The fermionic Ising spin glass is analyzed for arbitrary filling and for all temperatures. A self-organized three-band structure of the model is obtained in the magnetically ordered phase. Deviation from half-filling generates a central nonmagnetic band, which becomes sharply separated at
T=0 by (pseudo)gaps from upper and lower magnetic bands. Replica symmetry breaking effects are derived for several observables and correlations. They determine the shape of the three-band density of states, and, for given chemical potential, influence the fermion filling strongly in the low-temperature regime.
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(T0) 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
Tc3 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(T0) 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.
We present details of the phase diagrams of fermionic systems with random and frustrated interactions, emphasizing the important role of the chemical potential. 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, being controlled by quantum statistics. We compare the spin density diluted Sherrington-Kirkpatrick spin glass with classical spin 1 models such as the BEG model. We analyse in detail the infinite range model and show that spin glass order must decay discontinuously as the chemical potential exceeds a critical value, provided the temperature is below the tricritical one, and that the T=0 transition is of classical type. Parisi replica permutation symmetry breaking (RPSB) governs the thermal spin glass transitions and fermionic modifications of the SK-models AT-line emerge. RPSB takes place everywhere within the 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 aspects of models with additional spin and charge quantum--dynamics such as metallic spin glasses.
We determine the low-energy spectrum and Parisi replica-symmetry-breaking function for the spin-glass phase of the quantum Ising model with infinite-range random exchange interactions and transverse and longitudinal ( h ) fields. We show that, for all h , the spin-glass state has full replica symmetry breaking and the local spin spectrum is gapless, with a spectral density that vanishes linearly with frequency. These results are obtained using an action functional—argued to yield exact results at low frequencies—that expands in powers of a spin-glass order parameter, which is bilocal in time, and a matrix in replica space. We also present the exact solution of the infinite-range spherical quantum p -rotor model at nonzero h : here, the spin-glass state has one-step replica symmetry breaking and gaplessness only appears after imposition of an additional marginal stability condition. Possible connections to experiments on random arrays of trapped Rydberg atoms are noted. Published by the American Physical Society 2024
In the presence of quenched disorder, the interplay between local magnetic-moment formation and Anderson localization for electrons at a zero-temperature, metal-insulator transition (MIT) remains a long unresolved problem. Here, we study the emergence of these phenomena in a power-law random banded matrix model of spin-1/2 fermions with repulsive Hubbard interactions. Focusing on the regime of weak interactions, we perform both analytical field theory and numerical self-consistent Hartree-Fock calculations. We show that interference-mediated effects strongly enhance the density of states and magnetic fluctuations upon approaching the MIT from the metallic side. These are consistent with results due to Finkel'stein obtained four decades ago. Our numerics further show that local moments nucleate from typical states as we cross the MIT, with a density that grows continuously into the insulating phase. We identify spin-glass order in the insulator by computing the overlap distribution between converged Hartree-Fock mean-field moment profiles. Our results indicate that itinerant interference effects can morph smoothly into moment formation and magnetic frustration within a single model, revealing a common origin for these disparate phenomena.
The stability of Parisi's solution (1979-80) of the infinite-range spin glass model is examined near the transition temperature. It is found that the quadratic terms in the variation of the free energy are all non-positive. The free energy is only a local maximum with respect to q(x) if q is constrained to be a monotonic function of x. The connection between replica symmetry breaking and failure of linear response theory for magnetic susceptibility is shown in more detail. It is shown that the order parameter susceptibility is divergent, as expected, in the spin glass phase.