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arXiv:2107.13572 · Physical review. B./Physical review. B

Sign-problem-free variant of the complex Sachdev-Ye-Kitaev model

Byungmin Kang, Junggi Yoon

Published Jan 13, 2022• 4 citations
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Abstract

We construct a sign-problem-free variant of the complex Sachdev-Ye-Kitaev (SYK) model which keeps all the essential properties of the SYK model, including the analytic solvability in the large-$N$ limit and being maximally chaotic. In addition to the number of complex fermions $N$, our model has an additional parameter $M$ controlling the number of terms in the Hamiltonian which we take $M\ensuremath{\rightarrow}\ensuremath{\infty}$ with keeping $M/N$ constant in the large-$N$ limit. While our model respects global $\text{U}(1)$ symmetry associated with the fermion number conservation, both the large-$N$ limit and the sign-problem-free nature become explicit in the Majorana representation. We present a detailed analysis on our model, i.e., the random matrix classification based on the symmetry analysis, analytic approach, and the quantum Monte Carlo simulations. All these analyses show that our model exhibits a non-Fermi-liquid physics, a gapless fermionic system lying beyond the conventional Fermi-liquid picture.

DOI
10.1103/physrevb.105.045117

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We construct a sign-problem-free variant of the complex Sachdev-Ye-Kitaev (SYK) model which keeps all the essential properties of the SYK model, including the analytic solvability in the large-NNN limit and being maximally chaotic. In addition to the number of complex fermions NNN, our model has an additional parameter MMM controlling the number of terms in the Hamiltonian which we take M\ensuremath→\ensuremath∞M\ensuremath{\rightarrow}\ensuremath{\infty}M\ensuremath→\ensuremath∞ with keeping M/NM/NM/N constant in the large-NNN limit. While our model respects global U(1)\text{U}(1)U(1) symmetry associated with the fermion number conservation, both the large-NNN limit and the sign-problem-free nature become explicit in the Majorana representation. We present a detailed analysis on our model, i.e., the random matrix classification based on the symmetry analysis, analytic approach, and the quantum Monte Carlo simulations. All these analyses show that our model exhibits a non-Fermi-liquid physics, a gapless fermionic system lying beyond the conventional Fermi-liquid picture.
The SYK model of Sachdev, Ye and Kitaev is a recent model of N Majorana fermions coupled via quartic random interaction. It has the extraordinary properties of being exactly solvable at large N , having emergent conformal symmetry in the IR, and exhibiting maximal quantum chaos. The confluence of these properties make SYK a highly attractive candidate for a solvable AdS/CFT correspondence. What’s more, it was found to capture elements of non-Fermi-liquid behavior in strange metals. This thesis introduces the SYK model and discusses a recent variant, the SYK∗ model of Marcus and Vandoren. In the latter, the effective action is derived and branches of conformal solutions are discussed. The parameter space of SYK∗ is investigated for large and small M . The former reveals a phase separation and a N critical point in the T−M plane is claimed to exist.
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.
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.
We present an exact solution in the large-NNN limit of the Lévy Sachdev-Ye-Kitaev (LSYK) model introduced in Ref. [1], wherein the couplings are drawn from a Lévy Stable distribution parameterized by a tail exponent μ∈[0,2]μ\in [0, 2]μ∈[0,2]. Starting from the Hamiltonian and its associated partition function, we highlight the key differences from the standard Gaussian SYK model and derive the large-NNN Schwinger-Dyson equations via a bosonic oscillator representation of the action. These equations are solved both numerically and analytically in the large-qqq and infrared limits. We subsequently analyze the chaotic properties of the model by computing the Krylov exponent from the large-qqq Green's function and extracting the Lyapunov exponent from the 444-point function. The parameter μμμ continuously interpolates between a free theory at μ=0μ= 0μ=0 and the conventional, maximally chaotic Gaussian SYK model at μ=2μ= 2μ=2, with non-maximal chaos persisting throughout the intermediate regime 0<μ<20 < μ< 20<μ<2. Thermodynamic quantities, including the entropy, free energy, average energy, and specific heat capacity, are computed and compared with their Gaussian SYK counterparts. The interpretations of the thermodynamics are discussed with respect to the holographic dual and non-Fermi liquid theory. Finally, we discuss an alternative representation of the LSYK model based on a distinct decomposition of the Lévy Stable distribution, which establishes a non-trivial connection to Gaussian SYK, and provide supporting analytical and numerical results in the appendices.
We introduce two disorder-free variants of the Sachdev-Ye-Kitaev (SYK) model, demonstrate their integrability, and study their static and dynamical properties. Unlike diagrammatic techniques, the integrability of these models allows us to obtain dynamical correlation functions even when the number of Majorana fermions is finite. From the solutions, we find that out-of-time-order correlators (OTOCs) in these models exhibit exponential growth at early times, resembling that of many-body quantum chaotic systems, such as those with disorder or external kick terms, despite their large N behavior differing from that of typical chaotic systems. Conversely, our analysis shows no evidence of random-matrix behavior in level spacing statistics or the spectral form factor. Our findings illustrate that the clean versions of the SYK models represent simple but nontrivial examples of disorder-free quantum many-body systems displaying chaoslike behavior of OTOCs.