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Path integral approach to quantum thermalization

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Pith's one-line read The paper introduces a quasiclassical Green function path integral that describes unitary yet irreversible thermalization dynamics, and derives a closed spectral form factor for chaotic brickwork circuits that extends the semiclassical resu

desk verdict The paper builds a serious quasiclassical framework, but Eq. (8) fails the decoupling limit because the nonperturbative replacement was applied to the large-t form instead of the exact Potts result. read the letter →

arxiv 2509.06028 v1 pith:U7KKTWVW submitted 2025-09-07 cond-mat.mes-hall quant-ph

Alexander Altland , Kun Woo Kim , Tobias Micklitz This is my paper · ORCID
classification cond-mat.mes-hallquant-ph
keywords quantumthermalizationspectralformfactorquasiclassicalGreenfunctionGΣ(Luttinger-Ward)formalismAltshuler-Andreevsaddlecolor-flavortransformmany-bodyHeisenbergtimebrickworkcircuit
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

The paper introduces a real-time path integral over slow 'quasiclassical Green functions' as a first-principles description of quantum thermalization in systems that act as their own environment. Its central result is a closed formula for the spectral form factor of a brickwork circuit, K(t) = (1 + K_1(t)e^{-Γt})^L + K_L(t)(1-e^{-Γt})^L, which interpolates from the short-time t^L ramp of independent qudits, through an exponential synchronization at rate Γ, to the plateau of a single ergodic system at the many-body Heisenberg time t ~ D^L. The same construction, adapted to energy-conserving quantum-dot arrays, yields a soft, diffusive approach to ergodicity on times ~L^2. If correct, this provides a transferable effective field theory for strongly entangled many-body chaos, with parameter-free agreement against numerics when the local Hilbert-space dimension is large.

What carries the argument

The central object is the quasiclassical Green function G^ab_tu, a collective variable formed by summing products of retarded and advanced wave amplitudes over a qudit's D internal states. Three pieces of machinery carry the argument: (1) a GΣ (Luttinger–Ward) construction that elevates G to an integration variable, turning the low-energy sector into a nonlinear sigma-model of Goldstone modes associated with a spontaneously broken causal symmetry; (2) the color–flavor transform, which converts Haar-averaged unitary evolution into a matrix B-field living on the symmetric space U(2t)/U(t)×U(t); and (3) Altshuler–Andreev saddle points, non-perturbative stationary points whose inclusion replaces

What would settle it

Run a brickwork-circuit simulation at L=3, D=64 with a coupling set so Γ≈1/D, resolving the spectral form factor around t=D and t=D^3. Equation (8) predicts a kink of height D at t=D and a plateau onset at t=D^3; if the Altshuler-Andreev contributions decay with a rate different from Γ, the kink height or the plateau onset will shift visibly without any fit parameter.

Watch

Extended reading notes

Core claim

The central discovery is that the unitary dynamics of an ergodic many-body system can be represented by a real-time path integral whose slow variables are quasiclassical Green functions G^ab_tu = -(ia/D)∑_μ ψ^a_{μt}ψ^b*_{μu}. Interactions between subsystems act as an environment: they damp all fluctuation modes except one globally synchronized 'Hadamard' mode X_mn = ∏_j B_{j,mn}, which carries the late-time ergodic phase. Inclusion of Altshuler-Andreev saddle points—the non-perturbative configurations that terminate the ramp at time D—yields the closed form factor formula Eq. (8), and a symmetry-restoring effective theory with D^L levels extends it to the many-body Heisenberg time. The paper

Load-bearing premise

The load-bearing premise is that interaction damps the rare Altshuler-Andreev saddle fluctuations at the same golden-rule rate Γ as ordinary fluctuations, a step the paper states as an expectation rather than a derivation; the late-time symmetry action is likewise inferred from symmetry, not derived from the microscopic action.

Editorial extensions

If this is right

  • For the brickwork model, the spectral form factor crosses over from t^L to t exponentially in Γt, and the non-perturbative replacements t→K_1 and t→K_L keep the formula valid at and beyond the single-qudit Heisenberg time t~D.
  • In symmetry-enriched brickwork circuits, the late-time form factor factorizes into the singlet form factor times an SU(N)-Heisenberg evolution, so conserved-charge diffusion sets a minimum thermalization (Thouless) time ~L^2.
  • For continuous Hamiltonian systems, energy diffusion replaces exponential relaxation, giving Eq. (9) with its characteristic max(...) factors and the same L^2 scaling of the slowest mode.
  • Because the construction requires only fast-thermalizing qudits with local pairwise correlations, it is modular and transferable to other system classes whose randomness breaks integrability.
  • The same path integral provides a diagnostic for quantum thermalization in engineered devices, distinguishing the ergodic, diffusive, and plateau regimes from a single analytical expression rather than from exact diagonalization.

Reading between the lines

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

  • A natural next test is to push the brickwork formula to L=3 or 4 with D significantly larger than 12 (e.g., via tensor-network methods), where the kink at t=D and the plateau at t=D^L become sharp enough to distinguish Eq. (8) from the semiclassical Eq. (6).
  • If the uniform damping of Altshuler-Andreev saddles holds generally, the same construction should extend to spatially non-uniform couplings, producing site-dependent rates Γ_j and a generalized synchronization mode that interpolates between local and global ergodicity.
  • The equality of the L-qudit propagator with a single D^L-level ergodic propagator suggests an attractor: any chaotic arrangement of pairwise interactions should flow to the same late-time universal spectral form factor, making the framework a practical probe for device thermalization without solving the full Schrödinger equation.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper introduces a real-time path-integral formalism for many-body quantum thermalization, built from quasiclassical Green functions in a tensor-product Hilbert space. The central objects are collective G-variables, whose fluctuations are controlled by a nonlinear sigma model; the GΣ/Luttinger-Ward construction is used to couple qudits. For a brickwork circuit model the authors derive the spectral form factor, first semiclassically as a t-state Potts model (Eqs. (49)–(50)) and then propose a nonperturbative generalization Eq. (8) designed to incorporate single-qudit and many-body Heisenberg-time cutoffs. For an array of capacitively coupled quantum dots they derive a diffusive late-time form factor Eq. (9). The results are compared with exact diagonalization data for L=2–5 and D=8–64, with one fit parameter (Γ) in the discrete case.

Significance. Should the central claims hold, this is a substantial step: an effective field theory for the crossover from short-time semiclassical dynamics to the many-body Heisenberg time, with a modular construction that could apply to other models. The paper is largely self-contained, recovers the known Potts-model result, and provides a clear semiclassical-exactness argument for the cancellation of higher-order D^{-1} corrections (Appendix D). Reproducible numerical data are deposited on Zenodo. However, the headline nonperturbative result Eq. (8) fails a basic consistency test in the decoupling limit, and the Heisenberg-time extension relies on unproved expectations about AA saddles and the universal ergodic sigma model. Because these are load-bearing, the paper in its present form does not fully establish the announced beyond-perturbation-theory framework.

major comments (2)
  1. [§V.D, Eq. (8)] Equation (8) fails the decoupling limit Γ→0. The exact semiclassical Potts result, Eq. (50), is K=(1+(t-1)e^{-Γt})^L+(t-1)(1-e^{-Γt})^L and reduces to K=t^L at Γ=0. Equation (8) is obtained by replacing t→K_1(t) and t→K_L(t) in Eq. (6), the large-t simplification of Eq. (50) in which every (t-1) is replaced by t. At Γ=0, Eq. (8) gives (1+K_1(t))^L, but the model with Λ=0 consists of L decoupled Haar qudits and must give K_1(t)^L. The discrepancy is O(L/K_1), not O(D^{-1}); for the D=8,12 systems in Figs. 2–3 it is tens of percent at t∼D where Γt≪1. Thus Eq. (8) is not the claimed non-perturbative generalization; it is at best an asymptotic large-t formula, and the fitted Γ values may be absorbing this systematic error. The extension should be applied to Eq. (50) (e.g. by replacing (t-1) with K_1(t)-1 and K_L(t)-1), or the precise domain of validity of Eq. (8) must be stated.
  2. [§V.D.1–D.2] The non-perturbative extension rests on two unproved inputs: (i) the AA-saddle contributions are damped at the same rate Γ as the standard fluctuations ('we thus expect', §V.D.1), and (ii) the late-time effective theory is the universal ergodic sigma model with D^L levels, Eq. (54), inferred from symmetry rather than derived from the microscopic action. Both are load-bearing for Eq. (8) and for the claim of describing dynamics up to the many-body Heisenberg time. The first is not a routine technicality: AA saddles are stationary points distinct from the standard saddle, and their coupling to interactions need not coincide with the Golden-rule damping of near-standard fluctuations. Since the paper defers the full technical execution to a forthcoming publication, the current manuscript does not yet provide a first-principles derivation of its central nonperturbative result. The authors sho
minor comments (5)
  1. [§II.B] The sentence 'we obtain a find the form factor' appears to have a missing word; it should likely read 'we find the form factor'.
  2. [Appendix B.1] The sentence fragment 'Here, the unit operators on the diagonal' is incomplete; the definition of G^{-1}(B) should be finished.
  3. [Eq. (21)] The displayed action contains an unreadable block of symbols; please check the typesetting of the fermion bilinear term.
  4. [§VII.D and Eq. (9)] The role of the two fit parameters (C and Γ) and the softening of the max-function over λ-scales should be stated explicitly in the main text; currently the lack of numerical control is mentioned only in the caption.
  5. [References] Reference [20] lists only an arXiv identifier; for a journal submission, please provide the published version if available.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the semiclassical Potts-model derivation is self-contained; Eq. (8) is an acknowledged interpolation with a decoupling-limit flaw, not a circular reduction.

full rationale

The central semiclassical derivation is self-contained: the path integral over Grassmann variables, Haar averaging via the color-flavor transform, the GΣ decoupling, and the Hubbard-Stratonovich gauge transformation lead to the Potts-model sum Eq. (49) and its exact evaluation Eq. (50), reproducing Ref. [5] by an independent route. The self-citations to Refs. [22] and [25] are technical supports (AA saddles in the Keldysh sigma model; semiclassical exactness) that are published, parameter-free, and do not contain Eq. (8); they therefore count as real evidence rather than circular dependencies. The nonperturbative extension Eq. (8) is not derived from a detailed AA-saddle computation: the paper explicitly labels the key step as 'we thus expect' and defers full execution to a forthcoming publication. This is a missing-derivation/correctness gap, not a circular reduction. The replacements t→K_1 and t→K_L are an interpolation ansatz; indeed the formula fails the exact decoupling limit Γ→0, giving (1+K_1)^L instead of K_1^L, because the substitution was applied to the large-t form Eq. (6) rather than the exact Eq. (50). A fitted Γ is used in the numerical comparisons, but the kink structure at t=D is a nontrivial structural prediction. Overall, no step reduces to its input by construction; the score of 2 reflects the minor self-citations and the acknowledged, partially ad hoc character of the nonperturbative extension, without constituting circularity.

Assumptions & free parameters 2 free parameters · 9 assumptions · 0 invented entities

The main inputs are standard techniques from random matrix theory and disordered systems. The load-bearing ad hoc elements are the conjectured damping of AA saddles, the symmetry-based late-time effective theory, and the tolerance window in the continuous model. The numerical comparisons introduce fitted parameters Γ and C, which are not part of the analytical derivation.

free parameters (2)
  • Damping rate Γ (fitted in numerical comparison) = For L=2, Λ=0.1: 0.0039, 0.0028, 0.0022, 0.0023 (D=8,16,32,64); Λ=0.2: 0.010, 0.0085, 0.0090, 0.0095; Λ=0.3: 0.020, 0.020
    The theory predicts Γ = Λ^2/4 from the interaction variance, but the numerical comparisons fit Γ as a free parameter. At large D the fits approach the predicted value; for L=4 and the continuous model the fitted values deviate significantly.
  • Overall constant C (continuous model, Eq. 9) = C = 0.86, 1.45, 2.34, 3.16 for L=2,3,4,5 with D=8.
    Eq. (9) contains an undetermined numerical constant C because the theory lacks control over high-energy scales. The authors fit C to the numerics.
assumptions (9)
  • domain assumption Qudits are ergodic on microscopic timescales and are statistically equivalent to Haar-distributed unitaries or random matrix Hamiltonians.
    Section II.A describes the building blocks as qudits that relax quasi-instantly. This is the foundational assumption for the quasiclassical Green function description.
  • domain assumption Interaction matrix elements are Gaussian distributed with variance Λ^2/(2D^2) (Eq. 17).
    This specific distribution is used to average the interaction and derive the out and in vertices in Section V.A.1.
  • domain assumption The semiclassical pair propagator has the form Π = (1/D) δ_{Δt,Δu} Θ(u-t) (Eq. 10 and Eq. 33).
    This expresses ergodicity and the independence of the propagator from specific states and times, used throughout the derivation.
  • standard math Color-flavor transform (Eq. 20) from Ref. [21] is valid.
    Used to convert Haar averages into integrals over B-matrices in Section IV.B.
  • standard math Semiclassical exactness of the nonlinear sigma model (Ref. [25]) extends to the interacting theory.
    Invoked in Section V.D.2 and Appendix D to justify that only quadratic fluctuations contribute around each saddle point.
  • standard math Altshuler-Andreev saddle points (Refs. [16,22]) describe the termination of the ramp at the Heisenberg time.
    Used to introduce K_1(t) and K_L(t) in Eq. (8).
  • ad hoc to paper Altshuler-Andreev saddle contributions are damped by interactions at the same rate Γ as the standard fluctuations.
    State as 'we thus expect' in Section V.D.1. This is a conjecture, not a derivation, and is load-bearing for Eq. (8).
  • ad hoc to paper The late-time effective theory is the universal ergodic sigma model with D^L levels (Eq. 54).
    Introduced by symmetry argument in Section V.D.2; not derived from the microscopic action in this paper.
  • ad hoc to paper In the continuous model, a small tolerance window in the weight function is assumed, and the max-function cutoff is imposed.
    Section VII.D: 'We therefore implicitly assume a small tolerance window in the definition of the weight function.' This fixes the normalization of the delta-function and modifies the action without a first-principles derivation.

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Cite this review

Pith. "Pith review of Path integral approach to quantum thermalization." pith.science (2026). https://pith.science/paper/U7KKTWVW

@misc{pith2026250906028,
  author       = {Pith},
  title        = {Pith review of: Path integral approach to quantum thermalization},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/U7KKTWVW}},
  note         = {Machine review of arXiv:2509.06028}
}
abstract

We introduce a quasiclassical Green function approach describing the unitary yet irreversible dynamics of quantum systems effectively acting as their own environment. Combining a variety of concepts of quantum many-body theory, notably the nonlinear $\sigma$-model of disordered systems, the $G \Sigma$-formalism for strong correlations, and real time path integration, the theory is capable of describing a wide range of system classes and disorder models. It extends previous work beyond perturbation theory (in inverse Hilbert space dimensions), enabling a description of thermalization dynamics from short scattering times, through the onset of ergodicity at an effective `Thouless time', up to the many-body Heisenberg time. We illustrate the approach with two case studies, (i) a brickwork model of unitarily coupled quantum circuits with and without conserved symmetries, and (ii) an array of capacitively coupled quantum dots. Using the spectral form factor as a test observable, we find good agreement with numerical simulations. We present our formalism in a self-contained and pedagogical manner, aiming to provide a transferable toolbox for the first-principles description of many-body chaotic quantum systems in regimes of strong entanglement.

Figures

Figures reproduced from arXiv: 2509.06028 by the authors.

Figure 1
Figure 1. FIG. 1. The three stages of quantum thermalization in an energy conserving medium defined by [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The SFF of a two-site system, [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The SFF for [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figures from the paper (12 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The product formula Eq [PITH_FULL_IMAGE:figures/full_fig_p005_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Left: coherent double sum over retarded (top) and ad [PITH_FULL_IMAGE:figures/full_fig_p005_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Left: identification of Goldstone mode generators [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. The pairing of four Goldstone mode generators to [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Left: interaction vertex coupling the quantum states [PITH_FULL_IMAGE:figures/full_fig_p007_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9. Color-flavor transform visualized: Bilinears of field [PITH_FULL_IMAGE:figures/full_fig_p009_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10. A brickwall design describing the alternating ap [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11. Top: The vertex [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The interaction vertex expanded to lowest order [PITH_FULL_IMAGE:figures/full_fig_p014_12.png]
Figure 13
Figure 13. Figure 13: FIG. 13. An array of random quantum dots coupled by pair [PITH_FULL_IMAGE:figures/full_fig_p017_13.png]
Figure 14
Figure 14. Figure 14: FIG. 14. Energy exchange between neighboring dots [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: FIG. 15. The cancellation of contributions to the [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]

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Forward citations

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Reference graph

Works this paper leans on

45 extracted references · 40 canonical work pages · cited by 2 Pith papers

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    (41) reads as 0=δ Σj S[ ¯G, ¯Σ, B]=D ( ¯Gj+iτ 3(G−1(Bj)−i ¯Σjτ3)−1) , 0=δ Gj S[ ¯G, ¯Σ, B]=D ¯Σj+δ Gj Sint[ ¯G]

    Stationary phase analysis Following the same protocol as in the single qudit case, we proceed by stationary phase analysis, where the general- ization of equations (26) to the action Eq. (41) reads as 0=δ Σj S[ ¯G, ¯Σ, B]=D ( ¯Gj+iτ 3(G−1(Bj)−i ¯Σjτ3)−1) , 0=δ Gj S[ ¯G, ¯Σ, B]=D ¯Σj+δ Gj Sint[ ¯G]. At first sight, the presence ofS int appears to significa...

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    While Eq

    Nonlinearσ-model Introducing the discrete derivative in the center time de- pendence, dBnm ≡(B−B T)nm =B nm−B (n−1)(m−1), the next step is an expansion of the logarithmic action to leading order in this difference: S0[B]≡Dtr( 1 1+B †B B†dB).(29) To establish contact with the literature, we note that a straightforward reordering of terms (always to leading...

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    Quadratic action To establish contact to the semiclassical principles dis- cussed in section III, consider the quadratic expansion, S(2) 0 [B]≡Dtr (B†dB) ,(31) This Gaussian weight defines the discrete time propagator of the theory, ⟨Bn+,n− B† m−,m+⟩≡Π n,m,(32) as the inverse D(dΠ)n,m =D(Π n,m−Π n−1,m)=δ n,m, wheren≡(n +, n−),n−1=(n +−1, n−−1), andδ n,m =...

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    Averaging overUyields the generalization of Eq

    Averaging over randomness Our system contains two sources of randomness, the Haar-distributed unitariesU, and the Gaussian distributed H. Averaging overUyields the generalization of Eq. (21) to multiple sites, S[U, ψ]→S[B, ψ]=D ∑ j tr ln(1+B jB† j)+(36) + ∑ j ( ¯ψ+ j−1BT jψ− j−1− ¯ψ− j1B† j ψ+ j1). Averaging overHamounts to a straightforward Gaussian inte...

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    How- ever, at this point the advantage of theGΣ-construction becomes evident: the interaction containsψin the form of color singlets Eq

    Luttinger-Ward functional The nonlinearity of theS i inψimplies that we can no longer integrate over these variables in closed form. How- ever, at this point the advantage of theGΣ-construction becomes evident: the interaction containsψin the form of color singlets Eq. (38). Lagrange multipliers Eq. (24) im- plemented locally at each site allow us to trad...

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    Hubbard-Stratonovich decoupling of the interaction by introduction of a time-bilocal auxiliary field, ϕjn+n− ≡ϕ jn

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    Reminiscing a scalar potential coupling, it suggests removingϕfrom the action by a time-dependent gauge transformation

    The quadraticB-action including this field then reads (symbolic notation), ∑jn B† jn(d−ϕ j,n)Bjn, with the discrete time derivativedB=B−B T . Reminiscing a scalar potential coupling, it suggests removingϕfrom the action by a time-dependent gauge transformation. With the phase factorΘ n = ∑m ϕn−m, this is achieved by the change of variablesB n→exp(Θ n)Bn

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    However, it reappears in opera- tors representing correlation functions in the form of gauge-phase factors containing discrete time integrals overΘ n

    This transformation removes the interaction poten- tial from the action. However, it reappears in opera- tors representing correlation functions in the form of gauge-phase factors containing discrete time integrals overΘ n

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    The final Gaussian integration over these variables then effectively sums over interaction processes to in- finite order in perturbation theory. C. Form factor (semiclassical) Applied to the form factor Eq. (44), the protocol above leads to K(t)→D 2L ∑ {m,n} ∏ j ⟨(BtnB† n1B† 1...

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    The representation K1(t)=t−(t−D)Θ(t−D), indicates that the termination of the semiclassical rampK 1,semicl

    Time scales∼D At the timet=D, the form factorK 1(t)of a single qu- dit levels off at the valueK 1(t)=D. The representation K1(t)=t−(t−D)Θ(t−D), indicates that the termination of the semiclassical rampK 1,semicl. =tis due to contributions to the path integral non-vanishing at t...

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    particle-hole excitation

    Time scales∼D L The analysis of non-perturbative effects at late scales re- quires a different strategy, which we here outline with addi- tional details provided in Appendix D. Semiclassical exactness:To prepare the analysis of the survivor modes, we notice that in the non-int...

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    It implies the vector integral generalization δµρ = ∫ Dψ e− ¯ψψ ψµ ¯ψρ (A1) whereψ={ψ µ}now is aD-component Grassmann vector, Dψ= ∏µ d ¯ψµdψµ and ¯ψψ= ∑µ ¯ψµψµ

    Single qudit path integral For a single Grassmann variable,ψ, [8] consider the defi- nition of the integral ∫ dψ ψm =δ m,1. It implies the vector integral generalization δµρ = ∫ Dψ e− ¯ψψ ψµ ¯ψρ (A1) whereψ={ψ µ}now is aD-component Grassmann vector, Dψ= ∏µ d ¯ψµdψµ and ¯ψψ= ∑µ...

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    Qudit network The interaction contribution can be treated analogously, by insertion of the qudit generalization of Eq. (A1), δµν,ρσ = ∫ Dψ e− ¯ψψ ψ2νψ1µ ¯ψ1ρ ¯ψ2σ.(A2) (Notice the ‘anti-normal ordering’, grouping creation and annihiliation ‘operators’ together.) If we now plac...

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    Collecting the quadratic terms in Eqs

    The matrixG(B) The construction of the path integral starts with the inte- gration overψ-variables, Gaussian after all nonlinear terms have been Lagrange multiplier locked toG. Collecting the quadratic terms in Eqs. (35), (36), we obtain the bilinear form ¯ψG−1ψwith G−1(B)= ⎛ ...

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    Our starting point is the configurations Eq

    Interaction vertex In this Appendix, we discuss the expansion of the discrete time interaction vertex inB-fluctuations. Our starting point is the configurations Eq. (45). With Eq. (B1), and (13), a breakdown of the individual components we need in the fol- lowing is given by (...

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    Interaction vertex beyond leading order perturbation theory We here demonstrate the vanishing of the interaction vertex on field configurations satisfying the synchronization condition. To this end, consider the expansion Eq. (B3) of the interaction between two neighboring qud...

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    Noting that theT-rotations do not couple to the second and third term in the first line due cyclic invariance, we focus on the first term

    Non-interacting theory To obtain the non-interacting part of the continuous time action, we substitute the stationary phase configurations G=T ¯GT−1 andΣ=λ 2Ginto the action (59). Noting that theT-rotations do not couple to the second and third term in the first line due cycli...

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    We substitute the representationG=T ¯GT−1 with the mean field solution Eq

    Interaction vertex Next, consider the interaction Si[G]= Λ2D2 2 ∑ ⟨j,k⟩ tr((G j⊙G k)τ3(Gj⊙G k)τ3)= = Λ2D2 2 ∑ ⟨j,l⟩ ∑ s ∫tt′ ((Gj⊙G k)ss tt′(Gj⊙G k)ss t′t−(G j⊙G k)s¯s tt′(Gj⊙G k)s¯s t′t) . We substitute the representationG=T ¯GT−1 with the mean field solution Eq. (61) and the...

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