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arXiv:2010.01142 · Physical Review Research

Quantum discontinuity fixed point and renormalization group flow of the Sachdev-Ye-Kitaev model

Roman Smit, Davide Valentinis, Jörg Schmalian, Peter Kopietz

Published Jul 23, 2021• 6 citations
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Abstract

We determine the global renormalization group (RG) flow of the Sachdev-Ye-Kitaev (SYK) model. From a controlled truncation of the infinite hierarchy of the exact functional RG flow equations, we identify several fixed points. Apart from a stable fixed point, associated with the celebrated non-Fermi liquid state of the model, we find another stable fixed point related to an integer-valence state. These stable fixed points are separated by a discontinuity fixed point with one relevant direction, describing a quantum first-order transition. Most notably, the fermionic spectrum continues to be quantum critical even at the discontinuity fixed point. This rules out a description of the transition in terms of a local effective Ising variable as is established for classical transitions. We propose an entangled quantum state at phase coexistence as a possible physical origin of this critical behavior.

DOI
10.1103/physrevresearch.3.033089

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We determine the global renormalization group (RG) flow of the Sachdev-Ye-Kitaev (SYK) model. From a controlled truncation of the infinite hierarchy of the exact functional RG flow equations, we identify several fixed points. Apart from a stable fixed point, associated with the celebrated non-Fermi liquid state of the model, we find another stable fixed point related to an integer-valence state. These stable fixed points are separated by a discontinuity fixed point with one relevant direction, describing a quantum first-order transition. Most notably, the fermionic spectrum continues to be quantum critical even at the discontinuity fixed point. This rules out a description of the transition in terms of a local effective Ising variable as is established for classical transitions. We propose an entangled quantum state at phase coexistence as a possible physical origin of this critical behavior.
Using a functional renormalization group approach we derive the renormalization group (RG) flow of a dissipative variant of the Yukawa-Sachdev-Ye-Kitaev model describing NNN fermions on a quantum dot, which interact via a disorder-induced Yukawa coupling with MMM bosons. The inverse Euclidean propagator of the bosons is assumed to exhibit a nonanalytic term proportional to the modulus of the Matsubara frequency. We show that, to leading order in 1/N1/N1/N and 1/M1/M1/M, the hierarchy of formally exact flow equations for the irreducible vertices of the disorder-averaged model can be closed at the level of the two-point vertices. We find that the RG flow exhibits a non-Fermi liquid fixed point characterized by a finite fermionic anomalous dimension $\ensuremath{\eta}$, which is related to the bosonic anomalous dimension $\ensuremath{\gamma}$ via the scaling law $2=2\ensuremath{\eta}+\ensuremath{\gamma}$ with $0<\ensuremath{\eta}<1/2$. We explicitly calculate $\ensuremath{\eta}$ and the critical exponents characterizing the linearized RG flow in the vicinity of the fixed point as functions of N/MN/MN/M.
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 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 develop a theoretical framework based on the nonperturvative renormalization group (RG) in the one-particle irreducible (Wetterich) formulation to tackle the interplay of coupled fermionic and order-parameter fluctuations at metallic quantum critical points (QCPs) with ordering wavevector Q⃗=0⃗\vec{Q}=\vec{0}Q​=0. We consistently treat the dynamical emergence of the Landau damping of the bosonic mode and non-Fermi liquid scaling of fermions upon lowering the cutoff scale. The loop integrals of the present theory involve only contributions from fluctuations above the cutoff scale, which protects the system from developing singular bosonic interactions. We emphasize the importance of the nontrivial relative scaling of the bosonic and fermionic cutoffs ΛΛΛ and ΛfΛ_fΛf​, which we fix by analyzing the RG flow of the scale-dependent ordering wave-vector Q⃗Λ\vec{Q}_ΛQ​Λ​. Upon neglecting Fermi self-energy in the loop integrals of the functional RG flow, we identify a non-Fermi liquid RG fixed point and recover the features obtained earlier within RPA-type approaches. In a subsequent step, we self-consistently include the scaling of the self-energy and the Yukawa coupling. We find a generic instability of the non-Fermi liquid RG fixed point. This implies, at least at this truncation level, absence of the QCP with Q⃗=0⃗\vec{Q}=\vec{0}Q​=0 and development of a first-order phase transition or a phase characterized by Q⃗≠0⃗\vec{Q}\neq\vec{0}Q​=0.
We study the Schwinger-Dyson equations of a fermionic planar matrix quantum mechanics [or tensor and Sachdev-Ye-Kitaev (SYK) models] at leading melonic order. We find two solutions describing a high entropy, SYK black-hole-like phase and a low entropy one with trivial IR behavior. There is a line of first order phase transitions that terminates at a new critical point. Critical exponents are nonmean field and differ on the two sides of the transition. Interesting phenomena are also found in unstable and stable bosonic models, including Kazakov critical points and inconsistency of SYK-like solutions of the IR limit.
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 ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\rightarrow}2\ensuremath{\pi}T$ in the low-temperature limit; however, the temperature scale marking the crossover to the conformal regime vanishes continuously at the critical point pc{p}_{c}pc​. The residual entropy at $T\ensuremath{\rightarrow}0$, 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 ${\ensuremath{\lambda}}_{\mathrm{L}}\ensuremath{\propto}{T}^{2}$.