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Journal of Physics C Solid State Physics

Ground-state properties of the Sherrington-Kirkpatrick model with a transverse field

Takahisa Yamamoto

Published Aug 20, 1988• 12 citations
Publisher

Abstract

The effect of a transverse field Delta on the ground state in the Sherrington-Kirkpatrick model is studied numerically for systems of finite size on the basis of a mean-field approach in which fluctuations from the mean field are included in the second-order perturbation. Simple mean-field equations are solved numerically and the ground-state energy, the order parameter, and the distribution of the magnetisation for each spin are calculated as a function of Delta . Although these results show characteristic features of the spin glass, the value obtained for Delta c , the field above which the system remains paramagnetic at T=0, is much larger than the rigorous value obtained by the perturbation method with respect to the exchange couplings. To improve the mean-field approach, an attempt is made to include a reaction field in the simple mean-field equations approximately. The value of Delta c becomes considerably closer to the rigorous value.

DOI
10.1088/0022-3719/21/23/019

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The effect of a transverse field Delta on the ground state in the Sherrington-Kirkpatrick model is studied numerically for systems of finite size on the basis of a mean-field approach in which fluctuations from the mean field are included in the second-order perturbation. Simple mean-field equations are solved numerically and the ground-state energy, the order parameter, and the distribution of the magnetisation for each spin are calculated as a function of Delta . Although these results show characteristic features of the spin glass, the value obtained for Delta c , the field above which the system remains paramagnetic at T=0, is much larger than the rigorous value obtained by the perturbation method with respect to the exchange couplings. To improve the mean-field approach, an attempt is made to include a reaction field in the simple mean-field equations approximately. The value of Delta c becomes considerably closer to the rigorous value.
In the Sherrington-Kirkpatrick model the effect of a transverse field Delta on spin glass freezing is studied. After the static approximation of the replica theory has been analysed, the nonlinear susceptibility is calculated by the perturbation method with respect to exchange coupling J ij up to the tenth order. The freezing temperature and the critical index of the nonlinear susceptibility are evaluated. The result shows that the system remains paramagnetic for Delta >1.506 J (the variance of (J ij ) is J 2 /N), which corrects the conjecture made before by the present authors. It is also shown that the static approximation in the replica theory overestimates T c for a large value of Delta .
In the Sherrington-Kirkpatrick (SK) model the effect of a transverse field Delta on the spin glass freezing is studied on the basis of the theory of Thouless, Anderson and Palmer (1977). It is shown from the logarithmic singular point of the free energy in the high-temperature region that T c decreases with increasing Delta . To be consistent with this result the mean-field equation is set up for T<T c . Due to the correlation effect spin flips associated with the transverse field are unlikely to occur at low temperatures and the system eventually undergoes the spin glass transition irrespective of the magnitude of Delta , which is characteristic of the SK model.
We present an extensive numerical study of the Sherrington-Kirkpatrick model in a transverse field. Recent numerical studies of quantum spin glasses have focused on exact diagonalization of the full Hamiltonian for small systems ($\ensuremath{\approx}20$ spins). However, such exact numerical treatments are difficult to apply on larger systems. We propose making an approximation by using only a subspace of the full Hilbert space spanned by low-lying excitations consisting of one-spin-flipped and two-spin-flipped states. The approximation procedure is carried out within the theoretical framework of the Hartree-Fock approximation and configuration interaction. Although not exact, our approach allows us to study larger system sizes comparable to that achievable by state-of-the-art quantum Monte Carlo simulations. We calculate two quantities of interest due to recent advances in quantum annealing, the ground-state energy and the energy gap between the ground and first excited states. For the energy gap, we derive a formula that enables it to be calculated using just the ground-state wave function, thereby circumventing the need to diagonalize the Hamiltonian. We calculate the scalings of the energy gap and the leading correction to the extensive part of the ground-state energy with system size, which are difficult to obtain with current methods.
We develop a mean-field theory for random quantum spin systems using the spin coherent state path integral representation. After the model is reduced to the mean-field one-body Hamiltonian, the integral is analyzed with the aid of several methods such as the semiclassical method and the gauge transformation. As an application we consider the Sherrington-Kirkpatrick model in a transverse field. Using the Landau expansion and its improved versions, we give a detailed analysis of the imaginary-time dependence of the order parameters. Integrating out the quantum part of the order parameters, we obtain the effective renormalized free energy written in terms of the classically defined order parameters. Our method allows us to obtain the spin glass-paramagnetic phase transition point $\ensuremath{\Gamma}∕J\ensuremath{\sim}1.62$ at T=0T=0T=0.
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 calculate the real-time-correlation function of the Sherrington-Kirkpatrick spin-glass model in a transverse field. Using a careful analysis of the perturbative expansion of the functional-integral representation, we derive the asymptotic form of the correlation function. In contrast to the previous works, we find for large transverse field a power-law decay of the correlation with time ttt as ${t}^{\ensuremath{-}3/2}$ at zero temperature and ${t}^{\ensuremath{-}2}$ at infinite temperature. At the small field region, we also find a significant change in the correlation which comes from the structural difference between two paramagnetic phases.