Parisi symmetry of the many-body quantum theory of randomly interacting fermionic systems

Abstract

We show that fermion systems with random and frustrated interactions display a strong coupling between glassy order and fermionic correlations, which culminates in the implementation of Parisi replica permutation symmetry breaking (RPSB) in their zero-temperature quantum field theories. RPSB effects, setting in below fermionic de Almeida--Thouless (dAT) lines, become stronger as the temperature T decreases and play a crucial role for many physical properties within the entire low-$T$ regime. The Parisi ultrametric structure is shown to determine the dynamic behavior of fermionic correlations (Green's functions) for large times and for the corresponding low-energy excitation spectra, which is predicted to affect transport properties in metallic (and superconducting) spin glasses. Thus we reveal the existence and the detailed form of a number of quantum-dynamical fingerprints of the Parisi scheme. These effects, being strongest as $\stackrel{\ensuremath{\rightarrow}}{T}0,$ are contrasted with the replica-symmetric nature of the critical field theory of quantum spin glass transitions at $T=0,$ which display only small corrections at low T from RPSB. RPSB effects moreover appear to influence the loci of the ground state transitions at ${O(T}^{0})$ and hence the phase diagrams. From explicit solutions for arbitrary T we find a representation of the Green's function in the $T=0$ limit. This leads to a map of the fermionic (insulating) spin glass solution to the local limit of a Hubbard model with random repulsive interaction. This map holds for any number of replica-symmetry-breaking steps K. We obtain the distribution of the Hubbard interaction U and its dependence on the order of RPSB. A generalized mapping between metallic spin glass and random U Hubbard model is conjectured. We also suggest that the new representation of the Green's function at $T=0$ can be used for generalizations to superconductors with spin glass phases. Further generalizations due to Coulomb effects including a crossover from four-state per site to effectively three-state per site models in the $\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}$ limit are briefly considered. We compare our spin glass results with recent $d=\ensuremath{\infty}$ (clean) Hubbard model analyses, paying particular attention to the common role of the corresponding Onsager reaction fields. We also present details of the phase diagrams, emphasizing the important role of the chemical potential $\ensuremath{\mu}.$ The insulating fermionic Ising spin glass model is shown to reveal different entangled magnetic instabilities and phase transitions. We review tricritical phenomena related to the strong correspondence between charge and spin fluctuations and controlled by quantum statistics. A comparison with the diluted Sherrington-Kirkpatrick (SK) spin glass and with classical spin 1 models such as the Blume-Emery-Griffiths model is given. Our detailed analysis for the infinite-range model shows that spin glass order must decay discontinuously as $\ensuremath{\mu}$ exceeds a critical value, provided T is below the tricritical ${T}_{c3}$ and that the $T=0$ transition is of classical type. RPSB occurs in any case on the irreversible side of the (modified) dAT lines for the fermionic SK model and hence at least everywhere within a fermionic spin glass phase. Although the critical field theory of the quantum paramagnet to spin glass transition in metallic systems remains replica symmetric at $T=0,$ with only small corrections at low T from RPSB, the phase diagram is affected at ${O(T}^{0})$ by RPSB. Generalizing our results for the fermionic Ising spin glass we consider modifications in phase diagrams of models with spin and charge quantum dynamics such as metallic spin glasses.