Gauge theory for mixed pp-spin glasses

Physical quantities in the mixed $p$-spin glasses are evaluated with Nishimori's gauge theory and several variance inequalities. The $\mathbb Z_2$-symmetry breaking and the replica-symmetry breaking are studied in finite and infinite dimensions. Obtained bounds on the expectation of the square of the magnetization and spontaneous magnetization enable us to clarify properties of paramagnetic and spin glass phases. It is proven that variances of ferromagnetic and spin glass order parameters vanish on the Nishimori line in the infinite volume limit. These results imply the self-averaging of these order parameters on the Nishimori line. The self-averaging of the spin glass order parameter rigorously justifies already argued absence of replica-symmetry breaking on the Nishimori line.Comment: 13 page

Gauge theory for quantum XYZ spin glasses

Nishimori's gauge theory is extended to the quantum XYZ $p$-spin glass model in finite dimensions. This enables us to obtain useful correlation equalities, which show also that Duhamel correlation functions at an arbitrary temperature are bounded by those in the corresponding classical model on the Nishimori line. These bounds give that the spontaneous magnetization vanishes in any low temperature even if the model enters the $\mathbb Z_2$-symmetry broken spin glass phase. This theory explains well-known fact from experiments and numerical calculations that the magnetic susceptibility does not diverge in the spin glass transition. The new gauge theory together with the known phase diagram of the Edwards-Anderson model can specify the spin glass region in the coupling constant space of the quantum Heisenberg XYZ spin glass model.Comment: 13 pages, 1 figure. arXiv admin note: text overlap with arXiv:2305.1963

Existence of de Almeida-Thouless-type instability in the transverse field Sherrington-Kirkpatrick model

The interpolation method for mean field spin glass models developed by Guerra and Talagrand is extended to a quantum mean field spin glass model. This extension enables us to obtain both replica-symmetric (RS) and one step replica-symmetry breaking (1RSB) solutions of the free energy density in the transverse field Sherrington-Kirkpatrick model. It is shown that the RS solution is exact in the paramagnetic phase. We provide a sufficient condition on coupling constants where the 1RSB solution gives better bound than the RS one. This condition reduced to physical quantities in disordered single spin systems allows a simple computer-assisted proof for the existence of the de Almeida-Thouless-type instability.Comment: 12 page

Deformation of a renormalization-group equation applied to infinite-order phase transitions

By adding a linear term to a renormalization-group equation in a system exhibiting infinite-order phase transitions, asymptotic behavior of running coupling constants is derived in an algebraic manner. A benefit of this method is presented explicitly using several examples.Comment: 6 pages, 5 figures, revtex4, typo corrected, references adde