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Muxin Han

Publications and source records attributed to Muxin Han.

At least 73 records · Page 4Linked to original sources
arXiv2017

Random Invariant Tensors

Invariant tensors are states in the (local) SU(2) tensor product representation but invariant under global SU(2) action. They are of importance in the study of loop quantum gravity. A random tensor is an ensemble of tensor states. An average over the ensemble is carried out when computing any physical quantities. The random tensor exhibits a phenomenon of `concentration of measure', saying that for any bipartition, the expected value of entanglement entropy of its reduced density matrix is asymptotically the maximal possible as the local dimension goes to infinity. This is also true even when the average is over the invariant subspace instead of the whole space for $4-$valent tensors, although its entropy deficit is divergent. One might expect that for $n\geq 5$, $n-$valent random invariant tensor would behavior similarly. However, we show that, the expected entropy deficit of reduced density matrix of such $n-$valent random invariant tensor from maximum, is not divergent but a finite number. Under some special situation, the number could be even smaller than half a bit, which is the deficit of random pure state over the whole Hilbert space from maximum.

quant-ph↗
arXiv2017

Discrete Gravity on Random Tensor Network and Holographic Rényi Entropy

In this paper we apply the discrete gravity and Regge calculus to tensor networks and Anti-de Sitter/conformal field theory (AdS/CFT) correspondence. We construct the boundary many-body quantum state $|Ψ\rangle$ using random tensor networks as the holographic mapping, applied to the Wheeler-deWitt wave function of bulk Euclidean discrete gravity in 3 dimensions. The entanglement Rényi entropy of $|Ψ\rangle$ is shown to holographically relate to the on-shell action of Einstein gravity on a branch cover bulk manifold. The resulting Rényi entropy $S_n$ of $|Ψ\rangle$ approximates with high precision the Rényi entropy of ground state in 2-dimensional conformal field theory (CFT). In particular it reproduces the correct $n$ dependence. Our results develop the framework of realizing the AdS$_3$/CFT$_2$ correspondence on random tensor networks, and provide a new proposal to approximate CFT ground state.

hep-th↗
arXiv2017

Einstein Equation from Covariant Loop Quantum Gravity in Semiclassical Continuum Limit

In this paper we explain how 4-dimensional general relativity and in particular, the Einstein equation, emerge from the spinfoam amplitude in loop quantum gravity. We propose a new limit which couples both the semiclassical limit and continuum limit of spinfoam amplitudes. The continuum Einstein equation emerges in this limit. Solutions of Einstein equation can be approached by dominant configurations in spinfoam amplitudes. A running scale is naturally associated to the sequence of refined triangulations. The continuum limit corresponds to the infrared limit of the running scale. An important ingredient in the derivation is a regularization for the sum over spins, which is necessary for the semiclassical continuum limit. We also explain in this paper the role played by the so-called flatness in spinfoam formulation, and how to take advantage of it.

gr-qc↗
arXiv2017

Loop-Quantum-Gravity Simplicity Constraint as Surface Defect in Complex Chern-Simons Theory

The simplicity constraint is studied in the context of 4d spinfoam models with cosmological constant. We find that the quantum simplicity constraint is realized as the 2d surface defect in SL(2,$\mathbb{C}$) Chern-Simons theory in the construction of spinfoam amplitudes. By this realization of simplicity constraint in Chern-Simons theory, we are able to construct the new spinfoam amplitude with cosmological constant for arbitrary simplicial complex (with many 4-simplices). The semiclassical asymptotics of the amplitude is shown to reproduce correctly the 4-dimensional Einstein-Regge action with cosmological constant term.

gr-qc↗
arXiv2017

SU(2) Flat Connection on Riemann Surface and Twisted Geometry with Cosmological Constant

SU(2) flat connection on 2D Riemann surface is shown to relate to the generalized twisted geometry in 3D space with cosmological constant. Various flat connection quantities on Riemann surface are mapped to the geometrical quantities in discrete 3D space. We propose that the moduli space of SU(2) flat connections on Riemann surface generalizes the phase space of twisted geometry or Loop Quantum Gravity to include the cosmological constant.

gr-qc↗
arXiv2017

On Spinfoams Near a Classical Curvature Singularity

We apply the technique of spinfoam to study the space-time which, classically, contains a curvature singularity. We derive from the full covariant Loop Quantum Gravity (LQG) that the region near curvature singularity has to be of strong quantum gravity effect. We show that the spinfoam configuration describing the near-singularity region has to be of small spins $j$, in order that its contribution to the full spinfoam amplitude is nontrivial. The spinfoams in low and high curvature regions of the space-time may be viewed as in two different {phases} of covariant LQG. There should be a phase transition as the space-time described by spinfoam becomes more and more curved. A candidate of order parameter is proposed for understanding the phase transition. Moreover, we also analyze the spin-spin correlation function of spinfoam, and show the correlation is of long-range in the low curvature phase. This work is a first step toward understanding the physics of black hole and early universe from the full covariant LQG theory.

gr-qc↗
arXiv2017

Loop Quantum Gravity, Exact Holographic Mapping, and Holographic Entanglement Entropy

The relation between Loop Quantum Gravity (LQG) and tensor network is explored from the perspectives of bulk-boundary duality and holographic entanglement entropy. We find that the LQG spin-network states in a space $Σ$ with boundary $\partialΣ$ is an exact holographic mapping similar to the proposal in arXiv:1309.6282. The tensor network, understood as the boundary quantum state, is the output of the exact holographic mapping emerging from a coarse graining procedure of spin-networks. Furthermore, when a region $A$ and its complement $\bar{A}$ are specified on the boundary $\partialΣ$, we show that the boundary entanglement entropy $S(A)$ of the emergent tensor network satisfies the Ryu-Takayanagi formula in the semiclassical regime, i.e. $S(A)$ is proportional to the minimal area of the bulk surface attached to the boundary of $A$ in $\partialΣ$.

hep-th↗
arXiv2016

Invariant Perfect Tensors

Invariant tensors are states in the SU(2) tensor product representation that are invariant under the SU(2) action. They play an important role in the study of loop quantum gravity. On the other hand, perfect tensors are highly entangled many-body quantum states with local density matrices maximally mixed. Recently, the notion of perfect tensors recently has attracted a lot of attention in the fields of quantum information theory, condensed matter theory, and quantum gravity. In this work, we introduce the concept of an invariant perfect tensor (IPT), which is a $n$-valent tensor that is both invariant and perfect. We discuss the existence and construction of IPT. For bivalent tensors, the invariant perfect tensor is the unique singlet state for each local dimension. The trivalent invariant perfect tensor also exists and is uniquely given by Wigner's $3j$ symbol. However, we show that, surprisingly, there does not exist four-valent invariant perfect tensors for any dimension. On the contrary, when the dimension is large, almost all invariant tensors are perfect asymptotically, which is a consequence of the phenomenon of concentration of measure for multipartite quantum states.

quant-ph↗
arXiv2016

4d Quantum Geometry from 3d Supersymmetric Gauge Theory and Holomorphic Block

A class of 3d $\mathcal{N}=2$ supersymmetric gauge theories are constructed and shown to encode the simplicial geometries in 4-dimensions. The gauge theories are defined by applying the Dimofte-Gaiotto-Gukov construction in 3d/3d correspondence to certain graph complement 3-manifolds. Given a gauge theory in this class, the massive supersymmetric vacua of the theory contain the classical geometries on a 4d simplicial complex. The corresponding 4d simplicial geometries are locally constant curvature (either dS or AdS), in the sense that they are made by gluing geometrical 4-simplices of the same constant curvature. When the simplicial complex is sufficiently refined, the simplicial geometries can approximate all possible smooth geometries on 4-manifold. At the quantum level, we propose that a class of holomorphic blocks defined in arXiv:1211.1986 from the 3d $\mathcal{N}=2$ gauge theories are wave functions of quantum 4d simplicial geometries. In the semiclassical limit, the asymptotic behavior of holomorphic block reproduces the classical action of 4d Einstein-Hilbert gravity in the simplicial context.

hep-th↗
arXiv2015

Encoding Curved Tetrahedra in Face Holonomies: a Phase Space of Shapes from Group-Valued Moment Maps

We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both spherical and hyperbolic tetrahedra within a unified framework. A new type of hyperbolic simplex is introduced in order for all the sectors encoded in the algebraic data to be covered. Generalizing the phase space of shapes associated to flat tetrahedra leads to group valued moment maps and quasi-Poisson spaces. These discrete geometries provide a natural arena for considering the quantization of gravity including a cosmological constant. A concrete realization of this is provided by the relation with the spin-network states of loop quantum gravity. This work therefore provides a bottom-up justification for the emergence of deformed gauge symmetries and quantum groups in 3+1 dimensional covariant loop quantum gravity in the presence of a cosmological constant.

math-ph↗
arXiv2014

SL(2,C) Chern-Simons Theory, a non-Planar Graph Operator, and 4D Loop Quantum Gravity with a Cosmological Constant: Semiclassical Geometry

We study the expectation value of a nonplanar Wilson graph operator in SL(2,C) Chern-Simons theory on $S^3$. In particular we analyze its asymptotic behaviour in the double-scaling limit in which both the representation labels and the Chern-Simons coupling are taken to be large, but with fixed ratio. When the Wilson graph operator has a specific form, motivated by loop quantum gravity, the critical point equations obtained in this double-scaling limit describe a very specific class of flat connection on the graph complement manifold. We find that flat connections in this class are in correspondence with the geometries of constant curvature 4-simplices. The result is fully non-perturbative from the perspective of the reconstructed geometry. We also show that the asymptotic behavior of the amplitude contains at the leading order an oscillatory part proportional to the Regge action for the single 4-simplex in the presence of a cosmological constant. In particular, the cosmological term contains the full-fledged curved volume of the 4-simplex. Interestingly, the volume term stems from the asymptotics of the Chern-Simons action. This can be understood as arising from the relation between Chern-Simons theory on the boundary of a region, and a theory defined by an $F^2$ action in the bulk. Another peculiarity of our approach is that the sign of the curvature of the reconstructed geometry, and hence of the cosmological constant in the Regge action, is not fixed a priori, but rather emerges semiclassically and dynamically from the solution of the equations of motion. In other words, this work suggests a relation between 4-dimensional loop quantum gravity with a cosmological constant and SL(2,C) Chern-Simons theory in 3-dimensions with knotted graph defects.

hep-th↗
arXiv2014

Covariant Loop Quantum Gravity, Low Energy Perturbation Theory, and Einstein Gravity with High Curvature UV Corrections

A low-energy perturbation theory is developed from the nonperturbative framework of covariant Loop Quantum Gravity (LQG) by employing the background field method. The resulting perturbation theory is a 2-parameter expansion in the semiclassical and low-energy regime. The two expansion parameters are the large spin and small curvature. The leading order effective action coincides with the Einstein-Hilbert action. The subleading corrections organized by the two expansion parameters give the modifications of Einstein gravity in quantum and high-energy regime from LQG. The perturbation theory developed here shows for the first time that covariant LQG produces the high curvature corrections to Einstein gravity. This result means that LQG is not a naive quantization of Einstein gravity, but rather provides the UV modification. The result of the paper may be viewed as the first step toward understanding the UV completeness of LQG.

gr-qc↗
arXiv2014

Black Hole Entropy in Loop Quantum Gravity, Analytic Continuation, and Dual Holography

A new approach to black hole thermodynamics is proposed in Loop Quantum Gravity (LQG), by defining a new black hole partition function, followed by analytic continuations of Barbero-Immirzi parameter to $γ\in i\mathbb{R}$ and Chern-Simons level to $k\in i\mathbb{R}$. The analytic continued partition function has remarkable features: The black hole entropy $S=A/4\ell_P^2$ is reproduced correctly for infinitely many $γ= iη$, at least for $η\in\mathbb{Z}\setminus\{0\}$. The near-horizon Unruh temperature emerges as the pole of partition function. Interestingly, by analytic continuation the partition function can have a dual statistical interpretation corresponding to a dual quantum theory of $γ\in i\mathbb{Z}$. The dual quantum theory implies a semiclassical area spectrum for $γ\in i\mathbb{Z}$. It also implies that at a given near horizon (quantum) geometry, the number of quantum states inside horizon is bounded by a holographic degeneracy $d= e^{A/4\ell_P}$, which produces the Bekenstein bound from LQG. On the other hand, the result in arXiv:1212.4060 receives a justification here.

gr-qc↗
arXiv2013

On Spinfoam Models in Large Spin Regime

We study the semiclassical behavior of Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model, by taking into account the sum over spins in the large spin regime. We also employ the method of stationary phase analysis with parameters and the so called, almost-analytic machinery, in order to find the asymptotic behavior of the contributions from all possible large spin configurations in the spinfoam model. The spins contributing the sum are written as ${J}_f=λ{j}_f$ where $ł$ is a large parameter resulting in an asymptotic expansion via stationary phase approximation. The analysis shows that at least for the simplicial Lorentzian geometries (as spinfoam critical configurations), they contribute the leading order approximation of spinfoam amplitude only when their deficit angles satisfy $γΘ_f\leqλ^{-1/2}$ mod $4π\mathbb{Z}$. Our analysis results in a curvature expansion of the semiclassical low energy effective action from the spinfoam model, where the UV modifications of Einstein gravity appear as subleading high-curvature corrections.

gr-qc↗
arXiv2013

Spinfoam fermions

We describe a minimal coupling of fermions and Yang Mills fields to the loop quantum gravity dynamics. The coupling takes a very simple form.

gr-qc↗
arXiv2013

Commuting Simplicity and Closure Constraints for 4D Spin Foam Models

Spin Foam Models are supposed to be discretised path integrals for quantum gravity constructed from the Plebanski-Holst action. The reason for there being several models currently under consideration is that no consensus has been reached for how to implement the simplicity constraints. Indeed, none of these models strictly follows from the original path integral with commuting B fields, rather, by some non standard manipulations one always ends up with non commuting B fields and the simplicity constraints become in fact anomalous which is the source for there being several inequivalent strategies to circumvent the associated problems. In this article, we construct a new Euclidian Spin Foam Model which is constructed by standard methods from the Plebanski-Holst path integral with commuting B fields discretised on a 4D simplicial complex. The resulting model differs from the current ones in several aspects, one of them being that the closure constraint needs special care. Only when dropping the closure constraint by hand and only in the large spin limit can the vertex amplitudes of this model be related to those of the FK Model but even then the face and edge amplitude differ. Curiously, an ad hoc non-commutative deformation of the $B^{IJ}$ variables leads from our new model to the Barrett-Crane Model in the case of Barbero-Immirzi parameter goes to infinity.

gr-qc↗
arXiv2013

Semiclassical Analysis of Spinfoam Model with a Small Barbero-Immirzi Parameter

We study the semiclassical behavior of Lorentzian Engle-Pereira-Rovelli-Livine (EPRL) spinfoam model, by taking into account of the sum over spins in the large spin regime. The large spin parameter λand small Barbero-Immirzi parameter γare treated as two independent parameters for the asymptotic expansion of spinfoam state-sum (such an idea was firstly pointed out in arXiv:1105.0216). Interestingly, there are two different spin regimes: 1<<γ^{-1}<<λ<<γ^{-2} and λ>γ^{-2}. The model in two spin regimes has dramatically different number of effective degrees of freedom. In 1<<γ^{-1}<<λ<<γ^{-2}, the model produces in the leading order a functional integration of Regge action, which gives the discrete Einstein equation for the leading contribution. There is no restriction of Lorentzian deficit angle in this regime. In the other regime λ>γ^{-2}, only small deficit angle is allowed |Θ_f|<<γ^{-1}λ^{1/2}$ mod 4πZ. When spins go even larger, only zero deficit angle mod 4πZ is allowed asymptotically. In the transition of the two regimes, only the configurations with small deficit angle can contribute, which means one need a large triangulation in order to have oscillatory behavior of the spinfoam amplitude.

gr-qc↗