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arXiv:cond-mat/9803239 · Physical review. B, Condensed matter

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

R. Oppermann, Bernd Rosenow

Published Oct 1, 1999• 11 citations
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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.

DOI
10.1103/physrevb.60.10325

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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-TTT 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 T\ensuremath→0,\stackrel{\ensuremath{\rightarrow}}{T}0,T\ensuremath→​0, are contrasted with the replica-symmetric nature of the critical field theory of quantum spin glass transitions at T=0,T=0,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){O(T}^{0})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=0T=0T=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=0T=0T=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 U\ensuremath→\ensuremath∞\stackrel{\ensuremath{\rightarrow}}{U}\ensuremath{\infty}U\ensuremath→​\ensuremath∞ limit are briefly considered. We compare our spin glass results with recent d=\ensuremath∞d=\ensuremath{\infty}d=\ensuremath∞ (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 \ensuremathμ.\ensuremath{\mu}.\ensuremathμ. 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 \ensuremathμ\ensuremath{\mu}\ensuremathμ exceeds a critical value, provided T is below the tricritical Tc3{T}_{c3}Tc3​ and that the T=0T=0T=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,T=0,T=0, with only small corrections at low T from RPSB, the phase diagram is affected at O(T0){O(T}^{0})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.
Models with range-free frustrated Ising spin interaction and additional Hubbard interaction are treated exactly by means of the discrete time slicing method of Grassmann-field theory. Critical and tricritical points, spin and charge correlations, and the fermion propagator are derived as a function of temperature and chemical potential $\ensuremath{\mu},$ of the Hubbard coupling U and the spin-glass energy J. U is allowed to be either repulsive (U>0)(U>0)(U>0) or attractive (U<0).(U<0).(U<0). Cuts through the multidimensional phase diagram are obtained. Analytical and numerical evaluations take important replica symmetry-breaking (RSB) effects into account. Results for the ordered phase are given at least in a one-step approximation (1RSB), and for T=0T=0T=0 we report the first two-, three-, and four-step calculations (4RSB) for fermionic spin glasses. The use of exact relations and invariances under RSB together with 2RSB calculations for all fillings and 4RSB solutions for half filling allow to model exact solutions by interpolation. For T=0,T=0,T=0, our numerical results provide strong evidence that the exact spin-glass pseudogap obeys $\ensuremath{\rho}{(E)=c}_{1}|E\ensuremath{-}{E}_{F}|$ for energies close to the Fermi level with ${c}_{1}\ensuremath{\approx}0.13.$ Rapid convergence of ${\ensuremath{\rho}}^{\ensuremath{'}}{(E}_{F})$ under increasing orders of RSB is observed and ${\ensuremath{\rho}}^{\ensuremath{''}}(E)$ is evaluated to estimate subleading powers. Over a wide range of the pseudogap and after a small transient regime $\ensuremath{\rho}(E)$ regains a linear shape with a larger slope and a small S-like perturbation. The leading term resembles the Efros-Shklovskii Coulomb pseudogap of two-dimensional localized disordered fermionic systems. Beyond half filling we obtain a $\ensuremath{\nu}\ensuremath{-}1\ensuremath{\sim}(\ensuremath{\mu}\ensuremath{-}{U)}^{2},$ $\ensuremath{\mu}>~U$ dependence of the fermion filling factor $\ensuremath{\nu}.$ We find a half filling transition between a phase for $U>\ensuremath{\mu},$ where the Fermi level lies inside the Hubbard gap, into a phase where $\ensuremath{\mu}(>U)$ is located at the center of the upper spin-glass (SG) pseudogap (SG gap). For $\ensuremath{\mu}>U$ the Hubbard gap combines with the lower one of two SG gaps (phase I), while for $\ensuremath{\mu}<U$ it joins the sole SG gap which exists in this half-filling regime (phase II). Shoulders of the combined gaps are shaped by RSB due to spin-glass order. We predict scaling behavior at the half-filling transition which becomes continuous due to $\ensuremath{\infty}\mathrm{RSB}.$ Implications of the half-filling transition between the deeper insulating phase II and phase I for the eventual delocalization by additional hopping processes in itinerant model extensions are discussed. Possible metal-insulator transition scenarios are described. Generalizations to random Hubbard coupling and alloy models as well as frustrated magnetic interactions with ferro- or antiferromagnetic components are discussed.
This dissertation investigates the role of disorder in quantum many-body systems, focusing on strange metals and spin glass systems. For strange metals, we analyze the low-frequency spectra of complex Sachdev-Ye-Kitaev (SYK) models at arbitrary densities, combining analytical and numerical methods. Building on these insights, we study the overdoped random t-J model, demonstrating that strange metal behavior persists across the entire doping range. We also show that the critical metal develops an instability to a low-doping spin-glass phase, and compute the critical doping value. Furthermore, we explore the chaotic properties of a distinct strange metal model—the large N theory of a critical Fermi surface—and establish that it exhibits maximal quantum chaos. For spin glass systems, we examine the low-energy spectrum and Parisi replica-symmetry-breaking function in the quantum Ising model with infinite-range random exchange interactions with the presence of transverse and longitudinal fields. In the spin glass phase, we find that the local spin spectra are gapless and provide a detailed analysis of the phase diagram. Additionally, we investigate the quantum spherical p-rotor model in an external field, showing that the spin glass phase remains robust across a wide range of field strengths. The potential experimental applications of these findings are also discussed.
We consider quantum rotors or Ising spins in a transverse field on a d-dimensional lattice, with random, frustrating, short-range, exchange interactions. The quantum dynamics are associated with a finite moment of inertia for the rotors, and with the transverse field for the Ising spins. For a suitable distribution of exchange constants, these models display spin-glass and quantum paramagnet phases and a zero-temperature (T) quantum transition between them. An earlier exact solution for the critical properties of a model with infinite-range interactions cna be reproduced by minimization of a Landau effective-action functional for the model in finite d with short-range interactions. The functional is expressed in terms of a composite spin field which is bilocal in time. The mean-field phase diagram near the T=0 critical point is mapped out as a function of T, strength of the quantum coupling, and applied fields. The spin-glass phase has replica symmetry breaking; but, as in the classical Ising spin glass, the order parameter becomes replica symmetric as T\ensuremath{\rightarrow}0. Next we examine the consequences of fluctuations about the mean field for the critical properties. Above d=8, and with certain restrictions on the values of the Landau couplings, we find that the transition is controlled by a Gaussian fixed point with mean-field critical exponents. For couplings not attracted by the Gaussian fixed point above d=8, and for all physical couplings below d=8, we find runaway renormalization-group flows to strong coupling. General scaling relations that should be valid even at the strong-coupling fixed point are proposed and compared with Monte Carlo simulations.
We report progress in understanding the fermionic Ising spin glass with arbitrary filling. A crossover from a magnetically disordered single band phase via two intermediate bands just below the freezing temperature to a 3-band structure at still lower temperatures - beyond an almost random field instability - is shown to emerge in the magnetic phase. An attempt is made to explain the exact solution in terms of a quantum Parisi phase. A central nonmagnetic band\nis found and seen to become sharply separated at T = 0 by gaps from upper and lower magnetic bands. The gap sizes tend towards zero as the number of replica symmetry breaking steps increases towards infinity. In an extended model, the competition between local pairing superconductivity and spin glass order is discussed.
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