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Replica symmetry breaking in spin glasses in the replicafree Keldysh formalism
by Johannes Lang, Subir Sachdev, Sebastian Diehl
Submission summary
Authors (as registered SciPost users):  Johannes Lang · Subir Sachdev 
Submission information  

Preprint Link:  scipost_202406_00046v1 (pdf) 
Date submitted:  20240619 15:13 
Submitted by:  Lang, Johannes 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approach:  Theoretical 
Abstract
At asymptotically late times ultrametricity can emerge from the persistent slow aging dynamics of the glass phase. We show that this suffices to recover the breaking of replica symmetry in meanfield spin glasses from the late time limit of the time evolution using the Keldysh path integral. This provides an alternative approach to replica symmetry breaking by connecting it rigorously to the dynamic formulation. Stationary spin glasses are thereby understood to spontaneously break thermal symmetry, or the KuboMartinSchwinger relation of a state in global thermal equilibrium. We demonstrate our general statements for the spherical quantum $p$spin model and the quantum SherringtonKirkpatrick model in the presence of transverse and longitudinal fields. In doing so, we also derive their dynamical GinzburgLandau effective Keldysh actions starting from microscopic quantum models.
Author indications on fulfilling journal expectations
 Provide a novel and synergetic link between different research areas.
 Open a new pathway in an existing or a new research direction, with clear potential for multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Current status:
Reports on this Submission
Strengths
1 General approach using the Keldysh formalism
Weaknesses
1 It does not consider already known results in the field.
2 Not easy for people not familiar with Keldysh formalism
3 Some definitions are missing
4 Notation not always clear
Report
The authors present a Keldysh formalism in which replica symmetry breaking in spinglasses is analysed in terms of a dynamic theory. They show that ultrametricity emerges in the asymptotic limit of infinite long times and they study the slow aging dynamics in the reparametrization invariance regime of Cugliandolo and Kurchan. They apply their formalism to the classical and quantum SherringtonKirkpatrick and spherical pspin spinglass models.
As a general comment on the manuscript it seem that the Authors are not aware of some results in the field. The emergence of ultrametricity in the asymptotic limit of infinite long times is not a new results, see e.g. Crisanti and Ritort J. Phys. A: Math. Gen. 36, R181 (2003) Sec. 4 and 6. The link between dynamics and statics (in meanfield models) is well established for almost 20 years. In particular, how to interpret r steps in replica symmetry breaking in terms of diverging (and ordered) time scales (we guess what they refer to as “ultrametric dynamics”?) has been addressed by Crisanti and Leuzzi in PRB 75 144301 (2007).
Beside this there are a number of technical comments on the manuscript and a couple of more general observations on some of the claims carried out in the work. We report them in the following in order of appearance in the manuscript.
Based on this we cannot recommend the manuscript for publication in the present form. The Authors must go through a substantial rewriting before the manuscript can be considered for publication to give credit to previous works and fit their results within the already known results.
Requested changes
1) Refs. 39 and 48 are the same reference.
2) Lines 99100
Concerning the related connection between supersymmetry and thermal symmetry in the paramagnetic phase of spin glasses that was previously found by Kurchan [48] an
important follow up, linking the multiscale dynamics to modecoupling theory also beyond schematic theories for continuous variables was developed and presented in a series of papers in the 2010’s. See Caltagirone at al. PRL 108, 085702 (2012), Parisi, Rizzo PRE 87, 012101
(2013) and references therein. In particular, one can look at Ferrari et al. PRB 86 014204 (2012) as a specific application to pspin models analysed in this manuscript (see also lines 601603).
3) Line 145: the relationship between distance (in the ultra metric space) and Parisi matrix overlap elements $P_{ab}$ is written as 1/P_{ab}. In this way a zero overlap corresponds to an infinite distance, i.e., an infinite edge of a triangle in the ultra metric space. Is this more conveniente than considering d_{ab} = 1  P_{ab} (thus having limited distances \in [0,1])?
4) Section 2.1.
About Parisi matrix manipulations the authors might consider or at least acknowledge the approach of Replica Fourier Transform of Crisanti and De Dominicis, Nuclear Physics B, 891 (2015).
Also what the Authors call “u” is usually indicated with “x”. To avoid possible confusion the Authors should use the now standard notation “x”. The letter “u” is also used un Sec. 3.1 with a complete different meaning.
5) Section 2.2.
The dynamics of disorder systems was first addressed in the late 70’s in the seminal works of Martin, Siggia, Rose, de Dominicis, see eq. de Dominics Phys. Rev B19, 4913 (1978). This approach leads naturally to a dynamic theory with the same structure of Keldysh dynamics. The Authors should comment somewhere about how the Keldysh formalism for spinglass dynamics is related to the DeDominicisPelitiMartinSiggiaRose dynamic meanfield theory. Here or in Sec. 4.1, or both.
6) Line 178
Please report after Eq. (6) the definition od G and \Sigma and their components.
Also report what the apices K, R and A stand for.
7) Lines 181,186 “s” is both the spin variable and the “slow” subindex.
8) I do not find a definition for G^A, see also comment 6).
9) Lines 189192.
In the aging regime is b << 1. That is, b/\Lambda < 1/ Λ.
In the fast regime b \to 1, i.e,. b/\Lambda \to 1/\Lambda.
Is b sent to 1 from left or right? The $b \to 1^$ limit might seem reasonable to interpolate between slow and fast, and would be consistent with the decomposition of integral (8), but in this case it would never be 1/\Lambda \leq t \laq b/\Lambda.
The Authors should clarify how this limit is taken and how the two times scales are conceived in terms of the regularizer b and of the cutoff Λ.
10) Lines 214215.
The Ansatz of CHS, Ref 35, initially put forward in the pspin spherical model (i.e. 1RSB stable at low T) has been extended to a generic number of time scales in the context of the spherical 2+p spin model (displaying also spinglass phases with a infinite number of breakings) in “Equilibrium dynamics of spinglass systems” by Crisanti. Leuzzi PRB 75 144301 (2007). That extension solved the issue of the instability of Sompolinsky dynamics in the static limit. The author should comment whether Ansatz (10) is consistent with this.
Also later one, on Lines 298299, the authors write
“Despite several attempts at recovering the results of replica theory from the dynamical
equations at late times [30–33], discrepancies remain [48].”
Here, as well, the above mentioned work “Equilibrium dynamics of spinglass systems” should be considered, in which such discrepancies are overcome.
See, furthermore, lines 641642 where it is written “Consequently, the aging dynamics of the spherical pspin model never reaches equilibrium.”
Also, the replica fieldtheory approach to the dynamic critical slowing down might be interesting to be compared to. See e.g., the above mentioned Paris, Rizzo PRE 87, 012101 (2013) as well as Leuzzi, Rizzo PRB 109, 174211 (2024). This stems from the structure of the superfield built in Ref. 39/48 by Kurchan.
11) Line 321, Eq (23). Please mention that the sigma’s are Pauli matrices.
12) Section 3.1 Line 3345.
The authors write: “An important distinction between the new platforms and classical glasses is the finite lifetime of the excited states due to spontaneous emission”.
How long is such lifetime? Is it long enough for aging to be experimentally detected?
13) Eq. (24). Please specify all the elements in the action: m, h_\sigma, etc..
The author might consider a Greek index different from \sigma that takes the values +, to denote the branch of the contour.
Indeed, \sigma denote the Pauli matrices here.
14) Eq. (31).
What does the superscript V stand for?
15) Line 425
In Eq. (24), in order to describe the critical behavior the authors move from Pauli matrices to soft spins S, i.e., scalar commuting variables in a quartic potential. However, in s_\kappa the Pauli matrices stay and are traced over. Some easy going description might perhaps help the reader to follow the manuscript without compulsorily resorting to ref. 45.
16) Line 491499.
How is the ferromagnetic phase mentioned here characterized in terms of the correlation functions involved in Eq. (49)?
When talking about instability in this dynamic equations, where slow and fast decomposition is assumed, the authors cite the work of de AlmeidaThouless and the Parisi solution, i.e., the instability of the equilibrium solution. The authors should clarify better what they mean by instability in this dynamic case and how it can be analyzed in the dynamic Keldysh theory context, apart from analogies with the statics.
17) Lines 518520: the authors write
“Physically, we can say that replica symmetry breaking corresponds to the inability of the system to fully thermalize to a single global inverse temperature β even at arbitrarily late times/the steady state.”
Is the sentence complete? Are the authors trying to say that if a RSB occurs then there is no single state equilibrium phase? Equivalently, that RSB signals multiequilibria phases?
18) Lines 542544
In Ref. [75] Derrida introduces the Hamiltonian of the random pspin model with Ising spins and shows that in the limit p>\infty it is equivalent to the random energy model. This was further discussed by
Gross and Mozart, Nucl. Phys. B240, 431 (1988). The soft spin version of the Ising pspin model introduced by Derrida was introduced and discussed by Kirkpatrick and Thirumalai Ref[21]. The spherical pspin spin glass model was first introduced in Ref. [77].
19) Line 545.
The SGPM transition in the spherical pspin model is not a first order one in the thermodynamic sense: there is no latent heat and no phase coexistence. The order parameter has a discontinuous transition and a dynamic transition (Ref [35]) arises next to the static one. This transition has been termed Random First Order Transition by Wolynes and coworkers in the 80's, including Ref. [21].
The authors should comment somewhere about how the Keldysh formalism for spinglass dynamics is related to the DeDominicisPelitiMartinSiggiaRose dynamic meanfield theory. For instance in Sec. 4.1.
20) Line 586.
“..and in the case of $\Sigma^K$ nonnegative.” Sentence unclear, please complete.
21) Lines 627628.
The authors should clarify what it is meant by saying that the spinglass phase is indistinguishable from a ferromagnet, in connection to mode coupling theory.
In glass forming liquids at the Mode Coupling temperature the correlation of density fluctuations does not decay to zero anymore but density fluctuations have, nevertheless, still zero mean. Here the analogy should be: non trivial nonzero spinspin correlation at long times but zero magnetization. Or it is something else occurring?
22) Lines 638642
The connection between dynamics and statics has been discussed in
Crisanti. Leuzzi PRB 75 144301 (2007) where it is shown that the CHS dynamics of Ref [35] generalized to any number of relaxation (diverging) times scale does lead asymptotically to the static RSB in the Parisi scheme. See also Crisanti de Dominics J. Phys A: Math. Theor. 44 115006 (2011) where it is shown that the static susceptibility in the Parisi RSB scheme coincides with the static limit of the CHS dynamics.
23) Line 721.
About experimental realisations. What are the experimental timescales in the experiments with dephasing? Is it practically feasible to probe the aging regime in those systems?
24) Eventually it is important to notice that reparametrization invariance is proved untrue as soon as the model is not “pure”, that is when more than a single pspin interaction term is present in the spinglass Hamiltonian. This is never mentioned throughout the manuscript where only the pure pspin model is considered but is a crucial issue of critical slowing down and aging in spinglass systems. See “Rethinking MeanField Glassy Dynamics and Its Relation with the Energy Landscape: The Surprising Case of the Spherical Mixed 𝑝Spin Model" by Folena, Franz, RicciTersenghi PRX 10, 031045 (2020)
25) In discussing the pspin model the Authors should mention that within the 1RSB solution there is no ultrametricity.
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