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arXiv:1604.01145 · Physical review. B./Physical review. B

Effects of low-lying excitations on ground-state energy and energy gap of the Sherrington-Kirkpatrick model in a transverse field

Yang Wei Koh

Published Apr 4, 2016• 5 citations
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Abstract

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.

DOI
10.1103/physrevb.93.134202

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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≈20\ensuremath{\approx}20\ensuremath≈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.
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.
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.
In the path integral formulation of the partition function of quantum spin models, most current treatments employ the so-called static approximation to simplify the process of summing over all possible paths. Although sufficient for studying the thermodynamic aspects of the system, static approximation ignores the contributions made by time-dependent, or non-static, fluctuations in the paths of the path integral. This non-static component is very small relative to the static part, and its careful treatment is necessary for the calculation of small non-extensive quantities such as the energy gap within the path integral framework. We propose a formalism for incorporating non-static effects into the path integral calculation of a class of spin models whose partition functions are reducible to the trace of a single spin (often known as the effective Hamiltonian). We first show that the time-dependent behavior of the single spin trace is governed by the Pauli equation. Time-dependent perturbation theory is used to obtain a perturbative expansion of the solution of the Pauli equation, and then for the single spin trace. This gives us a perturbative expansion of the path integral which can be integrated systematically using standard techniques. In this paper, we develop the theoretical framework outlined above in detail and apply it to a simple ordered spin model, the infinite-range ferromagnetic Ising model in a transverse field. We calculated two non-extensive quantites with this non-static approach: the N0N^0N0 and N−1N^{-1}N−1 terms of the ground-state energy (NNN=number of spins) and the energy gap between the ground and first-excited states. We checked our results by comparing with those of Holstein-Primakoff transform and numerical diagonalization of the Hamiltonian. The application of the method to quantum spin-glasses is briefly discussed.
In current studies of mean-field quantum spin systems, much attention is placed on the calculation of the ground-state energy and the excitation gap, especially the latter, which plays an important role in quantum annealing. In pure systems, the finite gap can be obtained by various existing methods such as the Holstein-Primakoff transform, while the tunneling splitting at first-order phase transitions has also been studied in detail using instantons in many previous works. In disordered systems, however, it remains challenging to compute the gap of large-size systems with specific realization of disorder. Hitherto, only quantum Monte Carlo techniques are practical for such studies. Recently, Knysh [Nature Comm. 7, 12370 (2016)] proposed a method where the exponentially large dimensionality of such systems is condensed onto a random potential of much lower dimension, enabling efficient study of such systems. Here we propose a slightly different approach, building upon the method of static approximation of the partition function widely used for analyzing mean-field models. Quantum effects giving rise to the excitation gap and nonextensive corrections to the free energy are accounted for by incorporating dynamical paths into the path integral. The time-dependence of the trace of the time-ordered exponential of the effective Hamiltonian is calculated by solving a differential equation perturbatively, yielding a finite-size series expansion of the path integral. Formulae for the first excited-state energy are proposed to aid in computing the gap. We illustrate our approach using the infinite-range ferromagnetic Ising model and the Hopfield model, both in the presence of a transverse field.
We perform large scale quantum annealing of the Sherrington-Kirkpatrick (SK) spin glass up to a system size N=40000N=40000N=40000 to estimate its ground state energy using the deGennes-Suzuki-Kubo mean-field quantum Ising dynamics, extending the earlier results (reported in Eur. Phys. J. B {\bf 98}, 226 (2025)). Here we numerically solve the deGennes-Suzuki-Kubo annealing dynamics to obtain the spin configurations and subsequently the ground state energy for a given system size at the end of the annealing, starting from a quantum paramagnetic state. The method shows high efficiency, with an overall algorithmic cost of O(N3)O(N^3)O(N3) in estimating the energy of the ground state. We later extend this quantum annealing study to estimate the ground state energies (starting again from the quantum paramagnetic phase, annealing down to any desired low value of the transverse field) for the Edwards-Anderson (EA) spin glass model on a square lattice.
The transverse-field Ising model is a fundamental quantum spin system that captures the competition between quantum fluctuations and interactions, playing a central role in studies of quantum phase transitions and non-equilibrium dynamics. However, classical computations of ground and excited states in large-scale or high-dimensional systems are severely limited by the exponential growth of the Hilbert space. Here, we propose a novel approach using a D-Wave quantum annealer, where a triangular-wave oscillating magnetic field is applied to induce Rabi oscillations, allowing the estimation of energy gaps between the ground and excited states. Unlike conventional quantum annealing methods limited to ground-state searches, this approach can directly access excited-state information. It is potentially applicable to larger systems, providing a new avenue for quantum-device-based simulation. The validity of the method is demonstrated through numerical simulations of relatively small systems.