Seeking to Develop Global SYK-Ness

Round  1

Reviewer 1 Report

Spatial locality is absolutely a crucial ingredient in both condensed matter systems and quantum field theories. It is not very clear what are the physical motivations of the generalized SYK models provided in the current paper. 

The definition of the model in Eq. (1) is not clear where are the daggers. For example for i2 and j2, which one has a dagger on them?

After Eq. 12, the author claimed that somehow the system spontaneously breaks fermion parity symmetry which is extremely wired. What is the signature of fermion parity symmetry breaking? Why this system tends to break fermion parity?

It is not at all clear why the scaling form like Eq. (9) would work for solving the saddle point equation. What is the physical significance of the on-shell regime? Why on-shell regime should have a conformal solution?

Can the result of optical conductivity be applied to any physical systems?

Author Response

Author's reply to the comments by Referee I:

Q1."Spatial locality is absolutely a crucial ingredient in both condensed matter systems and quantum field theories. It is not very clear what are the physical motivations of the generalized  SYK models provided in the current paper."

A1: We surmise that the referee might have confused the locality  of an abstract field theory (as a general possibility to  formulate it in terms of local variables) with the locality  of its correlations (as would be manifested by its various  correlation functions), the latter one being of essence to  the present manuscript. If, however, the referee meant  precisely the latter then the answer to his/her question should be as obvious  as, e.g., the need to consider (spatially non-local) Coulomb  interactions when analyzing any realistic condensed matter  system.As to the motivation for introducing non-local couplings  between the different sites of a multi-dimensional network  built out of many (non-random) systems with the SYK-like  dynamics, one immediate answer is that it extends the  earlier ideas of Refs.[24-26] to a much greater variety  of asymptotically solvable models. Presumably, the  undeniable scientific value of any such extension would  not need to be spelled out to the readers of a physics journal. Another (arguably, even more fundamental) reason  derives from the main goal of the earlier study of Ref.[27]  which set out to understand as to whether or not the  'holography-friendly' properties of the original SYK model  (where the 2q-fermion 'all-to-all' couplings  are strictly independent of any distances and/or time  intervals) can survive under a similar generalization -  that is, non-trivial  time/space dependence. In that case the SYK model would  describe generic properties of a whole universality class  of systems, thereby providing an important argument towards  a (much needed) justification of the (otherwise, largely  speculative) holographic approach to the quantum many-body  systems.Lastly, the specific choice of algebraically-decaying  interaction functions was made because it enables explicit  analytic solutions.

Q2."The definition of the model in Eq. (1) is not  clear where are the daggers. For example for i2 and j2,  which one has a dagger on them?"

A2: Eq.(1) was re-written in order to eliminate any future  confusion.

Q3."After Eq. 12, the author claimed that somehow the system  spontaneously breaks fermion parity symmetry which is  extremely wired. What is the signature of fermion parity  symmetry breaking? Why this system tends to break fermion  parity?"

A3: Assuming that the referee (who, according to the report,  finds the English in the manuscript to be at some need of  improvement) actually meant "weird" instead of "wired"  (which term could have matched the context, too, provided  that the IT slang were used), we surmise that she/he might  have again been thinkingof fermion parity in its orthodox (i.e.,  relativistic field-theoretical) sense.Instead, in the (non-relativistic) lattice fermion systems -  such as the nearest-neighbor lattice-SYK model - the notion  of fermion parity symmetry pertains merely to the changes  under the transformation $\chi_i\to -\chi_i$.  If such symmetry indeed remained intact it could have been  used as the reason for not considering any solutions where  the one-fermion function$<\chi^{\dagger}_i\chi_j>$ is non-zero for $i\neq j$.Therefore, a non-vanishing coefficient B in Eq.(9)  implies that the propagator $G(\epsilon,{\bf p})$ acquires  a non-trivial momentum dependence, thus signaling that  its real-space Fourier transform is no longer ultra-local.As to the general possibility of such symmetry breaking it  is worth noting that this symmetry is, in fact, specific to  the two-point interaction function given by Eq.(2), whereas  for a generic  $F_{i_1\dots i_qj_1,\dots j_q}^{\alpha_1\dots\alpha_q \beta_1\dots\beta_q}$ Eq.(1) would not possess such symmetry  in the first place.The phenomenon of spontaneous symmetry breaking where a  solution has lower symmetry than the underlying action is  common to the interacting many-body systems. Moreover, even  in the presence of an intact local symmetry (e.g., in a  gauge theory) a non-invariant amplitude can be used to  deduce gauge-invariant information (e.g., the gauge-invariant  poles and cuts of a gauge-non-invariant fermion propagator  in QED, QCD, etc.)

Q4."It is not at all clear why the scaling form like Eq. (9)  would work for solving the saddle point equation. What is  the physical significance of the on-shell regime?  Why on-shell regime should have a conformal solution?"

A4: The ansatz of Eq.(9) is, in fact, mostgeneral and includes all the four ingredients that would  be customarily used in any discussion of  the non-Fermi-liquid behavior: renormalized dispersion relation characterized by the critical exponent $z$ and prefactor $B$,anomalous exponent $\eta$ controlling the 'on-shell'  singularity, and the 'wave-function renormalization' $A(p)$. As is well known from the elementary textbooks the 'on-shell'  behavior of a single-fermion propagator signifies the  presence of well-defined quasiparticle excitations -  or a lack thereof.Away from the 'on-shell' regime (be it at a continuous  Fermi surface or near an isolated nodal Dirac point) the  ansatz (9) ceases to be applicable, so its use would only  be justified if the integrals in the SD equation were  dominated by the 'on-shell'  contributions - which indeed appears to be the case.Then, in the absence of any energy scale other than $\epsilon$  and $Bp^z$, the solution of the SD equations is simply a  function of these two dynamical scales - although it is  not necessarily conformal (the latter would be a stronger  property which is not to be confused with the mere scaling  with powers of $\epsilon$ and $p$). However, the validity  of Eq.(7) itself is provided by the same condition of a  large $N$ as that giving rise to the genuine conformal  saddle-point solution in the case of the original SYK model.

Q5."Can the result of optical conductivity be applied to  any physical systems?"

A5: In the (by now, published) Refs.[28-30] the  finite-temperature counterparts of the optical conductivity  were indeed used to speculate about the relevance of  the SYK-type physics to the data on some popular  'strange metals'.In contrast, in the present manuscript it was clearly  stated that in the absence of a specified (non-universal)  mechanism of momentum relaxation no general relation would  exist between the optical and finite-temperature D.C.   conductivities. However, once such a mechanism  (umklapp, impurities, phonons) is introduced, the  entire temperature- and frequency-dependent conductivity  could be evaluated and the results contrasted against  experiment. These tasks are beyond the scopes of the  exploratory work presented in the manuscript that is meant  to be published as brief report.

Reviewer 2 Report

The author considers a non-local generalization of the SYK model referred to as “global SYK” with one--species of multi-colored fermions with q-range interactions, and  two-species or “hybrids” with “itinerant” fermions. The  non-locality in space-time is assumed either power like or  Lorentzian. Unlike the standard SYK models the q-range  interaction is not random.  The author derives and analyzes the induced fermionic propagator and self energy in the large N limit through standard bosonization. For some range of  parameters for the power like interaction, the fermions develop an emergent dispersion relation. The same is observed for the Lorentzian interaction.  The class of models addressed in this ms are interesting, and the author points at various limits which relate to the standard SYK limit, as well as the relevance of  his class of models to highly correlated fermionic systems. Also the potential relation of this class of models to holography is intriguing. Most of the derivations in this ms are easily reproducible, although some parts need further elaboration in relation to specific physical applications, e.g. heavy fermions, localized insulators as the author says. Overall, I support the publication of this ms in your journal.

Author Response

Author's reply to comments by Referee II:

"The author considers a non-local generalization  of the SYK model referred to as “global SYK” with  one--species of multi-colored fermions with q-range  interactions, and  two-species or “hybrids” with  “itinerant” fermions. The  non-locality in space-time  is assumed either power like or  Lorentzian.  Unlike the standard SYK models the q-range  interaction is not random.                             

The author derives  and analyzes the induced fermionic propagator  and self energy in the large N limit through  standard bosonization. For some range of   parameters for the power like interaction,  the fermions develop an emergent dispersion  relation. The same is observed for the  Lorentzian interaction.            

The class of models  addressed in this ms are interesting, and the  author points at various limits which relate  to the standard SYK limit, as well as the  relevance of  his class of models to highly  correlated fermionic systems.                 

Also the potential  relation of this class of models to holography  is intriguing. Most of the derivations in this ms  are easily reproducible, although some parts need  further elaboration in relation to specific physical  applications, e.g. heavy fermions, localized  insulators as the author says. Overall,  I support the publication of this ms in your journal."

We thank the referee for supporting publication  of the manuscript. As to the concrete physical applications,  this exploratory work (in the brief report format) on the  long-ranged generalizations of the (non-random) SYK-type  models is only meant to demonstrate the existence of a  whole new direction in the SYK-related research by  questioning such wide-spread assumptions about the SYK  physics as its invariable 'ultra-locality'.  As such, the manuscript avoids making any overly optimistic  claims about and strained references to the actual  physical systems as in, e.g., Refs.[28-30]. That  would be premature before first resolving the more  fundamental universal aspects of the generalized SYK physics.

Round  2

Reviewer 1 Report

I accept all the modifications provided in the new version except a minor complain about the response to the question 3 in my original report. 

I finally understand what the author means by the Z2 symmetry, which is NOT the fermion parity symmetry which flips the sign of ALL fermion operators in the hamiltonian. The Z2 symmetry in the current paper is only flipping the fermion operators for a give site i, or can be viewed as fermion parity of the cluster on a given site. The author should be cautious about the way of presenting the symmetry. Because it is very easy to confuse the "local" Z2 symmetry with the global fermion parity. Fermion parity of an isolated system can never be broken while the local fermion parity can. 

Can the author provide some physical intuition for why this interaction can induce non-local green's function while the previous models (for example the coupled chain model in Gu et al.) cannot?

Author Response

We appreciate the quick response by Referee I and his/her constructive attitude 

that helps to improve on the presentation in the manuscript. In response to his/her new question 

We appreciate the quick response by Referee I and his/her constructive attitude 

that helps to improve on the presentation in the manuscript. In response to his/her new question we added a clarification to the manuscript,

we added a clarification to the manuscript, following Eq.(8).

The truth of the matter is that the commonly assumed applicability of ultra-local solutions is 

yet to be fully justified even in the case of short-ranged correlations, 

such as the SYK-lattice models of Refs.[5,11]. An attentive reader of 

the previous publications on this topic could have noticed 

(and, probably, would have been disturbed by the fact) that the popular 

ultra-local ansatz for the singe-fermion propagator [as in the original (space-less) SYK model] 

was POSTULATED, rather than proved, in all (repeat: ALL) of those works. 

Other than its technical convenience, the only 

argument invoked in a couple of publications was the aforementioned 

local $Z_2$ symmetry (which, if it indeed remained intact, would have prohibited 

any inter-site fermion correlations, thereby enforcing $<\chi^{\dagger}_i\chi_j>=0$ for $i\neq j$).

A systematic investigation into all the viable solutions of the saddle-point equations 

in the generalized SYK-type models is yet to be carried out. Such solutions 

should ultimately be selected by comparing their energies - for a reliable evaluation 

of which a proper ansatz first needs to be chosen.  

However, such choice is likely to depend on the details of the action (1) and, therefore, 

may not be universally applicable (or, for that matter, allow for a universal 

physical explanation).

As was first argued in the case of the random SYK-type models of Ref.[27],  

the ultra-local solution would generally be favored by the Hartree-type terms 

in the overall fermion energy, whereas the Fock-type ones tend to support the non-local solutions 

of the kind proposed in Ref.[27] and the present manuscript.   

Moreover, while being finite when evaluated on the 

ultra-local solution in the case of short-ranged couplings, 

the Hartree terms develop IR divergences, once the fermion interactions 

become sufficiently long-ranged.

For instance, the lattice sum $\sum_k F_{ik}(\tau)$ 

appearing in the Hartree terms with $F_{ik}(\tau)$ given 

by Eq.(3) diverges for all $\beta\leq d/2$ 

(in contrast, a spurious UV divergence for $\beta>d/2$ 

is absent as long as the separately introduced amplitude $F_{ii}(\tau)$ remains finite). 

This observation alone suggests that, at least, 

for $\beta\leq d/2$ the ultra-local solution becomes unstable,  

as compared to a non-local one.