Breakpoint selection and the gapped-versus-gapless spectrum of the infinite-range spherical quantum p-rotor spin glass in a field

The infinite-range (mean-field) spherical quantum $p$-rotor model with $p\ge 3$, a longitudinal field $h$, and quantum coupling $g$ has a spin-glass phase with one-step replica symmetry breaking, but the $N\to\infty$ saddle-point equations leave the breakpoint $x$ of the Parisi order parameter undetermined. Two prescriptions complete the solution: extremizing the free energy with respect to $x$ — the rigorously established equilibrium rule in the classical $g=0$ limit, which yields a local excitation spectrum with an energy gap — or imposing the marginal-stability condition of glassy dynamics, which yields a gapless spectrum whose density vanishes linearly with frequency. Which prescription the thermodynamic equilibrium state obeys at $g\ne 0$, $h\ne 0$, and which one is realized in physically or experimentally prepared states (for example in programmable Rydberg-atom simulators), is unresolved. The problem is to determine the selection principle for $x$, in equilibrium and under dynamical preparation, and thereby to decide whether the realized quantum glass is gapped or marginally stable and gapless, together with observable diagnostics that distinguish the two.

Background

Mean-field spin glasses and replica symmetry breaking. In an infinite-range (fully connected) spin glass, $N$ degrees of freedom are coupled by independent random exchanges; the paradigmatic example is the Sherrington–Kirkpatrick (SK) model of Ising spins. After averaging the free energy over the disorder using the replica trick — computing the disorder-averaged $n$-replica partition function $\overline{Z^n}$ and sending $n\to 0$ — the mean-field spin-glass solution requires replica symmetry breaking (RSB): the order parameter, the matrix of overlaps $Q_{ab}$ between replicas $a$ and $b$, is no longer diagonal. In Parisi's solution the overlaps are organized by a monotone function $q(u)$ on $0\le u\le 1$. With full RSB, $q(u)$ varies continuously; with one-step RSB, $q(u)$ takes only two values, $q_0$ for $u<x$ and $q_1$ for $u>x$, and the breakpoint $x$ is an additional variational parameter. The spherical $p$-spin model replaces Ising spins by real variables $\sigma_i$ obeying the global spherical constraint $\sum_i \sigma_i^2 = N$ and interacting through random $p$-spin couplings; its classical statics were solved by Crisanti and Sommers, and Talagrand proved rigorously that the Parisi formula with the equilibrium breakpoint fixed by stationarity of the free energy in $x$ gives the exact classical free energy.

Quantum spin glasses in a field. Quantum fluctuations can be introduced either by a transverse field (Ising case, a term $-g\sum_i X_i$ flipping spins) or, for the spherical model, by letting the $\sigma_i$ be quantum rotors with canonically conjugate momenta $\pi_i$, $[\sigma_i,\pi_j]=i\delta_{ij}$, and a kinetic energy $\frac{g}{2}\sum_i \pi_i^2$; larger $g$ means stronger quantum fluctuations, and at large $g$ both models are gapped quantum paramagnets (non-glassy phases without spin-glass order). The equilibrium dynamics of both infinite-range quantum spin glasses at nonzero longitudinal field $h$ — an essential ingredient in optimization applications — were obtained at $N=\infty$ by Tikhanovskaya, Sachdev, and Samajdar (2024). For the quantum Ising (SK) model they found full RSB and a gapless local spectrum, with a spectral density vanishing linearly with frequency, for all $h$. For the spherical quantum $p$-rotor model with $p\ge 3$, in contrast, the spin-glass phase has only one-step RSB, and the $N\to\infty$ saddle-point equations for the order parameter $Q_{ab}(\tau_1,\tau_2)$ (a bilocal function of two imaginary times, and a matrix in replica space) determine all quantities except the breakpoint $x$. At the free-energy-stationary $x$ — and indeed at generic $x$ — the local rotor spectrum $\mathrm{Im}\,Q_r(\omega)$ (the spectral density of the site-diagonal rotor correlation function) is gapped; the marginal-stability condition selects precisely the special $x$ at which the gap closes and $\mathrm{Im}\,Q_r(\omega)\propto\omega$ at small $\omega$, an equation that coincides, at $h=0$, with the one obtained from the real-time aging dynamics of the quantum spherical model by Cugliandolo and Lozano.

Marginal stability and why the choice matters physically. In classical mean-field glasses, relaxational dynamics does not reach equilibrium: it is trapped at marginally stable 'threshold' states whose energy density lies above the equilibrium value and whose escape barriers diverge with $N$; the aging dynamics — so called because relaxation keeps slowing down as the system ages — is then described by an effective temperature that violates the fluctuation–dissipation relation (the linear-response relation between correlation and response that holds in equilibrium). Whether the same dichotomy — a gapped, stationarity-selected equilibrium glass versus marginally stable, gapless dynamically prepared states — persists once quantum fluctuations ($g\ne 0$) and a longitudinal field ($h\ne 0$) are present is unknown. The question bears directly on programmable quantum simulators: random arrays of trapped Rydberg atoms (atoms excited to states of large principal quantum number) realize disordered, long-range-interacting effective spins in which the laser detuning acts as a longitudinal field, and the presence or absence of a spectral gap in the glassy phase governs the feasibility of adiabatically preparing low-energy states.

Real-time formulation. The replica equilibrium results can be recast in the Schwinger–Keldysh real-time path-integral formalism, a closed-time-contour construction for nonequilibrium quantum dynamics. Lang, Sachdev, and Diehl (2024) showed that the replica-free Keldysh Dyson equations recover the Parisi algebra — the hierarchical (ultrametric) organization of replica overlaps — in the late-time glassy limit, with replica symmetry breaking appearing as spontaneous breaking of the thermal (Kubo–Martin–Schwinger) symmetry, the imaginary-time periodicity that characterizes equilibrium correlation functions. They derived this for the quantum Ising spin glass in transverse and longitudinal fields, with full RSB, and applied the same formalism to the spherical $p$-spin model, whose glassy phase has only one-step RSB. However, in that framework the one-step breakpoint $x$ is still fixed only 'by an additional criterion, requiring either marginal stability or minimization of the free energy': the selection problem has been transposed into the real-time formalism, not solved.

Problem Statement

The single objective is to determine the physical selection principle for the one-step replica-symmetry-breaking breakpoint — and hence the low-frequency spectrum — of the infinite-range spherical quantum $p$-rotor glass, both in thermodynamic equilibrium and in physically prepared states.

Consider the spherical quantum $p$-rotor model with $p\ge 3$, rotor variables $[\sigma_i,\pi_j]=i\delta_{ij}$, kinetic term $\frac{g}{2}\sum_i \pi_i^2$ (so that $g$ tunes the strength of quantum fluctuations), independent Gaussian $p$-spin couplings with variance $\frac{p!}{2}J^2 N^{1-p}$, the global spherical constraint $\sum_i \sigma_i^2 = N$, and a longitudinal field $h\ne 0$:

$$H=\sum_{i_1<\cdots<i_p}J_{i_1\cdots i_p}\,\sigma_{i_1}\cdots\sigma_{i_p}+\frac{g}{2}\sum_i \pi_i^2-h\sum_i \sigma_i .$$

In the $N\to\infty$ spin-glass phase, the saddle-point equations for the bilocal order parameter $Q_{ab}(\tau_1,\tau_2)$ (the disorder-averaged correlation function of the rotors in replicas $a$ and $b$ at imaginary times $\tau_1$ and $\tau_2$) admit a one-step replica-symmetry-breaking solution in which every parameter is fixed except the breakpoint $x$.

(a) Equilibrium selection. Determine, without assuming the answer, which prescription selects $x$ in the thermodynamic equilibrium state at $g\ne 0$, $h\ne 0$: free-energy stationarity $\partial\mathcal{F}/\partial x=0$, which yields a local spectrum with an energy gap, or the marginal-stability condition of glassy dynamics, which yields a gapless spectrum with $\mathrm{Im}\,Q_r(\omega)\propto\omega$ as $\omega\to 0$ (here $\mathrm{Im}\,Q_r(\omega)$ is the spectral density of the local rotor correlation function). A complete answer derives the selection rule within a controlled framework (replica free-energy analysis made rigorous, a Thouless–Anderson–Palmer (TAP) and complexity analysis — the mean-field treatment of the exponentially many metastable glassy states, with the complexity, i.e. the configurational entropy counting those states — or a real-time Schwinger–Keldysh formulation) and must reduce, at $g=0$, to the established classical results: the stationary $x$ with a gapped equilibrium spectrum (Talagrand), and the marginal threshold states of the dynamics (Cugliandolo–Kurchan).

(b) Physical preparation. Determine which selection principle governs the states actually prepared by physical protocols at $g\ne 0$, $h\ne 0$ — quench, annealing, aging — that is, whether prepared states relax to the stationary (gapped) glass, remain at marginally stable threshold states as classical dynamics does, or reach other states accessible through quantum tunneling; and specify observable diagnostics — for example, the low-frequency behavior of the local spectral function, or fluctuation–dissipation-relation violations and aging effective temperatures — that distinguish the gapped glass from the marginally stable, gapless glass and can be measured in platforms such as Rydberg-atom arrays.

The problem is solved when both (a) and (b) are settled for the stated model — by derivation, rigorous argument, or thermodynamic-limit computation with controlled errors — with the classical $g\to 0$ limits reproduced and with quantitative diagnostic predictions.

References

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