and found, with correlations decaying as ${e}^{{\ensuremath{-}[\frac{(T\ensuremath{-}{T}_{c})}{T}]}^{2}t}$ for T greater than Tc, the spin-glass transition temperature. At and below Tc, spin-spin correlations are observed to decay to their long-time limit as ${t}^{\ensuremath{-}\frac{1}{2}}$.