Numerical study of fermion and boson models with infinite-range random interactions (1603.05246v2)

Abstract: We present numerical studies of fermion and boson models with random all-to-all interactions (the SYK models). The high temperature expansion and exact diagonalization of the $N$-site fermion model are used to compute the entropy density: our results are consistent with the numerical solution of $N=\infty$ saddle point equations, and the presence of a non-zero entropy density in the limit of vanishing temperature. The exact diagonalization results for the fermion Green's function also appear to converge well to the $N=\infty$ solution. For the hard-core boson model, the exact diagonalization study indicates spin glass order. Some results on the entanglement entropy and the out-of-time-order correlators are also presented.

Numerical Study of Fermion and Boson Models with Infinite-Range Random Interactions

The paper "Numerical study of fermion and boson models with infinite-range random interactions" conducts a thorough examination of the Sachdev-Ye-Kitaev (SYK) models that incorporate fermions and bosons with random all-to-all interactions. Leveraging numerical methods such as the high temperature expansion, exact diagonalization, and the $N=\infty$ saddle point approximation, the study explores the entropy density at both high and zero temperature limits, the fermionic Green's function behavior, and attributes of spin glass order in bosonic configurations.

Key Numerical Results

1. Entropy Density: A significant finding is the consistency in results obtained from exact diagonalization with the anticipated outcomes at the $N=\infty$ limit, revealing a non-zero entropy density as temperature approaches zero for fermionic SYK systems. This non-Fermi liquid entropy behavior is crucial as it implies compressible states not exhibiting traditional Fermi liquid characteristics.
2. Fermionic Green's Function: The study demonstrates good convergence of the ED fermion Green's function data to the theoretical $N=\infty$ solution, highlighting the robustness of the SYK model in representing non-Fermi liquid states.
3. Spin Glass Order in Bosonic Models: The exact diagonalization results present compelling evidence of spin glass order manifesting within the hard-core boson model—a stark contrast to the fermionic systems. This suggests that bosonic SYK models, under similar random interaction paradigms, are predisposed to spin-glass states, necessitating analytic approaches such as replica symmetry breaking.
4. Entanglement and Out-of-Time-Order Correlators: The numerical exploration also ventures into entanglement entropy in ground states, finding adherence to a volume law indicative of eigenstate thermalization, and scrutinizes the chaotic properties via out-of-time-order correlators (OTOCs), reiterating the quantum chaotic nature intrinsic to these models.

Implications and Future Directions

This paper advances the theoretical understanding of holographic connections to black hole dynamics and quantum chaos by reinforcing the credibility of the SYK model in numeraically replicating non-Fermi liquid states, often suggested by AdS/CFT correspondences. Moreover, the delineation of spin glass characteristics within bosonic configurations beckons further investigation into symmetry-breaking methodologies, broadening applications in understanding many-body localization phenomena.

Looking ahead, the pursuit of model extensions could entail heavier reliance on numerical techniques such as tensor networks or quantum Monte Carlo methods to tease apart finer distinctions between fermionic and bosonic random interactions, enriching theoretical models with practical variances observed at finite $N$ limits.

In conclusion, this study provides substantial numerical and theoretical insights into the structural dynamics of random interaction models, acting as a foundational basis for exploring broader quantum matter and gauge/gravity duality scenarios.