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arXiv:1610.04619 · Physical review. B./Physical review. B

Solvable model for a dynamical quantum phase transition from fast to slow scrambling

Sumilan Banerjee, Ehud Altman

Published Apr 19, 2017• 224 citations
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Abstract

We propose an extension of the Sachdev-Ye-Kitaev (SYK) model that exhibits a quantum phase transition from the previously identified non-Fermi-liquid fixed point to a Fermi-liquid-like state, while still allowing an exact solution in a suitable large-$N$ limit. The extended model involves coupling the interacting $N$-site SYK model to a new set of $pN$ peripheral sites with only quadratic hopping terms between them. The conformal fixed point of the SYK model remains a stable low-energy phase below a critical ratio of peripheral sites $p<{p}_{c}(n)$ that depends on the fermion filling $n$. The scrambling dynamics throughout the non-Fermi-liquid (NFL) phase is characterized by a universal Lyapunov exponent ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\rightarrow}2\ensuremath{\pi}T$ in the low-temperature limit; however, the temperature scale marking the crossover to the conformal regime vanishes continuously at the critical point ${p}_{c}$. The residual entropy at $T\ensuremath{\rightarrow}0$, nonzero in the NFL, also vanishes continuously at the critical point. For $p>{p}_{c}$ the quadratic sites effectively screen the SYK dynamics, leading to a quadratic fixed point in the low-temperature and low-frequency limit. The interactions have a perturbative effect in this regime leading to scrambling with Lyapunov exponent ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\propto}{T}^{2}$.

DOI
10.1103/physrevb.95.134302

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We propose an extension of the Sachdev-Ye-Kitaev (SYK) model that exhibits a quantum phase transition from the previously identified non-Fermi-liquid fixed point to a Fermi-liquid-like state, while still allowing an exact solution in a suitable large-NNN limit. The extended model involves coupling the interacting NNN-site SYK model to a new set of pNpNpN peripheral sites with only quadratic hopping terms between them. The conformal fixed point of the SYK model remains a stable low-energy phase below a critical ratio of peripheral sites p<pc(n)p<{p}_{c}(n)p<pc​(n) that depends on the fermion filling nnn. The scrambling dynamics throughout the non-Fermi-liquid (NFL) phase is characterized by a universal Lyapunov exponent \ensuremathλL\ensuremath→2\ensuremathπT{\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\rightarrow}2\ensuremath{\pi}T\ensuremathλL​\ensuremath→2\ensuremathπT in the low-temperature limit; however, the temperature scale marking the crossover to the conformal regime vanishes continuously at the critical point pc{p}_{c}pc​. The residual entropy at T\ensuremath→0T\ensuremath{\rightarrow}0T\ensuremath→0, nonzero in the NFL, also vanishes continuously at the critical point. For p>pcp>{p}_{c}p>pc​ the quadratic sites effectively screen the SYK dynamics, leading to a quadratic fixed point in the low-temperature and low-frequency limit. The interactions have a perturbative effect in this regime leading to scrambling with Lyapunov exponent \ensuremathλL\ensuremath∝T2{\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\propto}{T}^{2}\ensuremathλL​\ensuremath∝T2.
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.
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.
We compute the Lyapunov exponent characterizing quantum scrambling in a family of generalized Sachdev-Ye-Kitaev models, which can be tuned between different low temperature states from Fermi liquids, through non-Fermi liquids, to fast scramblers. The analytic calculation, controlled by a small coupling constant and large NNN, allows us to clarify the relations between the quasiparticle relaxation rate $1/\ensuremath{\tau}$ and the Lyapunov exponent ${\ensuremath{\lambda}}_{L}$ characterizing scrambling. In the Fermi liquid states we find that the quasiparticle relaxation rate dictates the Lyapunov exponent. In non-Fermi liquids, where $1/\ensuremath{\tau}\ensuremath{\gg}T$, we find that ${\ensuremath{\lambda}}_{L}$ is always TTT linear with a prefactor that is independent of the coupling constant in the limit of weak coupling. Instead it is determined by a scaling exponent that characterizes the relaxation rate. ${\ensuremath{\lambda}}_{L}$ approaches the general upper bound $2\ensuremath{\pi}T$ at the transition to a fast scrambling state. Finally, in a marginal Fermi liquid state the exponent is linear in temperature with a prefactor that vanishes as a nonanalytic function $\ensuremath{\sim}gln(1/g)$ of the coupling constant ggg.
We study a series of perturbations on the Sachdev-Ye-Kitaev (SYK) model. We show that the maximal chaotic non-Fermi-liquid phase described by the ordinary q=4q=4q=4 SYK model has marginally relevant or irrelevant (depending on the sign of the coupling constants) four-fermion perturbations allowed by symmetry. Changing the sign of one of these four-fermion perturbations leads to a continuous chaotic-nonchaotic quantum phase transition of the system accompanied by a spontaneous time-reversal symmetry breaking. Starting with the SYKq{\mathrm{SYK}}_{q}SYKq​ model with a qqq-fermion interaction, similar perturbations can lead to a series of new fixed points with continuously varying exponents.
We study the thermalization, after sudden and slow quenches, in an interacting model having a quantum phase transition from a Sachdev-Ye-Kitaev (SYK) non-Fermi liquid (NFL) to a Fermi liquid (FL). The model has SYK fermions coupled to noninteracting lead fermions and can be realized in a graphene flake connected to external leads. A sudden quench to the NFL leads to rapid thermalization via collapse-revival oscillations of the quasiparticle residue of the lead fermions. In contrast, the quench to the FL shows multiple prethermal regimes and much slower thermalization. In the slow quench performed over a time τ, we find that the excitation energy generated has a remarkable intermediate-τ nonanalytic power-law dependence, τ−η with η<1, which seemingly masks the dynamical manifestation of the initial residual entropy of the SYK fermions. Our study gives an explicit demonstration of the intriguing contrasts between the out-of-equilibrium dynamics of a NFL and a FL in terms of their thermalization and approach to adiabaticity.
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, ${\ensuremath{\sim}|\ensuremath{\omega}|}^{\ensuremath{-}1/3}$, scaling of the low-energy spectral density, despite being a noninteracting model. By relaxing (ii), we obtain close relatives of the 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.