Condensed matter physics, area laws & LQG?

[atyy]

http://pirsa.org/10110076/

Glen Evenbly gave a very nice, understandable overview of this line of research at the Perimeter Institute last year. It's basically about approximations for fast simulations of condensed matter systems. There are different flavours of these methods, good for different situations. He brings up the possible relationship between the application of these methods to CFTs and AdS/CFT at around 16 minutes. One of the great mysteries he solves for me is how to pronouce "Guifre", but he also says "anne-setz" ...

Evenbly and Vidal have now written up the stuff in that talk:

Tensor network states and geometry
http://arxiv.org/abs/1106.1082

 

[atyy]

A http://arxiv.org/abs/1106.4501" from Raman Sundrum - now at Maryland, where maybe he can bump into Ted Jacobson and Michael Levin more easily - entertains starting from a discrete viewpoint: "In this way, one may have a sequence of emergent phenomena: strongly-coupled discrete quantum system → continuum quantum field theory → Special Relativistic field theory → CFT → AdS General Relativity + gauge theory. ... Reading in reverse, one might well suspect that our own Universe has a discrete but strongly-interacting “DNA”."

 

[marcus]

A http://arxiv.org/abs/1106.4501" from Raman Sundrum - now at Maryland, where maybe he can bump into Ted Jacobson and Michael Levin more easily - entertains starting from a discrete viewpoint: "In this way, one may have a sequence of emergent phenomena: strongly-coupled discrete quantum system → continuum quantum field theory → Special Relativistic field theory → CFT → AdS General Relativity + gauge theory. ... Reading in reverse, one might well suspect that our own Universe has a discrete but strongly-interacting “DNA”."

A considerable part of the introduction to Sundrum's paper was delightful and conceptual as well. Thanks for the reference.

 

[atyy]

More tensor network people at PI! They'll have good conversations about renormalization.

http://www.perimeterinstitute.ca/News/In_The_Media/Bianca_Dittrich_to_Join_PI_Faculty/
"Dr. Dittrich's present work seeks to understand how one could construct a new class of lattice models which are independent of one's choice of discretization, which should then display a discrete notion of diffeomorphism symmetry, beginning with the models we know so far. This work has many potential links to other fields of study at PI, such as condensed matter, quantum computing, and numerical relativity."

 

[atyy]

Newly posted by marcus at his bibliography. These guys are on the case! They cite and make use of the paper that started this thread.

http://arxiv.org/abs/1109.4927
Coarse graining methods for spin net and spin foam models
Bianca Dittrich, Frank C. Eckert, Mercedes Martin-Benito
(Submitted on 22 Sep 2011)
We undertake first steps in making a class of discrete models of quantum gravity, spin foams, accessible to a large scale analysis by numerical and computational methods. In particular, we apply Migdal-Kadanoff and Tensor Network Renormalization schemes to spin net and spin foam models based on finite Abelian groups and introduce `cutoff models' to probe the fate of gauge symmetries under various such approximated renormalization group flows. For the Tensor Network Renormalization analysis, a new Gauss constraint preserving algorithm is introduced to improve numerical stability and aid physical interpretation. We also describe the fixed point structure and establish an equivalence of certain models.

 

[czes]

Newly posted by marcus at his bibliography. These guys are on the case! They cite and make use of the paper that started this thread.
http://arxiv.org/PS_cache/arxiv/pdf/1109/1109.4927v1.pdf
Coarse graining methods for spin net and spin foam models
Bianca Dittrich, Frank C. Eckert, Mercedes Martin-Benito
(Submitted on 22 Sep 2011)

I am desperately looking for a solution of the quantization paradox in the Quantum Gravity with regard to the observation of the GRB from the distant galaxy. Is the universe at the deepest level grainy?
We exclude the random walk model and most of the holographic models of the space-time foam.
http://www.centauri-dreams.org/?p=18718

May be , we have to distinguish the different kinds of the discretness ?

 

[atyy]

I hope they post lectures online! Bolding below is mine.

http://www.perimeterinstitute.ca/Events/Tensor_Networks_for_Quantum_Field_Theories/Tensor_Networks_for_Quantum_Field_Theories/
Tensor Networks for Quantum Field Theories
October 24 - 25, 2011
Perimeter Institute

Tensor network states, such as the matrix product state (MPS), projected entangled-pair states (PEPS), and the multi-scale entanglement renormalization ansatz (MERA), can be used to efficiently represent the ground state of quantum many-body Hamiltonians on a lattice. In this way, they provide a novel theoretical framework to characterize phases of quantum matter, while also being the basis for powerful numerical approaches to strongly interacting systems on the lattice.

The goal of this meeting is to discuss recent extensions of tensor network techniques to continuous systems. Continuous MPS and continuous MERA can tackle quantum field theories directly, without the need to put them on the lattice. Therefore they offer a non-perturbative, variational approach to QFT, with plenty of potential applications. On the other hand, the proposal of continuous MERA makes previous hand-waving arguments that the MERA is a lattice realization of the AdS/CFT correspondence ever more intriguing.

Pedagogical talks will be directed to introducing the subject to (PI resident) quantum field/string theorists. Discussions with the latter will aim at identifying future applications and challenges.

Scientific Organizers:
Guifre Vidal, Perimeter Institute
Frank Verstraete, University of Vienna

 

[atyy]

http://arxiv.org/abs/1109.5592
Connecting Entanglement Renormalization and Gauge/Gravity dualities
Javier Molina-Vilaplana
(Submitted on 26 Sep 2011)
I propose a connection between the Multi-Scale Entanglement Renormalization Ansatz (MERA) and holographic gravity duals. The relationship is provided by analyzing the renormalization group (RG) flow of correlation functions in MERA and showing their formal equivalence with the holographic RG flow of these correlation functions in Anti de Sitter (AdS) space. As a corollary, we argue that when considering correlations between disjoint regions, the holographic dual of the MERA procedure may be efficiently described by an AdS black hole.

 

[qsa]

I am desperately looking for a solution of the quantization paradox in the Quantum Gravity with regard to the observation of the GRB from the distant galaxy. Is the universe at the deepest level grainy?
We exclude the random walk model and most of the holographic models of the space-time foam.
http://www.centauri-dreams.org/?p=18718

May be , we have to distinguish the different kinds of the discretness ?

Nature is fundamentally a statistical system i.e. discrete at heart (must be), but let me give you a simple analogy. throwing a coin ,you might get 3045 heads and 89080 tails, that is discrete to be sure . But the ratio is real. so Nature is fundamentally discrete but you never measure that discreteness (you can't) , we can only measure the ratio like numbers to certain accuracy. So there it is, no conflict.

 

[atyy]

marcus posted in his bibliography a really interesting in GFT renormalization. The Dittrich et al paper in post #35 is aware of the tensor-network stuff in which the lattice is fixed, and this stuff, in which the several lattices are summed over. The Feynman diagrams of GFT are spin foams.

http://arxiv.org/abs/1111.4997
A Renormalizable 4-Dimensional Tensor Field Theory
Joseph Ben Geloun, Vincent Rivasseau
(Submitted on 21 Nov 2011)
We prove that an integrated version of the Gurau colored tensor model supplemented with the usual Bosonic propagator on U(1)4 is renormalizable to all orders in perturbation theory. The model is of the type expected for quantization of space-time in 4D Euclidean gravity and is the first example of a renormalizable model of this kind. Its vertex and propagator are four-stranded like in 4D group field theories, but without gauge averaging on the strands. Surprisingly perhaps, the model is of the φ6 rather than of the φ4 type, since two different φ6-type interactions are log-divergent, i.e. marginal in the renormalization group sense. The renormalization proof relies on a multiscale analysis. It identifies all divergent graphs through a power counting theorem. These divergent graphs have internal and external structure of a particular kind called melonic. Melonic graphs dominate the 1/N expansion of colored tensor models and generalize the planar ribbon graphs of matrix models. A new locality principle is established for this category of graphs which allows to renormalize their divergences through counterterms of the form of the bare Lagrangian interactions. The model also has an unexpected anomalous log-divergent (∫φ2)2 term, which can be interpreted as the generation of a scalar matter field out of pure gravity.
41 pages, 9 figures

 

[atyy]

MTd2 alerts us on marcus's bibliography to Rivasseau's latest manifesto. If we count TFT as LQG inspired, then it contains another explicit declaration of a search for AdS/LQG: "TFT should certainly benefit from this beautiful circle of ideas, for instance from the possibility of identifying the radial direction in AdS-CFT with the RG scale. There are some preliminary glimpses of a possible holographic nature of the boundary of colored tensor graphs."

Key points of the TFT manifesto:

"TFT can in particular include the study of renormalizable GFT models, which are similar to combinatorial models but with an additional gauge invariance."

"There is a strong link between the universal character of the central limit theorem in probability theory and the existence of a 1/N expansion"

"We saw already that there is a parallel between the hierarchy of central limit theorems in probability theory and the hierarchy of 1/N expansions in quantum field theory. There is also an associated hierarchy of renormalization group types: scalar, vector, matrix, tensors. They can be distinguished by their different notions of locality and the different power counting formulas to which they lead to."

"We know the renormalization group type can change along a given RG trajectory at a phase transition point. For instance at the BCS transition in condensed matter, the RG type changes form vector to scalar. There is therefore no reason the RG cannot change from tensor to lower-rank type at geometrogenesis."

 

[Physics Monkey]

"We know the renormalization group type can change along a given RG trajectory at a phase transition point. For instance at the BCS transition in condensed matter, the RG type changes form vector to scalar. There is therefore no reason the RG cannot change from tensor to lower-rank type at geometrogenesis."

Not sure what this means.

 

[atyy]

Not sure what this means.

I sent Vincent Rivasseau an email asking for some pointers to the literature that describes the RG type change in BCS theory. He sent me the following to post. He didn't post directly, because he was a bit afraid of spending too much time here, but indicated he might register if there's growing discussion. Vincent - thanks so much!

----------------------------------
Reply from Vincent Rivasseau
----------------------------------

The renormalization group in condensed matter was investigated in the 90's through modern field theoretic techniques by a group of mathematical physicists, including in particular Benfatto, Feldman, Gallavotti, Magnen, Trubowitz and myself.

We understood that in two space dimensions or more, the extended character of the Fermi surface singularity leads to a
RG very different from the (scalar) RG of ordinary QFT, which is governed by the point singularity of 1/p^2 at p=0. In particular the power counting is independent of the space-time dimension, and the leading graphs are chains of bubbles, similar to the ones leading the 1/N expansion of vector models.
This is because the leading elementary 4point graph is a certain type of bubble at zero external momentum.
Indeed at external momentum P the momenta q and q+P on the two lines of the bubble cannot run both over the full Fermi singularity; only at P=0 (for parity invariant Fermi singularities) there is maximal coincidence between the extended singularity on the two lines. There is also a related notion of locality, which works only for the leading graphs: indeed only for these graphs (at P=0) there is a phase cancellation which allows renormalization by a local counterterm of the initial Lagrangian type. Hence it is really a new RG type (in the sense used in the tensor track paper).

This was first explained in
An Intrinsic 1/N Expansion for Many Fermion System, avec J. Feldman, J. Magnen et E. Trubowitz, Europhys. Letters 24, 437 (1993). 35.
R. Shankar also wrote a pedagogic review on this, namely
Renormalization-group approach to interacting Fermions,
Rev Mod Phys 66 129-192 (1994).

There is in the BCS theory a phase transition namely the formation of the Cooper pair which is a Boson. Its propagator is the sum of the chain of bubbles of the Fermionic theory. But it has no Fermi surface. Hence the power counting for that resulting Boson behaves in the infrared as an ordinary 1/p^2 propagator, and this effect can be studied in detail. Therefore BCS is a well-understood case of change of RG type from vector to scalar type (see eg arXiv.cond-mat/9503047).

The hope is that the leading graphs of a suitable renormalizable TFT could generate the propagator of the graviton. If this is turns out to be true, the main problem of non-renormalizability of QG on ordinary space time would be solved in a satisfying way, ie without imposing an arbitrary cutoff on the theory. A more complicated and perhaps more realistic scenario would involve a cascade of transitions, eg from tensor to matrix (ie non commutative QFT's), then from matrix to vectors and scalars. Such a more complicated scenario could perhaps accommodate better the matter fields of the standard model and their interactions.

Best wishes
V. Rivasseau

 

[atyy]

I'm not sure Giddings's new paper is related to the tensor networks of condensed matter physics, but he does say "tensor network"! He also says that if AdS/CFT works, then maybe he is describing something that is part of AdS/CFT, which sounds a bit like this. With quantum mechanics maybe giving rise to statistical mechanics, Zurek's proposed derivation of the Born rule, and all the Bell's theorem stuff, I think it makes sense to imagine that evolution is still unitary for this round of the game.

http://arxiv.org/abs/1201.1037
Black holes, quantum information, and unitary evolution
Steven B. Giddings

 

[atyy]

Giddings has a talk about his stuff. In discussions with the audience it is mentioned that this seems similar to stuff from quantum information theory. Evenbly's thesis reviews the quantum circuit interpretation of MERA, as well as Swingle's idea that MERA and AdS/CFT are related.

Hilbert Space Networks and Unitary Models for Black Hole Evolution
http://online.itp.ucsb.edu/online/bitbranes12/giddings/

 

[atyy]

Hardy has remarks on quantum gravity in the final section of his essay. Markopoulou's quantum causal histories and Vidal's MERA are cited.

http://arxiv.org/abs/1201.4390
The Operator Tensor Formulation of Quantum Theory
Lucien Hardy

"The challenge of setting up quantum field theory is to work out how to take the limit of this situation to the infinitesimal (rather than discrete) case. However, this framework offers certain advantages as an approach to quantum field theory. Namely, it provides a formulation which is in keeping with the spirit of special relativity without necessary reference to any specific foliation.

This framework might also provide a good stepping stone to a theory of quantum gravity. Formalism locality, as a desirable property, was motivated by considerations from quantum gravity [29]."

One great challenge facing applying these techniques to quantum field theory and, possibly, to quantum gravity, is to know how to adapt or reproduce that relevant physics which is usually formulated in terms of differential equations using the Turing inspired ideas of computer science.

 

[atyy]

Jacobson gave an interesting talk Vacuum Entanglement Entropy, Horizon Thermodynamics and Gravitation. He mentions that entanglement is related to the rigidity of spacetime.

Before that van Raamsdonk had an equally interesting talk about Rindler quantum gravity. The paper by Czech et al The Gravity Dual of a Density Matrix says "Conversely, knowledge of the bulk geometry at successively greater distance from the boundary requires knowledge of entanglement at successively longer scales" with an explicit citation of Swingle's observations about MERA and holography.

 

[marcus]

That talk by Jacobson is great.
http://online.kitp.ucsb.edu/online/bitbranes_c12/jacobson/
In line with what you said, he relates the amount of entanglement across an horizon with 1/G the reciprocal of the Newton constant. G measures how easily the geometry can be deformed by stress-energy and so the reciprocal 1/G is a measure of "stiffness"

The talk itself is some 31 minutes, if I remember, but then with questions it runs to 44 minutes.
The essential, highly accessible portion I would say, is the first 18 or 19 minutes which REVIEWS the famous ideas of GR as the equation of state of unspecfied micro degrees of freedom. I would strongly recommend the first 18 or so minutes.

After that he talks about higher curvature terms and generalizations---newer work.

Rafael Sorkin is there and asks questions. Also Erik or Hermann Verlinde. Gary Gibbons also converses with TJ at the end.

 

[marcus]

That talk by Jacobson is great.
http://online.kitp.ucsb.edu/online/bitbranes_c12/jacobson/
In line with what you said, he relates the amount of entanglement across an horizon with 1/G the reciprocal of the Newton constant. G measures how easily the geometry can be deformed by stress-energy and so the reciprocal 1/G is a measure of "stiffness"

The talk is some 31 minutes, if I remember, but then with questions it runs to 44 minutes.
The essential, highly accessible portion I would say, is the first 18 or 19 minutes which REVIEWS the famous ideas of GR as the equation of state of unspecfied micro degrees of freedom. I would strongly recommend the first 18 or so minutes.

After that he talks about higher curvature terms and generalizations---newer work.

Sorkin is there and asks questions.

 

[atyy]

Basic question about the Jacobson stuff: in the Clausius relation dS=dQ/T, I think the heat flow must be reversible. Why is the energy flow across the horizon reversible?

 

[atyy]

Donnelly has a paper about the entanglement entropy of lattice gauge theory in the language of LQG - spin networks, intertwiners etc. It's interesting because of work on the holographic entanglement entropy, which is reviewed by Takayanagi.

http://arxiv.org/abs/1109.0036
Decomposition of entanglement entropy in lattice gauge theory
William Donnelly

"We note also that the Hilbert space of edge states in SU(2) lattice gauge theory is closely related to the Hilbert space of the SU(2) Chern-Simons theory whose states are counted in the loop quantum gravity derivation of black hole entropy [22, 23]."

http://arxiv.org/abs/1204.2450
Entanglement Entropy from a Holographic Viewpoint
Tadashi Takayanagi

"The upshot is that the area of a minimal surface in a (Euclidean) gravitational theory corresponds to the entanglement entropy in its dual non-gravitational theory"

"The lattice calculations [86, 87] (see also [88]) of pure Yang-Mills theory qualitatively confirm this prediction from AdS/CFT, though the order of phase transition is no longer first order for these finite N calculations."

 

[marcus]

That talk by Jacobson is great.
http://online.kitp.ucsb.edu/online/bitbranes_c12/jacobson/
In line with what you said, he relates the amount of entanglement across an horizon with 1/G the reciprocal of the Newton constant. G measures how easily the geometry can be deformed by stress-energy and so the reciprocal 1/G is a measure of "stiffness"

The talk is some 31 minutes, if I remember, but then with questions it runs to 44 minutes.
The essential, highly accessible portion I would say, is the first 18 or 19 minutes which REVIEWS the famous ideas of GR as the equation of state of unspecfied micro degrees of freedom. I would strongly recommend the first 18 or so minutes.

After that he talks about higher curvature terms and generalizations---newer work.

Sorkin is there and asks questions.

Basic question about the Jacobson stuff: in the Clausius relation dS=dQ/T, I think the heat flow must be reversible. Why is the energy flow across the horizon reversible?

I was curious about the same thing. He is invoking relations that assume equilibrium. Where is the equilibrium? Some of the horizons he describes are causal. How can stuff pass back and forth? I was hoping someone would take up your question. Maybe we can repeat the question in some other context or later in this thread.

I think everybody knows that Jacobson is beyond intuitive. What they want from him is a talk which is partly heuristic. Inventing and exploring concepts, looking at stuff in new ways. Perhaps they don't want him to spend time on rigorous detail. So maybe there actually are logical gaps in a Jacobson talk. Sometimes the gaps themselves could be inspirational? But there could be a clear answer to your question, that somebody else might point out.

I believe I saw David Gross, Gary Gibbons, one or both Verlinde at the talk, as well as Rafael Sorkin.

 

[atyy]

I was curious about the same thing. He is invoking relations that assume equilibrium. Where is the equilibrium? Some of the horizons he describes are causal. How can stuff pass back and forth? I was hoping someone would take up your question. Maybe we can repeat the question in some other context or later in this thread.

I think everybody knows that Jacobson is beyond intuitive. What they want from him is a talk which is partly heuristic. Inventing and exploring concepts, looking at stuff in new ways. Perhaps they don't want him to spend time on rigorous detail. So maybe there actually are logical gaps in a Jacobson talk. Sometimes the gaps themselves could be inspirational? But there could be a clear answer to your question, that somebody else might point out.

I believe I saw David Gross, Gary Gibbons, one or both Verlinde at the talk, as well as Rafael Sorkin.

Jacobson has some comments in his original paper. I don't follow the reasoning entirely. I guess the basic idea is that one can associate a temperature to any Rindler horizon, because of the Unruh effect from special relativistic QFT. If one can assign a temperature, presumably the system is quasi-static and close enough to equilibrium. Then reversibility would come down to the work done being "frictionless", which I guess I can buy.

"However, in general, such a system is not in “equilibrium” because the horizon is expanding, contracting, or shearing. Since we wish to apply equilibrium thermodynamics, the system is further specified as follows. The equivalence principle is invoked to view a small neighborhood of each spacetime point p as a piece of flat spacetime."

 

[atyy]

Takayanagi's talk Developments of Holographic Entanglement Entropy discusses Swingle's MERA/AdS/CFT conjecture, as well as Haegeman-Osborne-Verschelde-Verstraete's proposal for a continuum version of MERA starting at 30:00 minutes.

A questioner at around 37:42 asks whether MERA isn't more like dS/CFT! Takayanagi thinks not, but says he doesn't know much about dS/CFT.

Exciting! Looks like people are coming round to thinking LQG may contain string theory and hence gravity

 

[atyy]

Is Jacob Biamonte the best dressed physicist you've ever seen? He's got a lecture series on tensor network states. (Un-understandable! He's a Baez collaborator so it's terribly mathematical.)

As does Robert Pfeifer, whose paper in the OP mentioned LQG explicitly. (Seems quite accessible:)

PI's Tensor Networks for Quantum Field Theories Conference had lots of interesting talks, including one by Vidal on Tensor Networks and Geometry, the Renormalization Group and AdS/CFT. (All seem at the normal physics level and quite accessible:)

 

[atyy]

Here is a very interesting essay that uses Jacobson's argument.

http://arxiv.org/abs/1111.4948v2
Holographic Theories of Inflation and Fluctuations
Tom Banks, Willy Fischler
"The space-time geometry is encoded in the overlap rules, which supply both the conformal factor and the causal structure of the emergent metric, for large Hilbert spaces, which correspond to large causal diamonds. The Bekenstein-Hawking area law is built into our construction, so that, following Jacobson [5], we can assert that the geometry satisfies Einstein’s equations, with a stress tensor whose integrals are related to the thermodynamic averages of the Hamiltonian of local Rindler observers, with infinite acceleration."

In an earlier essay, Banks thinks the universe is a takeaway, in contrast, I think, to the restaurant at the end of the universe.

 

[atyy]

http://benasque.org/2012network/talks_contr/175_Molina.pdf
Holography, Tensor Networks and correlations between disjoint regions at criticality
Pasquale Sodano

http://arxiv.org/abs/1108.1277
Holographic View on Quantum Correlations and Mutual Information between Disjoint Blocks of a Quantum Critical System
Javier Molina-Vilaplana, Pasquale Sodano
(Submitted on 5 Aug 2011 (v1), last revised 23 Sep 2011 (this version, v2))
In (d+1) dimensional Multiscale Entanglement Renormalization Ansatz (MERA) networks, tensors are connected so as to reproduce the discrete, (d + 2) holographic geometry of Anti de Sitter space (AdSd+2) with the original system lying at the boundary. We analyze the MERA renormalization flow that arises when computing the quantum correlations between two disjoint blocks of a quantum critical system, to show that the structure of the causal cones characteristic of MERA, requires a transition between two different regimes attainable by changing the ratio between the size and the separation of the two disjoint blocks. We argue that this transition in the MERA causal developments of the blocks may be easily accounted by an AdSd+2 black hole geometry when the mutual information is computed using the Ryu-Takayanagi formula. As an explicit example, we use a BTZ AdS3 black hole to compute the MI and the quantum correlations between two disjoint intervals of a one dimensional boundary critical system. Our results for this low dimensional system not only show the existence of a phase transition emerging when the conformal four point ratio reaches a critical value but also provide an intuitive entropic argument accounting for the source of this instability. We discuss the robustness of this transition when finite temperature and finite size effects are taken into account.

http://arxiv.org/abs/1109.5592
Connecting Entanglement Renormalization and Gauge/Gravity dualities
Javier Molina-Vilaplana
(Submitted on 26 Sep 2011 (v1), last revised 24 Oct 2011 (this version, v2))
In this work we provide additional support for the proposed connection between the gauge/gravity dualities in string theory and the successful Multi-Scale-Entanglement-Renormalization-anstaz (MERA) method developed for the efficient simulation of quantum many body systems at criticality. This support comes by showing an explicit formal equivalence between the real space renormalization group (RG) flow of the two point correlation functions in different types of MERA states and the holographic RG flow of these correlation functions in asymptotically Anti de Sitter (AdS) spacetimes. These observations may be useful in order to formalize and make more precise the connection between the properties of different MERA states and their potential holographic descriptions.

 

[atyy]

Subir Sachdev gave a nice talk yesterday at the Perimeter.

http://pirsa.org/12070010
Entanglement, Holography, and the Quantum Phases of Matter

One of the papers discussed is Huijse, Sachdev & Swingle's Hidden Fermi surfaces in compressible states of gauge-gravity duality. The paper makes use of the null energy condition to obtain Eq (2.12), and at 58:00 there is a question from the audience whether the null energy condition is corrected by quantum effects. Unfortunately, I can't hear the discussion clearly enough to make out the conclusion they come to.

 

[atyy]

There's a new paper Holographic Geometry of Entanglement Renormalization in Quantum Field Theories from Nozaki, Ryu and Takayanagi.

They make an interesting comment on p23 about what it means in MERA when the gravity dual is classical:

"In AdS/CFT, we need to take the large N and strong coupling limit of gauge theories in order to realize the classical gravity limit (or equally Einstein gravity limit) where the holographic formula (1) can be applied. If we abandon the strong coupling limit, we expect higher derivative corrections to the Einstein gravity and the holographic entanglement entropy also includes higher derivative terms [43, 44]. If we do not take the large N limit, the gravity theory receives substantial quantum gravity corrections and the effective gravity action will become highly non-local, for which the holographic entanglement entropy has not been calculated at present. Therefore, one may wonder how these two limits can be seen in the MERA. Though we are not going to address a definite answer to this question, we can suggest a related important idea. In order to justify the identification (74), we need to assume that the all relevant bonds are (almost) maximally entangled. If this is not the case, the precise estimation of the entanglement entropy gets quite complicated, and we need the information of entanglement of bonds which is far from the minimal surface γA. Therefore, in such situations, calculations are expected to be “non-local” in the sense of tensor network geometry, which is identified with a AdS space. This may correspond to the fact that the bulk gravity becomes non-local if we do not take the large N limit."

 

[Physics Monkey]

Indeed, in my original paper I tried to emphasize that the identification of the minimal curve in the MERA graph really only gives a bound on the entanglement, although in numerical practice the bonds tend to all give the same contribution.

This issue has been a bit of a puzzle actually, since MERA seems to work too well i.e. is too like classical gravity even when we expect that the putative dual theory shouldn't be. On the other hand, maybe the gravity theory is always quasi-local on some scale. After all, the RG equations are local in any QFT. In general, it seems that there are many different length scales at which we can discuss locality, including the Planck length, the string length, and the AdS radius (approaching from the holographic side).