Stability Analysis of Quantum Spin Glasses

Quantum phase transitions exhibited by mean-field spin glass models have attracted growing attention because they allow analytical characterization of the possibilities and limitations of quantum annealing.

However, so far, detailed analysis has been almost exclusively restricted to the Sherrington-Kirkpatrick model.

In this work, we generalize the analysis to more general systems with random and rotationally invariant coupling matrices.

These quantum systems can be analyzed by mapping them to classical spin systems extended in the imaginary time direction using the Suzuki-Trotter decomposition, which makes the resultant order parameters generally dependent on the imaginary time. However, most of the previous studies assume that the order parameters except for the two-point correlation function are uniform in imaginary time. We term this treatment the quasi-static ansatz (qSA). To our knowledge, the validity of qSA has not been thoroughly examined.

We obtained the following results for this issue:

• We derived a self-consistent equation for determining the order parameters without using qSA. The solution of qSA represents a special solution of the derived self-consistent equation.

• For the solution, we derived the local stability condition against breaking of the imaginary-time uniformity. We also obtained the local stability condition for replica symmetry breaking (RSB).

• The critical mode for RSB is uniform in imaginary time. In addition, unless RSB occurs, the qSA-type solution is locally stable.

These support the treatment of qSA that has been employed in the previous studies.