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arXiv:2303.04822 · Physical Review Letters

Extracting Higher Central Charge from a Single Wave Function

Ryohei Kobayashi, Taige Wang, Tomohiro Soejima, Roger S. K. Mong, Shinsei Ryu

Published Jan 4, 2024• 16 citations
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Abstract

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.

DOI
10.1103/physrevlett.132.016602

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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.
A (2+1)D topological ordered phase with U(1) symmetry may or may not have a symmetric gapped edge state, even if both thermal and electric Hall conductivity are vanishing. It is recently discovered that there are "higher" versions of Hall conductivity valid for fermionic fractional quantum Hall (FQH) states, which obstructs symmetry-preserving gapped edge state beyond thermal and electric Hall conductivity. In this paper, we show that one can extract higher Hall conductivity from a single wave function of an FQH state, by evaluating the expectation value of the "partial rotation" unitary which is a combination of partial spatial rotation and a U(1) phase rotation. This result is verified numerically with the fermionic Laughlin state with ν=1/3ν=1/3ν=1/3, 1/51/51/5, as well as the non-Abelian Moore-Read state. Together with topological entanglement entropy, we prove that the expectation values of the partial rotation completely determines if a bosonic/fermionic Abelian topological order with U(1) symmetry has a symmetry-preserving gappable edge state or not. We also show that thermal and electric Hall conductivity of Abelian topological order can be extracted by partial rotations. Even in non-Abelian FQH states, partial rotation provides the Lieb-Schultz-Mattis type theorem constraining the low-energy spectrum of the bulk-boundary system. The generalization of higher Hall conductivity to the case with Lie group symmetry is also presented.
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.
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The existence of gapped boundaries of bosonic topological orders can be tested in terms of the vanishing of higher central charges, which can be easily computed in terms of the modular data. For fermionic topological orders, even the chiral central charge admits no simple expression in terms of the modular data. Using the congruence property of representations formed by the modular data, we develop a method that tests whether the higher central charges of a fermionic topological order, including the chiral central charge, vanish. The test can be carried out entirely in terms of the modular data of the super-modular tensor category describing the fermionic topological order, and does not require explicit computation of modular extensions. This leads to a stringent set of easily computable necessary conditions for a fermionic topological order to admit a gapped boundary. We apply our test to known examples of fermionic topological orders to determine which of them potentially admit a gapped boundary.
We investigate symmetry-preserving gapped boundary of (2+1)(2+1)(2+1)-dimensional [(2+1)D][(2+1)\mathrm{D}][(2+1)D] topological phases with global symmetry, which can be either bosonic or fermionic. We develop a general algebraic description for gapped boundary condition for symmetry-enriched or fermionic topological phases, extending the framework of Lagrangian algebra anyon for bosonic phases without symmetry. We then focus on application to the case with U(1) symmetry. We derive new obstructions to symmetry-preserving gapped boundary for U(1)f{}^{f}f-symmetric (2+1)D(2+1)\mathrm{D}(2+1)D fermionic topological phases, which are beyond chiral central charge ${c}_{\ensuremath{-}}$ and electric Hall conductivity ${\ensuremath{\sigma}}_{H}$. These obstructions are given by a simple Gauss-Milgram-type formula valid for supermodular category and regarded as a higher version of ${c}_{\ensuremath{-}}$ and ${\ensuremath{\sigma}}_{H}$.