Condensed matter physics, area laws & LQG?

[atyy]

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/talks/09QHtop.pdf.

http://arxiv.org/abs/0907.2994
Tensor network decompositions in the presence of a global symmetry
Sukhwinder Singh, Robert N. C. Pfeifer, Guifre Vidal
(Submitted on 17 Jul 2009)
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance also in the context of tensor network algorithms, thus setting the stage for cross-fertilization between these two areas of research.

http://arxiv.org/abs/0808.3773
Area laws for the entanglement entropy - a review
Authors: J. Eisert, M. Cramer, M.B. Plenio
(Submitted on 28 Aug 2008 (v1), last revised 16 Jan 2009 (this version, v3))
Physical interactions in quantum many-body systems are typically local: Individual constituents interact mainly with their few nearest neighbors. This locality of interactions is inherited by a decay of correlation functions, but also reflected by scaling laws of a quite profound quantity: The entanglement entropy of ground states. This entropy of the reduced state of a subregion often merely grows like the boundary area of the subregion, and not like its volume, in sharp contrast with an expected extensive behavior. Such "area laws" for the entanglement entropy and related quantities have received considerable attention in recent years. They emerge in several seemingly unrelated fields, in the context of black hole physics, quantum information science, and quantum many-body physics where they have important implications on the numerical simulation of lattice models. In this Colloquium we review the current status of area laws in these fields. Center stage is taken by rigorous results on lattice models in one and higher spatial dimensions. The differences and similarities between bosonic and fermionic models are stressed, area laws are related to the velocity of information propagation, and disordered systems, non-equilibrium situations, classical correlation concepts, and topological entanglement entropies are discussed. A significant proportion of the article is devoted to the quantitative connection between the entanglement content of states and the possibility of their efficient numerical simulation. We discuss matrix-product states, higher-dimensional analogues, and states from entanglement renormalization and conclude by highlighting the implications of area laws on quantifying the effective degrees of freedom that need to be considered in simulations.

http://arxiv.org/abs/0809.2393
Explicit tensor network representation for the ground states of string-net models
O. Buerschaper, M. Aguado, G. Vidal
(Submitted on 14 Sep 2008)
The structure of string-net lattice models, relevant as examples of topological phases, leads to a remarkably simple way of expressing their ground states as a tensor network constructed from the basic data of the underlying tensor categories. The construction highlights the importance of the fat lattice to understand these models.

 

[ensabah6]

Condensation and evolution of space-time network
Authors: Bi Qiao
(Submitted on 29 Sep 2008)

Abstract: In this work, we try to propose, in a novel way using the Bose and Fermi quantum network approach, a framework studying condensation and evolution of space time network described by the Loop quantum gravity. Considering quantum network connectivity features in the Loop quantum gravity, we introduce a link operator, and through extending the dynamical equation for the evolution of quantum network posed by Ginestra Bianconi to an operator equation, we get the solution of the link operator. This solution is relevant to the Hamiltonian of the network, and then is related to the energy distribution of network nodes. Showing that tremendous energy distribution induce huge curved space-time network, may have space time condensation in high-energy nodes. For example, in the black hole circumstances, quantum energy distribution is related to the area, thus the eigenvalues of the link operator of the nodes can be related to quantum number of area, and the eigenvectors are just the spin network states. This reveals that the degree distribution of nodes for space-time network is quantized, which can form the space-time network condensation. The black hole is a sort of result of space-time network condensation, however there may be more extensive space-time network condensation, for example, the universe singularity (big bang).


Quantum gravity as a Fermi liquid
Authors: Stephon H.S. Alexander, Gianluca Calcagni
(Submitted on 1 Jul 2008 (v1), last revised 21 Nov 2008 (this version, v2))

Abstract: We present a reformulation of loop quantum gravity with a cosmological constant and no matter as a Fermi-liquid theory. When the topological sector is deformed and large gauge symmetry is broken, we show that the Chern-Simons state reduces to Jacobson's degenerate sector describing 1+1 dimensional propagating fermions with nonlocal interactions. The Hamiltonian admits a dual description which we realize in the simple BCS model of superconductivity. On one hand, Cooper pairs are interpreted as wormhole correlations at the de Sitter horizon; their number yields the de Sitter entropy. On the other hand, BCS is mapped into a deformed conformal field theory reproducing the structure of quantum spin networks. When area measurements are performed, Cooper-pair insertions are activated on those edges of the spin network intersecting the given area, thus providing a description of quantum measurements in terms of excitations of a Fermi sea to superconducting levels. The cosmological constant problem is naturally addressed as a nonperturbative mass-gap effect of the true Fermi-liquid vacuum.

Comments: 45 pages, 1 figure; v2: discussion improved, version

Superconducting loop quantum gravity and the cosmological constant
Authors: Stephon H.S. Alexander, Gianluca Calcagni
(Submitted on 26 Jun 2008 (v1), last revised 23 Feb 2009 (this version, v2))

Abstract: We argue that the cosmological constant is exponentially suppressed in a candidate ground state of loop quantum gravity as a nonperturbative effect of a holographic Fermi-liquid theory living on a two-dimensional spacetime. Ashtekar connection components, corresponding to degenerate gravitational configurations breaking large gauge invariance and CP symmetry, behave as composite fermions that condense as in Bardeen-Cooper-Schrieffer theory of superconductivity. Cooper pairs admit a description as wormholes on a de Sitter boundary.

Comments: 10 pages; v2 matches the published version
Subjects: High Energy Physics - Theory (hep-th); Astrophysics (astro-ph); Superconductivity (cond-mat.supr-con); General Relativity and Quantum Cosmology (gr-qc)
Journal reference: Physics Letters B 672 (2009) 386
DOI: 10.1016/j.physletb.2009.01.046
Report number: IGC-08/6-5
Cite as: arXiv:0806.4382v2 [hep-th]

 

[ensabah6]

http://arxiv.org/abs/1002.1462

Embedding the Bilson-Thompson model in an LQG-like framework

Deepak Vaid
(Submitted on 8 Feb 2010)
We argue that the Quadratic Spinor Lagrangian approach allows us to approach the problem of forming a geometrical condensate of spinorial tetrads in a natural manner. This, along with considerations involving the discrete symmetries of lattice triangulations, lead us to discover that the quasiparticles of such a condensate are tetrahedra with braids attached to its faces and that these braid attachments correspond to the preons in Bilson-Thompson's model of elementary particles. These "spatoms" can then be put together in a tiling to form more complex structures which encode both geometry and matter in a natural manner. We conclude with some speculations on the relation between this picture and the computational universe hypothesis.

 

[Physics Monkey]

I personally find this whole area very exciting. For example, the spin networks used in loop quantum gravity can be greatly generalized and potentially even realized in condensed matter systems called string net states. Furthermore, there are some exciting hints relating the way one computes black hole entropy in loop quantum gravity and entanglement entropy in the tensor network approach. There are also connections between the tensor network approach and AdS/CFT as Wen notes at the end of his slides.

 

[atyy]

I personally find this whole area very exciting.

Me too!

For example, the spin networks used in loop quantum gravity can be greatly generalized and potentially even realized in condensed matter systems called string net states. Furthermore, there are some exciting hints relating the way one computes black hole entropy in loop quantum gravity and entanglement entropy in the tensor network approach. There are also connections between the tensor network approach and AdS/CFT as Wen notes at the end of his slides.

I wasn't aware of the link between string nets and spin networks until Buerschaper et al (string net -> tensor network) and Singh et al (tensor network -> spin network). Is there a more direct connection?

Also, what is the relationship between AdS/CFT and tensor networks? I remember reading a Horowitz and Polchinksi review that said AdS/CFT is an example of emergent gauge theory, which cited D'Adda 1978 - whom Levin and Wen also cite, so was a little aware that AdS/CFT and string nets had a common descent - but haven't any understanding beyond that.

Edit: Wow, I just saw you actually work on this stuff, unlike people like me who just read about it - very cool!

 

[atyy]

So it looks like Jal and I had a related conversation a while ago in posts 68-70 of his https://www.physicsforums.com/showthread.php?t=251509, with a quirky note by Michael Freedman pointing to papers by Brian Swingle and on to the entanglement entropy and holography.

 

[Physics Monkey]

Me too!

Outstanding!

I wasn't aware of the link between string nets and spin networks until Buerschaper et al (string net -> tensor network) and Singh et al (tensor network -> spin network). Is there a more direct connection?

The physical configurations in string net models are actually exactly like spin networks. The low energy physical subspace is the space of closed string states. However, closed string states may include branching with branching rules given by the analog of the vertex rules in SU(2) spin networks. For example, the state space of something like U(1) gauge theory can be thought of as trivalent graphs with edges labeled by integers and with vertices allowed when all the integers sum to zero at the vertex (with orientation). The branching rules for a theory like SU(2) are almost exactly the vertex rules for SU(2) spin networks. One subtlety is that in the string net models one is usually dealing with the so called quantum group. This is manifested in a limit to the size of the reps of SU(2) than can be used. SU(2) level k only allows reps up to $$j = k/2$$. The ground state of a string net model is some kind of superposition of all closed string states.

Also, what is the relationship between AdS/CFT and tensor networks? I remember reading a Horowitz and Polchinksi review that said AdS/CFT is an example of emergent gauge theory, which cited D'Adda 1978 - whom Levin and Wen also cite, so was a little aware that AdS/CFT and string nets had a common descent - but haven't any understanding beyond that.

Edit: Wow, I just saw you actually work on this stuff, unlike people like me who just read about it - very cool!

This is still an incompletely answered question. The development you refer to above is the notion that the low energy degrees of freedom may be quite different from the high energy degrees of freedom. For example, one may start with a lattice model of spins and obtain in the low energy description emergent fermions and gauge fields. Often, the emergent description is redundant (hence gauge theory) and invisible at high energies. AdS/CFT is an example of this in the sense that the useful emergent description of the $$\mathcal{N} = 4$$ theory is in terms of totally different variables. What is important in this comparison is the fact that the gravity theory is a redundant way (like a gauge theory) to compute physical quantities defined in the dual conformal field theory.

 

[Physics Monkey]

So it looks like Jal and I had a related conversation a while ago in posts 68-70 of his https://www.physicsforums.com/showthread.php?t=251509, with a quirky note by Michael Freedman pointing to papers by Brian Swingle and on to the entanglement entropy and holography.

That is me. The paper called "Entanglement Renormalization and Holography" is the beginnings of an attempt to see the emergence of AdS/CFT from the tensor network approach. In that paper I used a particular tensor network approach called the multi-scale entanglement renormalization ansatz (MERA) to argue for a holographic description of quantum states.

 

[atyy]

That is me. The paper called "Entanglement Renormalization and Holography" is the beginnings of an attempt to see the emergence of AdS/CFT from the tensor network approach. In that paper I used a particular tensor network approach called the multi-scale entanglement renormalization ansatz (MERA) to argue for a holographic description of quantum states.

Well, it's a pleasure to meet you! I'm a biologist, but I find this fascinating. I love Wen's work for its playfulness. Some time ago I noticed Wen began to distinguish his work from "old string theory", which meant, reading between the lines, that maybe it was related to new string theory, presumably AdS/CFT! Then last year, I noticed he began drawing lines between tensor networks and AdS/CFT in his final heuristic slide. I look forward to learning more about what you find out!

 

[ensabah6]

Outstanding!



The physical configurations in string net models are actually exactly like spin networks. The low energy physical subspace is the space of closed string states. However, closed string states may include branching with branching rules given by the analog of the vertex rules in SU(2) spin networks. For example, the state space of something like U(1) gauge theory can be thought of as trivalent graphs with edges labeled by integers and with vertices allowed when all the integers sum to zero at the vertex (with orientation). The branching rules for a theory like SU(2) are almost exactly the vertex rules for SU(2) spin networks. One subtlety is that in the string net models one is usually dealing with the so called quantum group. This is manifested in a limit to the size of the reps of SU(2) than can be used. SU(2) level k only allows reps up to $$j = k/2$$. The ground state of a string net model is some kind of superposition of all closed string states.




This is still an incompletely answered question. The development you refer to above is the notion that the low energy degrees of freedom may be quite different from the high energy degrees of freedom. For example, one may start with a lattice model of spins and obtain in the low energy description emergent fermions and gauge fields. Often, the emergent description is redundant (hence gauge theory) and invisible at high energies. AdS/CFT is an example of this in the sense that the useful emergent description of the $$\mathcal{N} = 4$$ theory is in terms of totally different variables. What is important in this comparison is the fact that the gravity theory is a redundant way (like a gauge theory) to compute physical quantities defined in the dual conformal field theory.

The old LQG article stated that LQG's spin networks can give rise to string nets, which Wen then shows can give rise to U(1), higgs, and SU(3)

SU(2) can be given but not chiral fermions. Is this true?

It was deleted out as "speculative" research

 

[Physics Monkey]

Well, it's a pleasure to meet you! I'm a biologist, but I find this fascinating. I love Wen's work for its playfulness. Some time ago I noticed Wen began to distinguish his work from "old string theory", which meant, reading between the lines, that maybe it was related to new string theory, presumably AdS/CFT! Then last year, I noticed he began drawing lines between tensor networks and AdS/CFT in his final heuristic slide. I look forward to learning more about what you find out!

It's nice to meet you too. Wen is my advisor, and it's very refreshing to be exposed to such different ways of thinking about things as well as being encouraged to do your own thing. I'm sure we'll be able to chat about this stuff more in the future.

PS What kind of biology do you do?

 

[Physics Monkey]

The old LQG article stated that LQG's spin networks can give rise to string nets, which Wen then shows can give rise to U(1), higgs, and SU(3)

SU(2) can be given but not chiral fermions. Is this true?

It was deleted out as "speculative" research

Chiral fermions are tough. A common trick in lattice gauge theory is to introduce an extra dimension which enables you to get chiral fermions in a sense. This mechanism is realized physically in the fractional quantum hall effect. Here there are gapless chiral fermions on the boundary of the sample, but there is a sense in which these fermions cannot live on their own, they need the bulk to exist even though it contains only gapped excitations. Also, the string net program does not describe these kinds of chiral phases and so they are less well understood.

 

[ensabah6]

Chiral fermions are tough. A common trick in lattice gauge theory is to introduce an extra dimension which enables you to get chiral fermions in a sense. This mechanism is realized physically in the fractional quantum hall effect. Here there are gapless chiral fermions on the boundary of the sample, but there is a sense in which these fermions cannot live on their own, they need the bulk to exist even though it contains only gapped excitations. Also, the string net program does not describe these kinds of chiral phases and so they are less well understood.

So what would be needed to generalize spinfoam/LQG spin networks to string net states and topological order, and could you use the theory to explain the 18 unexplained parameters of the SM?

 

[atyy]

It's nice to meet you too. Wen is my advisor, and it's very refreshing to be exposed to such different ways of thinking about things as well as being encouraged to do your own thing. I'm sure we'll be able to chat about this stuff more in the future.

PS What kind of biology do you do?

I had the great good fortune of having Wen supervise my undergraduate thesis quite some years ago. I was a clueless undergrad who wanted to learn a little physics before going off to neurobiology grad school, and he kindly made up something in quantum chaos that was accessible to me and lots of fun. Most of my work has been to use an experimental technique called "whole cell" recordings to study the synaptic inputs to neurons in the auditory cortex.

 

[space_cadet]

... One subtlety is that in the string net models one is usually dealing with the so called quantum group. This is manifested in a limit to the size of the reps of SU(2) than can be used. SU(2) level k only allows reps up to $$j = k/2$$ ...

When you consider that case of non-zero Cosmological constant $$\Lambda \neq 0$$ in LQG the spin-networks are also required to be labeled by reps of SU(2)_k. So that is not an obstacle. IMHO the string-net approach is equivalent to one containing spin-networks + many-body physics. It is the latter ingredient that is missing in most considerations of LQG, though that appears to be changing ...

 

[ensabah6]

When you consider that case of non-zero Cosmological constant $$\Lambda \neq 0$$ in LQG the spin-networks are also required to be labeled by reps of SU(2)_k. So that is not an obstacle. IMHO the string-net approach is equivalent to one containing spin-networks + many-body physics. It is the latter ingredient that is missing in most considerations of LQG, though that appears to be changing ...

Does this mean that Wen-Levin's string-net condensation gives rise directly to U(1) gauge bosons and electrons?

 

[space_cadet]

Does this mean that Wen-Levin's string-net condensation gives rise directly to U(1) gauge bosons and electrons?

I don't have a complete grasp on the physical picture Wen is proposing, but I would guess that is what should happen in the limit that $$\Lambda \rightarrow 0$$, i.e. as $$SU(2)_k \rightarrow SU(2)$$

 

[ensabah6]

I don't have a complete grasp on the physical picture Wen is proposing, but I would guess that is what should happen in the limit that $$\Lambda \rightarrow 0$$, i.e. as $$SU(2)_k \rightarrow SU(2)$$

If that's true, then some of the SM particles can be accounted for in LQG via topological order. Do you think you can get neutrinos from your Bilson-Thompson model, including lagrangian?

 

[space_cadet]

If that's true, then some of the SM particles can be accounted for in LQG via topological order.

I expect that to be true, only you need to move beyond LQG by including framed ribbons for eg.

Do you think you can get neutrinos from your Bilson-Thompson model ...

In his original paper, Sundance identified the neutrino as a particular braid, see http://arxiv.org/abs/hep-ph/0503213" . Assuming his model for charge (a $$\pm$$ twist of the ribbons corresponding to a charge of $$\pm \, e/3$$), and other details are correct, then the neutrino is already present.

... including Lagrangian?

The action for such a model, with braids attached to the faces of tetrahedra, can be written down in the Group Field Theory framework. I'm currently working on it. It involves writing a GFT action with the braid group as the symmetry group. This was also suggested to me by Etera Livine who has done a lot of work on GFT.

 

[ensabah6]

I expect that to be true, only you need to move beyond LQG by including framed ribbons for eg.

Do you think you can get neutrinos from your Bilson-Thompson model ...

In his original paper, Sundance identified the neutrino as a particular braid, see http://arxiv.org/abs/hep-ph/0503213" . Assuming his model for charge (a $$\pm$$ twist of the ribbons corresponding to a charge of $$\pm \, e/3$$), and other details are correct, then the neutrino is already present.



The action for such a model, with braids attached to the faces of tetrahedra, can be written down in the Group Field Theory framework. I'm currently working on it. It involves writing a GFT action with the braid group as the symmetry group. This was also suggested to me by Etera Livine who has done a lot of work on GFT.

1 True but it's not been proven that LQG's spin networks can give rise directly to Bilson-Thompsons braiding directly, with no modification. Bilson has his model, and there's spin foam, but it's not shown that braiding of spinfoam gives rise to exactly the properties of Bilson.

2 That would be interesting. Which SM particles and lagragians do you propose to first model? Bilson doesn't account for 2nd and 3rd generation, nor other properties like mass, charge, parity, color charge. Do you think you can also get charge, mixing angles, masses?

 

[space_cadet]

... Which SM particles and lagragians do you propose to first model? Bilson doesn't account for 2nd and 3rd generation, nor other properties like mass, charge, parity, color charge. Do you think you can also get charge, mixing angles, masses?

That's the general idea. I wouldn't pursue this scheme unless I felt that it could lead to a more complete description. There are many streams of thought which combine in this topic ~ quantum hall effect, black hole entropy and quantum computing to name three. The hope is that given some simple structures (braids for eg.) and some simple, local rules for the microscopic evolution, properties such as mass and charge would emerge in much the same way that they do in condensed matter systems. Of course, in the BT model charge is already included in the form of twists of on each ribbon $$\pm \pi$$ corresponding to a fractional charge of $$\pm 1/3$$ (modulo a factor of e, whose exact value we expect to be determined only in the thermodynamic limit, $$N \rightarrow \infty, V \rightarrow \infty, N/V \rightarrow \textmf{constant}$$)

 

[ensabah6]

That's the general idea. I wouldn't pursue this scheme unless I felt that it could lead to a more complete description. There are many streams of thought which combine in this topic ~ quantum hall effect, black hole entropy and quantum computing to name three. The hope is that given some simple structures (braids for eg.) and some simple, local rules for the microscopic evolution, properties such as mass and charge would emerge in much the same way that they do in condensed matter systems. Of course, in the BT model charge is already included in the form of twists of on each ribbon $$\pm \pi$$ corresponding to a fractional charge of $$\pm 1/3$$ (modulo a factor of e, whose exact value we expect to be determined only in the thermodynamic limit, $$N \rightarrow \infty, V \rightarrow \infty, N/V \rightarrow \textmf{constant}$$)

Do you think though it's physically plausible to map Bilson's ribbons onto spin networks?

 

[atyy]

A new paper in this set of ideas. Interestingly it cites Physics Monkey's paper (see post #8) in its concluding paragraph. Physics Monkey discussed the link to the old LQG formalism in his post #7.

http://arxiv.org/abs/1102.5524
Entanglement renormalization for quantum fields
Jutho Haegeman, Tobias J. Osborne, Henri Verschelde, Frank Verstraete
(Submitted on 27 Feb 2011)
p4, concluding para: Looking further afield, the cMERA constitutes a realization of the holographic principle. It is tempting to speculate, building on [19], that cMERA are a natural candidate to establish a link between entanglement renormalization and the best known realization of the holographic principle, namely the AdS/CFT correspondence.

 

[marcus]

This is a helpful thread for people like myself who aren't especially familiar either with tensor network or Wen's work. From my standpoint it would be nice if kept current---occasionally updated with new papers as Atyy just did a day ago. Here's a recap of some interesting excerpts.

...
http://arxiv.org/abs/0907.2994
Tensor network decompositions in the presence of a global symmetry
Sukhwinder Singh, Robert N. C. Pfeifer, Guifre Vidal
(Submitted on 17 Jul 2009)
Tensor network decompositions offer an efficient description of certain many-body states of a lattice system and are the basis of a wealth of numerical simulation algorithms. We discuss how to incorporate a global symmetry, given by a compact, completely reducible group G, in tensor network decompositions and algorithms. This is achieved by considering tensors that are invariant under the action of the group G. Each symmetric tensor decomposes into two types of tensors: degeneracy tensors, containing all the degrees of freedom, and structural tensors, which only depend on the symmetry group. In numerical calculations, the use of symmetric tensors ensures the preservation of the symmetry, allows selection of a specific symmetry sector, and significantly reduces computational costs. On the other hand, the resulting tensor network can be interpreted as a superposition of exponentially many spin networks. Spin networks are used extensively in loop quantum gravity, where they represent states of quantum geometry. Our work highlights their importance also in the context of tensor network algorithms, thus setting the stage for cross-fertilization between these two areas of research.

...

I personally find this whole area very exciting. For example, the spin networks used in loop quantum gravity can be greatly generalized and potentially even realized in condensed matter systems called string net states. Furthermore, there are some exciting hints relating the way one computes black hole entropy in loop quantum gravity and entanglement entropy in the tensor network approach. Thee end of his slides.

...
The physical configurations in string net models are actually exactly like spin networks. The low energy physical subspace is the space of closed string states. However, closed string states may include branching with branching rules given by the analog of the vertex rules in SU(2) spin networks. For example, the state space of something like U(1) gauge theory can be thought of as trivalent graphs with edges labeled by integers and with vertices allowed when all the integers sum to zero at the vertex (with orientation). The branching rules for a theory like SU(2) are almost exactly the vertex rules for SU(2) spin networks. One subtlety is that in the string net models one is usually dealing with the so called quantum group. This is manifested in a limit to the size of the reps of SU(2) than can be used. SU(2) level k only allows reps up to $$j = k/2$$. The ground state of a string net model is some kind of superposition of all closed string states.
...

So it looks like Jal and I had a related conversation a while ago in posts 68-70 of his https://www.physicsforums.com/showthread.php?t=251509, with a quirky note by Michael Freedman pointing to papers by Brian Swingle and on to the entanglement entropy and holography.

That is me. The paper called "Entanglement Renormalization and Holography" is the beginnings of an attempt to see the emergence of AdS/CFT from the tensor network approach. In that paper I used a particular tensor network approach called the multi-scale entanglement renormalization ansatz (MERA) to argue for a holographic description of quantum states.

A new paper in this set of ideas. Interestingly it cites Physics Monkey's paper (see post #8) in its concluding paragraph. Physics Monkey discussed the link to the old LQG formalism in his post #7.

http://arxiv.org/abs/1102.5524
Entanglement renormalization for quantum fields
Jutho Haegeman, Tobias J. Osborne, Henri Verschelde, Frank Verstraete
(Submitted on 27 Feb 2011)
p4, concluding para: Looking further afield, the cMERA constitutes a realization of the holographic principle. It is tempting to speculate, building on [19], that cMERA are a natural candidate to establish a link between entanglement renormalization and the best known realization of the holographic principle, namely the AdS/CFT correspondence.

There's more food for thought in the thread than this brief sample of excerpts indicates. Hoping you keep us posted.

 

[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" ...

 

[atyy]

John Baez and Jacob Biamonte are collecting networks and their diagrams, including tensor networks!

http://ncatlab.org/johnbaez/show/Diagrams

That links to Baez's post http://golem.ph.utexas.edu/category/2010/09/jacob_biamonte_on_tensor_netwo.html "In loop quantum gravity I learned a lot about “spin networks”. When I sailed up to the abstract heights of category theory, I discovered that these were a special case of “string diagrams” ". And now, going back down to earth, I see they have a special case called “tensor networks”. The post is about Biamonte, Clark and Jaksch's http://arxiv.org/abs/1012.0531 . The comments section of the post has lots of interesting things, including an overview by Biamonte, and a link to Penrose's paper about diagrams for tensorial terms. Apparently they are also useful for classical stochastic systems - that means I'm no longer goofing off when I read this stuff!

 

[atyy]

First, a link to Bahr, Dittrich and Ryan's http://arxiv.org/abs/1103.6264 who's reference [123] indicates they are on the look out for links between LQG and tensor networks.

Second, a discussion led by Steven White, one of whose topics is "Is the AdS/CFT - MERA correspondence just analogy? Or speculation? Or is there any quantitative relationship? What does the network mean in terms of AdS?" I haven't watched enough to know if they get round to it, but here it is just in case: http://online.itp.ucsb.edu/online/compqcm10/openprob2/. (Yes, they discuss it for about 10 minutes starting at 66:50.)

 

[atyy]

Here are a couple of proposals that use MERA to study AdS/CFT. I don't know if they are in the same spirit as Swingle's proposal.

http://arxiv.org/abs/1101.5993
Holographic phase space: $c$-functions and black holes as renormalization group flows
Miguel F. Paulos

http://arxiv.org/abs/1011.1474v2
Holographic description of large N gauge theory
Sung-Sik Lee

 

[marcus]

First, a link to Bahr, Dittrich and Ryan's http://arxiv.org/abs/1103.6264 who's reference [123] indicates they are on the look out for links between LQG and tensor networks.
...

You are certain right about that! Ryan just posted a paper on arxiv today which explores the QG link to tensor networks

http://arxiv.org/abs/1104.5471
Tensor models and embedded Riemann surfaces
James P. Ryan
9 pages, 7 figures
(Submitted on 28 Apr 2011)
"Tensor models and, more generally, group field theories are candidates for higher-dimensional quantum gravity, just as matrix models are in the 2d setting. With the recent advent of a 1/N-expansion for coloured tensor models, more focus has been given to the study of the topological aspects of their Feynman graphs. Crucial to the aforementioned analysis were certain subgraphs known as bubbles and jackets. We demonstrate in the 3d case that these graphs are generated by matrix models embedded inside the tensor theory. Moreover, we show that the jacket graphs represent (Heegaard) splitting surfaces for the triangulation dual to the Feynman graph. With this in hand, we are able to re-express the Boulatov model as a quantum field theory on these Riemann surfaces."

Ryan just gave a talk on 12 April at ILQGS
http://relativity.phys.lsu.edu/ilqgs/
His audio and slides PDF is available online.

Also you have expressed interest in Razvan Gurau's work (some with Rivasseau) and he just gave a talk on LQGS a couple of days ago (26 April). The audio and slides are now available. The two talks seem closely related---share some common terminology.

 

[atyy]

Are the systems that are well described by MERA also those which have the conjectured properties to have classical bulk?

AdS/CFT in which the bulk is classical is conjectured to hold only for some CFTs.
http://arxiv.org/abs/0903.4437 (appropriate singularity in boundary correlators)
http://arxiv.org/abs/0907.0151 (planar expansion and gap)
http://arxiv.org/abs/1101.4163 (some correlators factorize, hmmm, this naively sounds like MERA language)

I believe it is known that MERA give exact solutions for the ground state of models in the string net class. http://arxiv.org/abs/0712.0348
http://arxiv.org/abs/0806.4583

Incidentally, string net models connect to LQG in two (unrelated?) ways (i) string nets are spin networks as Physics Monkey details in post #7 (ii) string net models are related to Turaev-Viro or Barrett-Westbury spin foam models/TQFTs http://arxiv.org/abs/1102.0270, http://arxiv.org/abs/1004.1533. Rovelli's current spin foam model is supposed to be some sort of generalized TQFT http://arxiv.org/abs/1010.1939.

Apart from AdS/CFT which seems pretty general, there is a different, probably more restricted holography between 2+1D TQFTs and 1+1D RCFTs, which seems more relevant to the Levin Wen models. However, there may be some relationship between AdS/CFT and TQFT/CFT when the former is restricted to the appropriate dimensions. http://arxiv.org/abs/hep-th/0403225

 

[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.