# Condensed matter physics, area laws & LQG?

by atyy
Tags: condensed, laws, matter, physics
 Sci Advisor P: 7,917 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.
 Sci Advisor HW Helper P: 1,320 I never heard of Sun Wukong before, but I like him.
 Sci Advisor P: 7,917 @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.
 Sci Advisor HW Helper P: 1,320 @atyy, thanks a lot for your kind comment. I'm glad you found it vaguely comprehensible.
 Sci Advisor P: 7,917 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.
 Sci Advisor P: 7,917 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."
 Sci Advisor P: 7,917 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)
 Sci Advisor P: 7,917 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.
 Astronomy Sci Advisor PF Gold P: 22,679 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.
 P: 239 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).
P: 7,917
 Quote by 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. An Algebraic Perspective on Deep Learning
 P: 1 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.
P: 7,917
 Quote by 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.
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 Surya Ganguli's 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.
 Sci Advisor P: 7,917 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."
 Sci Advisor P: 7,917 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."
 Sci Advisor P: 7,917 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

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