Renormalization group constructions of topological quantum liquids and beyond

TL;DR

The paper introduces an RG-inspired framework, the s-source RG fixed point, to argue that conventional gapped phases obey an area law for entanglement entropy. It builds a classification and entropy/degen­eracy machinery, proves an area law for topological quantum liquids, and demonstrates that these phases admit MERA representations with bond dimensions growing sub-exponentially with system size. The authors also develop field-theory constructions, including Dirac fermions in expanding universes, to support the universality of the area law across gapped phases and conjecture broader implications for gapless systems. Overall, the work provides a physically motivated pathway toward understanding entanglement structure in a broad class of quantum many-body systems and links to tensor-network representations and holographic ideas.

Abstract

We give a detailed physical argument for the area law for entanglement entropy in gapped phases of matter arising from local Hamiltonians. Our approach is based on renormalization group (RG) ideas and takes a resource oriented perspective. We report four main results. First, we argue for the "weak area law": any gapped phase with a unique ground state on every closed manifold obeys the area law. Second, we introduce an RG based classification scheme and give a detailed argument that all phases within the classification scheme obey the area law. Third, we define a special sub-class of gapped phases, \textit{topological quantum liquids}, which captures all examples of current physical relevance, and we rigorously show that TQLs obey an area law. Fourth, we show that all topological quantum liquids have MERA representations which achieve unit overlap with the ground state in the thermodynamic limit and which have a bond dimension scaling with system size $L$ as $e^{c \log^{d(1+δ)}(L)}$ for all $δ>0$. For example, we show that chiral phases in $d=2$ dimensions have an approximate MERA with bond dimension $e^{c \log^{2(1+δ)}(L)}$. We discuss extensively a number of subsidiary ideas and results necessary to make the main arguments, including field theory constructions. While our argument for the general area law rests on physically-motived assumptions (which we make explicit) and is therefore not rigorous, we may conclude that "conventional" gapped phases obey the area law and that any gapped phase which violates the area law must be a dragon.

Figures (9)

• Figure 1: A $d=1$ version of the RG transformation.
• Figure 2: Coupling $H$ (blue disk) to its orientation reversed partner $H_{\text{rev}}$ (red disk) along their common boundary, we can produce the gapped sphere Hamiltonian.
• Figure 3: An array of wormholes in $d=2$.
• Figure 4: The transformation to a trivial state using an expanding wormhole array. The white spaces denote product states or just empty space. We have suppressed the wormholes and are effectively viewing the whole system as a composite of $H$ and $H_{\text{rec}}$ on a system with boundary.
• Figure 5: The red dot is local term in the Hamiltonian which is smeared into a quasi-local constraint. The dashed circle is a cutoff where we truncate the qausi-local constraint to a strictly local constraint. Gappability of an edge suggests the constraint can be chosen to live strictly within $A$.

Theorems & Definitions (15)

• Definition 1: $s$ source RG fixed point
• Conjecture 1: Fundamental Assumption
• Lemma 1: Entropy Lemma
• Lemma 2: Ground State Degeneracy Lemma
• Lemma 3: Restriction Lemma
• Theorem 1: TQL structure theorem
• Definition 2: Generalized $s$ source RG fixed point
• Lemma 4: Generalized Entropy Lemma
• Lemma 5: Generalized Ground State Degeneracy Lemma
• Lemma 6: Generalized Restriction Lemma