[Paper Review] Exactly Solvable Lattice Hamiltonians and Gravitational Anomalies

Yu-An Chen, Po-Shen Hsin|arXiv (Cornell University)|Oct 27, 2021

Topological Materials and Phenomena63 references4 citations

TL;DR

This paper presents a general construction of exactly solvable lattice Hamiltonians for bosonic beyond group cohomology invertible topological phases in any spacetime dimension, using commuting projector Hamiltonians on triangulated or hypercubic lattices. By gauging one-form and two-form symmetries in a parent group cohomology SPT phase, the authors realize a (4+1)D $bZ_2$-symmetric invertible phase with gravitational anomaly, including a non-trivial quantum cellular automaton of order two and a gapped symmetric boundary state.

ABSTRACT

We construct infinitely many new exactly solvable local commuting projector lattice Hamiltonian models for general bosonic beyond group cohomology invertible topological phases of order two and four in any spacetime dimensions, whose boundaries are characterized by gravitational anomalies. Examples include the beyond group cohomology invertible phase without symmetry in (4+1)D that has an anomalous boundary $\mathbb{Z}_2$ topological order with fermionic particle and fermionic loop excitations that have mutual $π$ statistics. We argue that this construction gives a new non-trivial quantum cellular automaton (QCA) in (4+1)D of order two. We also present an explicit construction of gapped symmetric boundary state for the bosonic beyond group cohomology invertible phase with unitary $\mathbb{Z}_2$ symmetry in (4+1)D. We discuss new quantum phase transitions protected by different invertible phases across the transitions.

Motivation & Objective

• To develop a general method for constructing exactly solvable Hamiltonians for bosonic beyond group cohomology invertible topological phases in arbitrary spacetime dimensions.
• To realize such phases via gauging of one-form and two-form symmetries in a parent group cohomology SPT model.
• To demonstrate the existence of gravitational anomalies in the boundary theory of these phases, particularly in (4+1)D.
• To construct a gapped symmetric boundary state for the $bZ_2$-symmetric (4+1)D beyond group cohomology phase.
• To identify new quantum phase transitions protected by invertible phases with distinct topological invariants.

Proposed method

• The construction begins with a parent group cohomology SPT phase described by a topological action involving $bZ_2$-valued 2- and 3-form gauge fields $A_2$ and $B_3$.
• The wavefunction is built using a phase factor $e^{i heta}$ with $ heta = rac{ heta}{2} heta$ derived from a 4-dimensional cocycle $ heta_4(a,b)$, which is a coboundary of a 5-dimensional action.
• The bulk Hamiltonian is derived via unitary transformation $H = U H_0 U^ $ from a trivial Hamiltonian $H_0 = - sum_e X_e - sum_f X_f$, where $U$ implements the topological phase.
• The action is modified via the Wu formula to express $B_3 igcup_1 B_3 = w_2 \bigcup B_3$, enabling the identification of the effective action $\pi \int w_2 \bigcup w_3$ after gauging.
• The resulting Hamiltonian is a sum of local commuting projectors, defined on any triangulated or hypercubic lattice with finite-dimensional local Hilbert spaces.
• The boundary state is obtained by truncating the bulk Hamiltonian, preserving symmetry and gapped structure, and realizing a $bZ_2$ topological order with fermionic statistics.

Experimental results

Research questions

• RQ1 Can exactly solvable lattice Hamiltonians be constructed for general bosonic beyond group cohomology invertible topological phases in arbitrary spacetime dimensions?
• RQ2 How do gravitational anomalies manifest in the boundary of such lattice models, particularly in the absence of global symmetry?
• RQ3 What is the role of one-form and two-form symmetries in realizing non-trivial topological order via gauging?
• RQ4 Can a gapped symmetric boundary state be explicitly constructed for a (4+1)D $bZ_2$-symmetric beyond group cohomology phase?
• RQ5 What new types of quantum phase transitions are protected by invertible phases with non-trivial cobordism invariants?

Key findings

• The authors construct an exactly solvable Hamiltonian for the (4+1)D beyond group cohomology invertible phase with effective action $\pi \int w_2 \bigcup w_3$, realized via gauging of one-form and two-form symmetries.
• The model realizes a non-trivial quantum cellular automaton of order two in (4+1)D, confirming a new type of topological order.
• The boundary of the (4+1)D $bZ_2$-symmetric phase exhibits a $\bbZ_2$ topological order with fermionic particle and loop excitations that mutually braided with $\pi$ statistics.
• A gapped symmetric boundary state is explicitly constructed for the (4+1)D $bZ_2$-symmetric phase, preserving the symmetry and topological order.
• The method generalizes to any spacetime dimension and applies to all bosonic invertible phases of order two or four under stacking, with finite-dimensional local Hilbert spaces.
• The construction confirms that the interface between distinct beyond group cohomology phases is protected by both 't Hooft and gravitational anomalies, even in lattice models without Poincaré symmetry.