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arXiv:2312.14026 · Physical review. B./Physical review. B

Non-Fermi liquid fixed point of the dissipative Yukawa-Sachdev-Ye-Kitaev model

Niklas Cichutek, Andreas Rückriegel, Max O. Hansen, Peter Kopietz

Published Apr 1, 2024• 2 citations
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Abstract

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 $N$ fermions on a quantum dot, which interact via a disorder-induced Yukawa coupling with $M$ 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 $1/N$ and $1/M$, 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 $N/M$.

DOI
10.1103/physrevb.109.155101

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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 \ensuremathη\ensuremath{\eta}\ensuremathη, which is related to the bosonic anomalous dimension \ensuremathγ\ensuremath{\gamma}\ensuremathγ via the scaling law 2=2\ensuremathη+\ensuremathγ2=2\ensuremath{\eta}+\ensuremath{\gamma}2=2\ensuremathη+\ensuremathγ with 0<\ensuremathη<1/20<\ensuremath{\eta}<1/20<\ensuremathη<1/2. We explicitly calculate \ensuremathη\ensuremath{\eta}\ensuremathη and the critical exponents characterizing the linearized RG flow in the vicinity of the fixed point as functions of N/MN/MN/M.
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.
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We construct exact functional renormalization group (RG) flow equations for nonrelativistic fermions in arbitrary dimensions, taking into account not only mode elimination, but also the rescaling of the momenta, frequencies, and fermionic fields. The complete RG flow of all relevant, marginal, and irrelevant couplings can be described by a system of coupled flow equations for the irreducible n-point vertices. Introducing suitable dimensionless variables, we obtain flow equations for generalized scaling functions which are continuous functions of the flow parameter, even if we consider quantities which are dominated by momenta close to the Fermi surface, such as the density-density correlation function at long wavelengths. We also show how the problem of constructing the renormalized Fermi surface can be reduced to the problem of finding the RG fixed point of the irreducible two-point vertex at vanishing momentum and frequency. We argue that only if the degrees of freedom are properly rescaled is it possible to reach scale-invariant non-Fermi-liquid fixed points within a truncation of the exact RG flow equations.
We analyze the relation between quantum chaotic behavior and phase transitions of the Yukawa--Sachdev-Ye-Kitaev model as a function of filling and temperature, which describes random Yukawa interactions between NNN complex fermions and MMM bosons in zero spatial dimensions for both the non-Fermi liquid and insulating states at finite temperature and chemical potential. We solve the ladder equations for the out-of-time-order correlator (OTOC) for both bosons and fermions. Despite the appearance of the chemical potential in the Hamiltonian, which explicitly introduces an additional energy scale, the OTOCs for the fermions and bosons in the non-Fermi liquid state turn out to be unaffected, and the Lyapunov exponents that diagnose chaos remain maximal. As the chemical potential increases, the system is known to experience a first-order transition from a critical phase to a gapped insulating phase. We postulate that the boundary of the region in parameter space where each phase is (meta)stable coincides with the curve on which the Lyapunov exponent is maximal. By calculating the exponent in the insulating phase and comparing to numerical results on the boundaries of stability, we show that this is plausible.
We study a model of NNN fermions in a quantum dot, coupled to MMM bosons by a disorder-induced complex Yukawa coupling [Yukawa Sachdev-Ye-Kitaev (SYK) model], in order to explore the interplay between non-Fermi liquid and superconductivity in a strongly coupled, (quantum-)critical environment. We analyze the phase diagram of the model for an arbitrary complex interaction and arbitrary ratio of N/MN/MN/M, with special focus on the two regimes of non-Fermi-liquid behavior: an SYK-like behavior with a power-law frequency dependence of the fermionic self-energy and an impuritylike behavior with frequency independent self-energy. We show that the crossover between the two can be reached by varying either the strength of the fermion-boson coupling or the ratio M/NM/NM/N. We next argue that in both regimes the system is unstable to superconductivity if the strength of time-reversal-symmetry-breaking disorder is below a certain threshold. We show how the corresponding onset temperatures vary between the two regimes. We argue that the superconducting state is highly unconventional with an infinite set of minima of the condensation energy at T=0T=0T=0, corresponding to topologically different gap functions. We discuss in detail similarities and differences between this model and the model of dispersion-full fermions tuned to a metallic quantum-critical point, with an effective singular dynamical interaction $V(\mathrm{\ensuremath{\Omega}})\ensuremath{\propto}1/{|\mathrm{\ensuremath{\Omega}}|}^{\ensuremath{\gamma}}$ (the $\ensuremath{\gamma}$ model).
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