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arXiv:2110.06932 · Physical Review Letters

Chiral Central Charge from a Single Bulk Wave Function

Isaac H. Kim, Bowen Shi, Kohtaro Kato, Victor V. Albert

Published Apr 28, 2022• 59 citations
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Abstract

A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real valued in any local product basis.

DOI
10.1103/physrevlett.128.176402

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A (2+1)-dimensional gapped quantum many-body system can have a topologically protected energy current at its edge. The magnitude of this current is determined entirely by the temperature and the chiral central charge, a quantity associated with the effective field theory of the edge. We derive a formula for the chiral central charge that, akin to the topological entanglement entropy, is completely determined by the many-body ground state wave function in the bulk. According to our formula, nonzero chiral central charge gives rise to a topological obstruction that prevents the ground state wave function from being real valued in any local product basis.
A (2+1)D topologically ordered phase may or may not have a gappable edge, even if its chiral central charge c_{-} is vanishing. Recently, it was discovered that a quantity regarded as a "higher" version of chiral central charge gives a further obstruction beyond c_{-} to gapping out the edge. In this Letter, we show that the higher central charges can be characterized by the expectation value of the partial rotation operator acting on the wave function of the topologically ordered state. This allows us to extract the higher central charge from a single wave function, which can be evaluated on a quantum computer. Our characterization of the higher central charge is analytically derived from the modular properties of edge conformal field theory, as well as the numerical results with the ν=1/2 bosonic Laughlin state and the non-Abelian gapped phase of the Kitaev honeycomb model, which corresponds to U(1)_{2} and Ising topological order, respectively. The Letter establishes a numerical method to obtain a set of obstructions to the gappable edge of (2+1)D bosonic topological order beyond c_{-}, which enables us to completely determine if a (2+1)D bosonic Abelian topological order has a gappable edge or not. We also point out that the expectation values of the partial rotation on a single wave function put a constraint on the low-energy spectrum of the bulk-boundary system of (2+1)D bosonic topological order, reminiscent of the Lieb-Schultz-Mattis-type theorems.
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.
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.
We derive a Kubo-like formula for the thermal Hall conductance of a 2d lattice systems which is free from ambiguities associated with the definition of energy magnetization. We use it to define a relative topological invariant of gapped 2d lattice systems at zero temperature. Up to a numerical factor, it can be identified with the difference of chiral central charges for the corresponding edge modes. This establishes the bulk-boundary correspondence for the chiral central charge. We also show that for any local commuting projector Hamiltonian, the relative chiral central charge vanishes, while for free fermionic systems, it is related to the zero-temperature electric Hall conductance via the Wiedemann-Franz law.
Recent years have witnessed substantial progress in the entanglement-based characterization of quantum phases of matter, with growing interest in real-space formulas due to their theoretical and experimental relevance. For two-dimensional gapped quantum systems, we propose a universal real-space formula for the chiral central charge based on the twist-operator correlator Zg,h=⟨ψ(N)∣TA(g)TB(h)TC(g)TC(h)∣ψ(N)⟩Z_{g,h}=\langleψ^{(N)}\rvert\mathcal{T}_A(g)\mathcal{T}_B(h)\mathcal{T}_C(g)\mathcal{T}_C(h)\lvertψ^{(N)}\rangleZg,h​=⟨ψ(N)∣TA​(g)TB​(h)TC​(g)TC​(h)∣ψ(N)⟩, where ∣ψ(N)⟩=∣ψ⟩⊗N\lvertψ^{(N)}\rangle=\lvertψ\rangle^{\otimes N}∣ψ(N)⟩=∣ψ⟩⊗N is the NNN-copy state and g,h∈SNg,h\in S_Ng,h∈SN​ are permutations. The twist operator TR(g)\mathcal{T}_R(g)TR​(g) permutes the NNN replicas within region RRR according to ggg. For pairs (g,h)(g,h)(g,h) satisfying the spherical condition, we show that the phase of the correlator is determined by the chiral central charge c−\mathfrak{c}_{-}c−​ through Zg,h/∣Zg,h∣=(θgθh/θgh)c−Z_{g,h}/\left\lvert Z_{g,h}\right\rvert=(θ_gθ_h/θ_{gh})^{\mathfrak{c}_{-}}Zg,h​/∣Zg,h​∣=(θg​θh​/θgh​)c−​, where θgc−θ_g^{\mathfrak{c}_{-}}θgc−​​ is the topological spin of the twist defect labeled by g∈SNg\in S_Ng∈SN​. The construction applies to both bosonic and fermionic systems. We derive the formula within a field-theoretic framework under suitable assumptions and further test it numerically in microscopic lattice models. Our results establish a universal connection between real-space entanglement structures and the chiral central charge, providing a systematic framework for characterizing chiral topological phases directly on the lattice.
The hallmark of two-dimensional chiral topological phases is the existence of anomalous gapless modes at the spatial boundary. Yet, the manifestation of this edge anomaly within the bulk ground-state wavefunction itself remains only partially understood. In this work, we introduce a family of multipartite entanglement measures that probe chirality directly from the bulk wavefunction. Our construction involves applying different permutations between replicas of the ground state wavefunction in neighboring spatial regions, creating "permutation defects" at the boundaries between these regions. We provide general arguments for the robustness of these measures and develop a field-theoretical framework to compute them systematically. While the standard topological field theory prescription misses the chiral contribution, our method correctly identifies it as the chiral conformal field theory partition function on high-genus Riemann surfaces. This feature is a consequence of the bulk-edge correspondence, which dictates that any regularization of the theory at the permutation defects must introduce gapless boundary modes. We numerically verify our results with both free-fermion and strongly-interacting chiral topological states and find excellent agreement. Our results enable the extraction of the chiral central charge and the Hall conductance using a finite number of wavefunction replicas, making these quantities accessible to Monte-Carlo numerical techniques and noisy intermediate-scale quantum devices.