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arXiv:2005.07205 · Physical Review Research

Quantum phase transition in the Yukawa-SYK model

Yuxuan Wang, Andrey V. Chubukov

Published Jul 16, 2020• 57 citations
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Abstract

We study the quantum phase transition upon varying the fermionic density ν in a solvable model with random Yukawa interactions between N bosons and M fermions, dubbed the Yukawa-SYK model.We show that there are two distinct phases in the model: an incompressible state with gapped excitations and an exotic quantumcritical, non-Fermi liquid state with exponents varying with ν.We show analytically and numerically that the quantum phase transition between these two states is first-order, as for some range of ν the NFL state has a negative compressibility.In the limit N/M → ∞, the first-order transition gets weaker and asymptotically becomes second-order, with an exotic quantum-critical behavior.We show that fermions and bosons display highly unconventional spectral behavior in the transition region.

DOI
10.1103/physrevresearch.2.033084

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We study the quantum phase transition upon varying the fermionic density ν in a solvable model with random Yukawa interactions between N bosons and M fermions, dubbed the Yukawa-SYK model.We show that there are two distinct phases in the model: an incompressible state with gapped excitations and an exotic quantumcritical, non-Fermi liquid state with exponents varying with ν.We show analytically and numerically that the quantum phase transition between these two states is first-order, as for some range of ν the NFL state has a negative compressibility.In the limit N/M → ∞, the first-order transition gets weaker and asymptotically becomes second-order, with an exotic quantum-critical behavior.We show that fermions and bosons display highly unconventional spectral behavior in the transition region.
Non-Fermi liquids (NFLs) are a class of strongly interacting gapless fermionic systems without long-lived quasiparticle excitations.An important group of NFL models feature itinerant fermions coupled to soft bosonic fluctuations near a quantum-critical point and are widely believed to capture the essential physics of many unconventional superconductors.However, numerically, the direct observation of a canonical NFL behavior in such systems, characterized by a power-law form in the Green's function, has been elusive.Here, we consider a Sachdev-Ye-Kitaev (SYK)-like model with random Yukawa interaction between critical bosons and fermions (dubbed the Yukawa-SYK model).We show that it is immune from the minus-sign problem and hence can be solved exactly via large-scale quantum Monte Carlo simulation beyond the large-N limit accessible to analytical approaches.Our simulation demonstrates that the Yukawa-SYK model features "self-tuned quantum criticality"; namely, the system is critical independent of the bosonic bare mass.We put these results to the test at finite N, and our unbiased numerics reveal clear evidence of these exotic quantum-critical NFL properties-the power-law behavior in the Green's function of fermions and bosons-which propels the theoretical understanding of critical Planckian metals and unconventional superconductors.
We show that a random interacting model exhibits solvable non-Fermi-liquid behavior and exotic pairing behavior. This model, dubbed as the Yukawa-SYK model, describes the random Yukawa coupling between M quantum dots each hosting N flavors of fermions and N^{2} bosons that self-tune to criticality at low energies. The diagrammatic expansion is controlled by 1/MN, and the results become exact in a large-M, large-N limit. We find that pairing only develops within a region of the (M,N) plane-even though the pairing interaction is strongly attractive, the incoherence of the fermions can spoil the forming of Cooper pairs, rendering the system a non-Fermi liquid down to zero temperature. By solving the Eliashberg equation and the renormalization group equation, we show that the transition into the pairing phase exhibits Kosterlitz-Thouless quantum-critical behavior.
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 analyze the phase diagram of the spin-12\frac{1}{2}21​ Yukawa--Sachdev-Ye-Kitaev model, which describes complex spin-12\frac{1}{2}21​ fermions randomly interacting with real bosons via a Yukawa coupling, at finite temperatures and varying fermion density. A recent work [Wang and Chubukov, Phys. Rev. Res. 2, 033084 (2020).] showed that, upon varying the filling or chemical potential, a first-order quantum phase transition exists between a non-Fermi-liquid (NFL) phase and an insulating phase. Here we show that in such a model with time-reversal symmetry this quantum phase transition is preempted by a pairing phase that develops as a low-temperature instability. As a remnant of the would-be NFL-insulator transition, the superconducting critical temperature rapidly decreases beyond a certain chemical potential. On the other hand, depending on the parameters, the first-order quantum phase transition extends to finite temperatures and terminates at a thermal critical point, beyond which the NFL and the insulator become the same phase, similar to that of the liquid-gas and metal-insulator transition in real materials. We determine the pairing phase boundary and the location of the thermal critical point via combined analytic and quantum Monte Carlo numeric efforts. Our results provide a model realization of the transition of NFLs towards superconductivity and insulating states and therefore offer a controlled platform for future investigations of the generic phase diagram that hosts a NFL, insulator, and superconductor and their phase transitions.
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 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).