D.V. Khveshchenko

Department of Physics and Astronomy, University of North Carolina, Chapel Hill, NC 27599, U. S. A.

Received 2 June 2022; revised 14 June 2022; accepted 14 June 2022

This paper discusses the examples of 0 + 1-dimensional Liouvillean dynamics instigated by the various deformations of the Sachdev–Ye–Kitaev (SYK) model. In reference to such deformations the main focus is on the regions of parameter space where the competing SYK couplings are of a comparable strength and cannot be treated as each other’s perturbations in the vicinity of the conformal fixed points corresponding to the pure SYKq models with different values of q. Crossovers between such fixed points (‘SYK transits’) can be efficiently studied in the equivalent framework of single-particle quantum mechanics.

Keywords: SYK model, Liouvillean quantum mechanics, Lyapounov exponent

1. The rise of SYK

The glorious rise of the celebrated SYK model [18] into one of the central themes in modern interdisciplinary theoretical studies was due to a rare confluence of such precious properties as its elegant solubility, maximally chaotic behaviour, asymptotic conformal symmetry, and more. The numerous in-depth analyses of the SYK model revealed a number of important connections between such seemingly disjoint subjects as random matrices, quantum black holes, disordered quantum dots, and, possibly, strange metallic behaviours in the various condensed matter systems.

One of these novel connections may have already contributed towards a resolution of the longstanding black hole information paradox by demonstrating that the  properly (re)defined Hawking radiation entropy can be unitary, following the previously predicted Page curve [9, 10].

In the condensed matter context, the SYK model has served as a powerful inspiration for a great many proposed non-Fermi-liquid (NFL) scenarios [1126]. However, the very existence of numerous plausible explanations of, e.g. the ubiquitous linear temperature dependence of resistivity in bad metals [2743], may seem to suggest that its ultimate explanation is yet to be found.

Nevertheless, alongside the renewed interest in hydrodynamics inspired by the holographic ideas, the SYK scenaria have been particularly important for pursuing the ad hoc field of ‘bottom-up’ (a.k.a. ‘non-AdS/non-CFT’) holography which purports to describe a variety of (allegedly) strongly correlated condensed matter systems [4450]. Indeed, with the once abundant and defiantly upbeat claims of ‘explaining’ high-Tc materials, heavy fermions, graphene, etc. by virtue of some uncontrolled calculations in the conveniently chosen (and/or previously studied) classical gravity theories all but gone, the SYK model has remained a rather unique theoretical playground for obtaining rigorous results.

In that regard, the SYK model would be often referred to as a  genuine example of low-dimensional holographic correspondence – even despite the  fact that, both being effectively one-dimensional, the low-energy sector of SYK and its dual (formally, two-dimensional) Jackiw–Teitelboim (JT) gravity present a form of equivalence between different realizations of the  quantized co-adjoint orbit of the (chiral) Virasoro group. Such equivalence does not quite rise to the level of full-fledged holographic duality, as the  JT bulk dual is non-dynamical and determined by the  boundary degrees of freedom. By contrast, in order to qualify as a true holographic scenario the bulk theory would have to have some non-trivial bulk dynamics that gets quenched and turns classical only in a certain (‘large-N’) limit [4450].

Moreover, similar remarks can also be made about the  (historically, somewhat less extensively discussed) correspondence between the  3d gravity with BTZ-like black hole backgrounds and the  various (KdV and alike) families of solvable 1 + 1-dimensional systems (see, e.g. Ref. [51] and references therein).

2. The SYK deformations

Since the beginning of the SYK era there have been attempts to explore deviations from the  original SYK4 model in order to assess the generality (or, conversely, uniqueness) of the behaviour that it represents. In particular, there has been much discussion of the conjectured NFL–Fermi liquid (FL) transition in the hybrid SYK4SYK2 model [18, 1126].

A renormalization flow between the  two fixed points has been mostly studied by means of perturbation theory operating in terms of the propagator G(τ1, τ2) of N  ≫ 1 spaceless Majorana fermions [1126]. In the conformal limit of the generalized SYKq of order q ≥ 4 the latter exhibits the fermion dimension Δ = 1/q, thus making the perturbation proportional to Gq/2 strongly relevant (dimension one) in the near vicinity of the  UV SYKq fixed point. Conversely, the formerly leading term Gq becomes strongly irrelevant (dimension four) near the IR fixed point of SYKq/2. A unique feature of the q = 4 case, though, is that the transition occurs not between two different NFLs but the SYK4 NFL and the disordered FL.

Notably, in the course of crossing over between the  different fixed-point regimes the  value of q of the dominant term plays a role akin to that of the central charge in 2d conformal field theories.

As far as the potential physical applications are concerned, some of the previous analyses [1126, 5255] suggest that the  putative phase transition may take place at critical couplings vanishing as powers of 1/N – which value would be practically indistinguishable from zero in a macroscopic system – while others yield values that remain finite in the  ∞ limit.

The common approach to a  SYK-type model starts out by integrating the fermions out, thereby arriving at the action in terms of the bi-local fields G(τ1, τ2) and the corresponding self-energy Σ(τ1, τ2),

S= N 2 Trln τ Σ + N 2  d τ 1  d τ 2 τ 1 , τ 2 G τ 1 , τ 2 F G τ 1 , τ 2 }, 1

where the functional F[G] results from averaging over the  Gaussian-correlated random amplitudes of all-to-all q-body entanglement. Moreover, such entangling couplings can be made non-uniform, thus introducing a  notion of spatial dimensions and further extending the class of attainable models to include those with ‘distance’-dependent entanglement [56, 57].

Solving for the self-energy, the Schwinger–Dyson equation derived from (1) takes the form

τ 1 G τ 1 , τ 2  d τ 3 τ 1 δF δG τ 3 G τ 3 , τ 2 =δ τ 1 τ 2 . 2

In the original SYKq model with F(G) = J2Gq Eq. (2) remains invariant under any diffeomorphisms τ → f(τ) of the thermodynamic time variable τ subject to the boundary condition f(τ + β) = f(τ) + β as long as the derivative term is neglected and provided that G and Σ transform as

G τ 1 , τ 2 G f = f τ 1 f τ 2 Δ G f τ 1 ,f τ 2 , Σ τ 1 , τ 2 Σ f = f τ 1 f τ 2 1Δ Σ f τ 1 ,f τ 2 . 3

At ≫ 1 a  representative power-law solution to Eq. (2) reads G0(τ1, τ2) ~ sgnτ/() (hereafter τ = τ1 – τ2).

Choosing a  particular mean-field solution reduces the  invariance under arbitrary diffeomorphisms down to the subgroup of the Mobius transformations SL(2, R). Correspondingly, a gradient expansion of the logarithm in Eq. (1yields the  (approximately) local effective action which describes the finite temperature dynamics of the reparametrization mode [5879]

S 0 f =O 1 N J q 2  Sch tan πf β ,τ dτ, 4

where Sch stands for the  Schwarzian derivative Sch f,x = f f 3 2 f f 2 obeying the  differential ‘chain rule’ Sch{F(f),  x}  =  Sch{F(f),  f} × ×f2 +  Sch{fx} and operating on the  manifold of (nearly) degenerate states related by virtue of the transformations (3).

3. Liouvillean quantum mechanics

Under the customary parametrization f′ (τ) = eϕ(τ) the  Schwarzian action (4) assumes the  (pseudo-) free form S0(ϕ) ∼   dτ(ϕ′)2. In the process of averaging the products of propagators

G f τ 1 , τ 2 G f τ 2p1 , τ 2p =  = Dϕ e S 0 ϕ i=1 p e Δ ϕ τ 2i1 +ϕ τ 2i τ 2i1 τ 2i  dτ e ϕ 2Δ 5

over the fluctuations of ϕ, the action S0(ϕ) gets augmented by the  Liouville term ΔS2(ϕ)  =  h2dτe2ϕ(τ) with h∼  J. Technically, upon promoting the  denominator in (5) to the exponent with the help of some auxiliary integration a la Feynman the overall effective potential acquires a piece-wise Liouville term acting during the time intervals between 2p consecutive insertions of the operator eiΔϕ [81, 82].

The resulting action S0 + ΔS2 can then be quantized by considering the corresponding (rescaled) Schroedinger equation [81, 82]

1 2 ϕ 2 + h 2 e 2ϕ ψ=Eψ, 6

the scattering states of which ψk(z)  ∼ K2ik(2 z ) (here z  =  λeϕ) belong to the  continuum with the  spectrum Ek  =  k2 and the density of states ρ E sinh2π 2NE/J [5879, 81, 82].

These exact expressions can be used to compute the matrix elements 0 e Δϕ k exactly. Such calculations reveal the universal limit of an arbitrary power of Gf averaged over the soft mode fluctuations in the late-time, τ > N/J – or, at finite temperatures, in the strong coupling, / ≳ 1 – regime [81, 82],

G f p 1 (Jτ) 3/2 . 7

This behaviour is markedly different from the (non-universal) mean-field one G 0 p 1/ τ 2p/q at shorter times (τ < N/J) or weak couplings (/ ≲ 1).

In that regard, the presence of the exponential term ΔS2(ϕ) in the overall effective action is instrumental. In its absence the Gaussian fluctuations of the  field ϕ governed by S0(ϕ) would have caused non-algebraic decay, thus being unable to deliver the universal power-law (7). In fact, such a behaviour could have never emerged out of the  purely Gaussian ϕ fluctuations even if the  correlator ⟨ϕ(τ)ϕ(0)⟩ were logarithmic, as ln G f p would still depend on p and Δ (in both cases, quadratically).

The effective action might also include the various intrinsically non-local terms

Δ S n = h n  d τ 1  d τ 2 f τ 1 f τ 2 f τ 1 f τ 2 2 Δ n , 8

that can dominate over (4) for Δn ≤ 3/2. In the previous analyses, such terms would be routinely substituted with the local operators ΔSn = hn ∫ dτ eΔ(τ) thus further modifying the  equivalent quantum mechanical Hamiltonian in (6).

4. Bi-quadratic SYK deformation

This important example of the deformed SYK model has been extensively discussed in the context of random tunnelling between two different SYK systems. For example, it arises in such, at first sight, unrelated fields as theoretical cosmology (‘traversable wormhole’) [5879] and coupled quantum dots [8393].

In most analyses, the  perturbed propagator would be taken in the form (3) of a ‘gauge-transformed’ mean-field solution G0, thereby accounting for the ‘soft’ reparametrization mode f(τ) while ignoring any potential changes to the  mean-field background field configuration itself.

In particular, adding the  SYK2 (‘tunnelling’) term with the  amplitude Γ replaces the  Liouville potential in the SYK4 action (written in the Euclidean signature) with the Morse-type one [9497]

S= dτ 1 2J ϕ 2 +J e 2ϕ + Γ 2 J e ϕ + 1 β 2 J e 4ϕ . 9

The corresponding Schroedinger equation with the (properly rescaled) Hamiltonian

H= 1 2 ϕ 2 + e 2ϕ +λ e ϕ + λ e 4ϕ , 10

that can be solved exactly in terms of the  wave functions (here z  =  2λeϕ) ψk(z) ∼ e–ϕ/2Wλ,ik(z) with the continuous spectrum Ek = k2 + 1/4 + λ2 if the last – subdominant at low temperatures (or large negative ϕ)  –  term in (10) can be dropped [9498].

Besides, for λ < 0 the Hamiltonian (10) appears to possess a finite number of bound states

ψ n z z λn1/2z/2 L n 2λ2n1 z 11

at the discrete energies En = –(λ – n + 1/2)2, n = 0, , [λ – 1/2]. Near its minimum this spectrum can be approximated by the oscillator one.

A somewhat different path leading up to the Morse-type action (9) was taken in Ref. [98]. The  effect of the  tunnelling term with Δ1  =  1/2 was argued to be two-fold: first, it contributes to (and/or refines) the  purely Schwarzian (or ‘hard’ mode) saddle-point solution and, second, controls the pseudo-Goldstone (or ‘soft’ mode) fluctuations. These roles would be separately played by the ‘longitudinal’ (or radial, eξ = 1 – f', in the holographically dual JT picture) and ‘transverse’ (or angular, ϕ) fluctuations, respectively. The former was argued to be strongly non-Gaussian and the effect of such fluctuations was claimed in Ref. [98] to strengthen (somewhat unexpectedly) the SYK4 conformal mean-field behaviour over a broader range of parameters.

More specifically, the strong coupling Schwarzian regime was argued to sustain the SYK2 perturbation at all couplings γ ≡ Γ/J below γc ∼ 1/N while at its higher values the  propagator was found to crossover to the q = 2 FL fixed point. This was argued to be suggestive of a zero-temperature phase transition taking place at γc, rather than at a much larger value of order 1/N1/2, as per the naive estimate. Such parametric reduction of γc was claimed to manifest a stabilizing effect of the SYK2 coupling on the  mean-field conformal solution against the Schwarzian fluctuations due to the formation of a  polaron-like non-perturbative field configuration.

Correspondingly, the earlier perturbative analysis by the same authors revealed that a weak SYK2 coupling does not alter the Schwarzian asymptotic (7) up to the values of γ of order γc [55].

Such observations appear to be generally consistent with those of Refs. [99, 100] that conjectured the existence of a chaotic-integrable transition in the SYKq–SYK2 model at finite temperatures. Above the  transition temperature the  system was found to behave chaotically while below it the chaos-related Lyapunov exponent (see below) dropped to zero, thus hinting at the FL nature of the underlying ground state.

On the  technical side, upon, first, introducing two Lagrange multipliers λ and Λ and, then, voluntarily relaxing the corresponding constraints by fixing their mean-field values, Ref. [98] arrived at the effective action

S= dτ 1 2 ϕ 2 +Λ e 2ϕ f +λ e ϕ χ 1 (βJ) 2 e 4ϕ 1 2  d τ 1 d τ 2 χ τ χ τ f τ 1 f τ 2 . 12

In this (perhaps, somewhat excessive) parametrization, the functional integration about the mean-field SYK4 fixed point factorizes into, first, taking a  quantum mechanical expectation value over the exact ground state ψ0(ξ) of the Hamiltonian (10) and, then, additionally averaging over the Gaussian (perturbative) ϕ fluctuations. In Ref [98], neither mechanism was found to have any profound effect on the correlators, though.

In particular, an arbitrary power of the mean-field propagator would still retain its bare mean-field form provided that the  ϕ-fluctuations were controlled by a  large parameter λ. Likewise, averaging over the  ground state of (10) adds the  square of a  non-singular expectation value ⟨0|epΔϕ|0⟩ =  dϕepΔϕψ20(ϕ) which does not give rise to any decaying power-law factor either. In that sense, the  largely negligible effect of, both, the  Gaussian fluctuations and the  ground state averaging may indeed be viewed as increased stability of the mean-field regime in the presence of even a small SYK2 coupling.

It should be noted, though, that under the assumption of λ < 0 the Morse potential in (10) appears to differ from that of Refs. [5254] which is strictly repulsive, monotonic (λ > 0), and lacks any bound states. It might also be concerning that if the potential in (10) were to support any bound states with En  <  0, then the  fluctuation-averaged two-point correlator ⟨ Gf(τ)⟩  =  Σne–EnτN(En) would be receiving  –  on top of the  universal term (7) that stems from the continuum of scattering states with Ek  >  0  – a  nonunitary (exponential) contribution, the potential divergence of which could only be arrested by the  squared matrix element N(En < 0) = |⟨0|eΔϕ|n|2.

Interestingly, for J  =  Γ the  aforementioned monotonic and non-monotonic Morse potentials represent two super-partners fitting into one super-symmetric pair W±(ϕ)  =  V2  ±  dV/dϕ with V(ϕ) ∝ eϕ. The ground state of the binding potential then takes the form ψ0(ϕ) ∝ exp(– ∫ V dϕ).

Conceivably, the  effective action S(ϕ) may develop other interesting regimes at the points of still higher symmetry. One such example would be provided by the Hulten potential

W ϕ =λ e ϕ 1 e ϕ , 13

first three terms of the  expansion of which in powers of eϕ coincide with the  ‘hyper-symmetric’ (or ‘tri-critical’) point J = Γ = 1/β in (9). Also, the 1/ϕ-behaviour at a small negative ϕ would be similar to that in the Coulomb potential, although the  potential (13) features only a  finite number ([λ]) of bound states at En = –[(λ2 – n2)/2λn)]2.

5. Large q limit

An alternate approach to the SYK models exploits the large-q approximation, where the propagator is sought out in the form

G τ = 1 2 sgnτ 1+ 2 q g τ + . 14

Higher order corrections in 1/q can also be evaluated, albeit at the  increasingly prohibitive costs [5879].

The action in the path integral over the field g then takes the form

S g = N q 2  d τ 1  d τ 2 1 2 dg  d τ 1 dg  d τ 2 +W g , 15

with the corresponding equation of motion

τ 2 g= W g q . 16

Formally solving (16) one obtains the classical trajectory

τ= g 0 0 dg W g 0 W g , 17

with the use of which thermodynamics of the system can be studied by putting τ  =  β/2 [5254]. In particular, the turning point g0 < 0 of the potential can be directly related to the  mean-field energy [54]

E= N 4 q 2 βW g 0 2 3/2 g 0 0  dg W 0 W g . 18

As already mentioned, one possible generalization of the  bi-quadratic (Schwarzian plus tunnelling) q = 4 action to the larger values of q is provided by the SYKq–SYKq/2 functional

F G = 2 q J 2 q 2 G q τ 1 , τ 2 + 2 q/2 Γ 2 q 2 G q/2 τ 1 , τ 2 . 19

The corresponding effective potential

W g = J 2 e 2g + Γ 2 e g 20

allows for the explicit saddle point solution [5254]

g τ =ln 1+ J 2 +4 Γ 2 τ+ Γ 2 τ 2 21

that gives rise to the mean-field propagator

G 0 τ = 1 2 sgnτ 1+ J 2 +4 Γ 2 τ+ Γ 2 τ 2 2/q . 22

For future reference, the  final-temperature counterpart of (21) reads g τ =ln v 2 J 2 / β 2 + Γ 4 × cos 2vτ/βv + Γ 2 β 2 /2 v 2 , where the  parameter v is to be determined from the relation 2 v 2 = Γ 2 β 2 +cosv J 2 v 2 β 2 + Γ 4 β 4 and becomes v = 1–O(1/βJ) for Γ = 0 [5254].

It is worth noting that in the look-alike equations (10) and (20) the  field variables ϕ and g depend on the  ‘centre-of-mass’ (cf. Eq. (8)) and relative times, respectively. Also, unlike the  approximate conformal propagator G0, the  expression (22) is UV-finite and naturally regularized at τ ∼  min [1/J, 1/Γ]. Hence, by contrast with the latter, the saddle-point solution (22) remains applicable at all γ, both large and small. Therefore, the fluctuations of g(τ) describe pseudo-Goldstone excitations about the fixed ‘valley’ in the space of field configurations which no longer needs to be adjusted.

6. Quadratic fluctuations

Small fluctuations about the  mean-field solution (22) are described by the Gaussian action

S 2 = N 2 q 2  d τ 1  d τ 2 δg τ 1 2 S g 2 g 0 δg τ 2 . 23

For a potential W(g) = Σncneng these fluctuations δg would then be governed by a functionally similar kernel 2W/∂g2 = Σn cnn(n – 1)eng. Albeit similar in its appearance to the previously discussed S(ϕ), this action is bi-local and cannot be readily used for deriving the Hamiltonian and quantizing it by means of the substitution g → –i/∂g.

In contrast to the Schwarzian action (4) the δg fluctuations are scale-invariant and their strength is independent of energy or temperature, being instead controlled by the numerical parameter N/q2. For a finite q the strength of such fluctuations decreases with increasing N, yet it remains fixed in the double-scaling limit,  ∞ and N/q2 = const.

Inverting the  Hessian operator evaluated at the saddle point (21) requires one to find the Green’s function of the retarded kernel D T,τ = δ g T+ τ 2 δ g T τ 2 = 12 K 1 12 that satisfies the equation

  d τ 5  d τ 6 1 2 δ 15 δ 26 + 12 2 W g 2 56  ×56 K 1 34= δ 13 δ 24 δ 14 δ 23 . 24

Upon Fourier transforming with respect to the ‘centre-of-mass’ time variable T one can use the spectral decomposition

D T,τ = n   dω 2π e iωT ψ n τ 2 ψ n * τ 2 ω 2 ω n 2 +i0  = n e i ω n T ω n ψ n τ 2 ψ n * τ 2 25

in terms of the eigenfunctions of the equation

τ 2 + 2 W g 2 g 0 ψ n τ = ω n 2 ψ n τ . 26

By analogy with the  aforementioned averaging over the ϕ-fluctuations the Gaussian average over δg in the vicinity of the saddle point (22) produces the ‘Debye–Waller’ factor

G p τ G 0 p τ = e 2pΔδg τ = exp 2 p 2 Δ 2 D 0,τ D 0,0 . 27

Notably, this averaging is to be performed over the entire function δg, thereby making no distinction between the ‘angular’ and ‘radial’ modes.

This might be somewhat similar to, e.g. the standard weak-coupling analysis of the two-dimensional nonlinear O(N) σ-model, that seems to emphasize a  distinction between the  longitudinal and transverse fluctuations of the order parameter (one gapped and N–1 Goldstone modes, respectively). By contrast, the exact solution demonstrates no such difference as the true O(N)-symmetric spectrum consists of the N identical gapped modes.

Evaluating (20) on the classical trajectory (21) at zero temperature one finds the effective potential that asymptotically decays at large τ as ∼ 1/τ2 in both cases of large and small γ. The one-dimensional Green’s function of the resulting eigenvalue equation

τ 2 + κ τ 2 ω 2 ψ=0, 28

with κ > −1/4, can be found in the closed form

D ω τ, τ = π 2i τ τ H v 1 ω τ > H v 1 ωa  × H v 1 ωτ J v ω τ < H v 1 ω τ < J v ωa , 29

where τ> and τ< stand for the larger/smaller τ and τ′, respectively, v= 1/4+κ , and a is the UV cutoff.

For ω  =  0 (29) amounts to the  previously derived expression the finite-temperature version of which reads [5879]

D 0 x, x = 1 Vπ 1+tan x < Vπ 2 + x < × 1tan x > Vπ 2 x > , 30

where x = πτ/β and V=v+ 2 π cotπv/2 .

Expanding the Bessel functions one once again finds only a mild effect of the Gaussian fluctuations (this time around, of δg), as the ensuing reduction of the  amplitude G p τ / G 0 p τ =exp O 1 Δ 2 p 2 does not alter the mean-field exponent of the power-law decay.

7. Quadratic fluctuations in g-space

As an alternative to (26) one can formulate the eigenvalue equation in terms of the g-variable [54]

2 W 0 W g W 0 W g + 2 W g 2 g 0 ψ n = ω n 2 ψ n , 31

whereW0  =  W(g0) without the  need to explicitly solve for the classical trajectory g(τ).

However, a generally non-trivial derivative τg precludes an immediate use of the known solutions such as (11) in the case of, e.g. the Morse potential W(g). Then treating (31) as a generic second-order equation

p x x 2 ψ+q x x ψ+ EV ψ=0 32

and eliminating the linear derivative term one can convert Eq. (31) into the  standard Schroedinger equation with the potential V =V+ x 2 Q Q the wavefunction χ = ψQ with Q(x) = exp(∫dx q/2p).

Using this equation in the classically accessible domain g0 < g < 0 one can study the system’s thermodynamics. For example, in the case of the Hulten potential (13) one obtains non-trivial temperature dependences of energy E = E0 – O(J4/3β1/3) and entropy S = S0 – O(()4/3) that suggest rather peculiar thermodynamic relations.

8. Ladder eigenfunctions and chaos exponents

A chaotic behaviour may develop in the  complementary (classically prohibited) regime g < g0. One popular quantifier of chaos is provided by the out-of-time-order correlator (OTOC) given by the averaged amplitude ⟨Gf (τ1, τ3)Gf (τ2, τ4)⟩ analytically continued from the domain τ4 < τ2 < τ3 < τ1 to the complex times τ1 = β/4 – it/2, τ2 = –β/4 – it/2, τ3 = it/2 and τ4 = –β/2 + it/2.

On top of a non-exponential regular part of the zeroth order in 1/N the  OTOC function demonstrates an exponentially growing first-order correction. In the case of the SYKq–SYKq/2 model it reads

OTOC t =O 1 βJ f γ B e λ L t , 33

where f(0) = O(1). It is controlled by the Lyapunov chaos exponent determined by the  ladder eigenstate equation constructed out of the  Wightman correlators Glr(t) = G(τ = it + β/2) [5879]. Taking ωn in Eq. (26) to imaginary values ωn → iλL yields the exponentially growing ansatz D(T, t) ∼ eλLTψ(t), where the  real-time eigenfunction ψ(t) solves the equation [54]

x 2 cosθ coshx+cosθ 2 sin 2 θ (coshx+cosθ) 2 ψ= λ L β 2πv 2 ψ 34

with θ  =  tan1(v/Jβγ2). Importantly, the  potential in Eq. (34) is monotonic and its sign is opposite of that in Eq. (26).

For γ = 0 this potential is the original SYK’s one, V0(x)  =  2/cosh2x, that supports no bound states other than the ground one, ψ0(x)  ∼  1/coshx, with the eigenvalue E0 = (λLβ/2πv)2 = 1 [5879]. As has been repeatedly pointed out in the  literature, this value of the  chaotic operator growth is (almost) maximally possible, its reduction at a strong coupling (Jβ  ≫  1) being solely due to the temperature-dependent factor v,

λ L = 2π β 1O 1 βJ . 35

In the  complementary weak coupling regime ( ≪ 1) the chaotic exponent is λL  ∼ J.

In principle, the rest of the spectrum in Eq. (34) could provide for some slower growing terms. However, for γ = 0 no such terms appear as the next (single-node, hence first excited) state would be given by the function ψ1  ∼ g' with the eigenvalue E1 = 0 [54].

Also, at longer times   β the  behaviour of the OTOC function is determined by the 2-particle density of states, resulting in another universal power-law, OTOC(t) ∝ 1/t6 [81, 82].

In Ref. [98], a weak SYK2 term in (20) was found not to drastically alter the strong-coupling behaviour, except for a  reduction of the  amplitude by a  factor O(1/γ2)  <  1 in the  entire interval 1/γ  ≲ 1/N1/2. At such parameter values the  Schwarzian fluctuations were found to be suppressed, thus extending the validity of the SYK4 mean-field solution beyond the energy scale J/N all the way down to 2 at which the FL behaviour finally sets in.

In Ref. [54], the chaotic exponent of the large-q biquadratic model was computed with the use of perturbation theory about the state ψ0 for a small γ, thereby finding

λ L = 2π β 1O m  Jβ γ 2 , 1 Jβ γ 2 . 36

For comparison, Refs. [99, 100] found the  exponent λ L = 2π β 1O β 2 Γ 2 in the  SYKq – SYK2 model, suggesting the  possibility of a  finite-temperature transition for an arbitrarily small Γ.

The latter should, however, be contrasted against the result of Ref. [101] which reported λL  ∼ 1/2γ3 for γ ≫ max [1, 1/]. Such a behaviour conforms to the  generic quadratic temperature dependence of λL in the disordered FL and could indicate the absence of a genuine finite-temperature phase transition for a sufficiently large Γ.

Adding to the  list of possibilities, in Ref. [54] some non-maximal (temperature-independent and growing with the increasing integer parameter n) values of λL were reported on the basis of a numerical solution of some other (‘variable scaling’) model with W(g)  ∝ 1/(–g)n.

As far as more general potentials W(g) are concerned, the  Hulten potential (13), for one, falls somewhere in between the  ‘super-symmetric’ (γ  =  1) point of the  SYKq–SYKq/2 model and the ‘variable scaling’ one. The corresponding eigenvalue equation now reads

x 2 2 δ 1 coshx 1 coshx+δ ψ= λ L β 2πv 2 ψ, 37

where δ= 1+4 γ 2 /Jβ γ 2 . At the  super-symmetric point, where δ = 51/2/and for low temperatures (δ  ≪ 1), the potential in Eq. (37) approaches the original SYK’s V0(x) and the maximally chaotic behaviour λ L 2π βv is once again restored. In the opposite limit of δ ≫ 1, the potential flattens out and the Lyapunov exponent decreases monotonically all the way to zero. It does not vanish at any finite temperature, though, thus calling for a closer look at any scenario of a finite-temperature phase transition – or a zero-temperature one predicted to occur at a critical γc vanishing as a power of 1/N.

9. Summary

This paper discussed various generalizations of the  SYK model that lead to the  one-dimensional Liouvillean quantum mechanics. Of a particular interest are the crossovers between the different conformal fixed points where all pertinent coupling constants are likely to be of the same order. Such ‘SYK transits’ are not directly amenable to perturbation theory in the vicinity of the fixed points in question but can still be explored in the  large-q limit. To that end, one can utilize the already available – and seek out new – non-perturbative mean-field solutions akin to (22) that interpolate between the distinct conformal regimes. This way one could advance the  previous studies of the  bi-quadratic model (20) and its further extensions within a broader class of the effective potentials W(g).

In particular, this preliminary analysis finds that the  Lyapunov exponent at the  ‘super-symmetric’ point of the  model (20) remains non-zero down to the lowest temperatures. This observation may call for inspection of the earlier conclusions about the onset of the non-chaotic FL phase at a critical coupling γc which could be as weak as O(1/N1/2) or even O(1/N) [1126, 98100].

Also, further generalizations of the  standard Liovillean action related to the various analytically solvable quantum mechanical Hamiltonians might be of interest well above and beyond the original SYK context.


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D.V. Khveshchenko

Šiaurės Karolinos universitetas, Čepel Hilas, JAV