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arXiv:1803.07197 · Physical review. B./Physical review. B

Family of Sachdev-Ye-Kitaev models motivated by experimental considerations

Étienne Lantagne-Hurtubise, Chengshu Li, Marcel Franz

Published Jun 14, 2018• 26 citations
arXivHTMLPDF

Abstract

Several condensed-matter platforms have been proposed recently to realize the Sachdev-Ye-Kitaev (SYK) model in their low-energy limit. In these proposed realizations, the characteristic SYK behavior is expected to occur under certain assumptions about the underlying physical system that (i) render all bilinear terms small compared to four-fermion interactions and (ii) ensure that the coupling constants are approximately all-to-all and independent random variables. In this work we explore, both analytically and numerically, the family of models that arises when we relax these assumptions in ways motivated by real physical systems. By relaxing (i) and allowing large bilinear terms, we obtain a novel, exactly solvable cousin of the $\mathrm{SYK}$ model. It exhibits two distinct phases separated by a quantum phase transition characterized by a power-law, ${\ensuremath{\sim}|\ensuremath{\omega}|}^{\ensuremath{-}1/3}$, scaling of the low-energy spectral density, despite being a noninteracting model. By relaxing (ii), we obtain close relatives of the $\mathrm{SYK}$ model which exhibit interesting behaviors, including a chaotic non-Fermi liquid phase with continuously varying fermion scaling dimension, and a phase transition to a disordered Fermi liquid as a function of interaction range and disorder length scale.

DOI
10.1103/physrevb.97.235124

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Several condensed-matter platforms have been proposed recently to realize the Sachdev-Ye-Kitaev (SYK) model in their low-energy limit. In these proposed realizations, the characteristic SYK behavior is expected to occur under certain assumptions about the underlying physical system that (i) render all bilinear terms small compared to four-fermion interactions and (ii) ensure that the coupling constants are approximately all-to-all and independent random variables. In this work we explore, both analytically and numerically, the family of models that arises when we relax these assumptions in ways motivated by real physical systems. By relaxing (i) and allowing large bilinear terms, we obtain a novel, exactly solvable cousin of the SYK\mathrm{SYK}SYK model. It exhibits two distinct phases separated by a quantum phase transition characterized by a power-law, \ensuremath∼∣\ensuremathω∣\ensuremath−1/3{\ensuremath{\sim}|\ensuremath{\omega}|}^{\ensuremath{-}1/3}\ensuremath∼∣\ensuremathω∣\ensuremath−1/3, scaling of the low-energy spectral density, despite being a noninteracting model. By relaxing (ii), we obtain close relatives of the SYK\mathrm{SYK}SYK model which exhibit interesting behaviors, including a chaotic non-Fermi liquid phase with continuously varying fermion scaling dimension, and a phase transition to a disordered Fermi liquid as a function of interaction range and disorder length scale.
We propose a simple solvable variant of the Sachdev-Ye-Kitaev (SYK) model which displays a quantum phase transition from a fast-scrambling non-Fermi liquid to disordered Fermi liquid. Like the canonical SYK model, our variant involves a single species of Majorana fermions connected by all-to-all random four-fermion interactions. The phase transition is driven by a random two-fermion term added to the Hamiltonian whose structure is inspired by proposed solid-state realizations of the SYK model. Analytic expressions for the saddle-point solutions at a large number NNN of fermions are obtained and show a characteristic scale-invariant ${\ensuremath{\sim}|\ensuremath{\omega}|}^{\ensuremath{-}1/2}$ behavior of the spectral function below the transition which is replaced by a ${\ensuremath{\sim}|\ensuremath{\omega}|}^{\ensuremath{-}1/3}$ singularity exactly at the critical point. These results are confirmed by numerical solutions of the saddle-point equations and discussed in the broader context of the field.
We present a review of the Sachdev-Ye-Kitaev (SYK) model of compressible quantum many-body systems without quasiparticle excitations, and its connections to various theoretical studies of non-Fermi liquids in condensed matter physics. The review is placed in the context of numerous experimental observations on correlated electron materials. Strong correlations in metals are often associated with their proximity to a Mott transition to an insulator created by the local Coulomb repulsion between the electrons. We explore the phase diagrams of a number of models of such local electronic correlation, employing a dynamical mean field theory in the presence of random spin exchange interactions. Numerical analyses and analytical solutions, using renormalization group methods and expansions in large spin degeneracy, lead to critical regions which display SYK physics. The models studied include the single-band Hubbard model, the ttt-JJJ model and the two-band Kondo-Heisenberg model in the presence of random spin exchange interactions. We also examine non-Fermi liquids obtained by considering each SYK model with random four-fermion interactions to be a multi-orbital atom, with the SYK-atoms arranged in an infinite lattice. We connect to theories of sharp Fermi surfaces without any low-energy quasiparticles in the absence of spatial disorder, obtained by coupling a Fermi liquid to a gapless boson; a systematic large NNN theory of such a critical Fermi surface, with SYK characteristics, is obtained by averaging over an ensemble of theories with random boson-fermion couplings. Finally, we present an overview of the links between the SYK model and quantum gravity and end with an outlook on open questions.
Motivated by recent proposals of experimental realization of fast scramblers, we study a family of solvable variants of the (q=4q=4q=4) Sachdev-Ye-Kitaev model in which the rank and eigenvalue distribution of the coupling matrix Jij,kl{J}_{ij,kl}Jij,kl​ are tuneable. When the rank is proportional to the number of fermions, the low temperature behavior is sensitive to the eigenvalue distribution. We obtain a complete classification of the possible non-Fermi liquid quantum phases. These include two previously studied phases whose fermion scaling dimension depends continuously on the rank; we show that they are maximally chaotic, but necessitate an extensively degenerate or negative semidefinite coupling matrix. More generic distributions give rise to ``almost Fermi liquids'' with a scaling dimension $\mathrm{\ensuremath{\Delta}}=1/2$, but which differ from a genuine Fermi liquid in quasiparticle decay rate, quantum Lyapunov exponent, and/or specific heat.
This thesis is devoted to the study of Sachdev-Ye-Kitaev (SYK) models, which describe matter without quasiparticles. The SYK model is an exactly solvable model that is hoped to bring insights into the understanding of strongly correlated materials.\nThe main part discusses different aspects of SYK models. Chapter 2 presents numerical studies of the SYK models, shows its non-Fermi liquid nature and verifies the analytical predictions on two-point function and thermal entropies. Chapter 3 considers the thermodynamic and transport properties of the complex SYK models in higher dimensional generalizations, we show some universal relations obtained from SYK models which are consistent with holographic theories. Chapter 4 discusses a supersymmetric generalization of the SYK model. We show a different a scaling from the normal SYK model both analytically and numerically. We also discuss a different origin of the extensive zero temperature entropy in mathcalN=2\\mathcal{N}=2mathcalN=2 theory. Chapter 5 considers a continuous set of boson-fermion SYK models with a tunable low-energy scaling dimension. From the explicit computation of out of time ordered correlators, we find the theory is less chaotic in the free fermion limit. Chapter 6 describes the phases of a solvable ttt-JJJ model of electrons with an SYK-like form. We find a mathbbZ2\\mathbb{Z}_2mathbbZ2​ fractionalization picture where the deconfined phase is similar to the pseudo gap regime of underdoped cuprates, while the `quasi-Higgs' phase is similar to the metallic phase of overdoped cuprates.
The Sachdev-Ye-Kitaev (SYK) model is a concrete solvable model to study non-Fermi liquid properties, holographic duality, and maximally chaotic behavior. In this work, we consider a generalization of the SYK model that contains two SYK models with a different number of Majorana modes coupled by quadratic terms. This model is also solvable, and the solution shows a zero-temperature quantum phase transition between two non-Fermi liquid chaotic phases. This phase transition is driven by tuning the ratio of two mode numbers, and a nonchaotic Fermi liquid sits at the critical point with an equal number of modes. At a finite temperature, the Fermi liquid phase expands to a finite regime. More intriguingly, a different non-Fermi liquid phase emerges at a finite temperature. We characterize the phase diagram in terms of the spectral function, the Lyapunov exponent, and the entropy. Our results illustrate a concrete example of the quantum phase transition and critical behavior between two non-Fermi liquid phases.
Inspired by recent developments in the study of the model of double scaled SYK (DSSYK), as elucidated in a recent paper, we embark on a re-evaluation of the Sachdev-Ye-Kitaev (SYK) model. Our motivation stems from the insights gained from the DSSYK model, particularly its ability to capture essential features of quantum dynamics and gravitational effects . In this work, we delve into the SYK model, uncovering precise solutions for the two-point function and self-energy that have not been previously reported. Building upon the advancements made in particle physics phenomenology, we extend the SYK model to encompass tensor field theory. Through the incorporation of a cutoff term to ensure convergence, we substantially advance our understanding of quantum many-body physics. Our investigation extends to experimental parameter estimation and the exploration of cutoff dependency in random couplings, providing invaluable insights into system dynamics. The introduction of a novel tensor field theory replaces conventional fermionic degrees of freedom with tensorial counterparts, leading to the discovery of intriguing phase transition phenomena characterized by a first-order transition. Furthermore, we elucidate a direct linear relationship between the coupling parameter and the cutoff scale. These findings not only shed light on emergent behavior across both high-energy physics and condensed matter systems but also pave the way for further theoretical and experimental exploration, inspired by the recent advancements in the SYK model.