Condensed matter physics, area laws & LQG?

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=mOV4J...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 [itex]\Delta \notin \frac{d}{2} + \mathbb{Z}[/itex], for sufficiently low momenta. We then clarify the relation between the saddle point approximation to the Wilsonian effective action ([itex]S_W[/itex]) and boundary conditions at a cutoff surface in AdS space. In particular, we interpret non-local multi-trace operators in [itex]S_W[/itex] 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

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. An Algebraic Perspective on Deep Learning