Condensed matter physics, area laws & LQG?

marcus

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

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/talk...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).

atyy

I naively think of renormalization as usually acting on the Hamiltonian or action of the theory, and it's not very obvious what entanglement is doing, which I think of more as a property of a wave function. OTOH, MERA seems something like "wave function renormalization"? Are these two sorts of renormalization related? Off the top of my head, I can think of this paper by Balasubramanian and colleagues, who mention at the end that their work might be related to the connection you drew.

audioloop

not LQG but related

http://arxiv.org/pdf/1203.5367.pdf

...Condensed matter physics provides a third conceptual framework...
...In this context, QG can be treated analogously to crystallographic models of condensed matter physics...
...Quantum graphity is a background independent model that provides an alternative viewpoint on the notion and structure of space, based on condensed matter concepts but extended to a dynamic quantum lattice...

http://arxiv.org/pdf/hep-th/0611197v1.pdf

...As the system cools and the temperature drops, however, one or more phase transitions may occur in which the j degrees of freedom will become frozen. How the system cools depends on the relations between different coupling constants...

atyy

The Gravity Dual of a Density Matrix by Bartlomiej Czech, Joanna L. Karczmarek, Fernando Nogueira and Mark Van Raamsdonk seems to address a similar question as Nozaki, Ryu and Takayanagi. They cite Ryu and Takayanagi's earlier work. On p11, they have a section about reconstructing bulk metrics from extremal surface areas. On p16, they discuss how spacetime could emerge from entanglement, and in footnote 25 explicitly cite Swingle's AdS/CFT/MERA paper as advocating a similar picture.

atyy

Spin networks in LQG have an interpretation as geometry.
http://arxiv.org/abs/1007.0402
http://arxiv.org/abs/1102.3660
"The mathematics needed to describe such quanta of space is provided by the theory of spin networks ... The other way around, the Hilbert space of SU(2) Yang-Mills lattice theory admits an interpretation as a description of quantized geometries, formed by quanta of space, as we shall see in a moment. This interpretation forms the content of the "spin-geometry" theorem by Roger Penrose, and an earlier related theorem by Hermann Minkowski. These two theorems ground the kinematics of LQG."

Some tensor networks also have an interpretation as geometry.
http://arxiv.org/abs/0905.1317
http://arxiv.org/abs/1106.1082
http://arxiv.org/abs/1208.3469
"By making a close contact with the holographic formula of the entanglement entropy, we propose a general definition of the metric in the MERA in the extra holographic direction, which is formulated purely in terms of quantum field theoretical data."

Are these two forms of geometry related?

atyy

Rovelli says "A generic state of the geometry is not a spin network state: it is a linear superposition of spin networks." And "Classically, each node represents a polyhedron, thanks to Minkowski's theorem, but the polyhedra picture holds only in the classical limit and cannot be taken literally in the quantum theory. ... In other words, these are "polyhedra" in the same sense in which a particle with spin is a "rotating body"."

Singh and Vidal have a new paper, in which they say "Consequently, a tensor network made of SU(2)-invariant tensors decomposes as a linear superposition of spin networks ... As a practical demonstration we describe the SU(2)-invariant version of the MERA ..."

Physics Monkey

There is definitely a relationship, not that I have a super clear idea what it is precisely. I'm actually struggling a lot with this at the moment, since as you point out there is a bit of a gulf between the very action oriented setup of holography and traditional field theory and the very state oriented setup of these modern quantum info methods. I would like to bridge this gulf a bit to bring the two closer together.

At a practical level, they seem to contain much of the same information. Certain scaling dimensions, operator product coefficients, central charges, and so on can be obtained from either method. In so far as these data define a conformal field theory, say, then the methods appear to contain the same information.

I think one very useful approach is to think about representing the operator [itex] e^{-\beta H}[/itex]. When thinking about this operator maybe its not so mysterious why Hamiltonian RG and wavefunction RG convey the same information.

marcus

Since you are exploring the relation between LQG and condensed matter physics, you might like to look over what I think in a sense replaces the 2011 paper of Rovelli you quote here (http://arxiv.org/abs/1102.3660 ).

This is a set of 26 slides from a talk given in Stockholm in July at the MG13 meeting.

http://www.cpt.univ-mrs.fr/~rovelli/...lmSpinFoam.pdf

It's an up-to-date condensed but fairly complete overview of Loop gravity, main results and open problems, as Rovelli sees it.

atyy

Hmm, apparently the string theorists also have more than one sense of renormalization. I was looking at the discussion in your paper, and you mentioned that Lawrence and Sever emphasize that the renormalization flow depends on the state, which was a surprise to me. I looked up citations to their work, and found that Heemskerk and Polchinski say that de Boer et al's renormalization is non-Wilsonian! In Heemskerk and Polchinski's terminology "Wilsonian couplings and evolution depend only on scales above the cutoff and are independent of the state."