Quantum Phase Transition in the Yukawa-SYK Model

We study the quantum phase transition upon variation of the fermionic density $\nu$ 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 quantum-critical, non-Fermi liquid state with exponents varying with $\nu$. We show analytically and numerically that the quantum phase transition between these two states is first-order, as for some range of $\nu$ the NFL state has a negative compressibility. In the limit $N/M\to \infty$ 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.

The theoretical understanding of a NFL remains a challenge.Most of earlier studies of NFLs considered itinerant fermions coupled to soft bosonic modes near a quantum-critical point (QCP).These models show nontrivial NFL behavior at the one-loop order, however in most cases the loop expansion is not controlled because of logarithmic singularities, even in the large N limit, [12,15], and one has to introduce additional modifications to the model [23][24][25], e.g., dimension regularization or matrix large-N , to keep the calculations under control.Another route to NFL, which has emerged recently [26][27][28][29][30], explores Sachdev-Ye-Kitaev (SYK)-type models [31][32][33][34][35][36][37].These models describe randomly interacting fermions in a quantum dot.The advantage of SYK model is that it is exactly solvable in the large-N limit and displays NFL behavior with a particular fractional exponent x = 1/2 for the self-energy.Besides, the SYK model has a hidden holographic connection to quantum black holes [32,33,38] and in this respect is a simple prototypical model for both NFL and quantum gravity.
In this communication, we consider the generalization of the SYK model, the Yukawa-SYK (Y-SYK) model [39][40][41], in which M flavors of dispersion-less fermions in a quantum dot randomly interact with N flavors of massive bosons, e.g., optical phonons or gapped collective spin or charge fluctuations.The interest to this model has been triggered by its rich and unconventional physics, and by recent experimental discoveries of strongly correlated behavior in flat band systems like magic angle twisted bilayer graphene [42,43] and d xy band in Fe-based superconductors [44].The Y-SYK model has been earlier studied at half-filling [39,40,45].It was shown that the interaction "self-tunes" the system into a NFL, quantumcritical (QC) regime, despite that a bare bosonic mass is finite.This QC regime may in turn become unstable towards non-BCS superconductivity.[28,[39][40][41][46][47][48][49][50][51] Like the SYK model, the Y-SYK model also saturates the upper bound for the onset rate of quantum chaos [52], indicating the existence a classical holographic dual.
We report the results of on the Y-SYK model away from half-filling, at fermionic density ν = 1/2.For small deviations from ν = 1/2, we analytically obtain lowenergy forms of the fermionic and bosonic Green's functions with NFL exponents and show that the fermionic self-energy and the spectral function become asymmetric in frequency.At ν = 1, we show that fermions form an incompressible state and bosons remain gapped.We then focus on the quantum phase transition between the compressible NFL state and the incompressible state.We show both analytically and numerically that the phase transition is generally first-order because the chemical potential is a non-monotonic function of ν, and the compressibility dν/dµ < 0 for a range of ν.We argue that this is due to robust low-energy properties of the Y-SYK model.In the transition region the fermionic and bosonic spectral functions displays a peculiar precursor behavior [53].In the particular limit, where the number of bosonic flavors well exceeds the number of fermionic ones, non-monotonicity disappears and the transition becomes second-order.Even in this case, bosons displays a highly non-trivial "gap filling" behavior: the bosonic mass gap remains finite on both sides of the transition, but on the NFL side of the transition the spectral weight arXiv:2005.07205v2[cond-mat.str-el]18 May 2020 develops around zero energy, and the width of the range, where this happens, increases as the system moves deeper into the NFL region.Taken together, these results reveal rich and universal behavior of zero-dimensional quantumcritical NFL systems.
Some features of the Y-SYK model, like the asymmetry of fermionic self-energy Σ(iω), are similar to those of the complex SYK model [37,54].Recent numerical results [55,56] for this model suggest that it may also undergo a first-order quantum phase transition between a NFL state and an insulating state.However, the analytical understanding of that transition is still lacking.In particular it remains unclear whether the first-order transition is a universal property of the complex SYK model, or it depends on non-universal aspects of the system behavior at larger frequencies.The model.The Y-SYK model describes M flavors of dispersion-less fermions, randomly coupled to N flavors of bosons, each with a finite mass m 0 .The dynamics of the model on the Matsubara axis is described by the Lagrangian where {i, j} are fermion flavor indices, {α} labels the bosons, and µ is the chemical potential.We have kept the spin indices implicit.In an open system µ is a free (input) parameter, while in a closed system its value is set by the fermionic density per flavor ν ≡ c † i c i .The Yukawa fermion-boson coupling is assumed to be random: We assume ω 0 to be positive.We have chosen the Yukawa coupling to be imaginary, such that the effective interaction in the Cooper channel is repulsive.[39,40] The model has an exact particle-hole symmetry, under which µ → −µ.For definiteness we set µ > 0. Previous studies have focused on the system at the half filling ν = 1/2, in which case µ = 0.
The model has three energy scales: the bare mass of a boson m 0 , the strength of the Yukawa coupling ω 0 , and the chemical potential µ.We will focus on the "weakcoupling limit" ω 0 m 0 .We will see that in this limit there are only two relevant energies, µ and ω F = ω 3 0 /m 2 0 .We emphasize that already at weak coupling, the system behavior at low energies is highly non-perturbative and includes self-tuned criticality and NFL.We work at T = 0 and take both M and N as large numbers, but keep the ratio N/M is a parameter.
At the bare level bosons are gapped, and fermions are free dispersionless quasiparticles.Our goal is to find the fully dressed bosonic and fermionic propagators We extended results of earlier analysis at half-filling [39][40][41] to µ = 0 and found that for M, N 1 the fermionic and bosonic self-energies are expressed selfconsistently via the Schwinger-Dyson equations where ω ≡ dω/(2π).
We first show that the system behavior is qualitatively different at larger µ and at smaller µ, and then consider the phase transition between the two phases by tuning µ(ν) in an open (closed) system.Incompressible gapped phase at large µ.
The point of departure for the analysis at large µ is the observation that within a direct perturbative expansion the bosonic polarization because the poles of the integrand are in the same frequency half-plane.Using bare D(iΩ) = 1/(Ω 2 + m 2 0 ) we obtain for the fermionic self-energy Substituting this into (2), we find G(iω) = 1/(iω + (µ − µ * )), where µ * = ω F /2.The self-energy comes from lowenergy fermions and remains the same if we compute it self-consistently.Similarly, Π(iΩ) still vanishes if we reevaluate it with dressed fermionic Green's functions [57].This self-consistent approach is valid as long as fermions are gapped, i.e., µ > µ * .At smaller µ, such solution does not exist, as one can easily verify.Because fermionic energies are all negative, the filling ν = 1 independent on µ > µ * , hence this phase is incompressible (the compressibility dν/dµ = 0).

NFL phase at smaller µ.
At half-filling (ν = 1/2, µ = 0), previous studies have found that Π(iΩ) + m 2 0 ∝ |Ω| 1−2x0 and Σ(iω)+µ ∝ i|ω| x0 sgn ω, where x 0 is a function of N/M , ranging from x 0 = 1/2 − (M/2πN ) 1/2 at N/M → ∞ to x 0 = 0 at N/M → 0 (x 0 = 0.16 for N = M , see Eq. ( 6) below).This NFL behavior holds at small frequencies for any m 0 and any non-zero ω F .Note that Π(0) = −m 2 0 , i.e., the dressed bosonic mass vanishes.This implies that the system self-tunes to quantum critical regime, despite that the bare mass is large compared to the strength of the interaction (m 0 ω 0 ).For a nonzero µ, we find that bosons remain massless and fermions retain NFL behavoir, but fermionic selfenergy becomes an asymmetric function of ω.Specifi-cally, for Ω, ω ω F , where α parametrizes spectral asymmetry [54] and ω f is the NFL energy scale, below which |Σ(iω)| < ω.Altogether we have four dimensionless parameters: x, α, β, and ω f /ω F .Substituting these forms into Eq.( 2) and matching the power-law parts Σ and Π, we obtain two equations: (see Ref. 58) These relations are exact as relevant fermionic and bosonic frequencies in (2) are comparable to external ω, Ω, which we set to be much smaller than ω f .Note that the matching the real and the imaginary parts of Σ(iω) gives the same equation.Physical values of the exponent x in (6, 7) are x ≤ 1/2.A larger x would lead to negative βω f , which violates the unitarity of the theory.For At half-filling, α = 0 and x = x 0 is a function of M/N (x 0 ≈ 0.16 at M/N = 1).The two other conditions are Σ(0) = −µ and Π(0) = −m 2 0 .Using Eqs. ( 2) we obtain Substituting Σ(iω) and Π(iΩ) from ( 5) we obtain The functions F 1,2 (x) are regular O(1) functions of the argument, and We present them in Ref. 58.The relations (9) are not exact, because relevant ω in ( 8) are of order ω f .For these ω, Eqs. ( 5) are valid up to corrections O(1), as the forms of Σ(iω) and Π(iΩ) change at ω, Ω > ω f : the bosonic self-energy gradually decreases and the fermionic Σ(ω) acquires a Fermi liquid form.[59] Nevertheless, we find that the relations F 1 (1/2) = 1 and F 2 (1/2) = 1/2 are actually exact [58], i.e., at x = 1/2, µ = µ * = ω F /2.While α cannot be universally expressed via the chemical potential, it can be exactly expressed via the density ν.Using the Luttinger relation between ν and properly regularized G(iω)dω, we obtain (see Ref. [58])  5) for the self-energies for all frequencies.The vertical dashed line in the lower panel represents the incompressible phase.We set N = M , in which case x0 = 0.16.The solid line actually has a minute dip at ν ≈ 0.98 (see [58]).It is not present in the full numerical solution and likely is an artifact of using Eq. ( 5) at all energies.
The same relation holds for the complex SYK model [37,54].From Eqs. ( 10) and (7) we see that as x → 1/2 and α → 1, the filling ν approaches 1 and ω f tends to zero.This implies that the range of NFL behavior vanishes at ν → 1.
Combining Eqs.(6, 9 ,10), we obtain α and µ as functions of the filling ν.We plot these two functions for M = N in Fig. 1.We see that both α and µ are nonmonotonic functions of ν, and there is a range of ν where the compressibility dν/dµ is negative.The relation α(ν) is exact, the other one, µ(ν), is approximate, as to get it we used Eq. ( 9).To verify that the nonmonotonic behavior of µ is not the artifact of our approximation, we iteratively solved the nonlinear integral equations (2) for Σ(iω) and Π(iΩ) for all frequencies, using the analytical power-law forms in (5) as an input.We show the numerical results [60] for µ(ν) for M = N in Fig. 1.We see that the non-monotonic behavior persists.

Quantum phase transition.
The existence of a range of densities, where ∂ν/∂µ is negative, implies that the NFL solution is unstable, and the transition between the NFL and the insulating state must be first order.In an open system, there is a discontinuous transition to the incompressible phase [61] at some critical µ c .In a closed system, there is a phase coexistence region, in which the system displays simultaneously gap features from the incompressible state and NFL features at small frequencies (see Fig. 3(a)).This resembles the "gap filling" behavior near a Mott transition at T = 0. [53] At N M the range, where µ(ν) is non-monotonic, shrinks (this follows from Eq. ( 6)), and the transition becomes weakly first-order.We show this behavior in Fig. 2 for N/M = 60.We expect that at N/M → ∞ the transition becomes second order at ν = 1 (and µ = ω F /2, x = 1/2, α = 1).In this case, there exists a quantumcritical point that separates a gapless NFL phase (which is by itself quantum critical) and an incompressible, insulating phase.This transition has an unconventional fea-ture on its own: the peak in the bosonic D(Ω) at Ω = m 0 is present on both sides of the transition.In addition, a nonzero bosonic spectral weight builds up at small frequencies in the NFL phase and progressively takes the spectral weight from the peak at m 0 .The behavior of the fermionic spectral function is more conventional: the gap in the fermionic spectral function µ − ω F /2 vanishes at the transition and an asymmetric spectral weight builds in the NFL phase.We show the behavior of the spectral functions in Fig. 3(b) and present more details in Ref. [58].

Summary.
In this communication we analyzed the behavior of M flavors of fermions, randomly interacting with N flavors of massive bosons (the Y-SYK model) away from half-filling.We showed that the system can be in one of the two phases -a NFL phase with asymmetrically broadened spectral weight, and an incompressible gapped phase.We studied the quantum phase transition between these two phases upon the variation of fermionic density.We showed by analytical and numerical calculations that the transition is in general first-order, but becomes second-order in the limit N/M → ∞.In the case of the first-order transition, there is a gap filling behavior in the transition region in both fermionic and bosonic sectors.For the second order transition, fermionic gap closes at the transition, but the bosonic spectral function still displays a gap filling behavior.
Recent numerical studies [55,56] of the complex SYK model also indicated that the system undergoes a firstorder transition upon varying the density.In distinction to our analysis, there the NFL exponent x is fixed, while in our case it varies with the filling.The possibility of second-order transition at N/M → ∞ was not addressed in these numerical studies.
We conclude by listing several open questions.First, in our analysis we focused on the case T = 0.It is possible that at a finite T the first-order transition extends to a line, which terminates at a classical critical point, like in a water-vapor phase diagram.Second, we analyzed the two-point Green's functions.It will be interesting to examine the behavior of four-point functions, possibly using the conformal reparametrization symmetry of the low-energy theory.This will shed light on the issue of the strength of superconducting and charge fluctuations.Third, we focused on the weak coupling case, ω 0 m 0 .At strong coupling, the analysis becomes more involved, even though the large M, N limit still guarantees the validity of the self-consistent Schwinger-Dyson equations.It has been pointed out that purely bosonic SYK-like models may exhibit glassy behavior at low temperatures [54,62,63].It would be interesting to see if that happens at strong coupling in Y-SYK model.and Betty Moore Foundation's EPiQS Initiative through Grant GBMF4302 and GBMF8686, and at the Aspen Center for Physics, supported by NSF PHY-1066293.AVC is supported by the NSF DMR-1834856.YW is supported by startup funds at University of Florida.

FIG. 2 .FIG. 3 .
FIG.2.Same as in Fig.1, but for N = 60M .In this case x0 = 0.45. ) FIG.1.Upper panel: the dependence of the spectral asymmetry parameter α on the filling ν.Lower panel: the qualitative (solid line) and numerical (red dots) dependence of the chemical potential µ on the filling ν.The solid line was obtained by using the low-energy form (