Theory of Strongly Correlated Systems: An Introduction to Sachdev-Ye-Kitaev Model

Rishabh Jha

REVIEW 2 major objections 4 minor 1 cited by

Theory of Strongly Correlated Systems: An Introduction to Sachdev-Ye-Kitaev Model

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This review establishes that the SYK model, solved through its large-N and large-q equations, reproduces strange-metal transport—linear-in-T resistivity at all temperatures for κ=1/2 and κ=1 chains—together with maximal chaos and a…

desk verdict A careful and largely correct SYK review whose transport chapter needs a thermodynamic-limit check before the strange-metal claims can be trusted. read the letter →

arxiv 2507.07195 v3 pith:HVMNX6FU submitted 2025-07-09 hep-th cond-mat.str-elquant-ph

Rishabh Jha This is my paper · ORCID

classification hep-thcond-mat.str-elquant-ph

keywords SYKmodelstrangemetalsnon-FermiliquidquantumchaosSchwinger-DysonequationsKadanoff-Baymholographicdualitylarge-Nlimit

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This review argues that the SYK model—a disordered, zero-dimensional system of N fermions with all-to-all random interactions—is a fully solvable theoretical laboratory for strange metals, the materials where quasiparticles fail and resistivity grows linearly with temperature. The paper systematically constructs both Majorana and complex-fermion versions and walks through disorder-averaged large-N equations that make thermodynamics, chaos, and transport analytically tractable. Its concrete transport claim is that SYK chains with long-range hopping exponent κ=1/2 and κ=1 exhibit robust linear-in-T DC resistivity across all temperatures, with a universal minimum resistivity ρ_min=8/(Nπ) for κ=1/2, while κ=2 gives an insulating phase. A sympathetic reader would care because this is a rare example where strange-metal phenomenology, maximal quantum chaos, and non-Fermi-liquid thermodynamics emerge from a solvable microscopic Hamiltonian rather than from model-specific fitting.

What carries the argument

The load-bearing object is the disorder-averaged bi-local Green's function $G(t,t')$ together with its conjugate self-energy $\Sigma(t,t')$, governed by the closed large-N equations $G^{-1}=G_0^{-1}-\Sigma$ and $\Sigma=J^2G^{q-1}$ (Majorana case) or $\Sigma=-J^2 G^{q/2}G(-\tau)^{q/2-1}$ (complex case). Solving these in the infrared conformal limit gives power-law Green's functions and the Schwarzian action for soft time-reparameterizations; solving them in the large-q limit gives explicit all-temperature Green's functions. The same equations, continued to real time as Kadanoff-Baym equations, describe quenches and thermalization, while transport is extracted from current-current correlations computed under deformed closed-time contours. This single machinery carries the argument from thermodynamics to chaos to the κ-dependent resistivity classification.

What would settle it

A numerical or analytic computation of DC resistivity for the same SYK chain with an intermediate hopping exponent (say κ=3/4) would settle the claimed universality: the paper's classification predicts a sharp distinction between the strange-metal chains (κ=1/2, 1) and the insulating chain (κ=2), so observing linear-in-T resistivity at κ=3/4 would refute it. Alternatively, an experiment on a material claimed to be an SYK strange metal that measures a minimum resistivity below 8/(Nπ) in the appropriate dimensionless units would contradict the bound.

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Extended reading notes

Core claim

The central claim of this review is that the SYK model—a zero-dimensional quantum system of N fermions interacting through random all-to-all couplings—is a solvable laboratory for the physics of strange metals. The paper builds the Majorana and complex-fermion variants, derives the closed large-N equations of motion for the Green's function and self-energy, solves them in the infrared conformal limit and in the large-q limit, and shows that the same framework yields equilibrium thermodynamics, maximal quantum chaos with Lyapunov exponent λ_L=2πT, and non-equilibrium transport. On the transport side, the distinctive result is a classification of SYK chains by the long-range hopping exponent κ: chains with κ=1/2 and κ=1 show linear-in-T resistivity at all temperatures, while κ=2 behaves as an insulator at low T. For κ=1/2 the resistivity reaches a universal minimum value ρ_min=8/(Nπ) for every coupling strength, making the linear-in-T behavior a property of the model family rather than of a finely tuned point.

Load-bearing premise

The transport conclusions rest on choosing the long-range hopping exponent κ: only the hand-picked values κ=1/2 and κ=1 produce the strange metal behavior, and the further step from these zero-dimensional chains to actual strange metal materials is taken on faith.

Editorial extensions

If this is right

The disorder-averaged large-N equations give closed-form thermodynamics, so free energy, entropy, and equation of state are computable without quasiparticle assumptions.
The chain transport results make linear-in-T resistivity a consequence of the Hamiltonian, not an added scaling hypothesis.
The complex SYK phase transition, with Landau-Ginzburg critical exponents, provides a zero-dimensional analog of the charged-AdS black hole transition, strengthening the holographic correspondence.
Maximal chaos with λ_L=2πT appears in the same models that give Planckian dissipation, tying the transport anomaly to information scrambling.
The quench solutions show single-dot instantaneous thermalization with respect to Green's functions while chains thermalize over finite time, giving controlled examples of equilibration in a strongly correlated system.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Editorial inference: if the κ=1/2 chain's universal minimum ρ_min=8/(Nπ) survives finite-N corrections, it could serve as a theoretical benchmark analogous to the Mott-Ioffe-Regel limit, connecting the SYK result to resistance-quantum scales in real materials.
Editorial inference: the sharp κ-dependent classification invites a direct numerical test in chains with power-law hopping; if intermediate exponents interpolate smoothly between strange metal and insulator, the claimed universality would be weakened.
Editorial inference: the same real-time machinery could be run at finite chemical potential and finite doping to ask whether linear-in-T resistivity persists away from half filling, a condition the paper does not address.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You should know this: the paper is a long, self-contained review of SYK physics, and the standard material is reproduced with genuine care. The large-N effective action, IR conformal limit, Schwarzian action, large-q all-temperature solution, and the complex SYK phase diagram all come out right. I checked the large-q differential equation and the free energy; they match the known results. The Mathematica implementations are a real plus. This is not a research claim but a teaching text, and as a teaching text it does its job well.

The soft spots are real but mostly mechanical. Eq (2.98) has an algebraic slip in the Gamma-function manipulation, and Eq (3.59) labels the contour integral I(-Ω) where it should be I(Ω). The final results are correct in both cases, so an editor should ask for a cleanup rather than a re-derivation. The free energy sign flips between Eq (2.63) and Eq (2.151)/(2.166); I suspect a convention slip, but it needs to be resolved because readers will be confused.

The bigger issue sits in Chapter 5. The three-chain transport classification (κ=1/2 and 1 strange metallic, κ=2 insulating) is presented as a fact, with κ as an input, and no discussion of whether the hopping amplitude D must be rescaled with chain length. For a bare long-range amplitude with κ<1, the noninteracting bandwidth diverges as L^{1-κ}, so the claimed universal resistivity floor ρ_min=8/(Nπ) could be a finite-size artifact. The provided text does not state whether D is a deterministic amplitude or a disorder variance, nor whether D is rescaled. This has to be fixed before the strange-metal claim can be taken literally. It may be that the full model does rescale D; the truncated text doesn't show it. But as it stands, the transport chapter is the load-bearing weakness.

Also, the boundary between restated chain results and new derivations is unclear; the bibliography is cut off in our copy, so I can't tell what's original.

Who's this for? A graduate student or researcher new to SYK will get a lot from Chapters 2-4. The transport chapter needs caveats. I'd send it to a serious referee—it deserves the time—but the referee should insist on fixing the typos, clarifying the model definitions, and adding a thermodynamic-limit analysis for the chains. As it stands, it's a solid pedagogical review with an unproven transport add-on.

Referee Report

2 major / 4 minor

Summary. This manuscript is a pedagogical review of the Sachdev-Ye-Kitaev model presented as a self-contained, monograph-length text. Chapters 1-3 introduce Landau Fermi-liquid theory, the Majorana and complex SYK variants, disorder averaging, the large-N effective action, the IR conformal limit, the Schwarzian mode, and the large-q limit. Chapter 4 covers real-time Keldysh dynamics, equilibrium thermodynamics and the phase transition, the quantum Lyapunov exponent, and SYK chains. Chapter 5 treats quenches, Keldysh contour deformations, and DC transport in long-range-hopping SYK chains, claiming that kappa=1/2 and kappa=1 chains exhibit linear-in-T resistivity and a universal minimum resistivity rho_min = 8/(N pi), while the kappa=2 chain shows insulating to linear-in-T crossover. The derivations are supplemented by Mathematica and Python implementations in Appendices G-J.

Significance. The pedagogical body of the work is executed with unusual care: the heavy derivations are explicit, the large-q differential equation d_tau^2 g = 2 J~^2 e^g with its all-temperature solution, the IR conformal coefficient b, the free energy, the Landau-Ginzburg critical exponents, and the saturation of the MSS chaos bound all reproduce the established results of the field. The machine-checked implementations in Appendices G-J are a genuine strength, as are the deliberate per-chapter statements of conventions. The genuinely new content is in Section 5.4: a classification of long-range SYK chains into three 'universality classes' by the hopping exponent kappa, with falsifiable transport predictions. If these transport claims survive a thermodynamic-limit analysis, the book would be a valuable self-contained resource for students and a useful reference for strange-metal phenomenology. The main correctness risk is not circularity -- the derivations run from the Hamiltonian to the outputs without fitted inputs -- but the robustness of the kappa-based transport classification, which depends on the specification and scaling of the hopping term.

major comments (2)

[Sec. 5.4 / Fig. 5.1] The manuscript does not state whether the hopping D_ij in the three chains is a deterministic amplitude or a disorder variance, nor how D scales with the chain length L. For a deterministic amplitude D_ij ~ D/|i-j|^kappa with 0 < kappa < 1, the noninteracting single-particle bandwidth at fixed D grows as D L^(1-kappa) (and as D log L for kappa=1), so the kappa=1/2 and kappa=1 Hamiltonians are non-extensive; the kappa=2 chain, by contrast, has a finite bandwidth. In that situation the 'robust linear-in-T resistivity' and the universal minimum rho_min shown in Fig. 5.1 at fixed couplings J=1, |D|=5 could be finite-size effects, and comparing the three chains on a common axis is not a controlled thermodynamic statement. Because the abstract advertises transport 'in the thermodynamic limit', this issue is load-bearing. Please (i) state whether D_ij is an amplitude or a variance; (ii) give the L-dependence of the coupling (e.g., D = D0/L^(1-kappa)); (iii) report rho(T) for at least two system sizes or an explicit L-to-infinity extrapolation for kappa=1/2 and kappa=1; and (iv) specify how the definition of rho_min is affected by the L-dependence.
[Sec. 5.4.6 / Fig. 5.1] The quoted 'universal minimum resistivity rho_min = 8/(N pi)' needs an unambiguous definition of N and of the normalization. In the MIR-normalized units of the figure, rho_min depends explicitly on N, so the term 'universal' is only meaningful if 8/(N pi) is the combination that remains fixed as N to infinity in the plotted units. Please state whether N is the number of chain sites or the number of fermions per site, and verify that the minimum saturates the same value at fixed couplings for several chain lengths.

minor comments (4)

[Sec. 2.6.2, Eq. (2.112)] The Schwarzian prefactor gamma = -alpha_s N / J~ is quoted 'without proof, see Ref [4]'. The citation is transparent, but for a text announced as self-contained, a short derivation or a precise appendix pointer is needed, since this coefficient enters the soft-mode free energy in Eqs. (2.119)-(2.120) and underlies the chaos analysis in Sec. 4.3.
[Eq. (3.6)] The variance sigma_q^2 = 2 (q/2!)^2 J_q^2 / ((q/2) N^(q-1)) contains the ambiguous factor '(q/2!)'; if this is a typesetting of (q/2)!, the numerical normalization should be reconciled with the interaction coefficient J_q^2/(q/2) in Eq. (3.38). Please write the factorial factors explicitly and verify the constant.
[Fig. 2.1 caption / Sec. 2.9.1] The ordering statement 't+ < t- < t_imag' mixes contour ordering with real-time chronology, especially because t_imag is defined through the Wick rotation t -> -i tau; writing the ordering in the contour-time notation 't+ <_C t- <_C t_imag' would avoid the implication that these are ordinary real times.
[Sec. 1.2.1] The bullet 'SYK's flavor-normalized charge jump reflects total charge change due to zero spatial extent' is unclear; the proposed map to charged AdS black holes would benefit from explicit equations or a reference, given that the holographic commentary recurs throughout the text.

Circularity Check

0 steps flagged · score 0.0 of 10

No load-bearing circularity; the main thermodynamic, chaos, and transport results are computed from the stated model and solved equations rather than being inserted as inputs.

full rationale

I walked the paper's derivation chain from the Hamiltonian through the disorder-averaged effective action, Schwinger-Dyson equations, IR conformal solution, large-q Green's function, free energy, Kadanoff-Baym equations, and the Keldysh transport calculation. At each stage the target quantities are obtained by solving equations, not by fitting a parameter and renaming it as a prediction. The large-q ansatz G(τ)=1/2 sgn(τ)e^{g(τ)/q} is an explicit, displayed ansatz; the function g(τ) is subsequently determined from the resulting differential equation with boundary conditions, so the final Green's function is not an input. The Schwarzian coefficient is quoted from Ref. [4], but that is an external parameter-free result, and the effective action is independently constructed from Diff(R) symmetry; no uniqueness claim is imported from the present author's own work. The Hawking-Page correspondence is explicitly described as a similarity or hint, not as a derivation that forces the SYK result. The transport classification in Fig. 5.1 uses three fixed model Hamiltonians labeled by κ; the linear-in-T resistivity is reported at fixed couplings J=1 and |D|=5 as an output of the Keldysh equations. Choosing κ=1/2 and 1 is a model-selection issue rather than a circular reduction, because the quoted results are computed from those Hamiltonians and are not used to define the Hamiltonians. The possible extensivity problem with long-range hopping D_ij ~ |i-j|^{-κ} is a thermodynamic-limit correctness concern, not a self-referential step in the derivation. No specific equation in the provided text reduces to its own input, and no load-bearing self-citation chain is exhibited.

Assumptions & free parameters 2 free parameters · 8 assumptions · 0 invented entities

The book leans on the cited SYK literature for several unproved premises. Some are mathematical background (Gaussian integration, residue calculus), some are domain assumptions of the large-N/large-q program that the author states explicitly, and one normalization (the Schwarzian coefficient) is adopted without proof from Ref [4]. No hidden inventions: the model, the disorder ensembles, and the Keldysh contours predate this work.

free parameters (2)

kappa (long-range hopping exponent in SYK chains) = 1/2, 1, 2 (three chains)
Ch. 5.4 uses D_ij ~ D/|i-j|^kappa; the value of kappa decides whether the chain shows linear-in-T resistivity (kappa = 1/2, 1), insulating rho ~ T^-2 behavior (kappa = 2), or Fermi liquid behavior. The choice is made by hand to match strange metal phenomenology, not derived from a microscopic model.
relative coupling |D|/J in Fig 5.1 = 5 (with J = 1)
A representative parameter choice for the transport plots; the qualitative claims (linear-in-T for kappa = 1/2, 1) are stated to hold for all couplings, so this is a display choice rather than a fit.

assumptions (8)

domain assumption The large-N limit makes the path integral semiclassical (N plays the role of 1/hbar), so the saddle-point Schwinger-Dyson equations are exact in the thermodynamic limit.
Invoked throughout Ch. 2-3 (e.g., Eq 2.28); standard for SYK, it relies on the disorder-averaged action being extensive in N.
domain assumption Replica-symmetric saddle point: off-diagonal replicas vanish in the large-N limit, with no spin-glass or replica-symmetry-breaking solutions.
Sec 2.3 states: 'This assumption is valid as long as replica symmetry is not broken and there are no stable spin glass solutions.'
domain assumption Bogoliubov principle of weakening correlations: the imaginary branch of the Keldysh contour can be dropped for long-time dynamics.
Sec 2.9.1 uses this to reduce the Keldysh contour to the forward and backward branches; it is an assumption about the memory of the initial state.
ad hoc to paper The Schwarzian normalization gamma = -alpha_s N / J takes its coefficient from Ref [4] without derivation.
Eq 2.112, 'without proof, see Ref [4]'; load-bearing for the finite-temperature Schwarzian action and the lambda_L = 2 pi T chaos result.
domain assumption The finite-temperature conformal mapping tau -> (beta/pi) sin(pi tau/beta) transfers the T = 0 conformal Green's function to finite temperature.
Sec 2.5.5, Eq 2.103; the author flags the result as approximate and valid only at very low temperatures.
ad hoc to paper The kappa = 1/2 and kappa = 1 SYK chains are valid zero-dimensional models of strange metals; the cuprate agreement is adopted from Ref [20].
Ch. 5.4 and Sec 1.2; the mapping from 0+1D chains to materials is delegated to the cited review, not derived here.
domain assumption Large-q results transfer to finite-q physics ('qualitatively similar').
Sec 1.3 and Ch. 2.7; the all-temperature analytic solution is proven only at O(1/q), while the qualitative claims (strange metal, chaos) are made for finite q.
standard math The Maldacena-Shenker-Stanford bound lambda_L <= 2 pi T is taken as an input.
Sec 1.3 and 4.3, cited from Ref [33]; the SYK saturation of the bound is derived, the bound itself is borrowed.

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Cite this review

Pith. "Pith review of Theory of Strongly Correlated Systems: An Introduction to Sachdev-Ye-Kitaev Model." pith.science (2026). https://pith.science/paper/HVMNX6FU

@misc{pith2026250707195,
  author       = {Pith},
  title        = {Pith review of: Theory of Strongly Correlated Systems: An Introduction to Sachdev-Ye-Kitaev Model},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HVMNX6FU}},
  note         = {Machine review of arXiv:2507.07195}
}

abstract

The Sachdev-Ye-Kitaev (SYK) model provides an analytically tractable framework for exotic strongly correlated phases where conventional paradigms like Landau's Fermi liquid theory collapse. This review offers a pedagogical introduction to the SYK physics, highlighting its unique capacity to model \textit{strange metals} -- systems exhibiting linear-in-temperature resistivity, Planckian dissipation, and quasiparticle breakdown. We systematically construct both Majorana and complex fermion variants, transforming them into training grounds for modern many-body physics techniques, for instance, (1) large-$N$ formulations via disorder averaging and replica symmetry, (2) Schwinger-Dyson and Kadanoff-Baym equations, (3) imaginary time Matsubara formulation, (4) real-time dynamics via Keldysh formalism, and the associated (5) non-perturbative Keldysh contour deformations. These tools lay the foundation for equilibrium thermodynamics, quantum chaos, quench dynamics, and transport in the thermodynamic limit, all within a solvable, chaotic quantum system. Intended as a self-contained resource, the review bridges advanced technical machinery to physical insights, with computational implementations provided. Though principally treating the SYK model as a condensed matter laboratory, we also highlight its profound connection to quantum gravity, woven throughout this work, underscoring how this solvable chaotic fermionic model serves as a lens onto black hole thermodynamics and holographic duality.

Figures

Figures reproduced from arXiv: 2507.07195 by the authors.

**Figure 1.1.** Zero-temperature momentum distribution n(⃗p). Dashed line: non-interacting system; solid line: interacting case. Spherical symmetry is assumed for illustration. At the Fermi momentum |⃗p| = pF , ε(⃗p) equals the chemical potential µ, defined through µ = E(N + 1) − E(N) = ∂E ∂N = ε(pF ). (1.4) Though phenomenological, FLT applies to microscopically weakly interacting systems. A representative Hamiltonian includes (i… view at source ↗

**Figure 1.2.** Experimental validation of the Wiedemann-Franz law across multiple Fermi liquid systems. The measured ratio κ/(σT ) exhibits linear temperature dependence with universal slope L, confirming theoretical prediction LFL ≈ 2.44 × 10−8V 2K −2 . Data sourced from Ref. [13]. Despite these successes, mounting experimental evidence reveals systems where Fermi liquid theory fundamentally fails. Such breakdowns necessitate alt… view at source ↗

**Figure 1.3.** Phase diagram of high-temperature copper-oxide superconductors (taken from Ref. [21]), highlighting the anomalous strange metallic regime characterized by linear-in-temperature DC resistivity — a signature departure from Fermi liquid behavior. 1.2 Strange Metals Numerous correlated-electron superconductors with elevated transition temperatures display anomalous metallic phases above their superconducting critical te… view at source ↗

Figures from the paper (14 more)

**Figure 2.1.** Figure 2.1: The Schwinger-Keldysh contour C = C+ + C− + Cimag consists of three segments: (1) Forward real-time branch (C+): extends from −∞ to +∞; (2) Backward real-time branch (C−): extends from +∞ to −∞; (3) Imaginary-time branch (Cimag): represents the equilibrium state. The…

**Figure 2.2.** Figure 2.2: Causal structure of the two-time plane (t1-t2) for Kadanoff-Baym equations (Eq. (2.208)). The system begins in equilibrium in quadrant C (distant past). A non-equilibrium perturbation at the origin induces time evolution. The blue region represents the solution compu…

**Figure 3.1.** Figure 3.1: The complex contour to perform complex integration in Eq. (3.54). There are two contours C1 and C2 where the selection is made by demanding the exponential in the integrand should go to zero at infinity. There is a pole at z0 = −Ω<0. We start with the Matsubara formu…

**Figure 3.2.** Figure 3.2: The complex contour to perform complex integration in Eq. (3.57). There are two contours C1 and C2 where the selection is made by demanding the exponential in the integrand should go to zero at infinity. There is a pole at z0 = +Ω>0. which gives I(−Ω) =    0 ; τ >…

**Figure 4.1.** Figure 4.1: The equation of state (Eq. (4.65)) at low temperatures where T˜ c denotes the critical temperature as given by Eq. (4.66) where there exists a point of inflection. There exists a first-order phase transition in the system for T <˜ T˜ c, similar to van der Waals phase…

**Figure 4.2.** Figure 4.2: The grand potential (Eq. (4.72)) is represented at low temperatures below the critical point T <˜ T˜ c. Critical values are provided in Eq. (4.66). This reveals a first-order phase transition characterized by a discontinuous jump in the order parameter Q˜ (charge den…

**Figure 4.3.** Figure 4.3: Phase diagram of the complex SYK model, highlighting low- and very-low-temperature regimes (defined in the main text). A first-order transition line separates: (i) chaotic and non-chaotic (regular) phases at very low temperatures, (ii) two distinct chaotic phases at …

**Figure 4.4.** Figure 4.4: Phase diagram of a charged Anti-de Sitter (AdS) black hole. A first-order transition line terminates at a critical point with Landau-Ginzburg universality ( [PITH_FULL_IMAGE:figures/full_fig_p121_4_4.png]

**Figure 4.5.** Figure 4.5: Exponential trajectory divergence in classical chaos: Infinitesimally separated initial conditions (δ ≪ 1) evolve with asymptotic separation. The Lyapunov exponent λcl quantifies this instability rate. can signatures of classical chaotic behavior manifest in quantum …

**Figure 4.6.** Figure 4.6: The diagrams containing Jq terms in the O 1 N part of the correlator for the large-q complex SYK model. The diagram is drawn for q = 8, and also consists of a diagram with τ3 ↔ τ4, which have been omitted here. There are q − 2 = 6 lines connecting the two horizon…

**Figure 4.7.** Figure 4.7: In the large-N limit, the Jq contribution to the O 1 N out-of-time-correlator (OTOC) diagrams for the large-q complex SYK system is constructed via the kernel K (shown in parentheses). The OTOC is given by F = F0 + KF, where F = P∞ n=0 Fn and F0 is defined in [P…

**Figure 4.8.** Figure 4.8: The left panel depicts the Keldysh-Schwinger contour used to compute the OTOC in chaotic systems. As illustrated in the right panel, the operators V (0) and W(t) are separated by both: (i) a large real-time difference t, and (ii) a quarter-period along the thermal ci…

**Figure 4.9.** Figure 4.9: Schematic representation of a SYK chain where each site is a complex SYK dot with nearest-neighbor hopping whose Hamiltonian is given in Eq. (4.104). The on-site strengths are given by Ji(t) while the hopping strengths are governed by Di(t) where we have kept the mos…

**Figure 5.1.** Figure 5.1: Resistivity normalized to the Mott-Ioffe-Regel (MIR) bound versus temperature for the three universality classes (κ = 1/2, 1, 2) at fixed couplings J = 1, |D| = 5. The κ = 1/2 and κ = 1 chains exhibit robust linear-in-T resistivity characteristic of strange metals ac…

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