Migdal-Eliashberg and SUS- Y squared-SYK

The gist The note addresses a number of subtle issues pertaining to the long-standing problem of strong phonon-like fermion-boson coupling, contrasting it against various (non-)supersymmetric variants of the Yukawa-Sachdev-Ye-Kitaev model and commenting on holographic aspects of fermion pairing in such models.

How it works

The paper discusses the customary Migdal-Eliashberg approximation in the pertinent Schwinger-Dyson gap equation and its solutions, contrasting it against various (non-)supersymmetric variants of the Yukawa-Sachdev-Ye-Kitaev model and commenting on pseudo-holographic aspects of fermion pairing in such models The central issue at stake is a competition between a possible nonFermi liquid (NFL) behavior and onset of superconductivity Built-in randomness of the coupling parameters in both models is often considered to be crucially important for demonstrating the NFL behaviors that have been found to agree with the experimental observations in the cuprates and other documented ’strange metals’ From the formal standpoint, ensemble averaging can greatly simplify matters by selecting the so-called ’melonic’ graphs as the dominant class of Feynman diagrams in the large-N limit In the condensed matter context, this selection corresponds to the celebrated Migdal-Eliashberg (ME) approximation in the underlying Schwinger-Dyson (SD) equations which neglects vertex corrections The ME approximation improves on the basic Bardeen-CooperSchrieffer (BCS) theory by incorporating possible electron mass renormalization and phonon retardation effects due to the electron polarization

The SD Equations

The matrix-valued (Nambu) fermion selfenergy is comprised of the normal-state Σ(ω, k) and pairing Φ(ω, k) components defined by equation (1) Together with the boson polarization Π(ω, k) it obeys the standard SD equations as shown in equations (2) and (3) The free energy difference between the normal and paired states (condensation energy) is computed as in equation (6) In the case of multiple solutions to Eqs.(1-5), comparing their free energies provides the way of identifying the most stable one - hence, the first (or mostly likely) one to develop The previous studies of the electron-phonon systems revealed a spurious instability at moderate (bare) electron-phonon couplings, at which point the effective coupling diverges while the phonon spectrum flattens out As an alternate scenario, it was observed that at sufficiently strong couplings the electronic specific heat turns negative, thus signaling the onset of intrinsically non-equilibrium behavior in the electron-phonon system

ME Approximation and Spin Chains

Under the conditions of applicability of the ME approximation a typical value of the ratio ω/ξk in phonon emission/absorption is of order vs/vF ≪ 1 for acoustic phonons and ωD/ϵF ≪ 1 (except for the extreme lowdensity systems) for their optical counterparts For fermions with a non-flat spatial dispersion and an extended Fermi surface, applying the ME argument about the smallness of vertex corrections, Λ(ω, ω′; k, k′) = g ≈ const facilitates the momentum integration in Eqs.(2,3) One then obtains the (renormalized) electron dispersion in the denominators of Eqs.(2,3), alongside the linear dependence on the Fermi surface density of states (DOS) νF, thus arriving at the coupled equations as in equations (7) and (8) Solutions to the SD equations (7,8) can be elegantly viewed as the minimal energy configurations of a 1d chain of (normalized) classical spins Sn = 1/∆(ωn)[iωn + Σ(ωn), ReΦ(ωn), ImΦ(ωn)] labeled by the site number n which corresponds to the Matsubara frequency ωn = 2πT(n + 1/2) The effective spin-chain Hamiltonian H = X BnSn - 1/2 X nm JnmSnSm is equivalent to the free energy (6), featuring the sitedependent magnetic field Bn = zωn and ferromagnetic exchange coupling Jnm = d(ωn - ωm)

Ultra-local Limit and Linearized Gap Equation

In the flat-band limit of ξk → const, the aforementioned ’radial’ integration in the momentum space becomes trivial, resulting in the different (greater by one) power of the ∆(ω) factors in the denominators of Eqs.(7,8) Correspondingly, the effective spin representation does not naturally emerge The onset of Cooper pairing in the NFL normal state can also be studied by linearizing Eq.(19) for the gap function, thus reducing it to some eigen-function (linear) integral equation Seeking algebraic solutions in the form Φ(ω) ∼ ω η-1, one then obtains a quadratic equation for the exponent with roots η± = 1 - 1/2γ ± i r gγ - 1/4 γ squared At small frequencies and/or temperatures one expects that the solution of the original integral equation (30) approaches a constant Further imposing the boundary conditions that force the gap function Φ0(ω) to vanish at the UV cut-off ωD and level off at ω <∼ Φ(0) results in its (approximate) expression as in equation (33)

SYK Model and SUSY Variants

The Hamiltonian of the original real/complex SY Kq model is given by the sum over equally-weighted products of an even number q of the Majorana [2] or Dirac [15] fermions At large N, summation of the dominant melonic graphs yields a self-energy Σ(τ) = qJ 2G q-1(τ) which, in turn, gives rise to the Luttinger-Ward (LW) (or ′G − Σ') potential Fsyk[G, Σ] = ln P f(∂τ − Σ) + 1/2 Z τ(J 2G q(τ) − Σ(τ)G(τ)) The non-Gaussian (quartic, etc.) fermion couplings can be alternatively described in terms of an auxiliary phonon-like boson field which mediates the interactions between Majorana/Dirac fermions, thus resulting in the real/complex YSYK model In the normal state, the fermion self-energy and boson polarization Σ(τ) = g squared qGq-1(τ)Dp(τ), Π(τ) = g squared pDp-1(τ)Gq(τ) determine the generalized LW (or ′G/D − Σ/Π') functional Fysyk[G, Σ, D, Π] as in equation (43) In the normal state, by putting Φ = 0 in Eq.(21) and applying the Ward identity one finds that the frequency integral in (21) vanishes identically if the propagator G(ω) has a pure pole structure

Pairing Ladder

Much of the SYK studies centered around computing four-point functions - both, ordinary retarded and out-oftime-order (OTOC) - in the ladder approximation The OTOC functions would often be utilized as markers and quantifiers of a chaotic behavior Diagrammatically, the sum over the ladder diagrams yields solutions to the integral eigenvalue equation Z K(1, 2; 3, 4)Ψη(3, 4) = λ(∆, η)Ψη(1, 2) for the bi-local operator given by the ladder kernel K(1, 2; 3, 4) The eigen-vectors of the kernel Eq.(64) can be divided onto even and odd ones under a permutation of their arguments (Ψ+/− η(12) = ±Ψ+/− η(21)), and so they can be sought out in the algebraic form as in equation (65) The possibility of singling out the ladder diagrams is justified by their dominance at N >> 1 and is consistent with the ME approximation

Holographic Mirages

The zero-spin boson with ηb = 2 corresponds to the (pseudo-)Goldstone soft mode which breaks reparametrization invariance, both, spontaneously and explicitly In the IR limit, it is described by the ’gravitational’ Schwarzian action which is native to the SYK model, including its SUSY variants Despite the popular claim, though, the underlying AdS2/CF T1 relationship does not quite rise to the level of genuine holographic correspondence Instead of providing an iron-clad example of genuine inter-dimensional correspondence, it then constitutes a mere example of ’Hall-o-graphy’, as per the definition in Refs.

1. Bold header: Improved Non-Fermi Liquid (NFL) Phase Prediction

This AI system can predict whether an observation of strange metallic NFL features in a normal state is a precursor of potentially higher critical temperature, magnetic field, current, etc., by analyzing the competition between non-Fermi liquid behavior and the onset of superconductivity in systems modeled by SYK and YSYK.

2. Bold header: Enhanced Phonon Coupling Analysis

The system can assess the limits of applicability for the Migdal-Eliashberg (ME) approximation by comparing its results against scenarios where disorder averaging in SYK may not be that important for achieving melonic dominance, or when the breakdown of the ME approximation in the normal state may occur even at moderate couplings due to intrinsic (bi-)polaronic effects.

3. Bold header: Superconducting Gap Function Determination

The system can determine if the "non-monotonic solutions with n additional nodes within the interval 0 < ω < ωD for oscillating gap functions are spurious and not readily realizable," based on comparisons between the results derived from linearization of Eq.(19) and the analysis of Eq.(63).

4. Bold header: Holographic Correspondence Verification

The AI can test whether a proposed holographic dual relationship is genuine inter-dimensional correspondence by checking if it reduce[s] both systems to the same mathematical description, specifically by contrasting the AdS2/CF T1 duality against the requirement that such a correspondence should involve pairs of systems operating in genuinely different dimensions.

5. Bold header: Critical Coupling Threshold Identification

The system can identify whether NFL pairing emerges only if the coupling exceeds a certain threshold, by analyzing the conditions under which the solution of Eq.(30) approaches a constant versus when it exhibits oscillatory behavior.

This note addresses a number of subtle issues pertaining to the long-standing problem of strong phonon-like fermion-boson coupling. Among the central topics are the customary Migdal-Eliashberg approximation in the pertinent Schwinger-Dyson gap equation and its solutions. The previously gained insight is assessed by contrasting it against the various (non-)supersymmetric variants of the Yukawa-Sachdev-Ye-Kitaev model. Also, some previously discussed (pseudo-)holographic aspects of fermion pairing in such models are commented upon.