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arXiv:2206.00027 · Physical Review Letters

Modular Commutators in Conformal Field Theory

Yijian Zou, Bowen Shi, Jonathan Sorce, Ian T. Lim, Isaac H. Kim

Published Dec 22, 2022• 23 citations
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Abstract

The modular commutator is a recently discovered entanglement quantity that quantifies the chirality of the underlying many-body quantum state. In this Letter, we derive a universal expression for the modular commutator in conformal field theories in 1+1 dimensions and discuss its salient features. We show that the modular commutator depends only on the chiral central charge and the conformal cross ratio. We test this formula for a gapped (2+1)-dimensional system with a chiral edge, i.e., the quantum Hall state, and observe excellent agreement with numerical simulations. Furthermore, we propose a geometric dual for the modular commutator in certain preferred states of the AdS/CFT correspondence. For these states, we argue that the modular commutator can be obtained from a set of crossing angles between intersecting Ryu-Takayanagi surfaces.

DOI
10.1103/physrevlett.129.260402

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The modular commutator is a recently discovered entanglement quantity that quantifies the chirality of the underlying many-body quantum state. In this Letter, we derive a universal expression for the modular commutator in conformal field theories in 1+1 dimensions and discuss its salient features. We show that the modular commutator depends only on the chiral central charge and the conformal cross ratio. We test this formula for a gapped (2+1)-dimensional system with a chiral edge, i.e., the quantum Hall state, and observe excellent agreement with numerical simulations. Furthermore, we propose a geometric dual for the modular commutator in certain preferred states of the AdS/CFT correspondence. For these states, we argue that the modular commutator can be obtained from a set of crossing angles between intersecting Ryu-Takayanagi surfaces.
In Phys. Rev. Lett. 128, 176402 (2022), we argued that the chiral central charge---a topologically protected quantity characterizing the edge theory of a gapped (2+1)(2+1)(2+1)-dimensional system---can be extracted from the bulk by using an order parameter called the modular commutator. In this paper, we reveal general properties of the modular commutator and strengthen its relationship with the chiral central charge. First, we identify connections between the modular commutator and conditional mutual information, time reversal, and modular flow. Second, we prove, within the framework of the entanglement bootstrap program, that two topologically ordered media connected by a gapped domain wall must have the same modular commutator in their respective bulk. Third, we numerically calculate the value of the modular commutator for a bosonic lattice Laughlin state for finite sizes and extrapolate to the infinite-volume limit. The result of this extrapolation is consistent with the proposed formula up to an error of about 0.70.7%0.7.
A recent surge of research on many-body quantum entanglement has uncovered intriguing properties of quantum many-body systems. A prime example is the modular commutator, which can extract a topological invariant from a single wave function. Here, we unveil geometric properties of many-body entanglement via a modular commutator of two-dimensional gapped quantum many-body systems. We obtain the geometric additivity of a modular commutator, which indicates that the modular commutator for a multipartite system may be an integer multiple of the one for tripartite systems. Using our additivity formula, we also derive a curious identity for the modular commutators involving disconnected intervals in a certain class of conformal field theories. We further illustrate this geometric additivity for both bulk and edge subsystems using numerical calculations of the Haldane and $\ensuremath{\pi}$-flux models.
We propose a ground state entanglement probe for gapped, two-dimensional quantum many-body systems that involves taking powers of reduced density matrices in a particular geometric configuration. This quantity, which we denote by ωα,βω_{α,β}ωα,β​, is parameterized by two positive real numbers α,βα, βα,β, and can be seen as a ``Rényi-like" generalization of the modular commutator -- another entanglement probe proposed as a way to compute the chiral central charge from a bulk wave function. We obtain analytic expressions for ωα,βω_{α,β}ωα,β​ for gapped ground states of non-interacting fermion Hamiltonians as well as ground states of string-net models. In both cases, we find that ωα,βω_{α,β}ωα,β​ takes a universal value related to the chiral central charge. For integer values of ααα and βββ, our quantity ωα,βω_{α,β}ωα,β​ can be expressed as an expectation value of permutation operators acting on an appropriate replica system, providing a natural route to measuring ωα,βω_{α,β}ωα,β​ in numerical simulations and potentially, experiments.
Recently, it was proposed that the chiral central charge of a gapped, two-dimensional quantum many-body system is proportional to a bulk ground state entanglement measure known as the modular commutator. While there is significant evidence to support this relation, we show in this Letter that it is not universal. We give examples of lattice systems that have vanishing chiral central charge, which nevertheless give nonzero "spurious" values for the modular commutator for arbitrarily large system sizes, in both one and two dimensions. Our examples are based on cluster states and utilize the fact that they can generate nonlocal modular Hamiltonians.
Recent years have witnessed significant progress in the entanglement-based characterization of quantum phases of matter. The primary objects of interest are the reduced density matrix and its associated entanglement Hamiltonian. As intrinsic properties of a quantum state, these quantities theoretically determine all experimentally accessible local observables. In this work, we investigate the response of two entanglement measures to the real-time dynamics driven by the entanglement Hamiltonian--a process known as modular flow. We demonstrate that our results can be unified into a single generating function, ⟨ρABαeλQABeμQBCρBCβ⟩\langleρ_{AB}^α\mathrm{e}^{λ{Q}_{AB}}\mathrm{e}^{μ{Q}_{BC}}ρ_{BC}^β\rangle⟨ρABα​eλQAB​eμQBC​ρBCβ​⟩. This function is of independent interest as it represents a generalization of the recently proposed Rényi modular commutator. In appropriate limits, this function yields the response of Rényi entropy and its charged version, which we find to be uniquely determined by chiral topological invariants, specifically the chiral central charge and the Hall conductance. Our analytical findings are validated through two independent approaches: (i) free fermion systems using the real-space Chern number formula, and (ii) an effective field theory treatment that regularizes the entanglement cut via chiral conformal field theory. Both methods yield consistent results.
We apply modular flow-entanglement generated dynamics-to characterize quantum orders of ground state wave functions. In particular, we study the linear response of the entanglement entropy of a simply connected region with respect to modular flow. First, we apply it to (1+1)D conformal field theories and demonstrate its relationship to the chiral central charge-or, equivalently, the perturbative gravitational anomaly-which is shown to vanish. Next, we apply it to (2+1)D gapped ground states where it reduces to a recently proposed formula by Kim et al. that is conjectured to compute the edge chiral central charge. Modular flow provides an intuitive picture for this conjecture based on bulk-edge correspondence. We also provide numerics on free-fermion models that lend support to our picture.