Mean-Field Quantum Spin Glasses and Replica Symmetry Breaking

Mean-field quantum spin glasses combine quenched random exchange with quantum dynamics and an infinite-range limit in which a replica saddle becomes exact. Their order parameter is a two-time overlap $Q_{ab}(\tau,\tau')$; its off-diagonal structure diagnoses frozen correlations, while replica-symmetry breaking resolves the multitude of pure states. This construction does not establish replica-symmetry breaking in a finite-dimensional material.

Required background. Replica methods supplies the $n\to0$ representation. Scaling directions supplies stability language. Thermal density operators supplies imaginary-time equilibrium correlators.

Infinite-range quantum model

For the transverse-field Sherrington–Kirkpatrick model, choose

$$H=-\frac{1}{\sqrt N}\sum_{i<j}J_{ij}\sigma_i^z\sigma_j^z -\Gamma\sum_i\sigma_i^x, \qquad \overline{J_{ij}}=0, \quad \overline{J_{ij}^2}=J^2.$$

The $N^{-1/2}$ scaling makes the energy extensive. Replicating the imaginary-time path integral and averaging $J_{ij}$ generates a term coupling every pair of replicas and times. A Hubbard–Stratonovich field becomes

$$Q_{ab}(\tau,\tau') =\frac1N\sum_i \left\langle\sigma_{i,a}^z(\tau)\sigma_{i,b}^z(\tau')\right\rangle.$$

The diagonal $Q_{aa}(\tau,\tau')$ contains genuine quantum dynamics. For $a\ne b$, a time-independent equilibrium component $q_{ab}$ measures the overlap between thermodynamic states. The Edwards–Anderson parameter is the long-time diagonal limit, equivalently an appropriate off-diagonal overlap at the saddle Edwards and Anderson 1975.

The original validity map locates the exact statement. Inspect the separation between the infinite-range saddle and the finite-dimensional evidence branch.

Mean-field glass order and its boundary. The $N\to\infty$ saddle can support replica symmetry breaking after a replicon instability; finite-dimensional aging or slow response requires separate spatial and dynamical evidence. Schematic.

Replica-symmetric saddle and instability

In the classical limit $\Gamma=0$ and zero field, the replica-symmetric ansatz gives the self-consistency equation

$$q=\int_{-\infty}^{\infty}\frac{dz}{\sqrt{2\pi}}e^{-z^2/2} \tanh^2(\beta J\sqrt q\,z).$$

The solution $q=0$ loses stability at $T=J$. Below the Almeida–Thouless stability boundary, a replica-symmetric ansatz has a negative replicon eigenvalue; it must not be used merely because it solves the saddle equation.

Parisi’s hierarchical ansatz replaces one off-diagonal number by a function $q(x)$, $0\le x\le1$, encoding an ultrametric overlap distribution Parisi 1980. The $n\to0$ order of limits is essential: solve the finite-$n$ combinatorics, continue, and only then minimize. Permuting these steps can select the wrong saddle.

Quantum dynamics and controlled scope

For $\Gamma\ne0$, $[\sigma^x,\sigma^z]\ne0$ makes the diagonal order parameter time dependent. A static approximation that replaces $Q_{aa}(\tau,\tau')$ by a constant can locate qualitative regimes but misses quantum dynamics and is not generally controlled near a quantum critical point. Solving the effective single-site retarded problem, its replica saddle, and the replicon spectrum is the full mean-field task.

Quantum infinite-range models can show critical local dynamics distinct from the classical SK limit; Miller and Huse 1993 analyzed the transverse-field transition, while Read, Sachdev, and Ye 1995 developed a quantum Heisenberg glass with nontrivial dynamics. Model, spin symmetry, and large-$N$ limit must therefore accompany any exponent.

The exact ceiling is $N\to\infty$ with the declared coupling distribution and replica continuation. Finite-range droplets, lower critical dimension, activated dynamics, baths, and experimental waiting times are separate questions. The disorder and glass claim test matrix records that mean-field-to-finite-dimensional boundary.

Exercise

Locate the classical instability. Expand the replica-symmetric equation for small $q$ and determine when a nonzero solution can first appear.

Solution

For small $q$, $\tanh(\beta J\sqrt q\,z)=\beta J\sqrt q\,z+O(q^{3/2})$. Therefore

$$q=(\beta J)^2q\int Dz\,z^2+O(q^2) =(\beta J)^2q+O(q^2).$$

The linearized eigenvalue crosses one at $\beta J=1$, or $T_c=J$. This finds the onset of the replica-symmetric order parameter; stability below it still requires the replicon test and ultimately replica symmetry breaking.

References

• S. F. Edwards and Philip W. Anderson, “Theory of Spin Glasses,” Journal of Physics F: Metal Physics 5 (1975) 965–974. DOI
• Jonathan Miller and David A. Huse, “Zero-Temperature Critical Behavior of the Infinite-Range Quantum Ising Spin Glass,” Physical Review Letters 70 (1993) 3147–3150. DOI
• Giorgio Parisi, “A Sequence of Approximated Solutions to the S-K Model for Spin Glasses,” Journal of Physics A: Mathematical and General 13 (1980) L115–L121. DOI
• N. Read, Subir Sachdev, and Jinwu Ye, “Landau Theory of Quantum Spin Glasses of Rotors and Ising Spins,” Physical Review B 52 (1995) 384–410. DOI