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

Me too!

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 http://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

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 [tex] j = k/2 [/tex]. 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 [tex] \mathcal{N} = 4 [/tex] 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

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

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

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

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

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

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

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

When you consider that case of non-zero Cosmological constant [tex] \Lambda \neq 0 [/tex] 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

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

space_cadet

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 [tex]\Lambda \rightarrow 0 [/tex], i.e. as [tex] SU(2)_k \rightarrow SU(2) [/tex]