Equilibrium Dynamics of Infinite-Range Quantum Spin Glasses in a Field

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.

Popular Summary

Recent progress in quantum technologies has today enabled the study of complex optimization problems. Using a variety of quantum simulators, qubits (or, equivalently, quantum spins) can be arranged in arbitrary geometries as required while the interactions between them encode the problem of interest. A particularly challenging class of such problems involves determining the ground states and dynamics of so-called spin glasses, which originate from the interplay of strong randomness and strong interactions.

A common starting point to describe such spin glasses is the celebrated quantum Ising model but with disordered (random) interactions that are infinite ranged, that is, each spin talks to all others. Although this model has been much explored, little is known about its properties in a longitudinal magnetic field, which is an essential ingredient in the optimization toolbox of modern-day quantum annealers. Here, we provide exact results for the long-time equilibrium dynamics of this model. We also study a related spin-glass model in which the Ising spins are replaced by quantum rotors interacting via multibody couplings and elucidate its dynamics as well.

These spin-glass states are characterized by a striking inequivalence of statistical properties between different copies (or “replicas”) of the system, reflecting a complex hierarchical landscape of energy minima. Our work analytically derives the distinct structures of such “replica symmetry breaking” for the Ising and rotor models, based on which, we obtain the low-frequency dynamic spin spectrum of both. Remarkably, in each case we find that the spectrum is independent of the field.

Our results on the nature of the spin-glass phase and its spectral properties pave the way for the study of out-of-equilibrium dynamics and provide direct insights into the potential performance of adiabatic algorithms in preparing low-energy states. Moreover, multibody-interacting spin-glass models are known to be hard to solve classically, thus affording new opportunities in the search for algorithms that could yield a quantum advantage.

Figure 1

The Parisi spin-glass order parameter for the Ising model at $h\ne 0$.

Figure 2

The Parisi spin-glass order parameter for the spherical quantum $p$-rotor model for $p\ge 3$.

Figure 3

The behavior of the zero-temperature spectral function $\rho (\omega )=\mathrm{Im}{Q}_r(\omega )/\pi$, as determined by Eqs. (110)–(117), for the one-step replica-symmetry-breaking solution of the $p=3$ spherical quantum $p$-rotor model at different values of the transverse field $g$ (with $J=1$). The marginal stability condition in Eq. (106) is imposed. The spectrum is independent of $h$ as long as we are in the replica-symmetry-breaking phase in Fig. 10.

Figure 4

The Parisi spin-glass order parameter for the Ising model at $h=0$.

Figure 5

The gap $\mathrm{\Delta }$ of the replica-symmetric solution of the Ising model as a function of $r$ and $h$ for various temperatures, as determined from Eq. (52): (a) $T={10}^{-3}$; (b) $T={10}^{-2}$; (c) $T=0.1$; (d) $T=0.2$. The parameters $U$, $\kappa$, and $\mathrm{\Lambda }$ have all been set to unity in these calculations.

Figure 6

The phase diagram of the Ising model in the $r$-$h$ plane for various values of $y$ at $T=0.2$, illustrating the replica-symmetric (RS) and replica-symmetry-breaking (RSB) phases: (a) $y=0.01$; (b) $y=0.1$; (c) $y=0.2$; (d) $y=0.5$. Here, the Landau parameter $r$ is interpreted as proportional to $g-g_c$, where $g$ is the transverse field and $g_c$ is the critical transverse field at $T=0$ and $h=0$. The value of $\mathrm{\Delta }$ at each point in parameter space is obtained from Fig. 5, and $U$, $\kappa$, and $\mathrm{\Lambda }$ have been set to unity, as previously.

Figure 7

The behavior of the order parameter $q$ for the replica-symmetric solution of the quantum spherical $p=3$ model as a function of the transverse field $g$ for different values of longitudinal fields $h$ (with $J=1$ and $T=0.005$). The phase transition between the replica-symmetric and paramagnetic solutions occurs only for $h=0$.

Figure 8

The spectral functions of the quantum spherical $p=3$ model for (a) $h=0$ and (b) $h=1$ at zero temperature (with $J=1$). (a) The phase transition between the RS phase and the quantum paramagnet occurs at a finite $g>3$ and the gap thus eventually saturates upon increasing $g$. (b) The replica-symmetric solution persists for all values of $g$ with a gap that grows with increasing $g$.

Figure 9

(a) The behavior of $q_0$ and $q_1$ as a function of $g$ for the spherical quantum $p=3$ rotor model in a longitudinal field with $T=0.005$ and $J=1$. Note that $q_1$ is independent of $h$ but $q_0$ is not. (b) The boundary of stability of the RSB phase as determined by setting $q_0(g,h^{\ast })=q_1(g)$. For $h>h^{\ast }(g)$, the necessary condition $q_0\le q_1$ required for the RSB solution no longer holds. Note that both plots are extrapolated around $q_0\approx 0$ and $h^{\ast }\approx 0$.

Figure 10

The phase diagram of the spherical quantum $p=3$ rotor model in a longitudinal field with $T=0.005$ and $J=1$. The phase diagram is evaluated by calculating the difference between the free energies of the replica-symmetric [RS, Eq. (120)] and replica-symmetry-breaking [RSB, Eq. (121)] solutions. The orange line on the phase boundary traces the function $h^{\ast }(g)$ plotted in Fig. 9.

Figure 11

The spin-glass order parameter $q$ of the classical spherical $p=2$ rotor model (with $J=1$), as a function of the temperature $T$ for several values of the longitudinal field $h$. For the replica-symmetric solution considered here, the spin-glass phase transition occurs at $T=1$.

Figure 12

(a) The gap in the spectrum Eq. (C13) as a function of the longitudinal field strength $h$ for the quantum spherical $p=2$ rotor model, taking $g=1$. The system is gapless in the limit $h\to 0$. (b) The order parameter $q$ as a function of the transverse magnetic field for several values of $h$, illustrating the absence of a sharp phase transition at any nonzero $g$ and $h$.

Figure 13

(a) The behavior of $\overline{\lambda }$ and ${\mathrm{\Sigma }}_r({\omega }_n=0)$ as a function of $g$ for the quantum spherical $p=3$ rotor model; $\overline{\lambda }$ grows significantly faster with $g$ than ${\mathrm{\Sigma }}_r$. In addition, ${\mathrm{\Sigma }}_r({\omega }_n=0)\ll \overline{\lambda }$, which justifies the leading-order approximation made in Eq. (D4). (b) The self-consistent solution of the paramagnetic Eqs. (D1)–(D3) for the self-energy, ${\mathrm{\Sigma }}_r$, for large values of $g$. The self-energy becomes a constant in the limit $g\to \mathrm{\infty }$.