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arXiv:1907.01609

Intermittency of dynamical phases in a quantum spin glass

Vadim Smelyanskiy, Kostyantyn Kechedzhi, Sergio Boixo, Hartmut Neven, B. L. Altshuler

Published Jul 2, 2019• 7 citations
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Abstract

Answering the question of existence of efficient quantum algorithms for NP-hard problems require deep theoretical understanding of the properties of the low-energy eigenstates and long-time coherent dynamics in quantum spin glasses. We discovered and described analytically the property of asymptotic orthogonality resulting in a new type of structure in quantum spin glass. Its eigen-spectrum is split into the alternating sequence of bands formed by quantum states of two distinct types ($x$ and $z$). Those of $z$-type are non-ergodic extended eigenstates (NEE) in the basis of $\{σ_z\}$ operators that inherit the structure of the classical spin glass with exponentially long decay times of Edwards Anderson order parameter at any finite value of transverse field $B_{\perp}$. Those of $x$-type form narrow bands of NEEs that conserve the integer-valued $x$-magnetization. Quantum evolution within a given band of each type is described by a Hamiltonian that belongs to either the ensemble of Preferred Basis Levi matrices ($z$-type) or Gaussian Orthogonal ensemble ($x$-type). We characterize the non-equilibrium dynamics using fractal dimension $D$ that depends on energy density (temperature) and plays a role of thermodynamic potential: $D=0$ in MBL phase, $0

DOI
10.48550/arxiv.1907.01609

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We review our earlier studies on the order parameter distribution of the quantum Sherrington-Kirkpatrick (SK) model. Through Monte Carlo technique, we investigate the behavior of the order parameter distribution at finite temperatures. The zero temperature study of the spin glass order parameter distribution is made by the exact diagonalization method. We find in low-temperature (high-transverse-field) spin glass region, the tail (extended up to zero value of order parameter) and width of the order parameter distribution become zero in thermodynamic limit. Such observations clearly suggest the existence of a low-temperature (high-transverse-field) ergodic region. We also find in high-temperature (low-transverse-field) spin glass phase the order parameter distribution has nonzero value for all values of the order parameter even in infinite system size limit, which essentially indicates the nonergodic behavior of the system. We study the annealing dynamics by the paths which pass through both ergodic and nonergodic spin glass regions. We find the average annealing time becomes system size independent for the paths which pass through the quantum-fluctuation-dominated ergodic spin glass region. In contrast to that, the annealing time becomes strongly system size dependent for annealing down through the classical-fluctuation-dominated nonergodic spin glass region. We investigate the behavior of the spin autocorrelation in the spin glass phase. We observe that the decay rate of autocorrelation towards its equilibrium value is much faster in the ergodic region with respect to the nonergodic region of the spin glass phase.
We review the numerical studies on the critical behaviour of the quantum Sherrington-Kirkpatrick (SK) spin glass model, which indicate that a quantum critical behaviour is observed up to a low but non-zero value of temperature. We revisit the numerical investigations on the spin glass order parameter distributions, which identify a low temperature along with high transverse field spin glass phase where the order parameter distribution becomes a delta function in the thermodynamic limit indicating the restoration of replica symmetry and ergodic nature of the system. In the remaining spin glass phase associated with high temperature and low transverse field, the observed distribution is broad akin to the Parisi order parameter distribution. This essentially indicates the non-ergodic behaviour of the system. We further discuss the annealing dynamics studies on the quantum SK model. Such investigations reveal the system size independence of annealing time when the annealing paths go through the ergodic spin glass region. Interestingly, when such dynamics are performed in the non-ergodic spin glass phase the annealing time becomes an increasing function of the system size. Spin autocorrelation shows faster relaxation in the ergodic spin glass region compared with that found in the non-ergodic spin glass region. This article is part of the theme issue 'Quantum annealing and computation: challenges and perspectives'.
Spin glasses and many-body localization (MBL) are prime examples of ergodicity breaking, yet their physical origin is quite different: the former phase arises due to rugged classical energy landscape, while the latter is a quantum-interference effect. Here, we study quantum dynamics of an isolated 1D spin glass under application of a transverse field. At high energy densities, the system is ergodic, relaxing via a resonance avalanche mechanism, that is also responsible for the destruction of MBL in nonglassy systems with power-law interactions. At low energy densities, the interaction-induced fields obtain a power-law soft gap, making the resonance avalanche mechanism inefficient. This leads to the persistence of the spin-glass order, as demonstrated by resonance analysis and by numerical studies. A small fraction of resonant spins forms a thermalizing system with long-range entanglement, making this regime distinct from the conventional MBL. The model considered can be realized in systems of trapped ions, opening the door to investigating slow quantum dynamics induced by glassiness.
For many real spin-glass materials, the Edwards-Anderson model with continuous-symmetry spins is more realistic than the rather better understood Ising variant. In principle, the nature of an occurring spin-glass phase in such systems might be inferred from an analysis of the zero-temperature properties. Unfortunately, with few exceptions, the problem of finding ground-state configurations is a non-polynomial problem computationally, such that efficient approximation algorithms are called for. Here, we employ the recently developed genetic embedded matching (GEM) heuristic to investigate the nature of the zero-temperature phase of the bimodal XY spin glass in two dimensions. We analyze bulk properties such as the asymptotic ground-state energy and the phase diagram of disorder strength vs. disorder concentration. The main focus of this work is on an investigation of the excitation spectrum as probed by changing the boundary conditions. Using appropriate finite-size scaling techniques, we consistently determine the stiffness of spin and chiral domain walls and the corresponding fractal dimensions. Results for the overlap distribution do not support the possibility of a multitude of thermodynamic pure states.
We first review, following our earlier studies, the critical behavior of the quantum Sherrington-Kirkpatrick (SK) model at finite as well as at zero temperatures. Through the analysis of the Binder cumulant we determined the entire phase diagram of the model and from the scaling analysis of the numerical data we obtained the correlation length exponent. For both the critical Binder cumulant and the correlation length exponent, we observed a crossover from classical- to quantum-fluctuation-dominated values at a finite temperature. We studied the behavior of the order parameter distribution of the model in the glass phase (at finite and zero temperatures). Along with a classical-fluctuation-dominated nonergodic region (where the replica symmetry is broken), we also found a quantum-fluctuation-dominated low-temperature ergodic region in the spin glass phase. In this quantum-fluctuation-dominated region, the order parameter distribution has a narrow peak around its most probable value, eventually becoming a delta function in the infinite-system-size limit (indicating replica symmetry restoration or ergodicity in the system). We also found that the annealing time (to reach a very low energy level of the classical SK model) becomes practically system-size-independent when the annealing paths pass through this ergodic region. In contrast, when such paths pass through the nonergodic region, the convergence time grows rapidly with the system size. We present a new study of the autocorrelation of the spins in both ergodic and nonergodic regions. We found a significant increase in the relaxation time (and also a change in the relaxation behavior) in the classical-fluctuation-dominated (nonergodic) region compared with that in the quantum-fluctuation-dominated (ergodic) region of the spin glass phase.
We address the long-standing problem of the structure of the low-energy eigenstates and long-time coherent dynamics in quantum spin-glass models.Below the spin-glass freezing transition, the energy landscape of the spin system is characterized by a proliferation of local minima where classical dynamics gets trapped.A theoretical description of quantum dynamics in this regime is challenging due to the complex nature of the distribution of the tunneling matrix elements between the local minima of the energy landscape.We study the transverse-field-induced quantum dynamics of the following "impurity band" (IB) spin model: zero energy of all spin configurations except for a small fraction of spin configurations ("marked states") that form a narrow band at a large negative energy.At a zero transverse field, the IB model demonstrates the freezing transition at inverse temperature β f ∼ 1 characterized by a nonzero value of the Edwards-Anderson order parameter.At a finite transverse field, the low-energy dynamics can be described by the effective downfolded Hamiltonian that acts in the Hilbert subspace involving only the marked states.We obtain in an explicit form the heavy-tailed probability distribution of the off-diagonal matrix elements of the down-folded Hamiltonian.This Hamiltonian is dense and belongs to the class of preferred basis Levy matrices.Analytically solving nonlinear cavity equations for the ensemble of down-folded Hamiltonians allows us to describe the statistical properties of the eigenstates.In a broad interval of transverse fields, they are nonergodic, albeit extended.It means that the band of marked states splits into a set of narrow minibands.Accordingly, the quantum evolution that starts from a particular marked state leads to a linear combination of the states belonging to a particular miniband.An analytical description of this qualitatively new type of quantum dynamics is a key result of our paper.Based on our analysis, we propose the population transfer (PT) algorithm: The quantum evolution under constant transverse field B ⊥ starts at a low-energy spin configuration and ends up in a superposition of Ω spin configurations inside a narrow energy window.This algorithm crucially relies on the nonergodic nature of delocalized low-energy eigenstates.In the considered model, the run-time of the best classical algorithm (exhaustive search) is t cl ¼ 2 n =Ω.For ffiffiffi n p ≫ B ⊥ ≫ 1, the typical runtime of the quantum PT algorithm ffiffiffiffi ffi t cl p e n=ð2B 2 ⊥ Þ scales with n and Ω as that of Grover's quantum search, except for the small correction to the exponent.Unlike the Hamiltonians proposed for analog quantum unstructured search algorithms, the model we consider is nonintegrable and the transverse field delocalizes the marked states.As a result, our PT protocol does not require fine-tuning of the transverse field and may be initialized in a computational basis state.We find that the run-times of the PT algorithm are distributed according to the alpha-stable Levy law with tail index 1.We argue that our approach can be applied to study the PT protocol in other transverse-field spin-glass models, with a potential quantum advantage over classical algorithms.