Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

Chen, Yuan; Hsin, Po-Shen

doi:10.48550/arxiv.2110.14644

Cited by 6 publications

(11 citation statements)

References 66 publications

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“…The fermionic loop in Refs. [30] and [31] is different from the dyonic loop discussed here: the fermionic loop there depends on the local w 3 structure [32], in contrast to the dyonic loops discussed here (and in Refs. [21]), and thus the fermionic loops in [30,31] have non-trivial statistics.…”

Section: Discussioncontrasting

confidence: 80%

“…Recently, Refs. [30] and [31] have constructed lattice Hamiltonian models for a beyond group cohomology invertible phase without symmetry in 4+1d that has a 3+1d boundary with fermionic particle and fermionic loop excitations with mutual π statistics. The fermionic loop in Refs.…”

Section: Discussionmentioning

confidence: 99%

“…Refs. [30,31], while the electric loop described by e i b (1) remains bosonic. 16 The twisted Z 2 two-form gauge theory can also be described by a local commuting projector Hamiltonian two variables u 1 and u 2 as (x 1 , x 2 ) = (φ 1 (u 1 , u 2 ), φ 2 (u 1 , u 2 )) ≡ φ(u 1 , u 2 ) ⊂ R 3 .…”

Section: Discussionmentioning

confidence: 99%

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Loops in 4+1d Topological Phases

Xie¹,

Dua²,

Hsin³

et al. 2021

Preprint

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2+1d topological phases are well characterized by the fusion rules and braiding/exchange statistics of fractional point excitations. In 4+1d, some topological phases contain only fractional loop excitations. What kind of loop statistics exist?We study the 4+1d gauge theory with 2-form Z 2 gauge field (the loop only toric code) and find that while braiding statistics between two different types of loops can be nontrivial, the self 'exchange' statistics are all trivial. In particular, we show that the electric, magnetic, and dyonic loop excitations in the 4+1d toric code are not distinguished by their self-statistics. They tunnel into each other across 3+1d invertible domain walls which in turn give explicit unitary circuits that map the loop excitations into each other. The SL(2, Z 2 ) symmetry that permutes the loops, however, cannot be consistently gauged and we discuss the associated obstruction in the process. Moreover, we discuss a gapless boundary condition dubbed the 'fractional Maxwell theory' and show how it can be Higgsed into gapped boundary conditions.We also discuss the generalization of these results from the Z 2 gauge group to Z N .

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Section: Discussioncontrasting

confidence: 80%

Section: Discussionmentioning

confidence: 99%

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Loops in 4+1d Topological Phases

Xie¹,

Dua²,

Hsin³

et al. 2021

Preprint

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“…In this article, we will derive the possible boundary theory from those topological invariants, especially the 4+1d invertible topological order characterized by the Stiefel-Whitney w 2 w 3 topological invariant in 5d. There are some earlier works in this direction [21,[27][28][29][30][31][32][33][34][35][36] which construct boundary theories of the w 2 w 3 invertible topological order. In this work, we will present a more complete and systematic derivation.…”

Section: Introductionmentioning

confidence: 99%

3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems

Wan,

Wang,

Wen

2021

Preprint

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We study bosonic systems on spacetime lattice (with imaginary time) defined by path integrals of commuting fields. We introduce a concept of branch-independent bosonic system, whose path integral is independent of the branch structure of the spacetime simplicial complex, even for a spacetime with boundaries. In contrast, a generic lattice bosonic system's path integral may depend on the branch structure. We find the 4+1d invertible topological order characterized by the Stiefel-Whitney cocycle w2w3 to be non-trivial for branch-independent bosonic systems, but this w2w3 topological order and a trivial gapped tensor product state belong to the same phase (via a smooth deformation without any phase transition) for generic lattice bosonic systems. This implies that the invertible topological orders in generic bosonic systems on spacetime lattice are not classified by oriented cobordism. The branch dependence on the lattice may relate to the orthonormal frame of smooth manifolds and the framing anomaly of continuum field theories. We construct branch-independent bosonic systems to realize the w2w3 topological order, as well as its 3+1d gapped or gapless boundaries. One of the gapped boundaries is a 3+1d Z2 gauge theory with (1) fermionic Z2 gauge charge particle which trivializes w2 and (2) "fermionic" Z2 gauge flux line which trivializes w3. In particular, if the flux loop's worldsheet is unorientable, then orientation-reversal 1d worldline must correspond to a fermion worldline that does not carry the Z2 gauge charge. We also explain why Spin and Spin c structures trivialize the w2w3 nonperturbative global gravitational anomaly to zero (which helps to construct the anomalous 3+1d gapped Z2 and gapless all-fermion U(1) gauge theories), but the Spin h and Spin×Z 2 Spin(n ≥ 3) structures modify the w2w3 into a nonperturbative global mixed gauge-gravitational anomaly, which helps to constrain Grand Unifications (e.g, n = 10, 18) or construct new models. CONTENTS1. Spacetime complex and branch structure 2. Chain, cochain, cycle, cocycle 3. Derivative operator on cochains 4. Cup product and higher cup product 5. Steenrod square and generalized Steenrod square 6. Branch structure dependence 7. Poincaré dual and pseudo-inverse of Poincaré dual B. Comparison with Standard Mathematical Conventions and Stiefel-Whitney Class C. Emergence of Half-Integer Spin and Fermi Statistics D. Cobordism Group Data and Anomaly Classification 1. Spacetime and gauge bundle constraint 2. Cobordism invariants a. Anomalies in SU(2) vs SO(3): cobordism vs homotopy group 3. Trivialization via group extension E. Oriented Bordism Groups and Manifold Generators F. Combinatorial Formula for Stiefel-Whitney Classes G. Compute w2w3 on Real Milnor, Wu, and Dold Manifolds H. Generalized Wu Relation I. Pullback Construction of Branch-Independent Bosonic Models

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“…In the last two decades, there have been many proposals of fermionto-qubit mappings for two dimensions [1][2][3][4][5][6][7][8][9] and three or arbitrary dimensions [10][11][12]. These fermion-to-qubit mappings play important roles in various topics of modern physics, such as exactly solvable models for topological phases [3,[13][14][15], fermionic quantum simulations [2,4,5,8,10], and quantum error correction [16][17][18][19][20][21]. In particular, the exact bosonizations in Refs.…”

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confidence: 99%

Equivalence between fermion-to-qubit mappings in two spatial dimensions

Chen¹,

Xu²

2022

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We argue that all locality-preserving mappings between fermionic observables and Pauli matrices on a two-dimensional lattice can be generated from the exact bosonization in Ref. [1], whose gauge constraints project onto the subspace of the toric code with emergent fermions. Starting from the exact bosonization and applying Clifford finite-depth generalized local unitary (gLU) transformation, we can achieve all possible fermion-to-qubit mappings (up to the re-pairing of Majorana fermions). In particular, we discover a new super-compact encoding using 1.25 qubits per fermion on the square lattice, which is lower than any method in the literature. We prove the existence of fermionto-qubit mappings with qubit-fermion ratios r = 1 + 1 2k for positive integers k, where the proof utilizes the trivialness of quantum cellular automata (QCA) in two spatial dimensions. When the ratio approaches 1, the fermion-to-qubit mapping reduces to the 1d Jordan-Wigner transformation along a certain path in the two-dimensional lattice. Finally, we explicitly demonstrate that the Bravyi-Kitaev superfast simulation, the Verstraete-Cirac auxiliary method, Kitaev's exactly solved model, the Majorana loop stabilizer codes, and the compact fermion-to-qubit mapping can all be obtained from the exact bosonization.

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Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

Cited by 6 publications

References 66 publications

Loops in 4+1d Topological Phases

Loops in 4+1d Topological Phases

3+1d Boundaries with Gravitational Anomaly of 4+1d Invertible Topological Order for Branch-Independent Bosonic Systems

Equivalence between fermion-to-qubit mappings in two spatial dimensions

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