Condensed matter physics, area laws & LQG?

[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).

 

[atyy]

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).

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]

I think Markopoulou and Oriti have been sniffing this out a long time. Note that Wen has heuristically linked tensor networks and AdS/CFT (strings!) in the final slide of http://dao.mit.edu/~wen//09QHtop.pdf.

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]

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.

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 e^{-\beta H}. When thinking about this operator maybe its not so mysterious why Hamiltonian RG and wavefunction RG convey the same information.

 

[marcus]

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 ..."

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/RovelliStockholmSpinFoam.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]

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 e^{-\beta H}. When thinking about this operator maybe its not so mysterious why Hamiltonian RG and wavefunction RG convey the same information.

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."

 

[atyy]

There is an interesting comment in McGreevy's notes that the locality in Wilsonian renormalization is not as local as the locality in AdS/CFT. He refers to a paper by Heemskerk and colleagues who say "Thus, energy-radius holography nicely explains part of the emergence of the bulk spacetime, but also misses a critical aspect. The existence of locality down to a fixed physical scale that can be parametrically smaller than the AdS length remains a mystery in the CFT. Thus, we refer to coarse holography and sharp holography, and it is the latter that we seek to explain".

Also, LQG doesn't seem to assign geometries to generic spin network states, except in the large spin limit. More generally the states seem to be twisted geometries. But I don't know if LQG's conception of geometry of a tensor network state is the same as that as proposed for condensed matter.

 

[atyy]

Regarding action and wave function renormalization, the other place where I've seen a statement about the state during action renormalization is in Tom Banks's QFT text. IIRC, I believe he says renormalization assumes all the high energy degrees of freedom are in their ground state. I don't know what exactly that means, maybe the Balasubramanian paper is explaining the same idea?

Edit: I looked up Banks's comment (p138): "In QFT we will always assume that the high frequency degrees of freedom are in their ground state, so that the Green function is the one defined by Feynman ..."

 

[Physics Monkey]

It's not completely clear to me what these sorts of statements mean since the high energy dof don't really exist on their own. From a condensed matter perspective the high energy modes are changing all the time and the type of low energy theory you have determines the type of RG you do e.g. scaling to a point in critical theories or scaling to a surface in fermi surface systems.

I don't know how much this really differs from the standard wilsonian picture (which we all certainly use regularly). Morally speaking it seems that the rg flow ofter wanders into difficult to navigate terrain where the effective degrees of freedom change dramatically e.g. electrons at high energy and anyons at low energy.

 

[Physics Monkey]

Area laws, condensed matter, and a little bit of holography/gravity make an appearance in an essay I wrote for FQXi http://fqxi.org/community/forum/topic/1559

Check it out and give me a vote if you like it.

 

[atyy]

I see your classic paper finally got accepted by PRD!

Here's another addition to the AdS/MERA literature:

Refined Holographic Entanglement Entropy for the AdS Solitons and AdS black Holes
Masafumi Ishihara, Feng-Li Lin, Bo Ning
"We consider the refinement of the holographic entanglement entropy on a disk region for the holographic dual theories to the AdS solitons ... based on AdS/MERA conjecture, we postulate that the IR fixed-point state for the non-extremal AdS soliton is a trivial product state."

I don't immediately see a relation between the geometrical interpretations of AdS/MERA and LQG's spin networks, but let me list the one LQG paper I know that makes an explicit but bizarre connection to AdS/CFT:

Holomorphic Factorization for a Quantum Tetrahedron
Laurent Freidel, Kirill Krasnov, Etera R. Livine
"Interestingly, the integration kernel that appears in the decomposition formula turns out to be the n-point function of bulk/boundary dualities of string theory. ... For the case n=4, the symplectic manifold in question has the interpretation of the space of “shapes” of a geometric tetrahedron with fixed face areas, and our results provide a description for the quantum tetrahedron in terms of holomorphic coherent states."

 

[atyy]

http://arxiv.org/abs/1209.3304
Constructing holographic spacetimes using entanglement renormalization
Brian Swingle
(Submitted on 14 Sep 2012)
We elaborate on our earlier proposal connecting entanglement renormalization and holographic duality in which we argued that a tensor network can be reinterpreted as a kind of skeleton for an emergent holographic space. Here we address the question of the large N limit where on the holographic side the gravity theory becomes classical and a non-fluctuating smooth spacetime description emerges. We show how a number of features of holographic duality in the large N limit emerge naturally from entanglement renormalization, including a classical spacetime generated by entanglement, a sparse spectrum of operator dimensions, and phase transitions in mutual information. We also address questions related to bulk locality below the AdS radius, holographic duals of weakly coupled large N theories, Fermi surfaces in holography, and the holographic interpretation of branching MERA. Some of our considerations are inspired by the idea of quantum expanders which are generalized quantum transformations that add a definite amount of entropy to most states. Since we identify entanglement with geometry, we thus argue that classical spacetime may be built from quantum expanders (or something like them).

Goes beyond the original AdS/MERA paper by using "we" - not sure whether that's royal or not Snow monkeys are Japanese, so it's probably the latter.

 

[Physics Monkey]

I never heard of Sun Wukong before, but I like him.

 

[atyy]

Me too! http://www.youtube.com/watch?v=mOV4JUOb9j0&feature=fvwrel

 

[atyy]

@Physics Monkey, I'm still reading your latest paper slowly, but just wanted to say that it's very nicely written that even a lay person like me can understand it! It formulates more sharply all the vague questions I've been having, and begins to answer them.

 

[Physics Monkey]

@atyy, thanks a lot for your kind comment. I'm glad you found it vaguely comprehensible.

 

[atyy]

http://arxiv.org/abs/1210.6759
Holographic Entanglement Entropy of AdS Solitons and Tensor Network States
Javier Molina-Vilaplana
(Submitted on 25 Oct 2012)
The recent proposal connecting the AdS/CFT correspondence and entanglement renormalization tensor network states (MERA) is investigated by showing that the entanglement entropy and the two point functions in a type of hybrid tensor network state composed by a finite number of MERA layers and a matrix product state (MPS) acting as a cap layer, imitate the behaviour of the holographic entanglement entropy and the two point functions in the AdS soliton geometry. Within the context of AdS/CFT, AdS solitons represent theories with a mass gap, i.e gapped systems. From these observations, an explicit connection between the entanglement structure of the tensor network and those parameters which define the AdS soliton geometry is provided.

 

[atyy]

http://arxiv.org/abs/1210.7244
Entanglement entropy in de Sitter space
Juan Maldacena, Guilherme L. Pimentel
"We then study the entanglement entropy of field theories with a gravity dual. When the dual is known, we use the proposal of [10,11] to calculate the entropy. It boils down to an extremal area problem. The answer for the entanglement entropy depends drasticallyon the properties of the gravity dual. In particular, if the gravity dual has a hyperbolic Friedman-Robertson-Walker spacetime inside, then there is a non-zero contribution at order N2 for the “interesting” piece of the entanglement entropy. Otherwise, the order N2 contribution vanishes."

 

[atyy]

It does seem that the relationship between renormalization flow and holography is not well understood. Here is an interesting article about scheme dependence.

http://arxiv.org/abs/1211.1729
Holographic interpretations of the renormalization group
Vijay Balasubramanian, Monica Guica, Albion Lawrence
(Submitted on 7 Nov 2012 (v1), last revised 27 Nov 2012 (this version, v2))
In semiclassical holographic duality, the running couplings of a field theory are conventionally identified with the classical solutions of field equations in the dual gravitational theory. However, this identification is unclear when the bulk fields fluctuate. Recent work has used a Wilsonian framework to propose an alternative identification of the running couplings in terms of non-fluctuating data; in the classical limit, these new couplings do not satisfy the bulk equations of motion. We study renormalization scheme dependence in the latter formalism, and show that a scheme exists in which couplings to single trace operators realize particular solutions to the bulk equations of motion, in the semiclassical limit. This occurs for operators with dimension \Delta \notin \frac{d}{2} + \mathbb{Z}, for sufficiently low momenta. We then clarify the relation between the saddle point approximation to the Wilsonian effective action (S_W) and boundary conditions at a cutoff surface in AdS space. In particular, we interpret non-local multi-trace operators in S_W as arising in Lorentzian AdS space from the temporary passage of excitations through the UV region that has been integrated out. Coarse-graining these operators makes the action effectively local.

Not directly related, but MERA fans may like to see how other people use the word "disentangle": http://techtalks.tv/talks/opening-remarks/57645/ (13:45)

 

[atyy]

marcus has listed an interesting new paper in his bibliography. It shows the LQG people are thinking about AdS/CFT and using MERA as a tool to understand it. Swingle's original paper is cited. Bianchi needs to read the new paper too, and link it up with Friedel, Krasnov, and Livine's mysterious observation I'm also glad they are thinking about induced gravity. Weinberg and Witten explicitly say it evades their no-go theorem.

http://arxiv.org/abs/1212.5183
On the Architecture of Spacetime Geometry
Eugenio Bianchi, Robert C. Myers
(Submitted on 20 Dec 2012)
We propose entanglement entropy as a probe of the architecture of spacetime in quantum gravity. We argue that the leading contribution to this entropy satisfies an area law for any sufficiently large region in a smooth spacetime, which, in fact, is given by the Bekenstein-Hawking formula. This conjecture is supported by various lines of evidence from perturbative quantum gravity, simplified models of induced gravity and loop quantum gravity, as well as the AdS/CFT correspondence.

 

[marcus]

As I recall, Physicsmonkey indicated he was Brian Swingle earlier in this thread, so there is a PF connection! Not only does Bianchi cite Brian's paper but he and coauthor thank him in the acknowledgments, for discussions.

 

[John86]

http://arxiv.org/abs/1212.5121
Modular transformation and bosonic/fermionic topological orders in Abelian fractional quantum Hall states
Xiao-Gang Wen
(Submitted on 20 Dec 2012)
The non-Abelian geometric phases of the degenerate ground states was proposed as a physically measurable defining properties of topological order in 1990. In this paper we discuss in detail such a quantitative characterization of topological order, using generic Abelian fractional quantum Hall states as examples. We show that the non-Abelian geometric phases not only contain information about the quasi-particle statistics, they also contain information about the Hall viscosity and the chiral central charge of the edge states. The chiral central charge appears as the universal 1/A correction to the Hall viscosity (where A is the area of the space). Thus, the non-Abelian geometric phases (both the Abelian part and the non-Abelian part) may provide a way to completely characterize 2D topological order. Also the non-Abelian part of the geometric phases gives rise to a projective representation of the modular group (or SL(2,Z)).

http://arxiv.org/abs/1212.4863
Boundary Degeneracy of Topological Order
Juven Wang, Xiao-Gang Wen
(Submitted on 19 Dec 2012)
We introduce the notion of boundary degeneracy of topologically ordered states on a compact orientable spatial manifold with boundaries, and emphasize that it provides richer information than the bulk degeneracy. Beyond the bulk-edge correspondence, we find the ground state degeneracy of fully gapped edge states depends on boundary gapping conditions. We develop a quantitative description of different types of boundary gapping conditions by viewing them as different ways of non-fractionalized particle condensation on the boundary. This allows us to derive the ground state degeneracy formula in terms of boundary gapping conditions, which reveals the fusion algebra of fractionalized quasiparticles. We apply our results to Toric code and Levin-Wen string-net models. By measuring the boundary degeneracy on a cylinder, we predict Z_k gauge theory and U(1)_k x U(1)_k non-chiral fractional quantum hall state at even integer k can be experimentally distinguished. Our works refine definitions of symmetry protected topological order and intrinsic topological order.

http://arxiv.org/abs/1212.2121
2D Lattice Model Construction of Symmetry-Protected Topological Phases
Peng Ye, Xiao-Gang Wen
(Submitted on 10 Dec 2012)
We propose a general approach to construct symmetry protected topological (SPT) states (ie the short-range entangled states with symmetry) in 2D spin/boson systems on lattice. In our approach, we fractionalize spins/bosons into different fermions, which occupy nontrivial Chern bands. After the Gutzwiller projection of the free fermion state obtained by filling the Chern bands, we can obtain SPT states on lattice. In particular, we constructed a U(1) SPT state, a SO(3) SPT state, and a SU(2) SPT state on lattice.

http://arxiv.org/abs/1212.1827
Quantized topological terms in weak-coupling gauge theories with symmetry and their connection to symmetry enriched topological phases
Ling-Yan Hung, Xiao-Gang Wen
(Submitted on 8 Dec 2012)
We study the quantized topological terms in a weak-coupling gauge theory with gauge group $G_g$ and a global symmetry $G_s$ in $d$ space-time dimensions. We show that the quantized topological terms are classified by a pair $(G,\nu_d)$, where $G$ is an extension of $G_s$ by $G_g$ and $\nu_d$ an element in group cohomology $\mathcal{H}^d(G,\R/\Z)$. When $d=3$ and/or when $G_s$ is finite, the weak-coupling gauge theories with quantized topological terms describe gapped symmetry enriched topological (SET) phases (ie gapped long-range entangled phases with symmetry). Thus those SET phases are classified by $\mathcal{H}^d(G,\R/\Z)$, where $G/G_g=G_s$. We also apply our theory to a simple case $G_s=G_g=Z_2$, which leads to 12 different SET phases where quasiparticles have different patterns of fractional $G_s=Z_2$ quantum numbers and fractional statistics. If the weak-coupling gauge theories are gapless, then the different quantized topological terms may describe different gapless phases of the gauge theories with a symmetry $G_s$, which may lead to different fractionalizations of $G_s$ quantum numbers and different fractional statistics (if in 2+1D).

 

[atyy]

Not directly related, but MERA fans may like to see how other people use the word "disentangle": http://techtalks.tv/talks/opening-remarks/57645/ (13:45)

I was only kidding there - but it turns out that Jason Morton works on both tensor networks and deep learning - apparently with the same mathematics!

Andrew Critch, Jason Morton. Polynomial constraints on representing entangled qubits as matrix product states

Jason Morton, Jacob Biamonte. Undecidability in Tensor Network States

Jason Morton. https://www.ipam.ucla.edu/publications/gss2012/gss2012_10605.pdf

 

[jasonmorton]

Hey, Jason here, thanks for the mention! I absolutely do think there is a connection between MERA and Deep Learning, as mentioned in the paper with Critch. I am trying to work out the details and hope to have more news this spring.

 

[atyy]

Hey, Jason here, thanks for the mention! I absolutely do think there is a connection between MERA and Deep Learning, as mentioned in the paper with Critch. I am trying to work out the details and hope to have more news this spring.

That is very cool! I see I gave the wrong link for your paper with Critch above, so let me correct that.

http://arxiv.org/abs/1210.2812
Polynomial constraints on representing entangled qubits as matrix product states
Andrew Critch, Jason Morton
"A conjectured dictionary between tensor network state models and classical probabilistic graphical models was presented in [11]. In this dictionary, matrix product states correspond to hidden Markov models, the density matrix renormalization group (DMRG) algorithm to the forward-backward algorithm, tree tensor networks to general Markov models, projected entangled pair states (PEPS) to Markov or conditional random fields, and the multi-scale entanglement renormalization ansatz (MERA) loosely to deep belief networks. In this work we formalize the first of these correspondences and use it to algebraically characterize quantum states representable by MPS and study their identifiability."

Incidentally, I came across your work via http://keck.ucsf.edu/~surya/ thesis. He's a fellow neurobiologist who had Sturmfels on his thesis committee. His website says "Although during my graduate work I played around with black holes, eleven dimensions, and little loops of string, I am now more fascinated by the world of biology which is full of incredible amounts of data but a relative paucity of theoretical frameworks within which to interpret and understand this data." But perhaps there are black holes in the brain after all To be honest, the deep architectures are genuinely inspired by biology, and although most of the learning rules seem unphysiological, experimental neurobiologists like me do find the DNN work informative. I have to admit I find DNNs more intuituitive, and I do wonder why it's ok to transfer the weights from a DBN to a DNN.

 

[atyy]

A new paper which says that the mutual information describes entanglement at finite temperature better than the entanglement entropy. Both of Physics Monkey's AdS/MERA papers are cited.

http://arxiv.org/abs/1212.4764
Holographic Mutual Information at Finite Temperature
Willy Fischler, Arnab Kundu, Sandipan Kundu
(Submitted on 19 Dec 2012)
Using the Ryu-Takayanagi conjectured formula for entanglement entropy in the context of gauge-gravity duality, we investigate properties of mutual information between two disjoint rectangular sub-systems in finite temperature relativistic conformal field theories in d-spacetime dimensions and non-relativistic scale-invariant theories in some generic examples. In all these cases mutual information undergoes a transition beyond which it is identically zero. We study this transition in details and find universal qualitative features for the above class of theories which has holographic dual descriptions. We also obtain analytical results for mutual information in specific regime of the parameter space. This demonstrates that mutual information contains the quantum entanglement part of the entanglement entropy, which is otherwise dominated by the thermal entropy at large temperatures.

Incidentally, there was an interesting result that despite correlations diverging at criticality, the mutual information in a classical stat mech Ising model peaks away from criticality. The result seems to have been confirmed. Hoefully this means that the brain isn't critical;)

http://arxiv.org/abs/1011.4421
Mutual information in classical spin models
Johannes Wilms, Matthias Troyer, Frank Verstraete
"The total many-body correlations present in finite temperature classical spin systems are studied using the concept of mutual information. As opposed to zero-temperature quantum phase transitions, the total correlations are not maximal at the phase transition, but reach a maximum in the high temperature paramagnetic phase."

http://arxiv.org/abs/1210.5707
Information theoretic aspects of the two-dimensional Ising model
Hon Wai Lau, Peter Grassberger
"All this suggests strongly that it is the slope of the mutual information, not the mutual information itself, that diverge at the critical point."

 

[atyy]

http://arxiv.org/abs/1302.5703
Holographic Local Quenches and Entanglement Density
Masahiro Nozaki, Tokiro Numasawa, Tadashi Takayanagi
(Submitted on 22 Feb 2013)
We propose a free falling particle in an AdS space as a holographic model of local quench. Local quenches are triggered by local excitations in a given quantum system. We calculate the time-evolution of holographic entanglement entropy. We confirm a logarithmic time-evolution, which is known to be typical in two dimensional local quenches. To study the structure of quantum entanglement in general quantum systems, we introduce a new quantity which we call entanglement density and apply this analysis to quantum quenches. We show that this quantity is directly related to the energy density in a small size limit. Moreover, we find a simple relationship between the amount of quantum information possessed by a massive object and its total energy based on the AdS/CFT.

"Now we would like to consider how to describe local quenches by using tensor networks. ...

Finally we would like to ask what is the holographic origin of gravitational force."

 

[atyy]

marcus highlighted this beautiful talk by Rivasseau in his bibliography.

http://pirsa.org/13020132/
Quantum Gravity as Random Geometry
Vincent Rivasseau
Abstract: Matrix models, random maps and Liouville field theory are prime tools which connect random geometry and quantum gravity in two dimensions. The tensor track is a new program to extend this connection to higher dimensions through the corresponding notions of tensor models, colored triangulations and tensor group field theories.
27/02/2013

 

[Physics Monkey]

Tensor networks get mentioned in the abstract!

http://arxiv.org/abs/1303.1080
Time Evolution of Entanglement Entropy from Black Hole Interiors
Thomas Hartman, Juan Maldacena
(Submitted on 5 Mar 2013)
We compute the time-dependent entanglement entropy of a CFT which starts in relatively simple initial states. The initial states are the thermofield double for thermal states, dual to eternal black holes, and a particular pure state, dual to a black hole formed by gravitational collapse. The entanglement entropy grows linearly in time. This linear growth is directly related to the growth of the black hole interior measured along "nice" spatial slices. These nice slices probe the spacelike direction in the interior, at a fixed special value of the interior time. In the case of a two-dimensional CFT, we match the bulk and boundary computations of the entanglement entropy. We briefly discuss the long time behavior of various correlators, computed via classical geodesics or surfaces, and point out that their exponential decay comes about for similar reasons. We also present the time evolution of the wavefunction in the tensor network description.

 

[atyy]

http://arxiv.org/abs/1303.6716
Symmetry protected entanglement renormalization
Sukhwinder Singh, Guifre Vidal
(Submitted on 27 Mar 2013)
Entanglement renormalization is a real-space renormalization group (RG) transformation for quantum many-body systems. It generates the multi-scale entanglement renormalization ansatz (MERA), a tensor network capable of efficiently describing a large class of many-body ground states, including those of systems at a quantum critical point or with topological order. The MERA has also been proposed to be a discrete realization of the holographic principle of string theory. In this paper we propose the use of symmetric tensors as a mechanism to build a symmetry protected RG flow, and discuss two important applications of this construction. First, we argue that symmetry protected entanglement renormalization produces the proper structure of RG fixed-points, namely a fixed-point for each symmetry protected phase. Second, in the context of holography, we show that by using symmetric tensors, a global symmetry at the boundary becomes a local symmetry in the bulk, thus explicitly realizing in the MERA a characteristic feature of the AdS/CFT correspondence.

http://arxiv.org/abs/1303.6772
Renormalization of an SU(2) Tensorial Group Field Theory in Three Dimensions
Sylvain Carrozza, Daniele Oriti, Vincent Rivasseau
(Submitted on 27 Mar 2013)
We address in this paper the issue of renormalizability for SU(2) Tensorial Group Field Theories (TGFT) with geometric Boulatov-type conditions in three dimensions. We prove that tensorial interactions up to degree 6 are just renormalizable without any anomaly. Our new models define the renormalizable TGFT version of the Boulatov model and provide therefore a new approach to quantum gravity in three dimensions. Among the many new technical results established in this paper are a general classification of just renormalizable models with gauge invariance condition, and in particular concerning properties of melonic graphs, the second order expansion of melonic two point subgraphs needed for wave-function renormalization.

 

[atyy]

I wonder if this is related to the line of thought in Jason Morton's work, mentioned above in #85-87.

http://arxiv.org/abs/1301.3124
Deep learning and the renormalization group
Cédric Bény
(Submitted on 14 Jan 2013 (v1), last revised 13 Mar 2013 (this version, v4))
Renormalization group (RG) methods, which model the way in which the effective behavior of a system depends on the scale at which it is observed, are key to modern condensed-matter theory and particle physics. We compare the ideas behind the RG on the one hand and deep machine learning on the other, where depth and scale play a similar role. In order to illustrate this connection, we review a recent numerical method based on the RG---the multiscale entanglement renormalization ansatz (MERA)---and show how it can be converted into a learning algorithm based on a generative hierarchical Bayesian network model. Under the assumption---common in physics---that the distribution to be learned is fully characterized by local correlations, this algorithm involves only explicit evaluation of probabilities, hence doing away with sampling.

For a comparison with standard ideas (but maybe this implementation is not so standard), how about:

http://www.cs.utexas.edu/~dana/nn.pdf
Predictive coding in the visual cortex
Rajesh P. N. Rao and Dana H. Ballard
"Lower levels operate on smaller spatial (and possibly temporal) scales, whereas higher levels estimate signal properties at larger scales because a higher-level module predicts and estimates the responses of several lower-level modules (for example, three in Fig. 1c). Thus, the effective RF size of units increases progressively until the highest level, where the RF spans the entire input image.

 

[democrito]

This is a new version of http://arxiv.org/abs/1210.6759, with a new title and new results on AdS/MERA

Holographic Geometries of one-dimensional gapped quantum systems from Tensor Network States

Javier Molina-Vilaplana

Abstract: We investigate a recent conjecture connecting the AdS/CFT correspondence and entanglement renormalization tensor network states (MERA). The proposal interprets the tensor connectivity of the MERA states associated to quantum many body systems at criticality, in terms of a dual holographic geometry which accounts for the qualitative aspects of the entanglement and the correlations in these systems. In this work, some generic features of the entanglement entropy and the two point functions in the ground state of one dimensional gapped systems are considered through a tensor network state. The tensor network is builded up as an hybrid composed by a finite number of MERA layers and a matrix product state (MPS) acting as a cap layer. Using the holographic formula for the entanglement entropy, here it is shown that an asymptotically AdS metric can be associated to the hybrid MERA-MPS state. The metric is defined by a function that manages the growth of the minimal surfaces near the capped region of the geometry. Namely, it is shown how the behaviour of the entanglement entropy and the two point correlators in the tensor network, remains consistent with a geometric computation which only depends on this function. From these observations, an explicit connection between the entanglement structure of the tensor network and the function which defines the geometry is provided.

 

[atyy]

http://arxiv.org/abs/1305.0856
The entropy of a hole in spacetime
Vijay Balasubramanian, Bartlomiej Czech, Borun D. Chowdhury, Jan de Boer
(Submitted on 3 May 2013)
We compute the gravitational entropy of 'spherical Rindler space', a time-dependent, spherically symmetric generalization of ordinary Rindler space, defined with reference to a family of observers traveling along non-parallel, accelerated trajectories. All these observers are causally disconnected from a spherical region H (a 'hole') located at the origin of Minkowski space. The entropy evaluates to S = A/4G, where A is the area of the spherical acceleration horizon, which coincides with the boundary of H. We propose that S is the entropy of entanglement between quantum gravitational degrees of freedom supporting the interior and the exterior of the sphere H.

http://arxiv.org/abs/1305.1064
On the Mutual Information between disconnected regions in AdS/CFT
Javier Molina-Vilaplana
(Submitted on 5 May 2013)
In this note, we compute the holographic mutual information between two separated circular regions in a 3+1 dimensional gauge theory dual to AdS$_5 \times S^5$ through the AdS/CFT correspondence, in the limit in which the separation $L$ between the regions is much larger than their sizes $a$. Our calculation uses some previous results concerning the holographic computation of the long distance correlator of two distant Wilson loops. We show that in these regimes, the holographic mutual information follows a power law decaying behaviour proportional to $\sqrt{\lambda}$ with $\lambda$ the t'Hooft coupling of the gauge theory. This result contradicts a conjectured sharp vanishing of the holographic mutual information between disjoint regions for the large separation regime and has been also compared with a recent result concerning a 3+1 dimensional free CFT.

 

[marcus]

This video appeared online yesterday:
http://pirsa.org/13050027/
Asymmetry protected emergent E8 symmetry
Speaker(s): Brian Swingle
Abstract: The E8 state of bosons is a 2+1d gapped phase of matter which has no topological entanglement entropy but has protected chiral edge states in the absence of any symmetry. This peculiar state is interesting in part because it sits at the boundary between short- and long-range entangled phases of matter. When the system is translation invariant and for special choices of parameters, the edge states form the chiral half of a 1+1d conformal field theory - an E8 level 1 Wess-Zumino-Witten model. However, in general the velocities of different edge channels are different and the system does not have conformal symmetry. We show that by considering the most general microscopic Hamiltonian, in particular by relaxing the constraint of translation invariance and adding disorder, conformal symmetry remerges in the low energy limit. The disordered fixed point has all velocities equal and is the E8 level 1 WZW model. Hence a highly entangled and highly symmetric system emerges, but only when the microscopic Hamiltonian is completely asymmetric.
Date: 06/05/2013 - 4:00 pm

 

[atyy]

http://arxiv.org/abs/1305.1045
A lattice non-perturbative definition of an SO(10) chiral gauge theory and its induced standard model
Xiao-Gang Wen
(Submitted on 5 May 2013)
The standard model is a chiral gauge theory where the gauge fields couple to the right-hand and the left-hand fermions differently. The standard model is defined perturbatively and describes all elementary particles (except gravitons) very well. However, for a long time, we do not know if we can have a non-perturbative definition of standard model as a Hamiltonian quantum mechanical theory. In this paper, we propose a way to give a modified standard model (with 48 two-component Weyl fermions) a non-perturbative definition by embeding the modified standard model into a SO(10) chiral gauge theory and then putting the SO(10) chiral gauge theory on a 3D spatial lattice with a continuous time. Such a non-perturbatively defined standard model is a Hamiltonian quantum theory with a finite-dimensional Hilbert space for a finite space volum. Using the defining connection between gauge anomalies and the symmetry-protected topological orders, we show that any chiral gauge theory can be non-perturbatively defined by putting it on a lattice in the same dimension, as long as the chiral gauge theory is free of all anomalies.

http://arxiv.org/abs/1305.3621
Statistical Inference and String Theory
Jonathan J. Heckman
(Submitted on 15 May 2013)
In this note we expose some surprising connections between string theory and statistical inference. We consider a large collective of agents sweeping out a family of nearby statistical models for an M-dimensional manifold of statistical fitting parameters. When the agents making nearby inferences align along a d-dimensional grid, we find that the pooled probability that the collective reaches a correct inference is the partition function of a non-linear sigma model in d dimensions. Stability under perturbations to the original inference scheme requires the agents of the collective to distribute along two dimensions. Conformal invariance of the sigma model corresponds to the condition of a stable inference scheme, directly leading to the Einstein field equations for classical gravity. By summing over all possible arrangements of the agents in the collective, we reach a string theory. We also use this perspective to quantify how much an observer can hope to learn about the internal geometry of a superstring compactification. Finally, we present some brief speculative remarks on applications to the AdS/CFT correspondence and Lorentzian signature spacetimes.

http://arxiv.org/abs/1305.3908
Quantum Renormalization Group and Holography
Sung-Sik Lee
(Submitted on 16 May 2013)
Quantum renormalization group scheme provides a microscopic understanding of holography through a general mapping between the beta functions of underlying quantum field theories and the holographic actions in the bulk. We show that the Einstein gravity emerges as a long wavelength holographic description for a matrix field theory which has no other operator with finite scaling dimension except for the energy-momentum tensor. We also point out that holographic actions for general large N matrix field theories respect the inversion symmetry along the radial direction in the bulk if the beta functions of single-trace operators are gradient flows with respect to the target space metric set by the beta functions of double-trace operators.

 

[atyy]

I came across this by reading Motl's Maldacena, Susskind: any entanglement is a wormhole of a sort.

http://arxiv.org/abs/1306.0533
Cool horizons for entangled black holes
Juan Maldacena, Leonard Susskind
(Submitted on 3 Jun 2013)
General relativity contains solutions in which two distant black holes are connected through the interior via a wormhole, or Einstein-Rosen bridge. These solutions can be interpreted as maximally entangled states of two black holes that form a complex EPR pair. We suggest that similar bridges might be present for more general entangled states. In the case of entangled black holes one can formulate versions of the AMPS(S) paradoxes and resolve them. This suggests possible resolutions of the firewall paradoxes for more general situations.

 

[Physics Monkey]

Another couple of papers that draw some connections to MERA and holography:

arXiv:1306.0515
Passing through the Firewall
Erik Verlinde, Herman Verlinde

We propose that black hole information is encoded in non-local correlations between microscopic interior and exterior degrees of freedom. We give a simple qubit representation of this proposal, and show herein that for every black hole state, the apparent firewall can be removed via a universal, state independent unitary transformation. A central element in our discussion is the distinction between virtual qubits, which are in a specified vacuum state, and real qubits, that carry the free quantum information of the black hole. We outline how our proposal may be realized in AdS/CFT.


arXiv:1305.6694
Entanglement entropy in higher derivative holography
Arpan Bhattacharyya, Apratim Kaviraj, Aninda Sinha

We consider holographic entanglement entropy in higher derivative gravity theories. Recently Lewkowycz and Maldacena arXiv:1304.4926 have provided a method to derive the equations for the entangling surface from first principles. We use this method to compute the entangling surface in four derivative gravity. Certain interesting differences compared to the two derivative case are pointed out. For Gauss-Bonnet gravity, we show that in the regime where this method is applicable, the resulting equations coincide with proposals in the literature as well as with what follows from considerations of the stress tensor on the entangling surface. Finally we demonstrate that the area functional in Gauss-Bonnet holography arises as a counterterm needed to make the Euclidean action free of power law divergences.

 

[atyy]

I learned via Motl that Kenneth Wilson has died. His marriage of the long lines of work on renormalization in high energy physics with Kadanoff's critical insight in condensed matter is surely one of the great peaks of theoretical physics. Interestingly, according to Wikipedia, Steve White was his student. White invented the DMRG which was a stepping stone to the MERA.