Motivated by recent work showing that a quantum error correcting code can be generated by hybrid dynamics of unitaries and measurements, we study the long time behavior of such systems. We demonstrate that even in the mixed phase, a maximally mixed i
nitial density matrix is purified on a time scale equal to the Hilbert space dimension (i.e., exponential in system size), albeit with noisy dynamics at intermediate times which we connect to Dyson Brownian motion. In contrast, we show that free fermion systems -- i.e., ones where the unitaries are generated by quadratic Hamiltonians and the measurements are of fermion bilinears -- purify in a time quadratic in the system size. In particular, a volume law phase for the entanglement entropy cannot be sustained in a free fermion system.
We construct an exactly solvable commuting projector model for a $4+1$ dimensional ${mathbb Z}_2$ symmetry-protected topological phase (SPT) which is outside the cohomology classification of SPTs. The model is described by a decorated domain wall con
struction, with three-fermion Walker-Wang phases on the domain walls. We describe the anomalous nature of the phase in several ways. One interesting feature is that, in contrast to in-cohomology phases, the effective ${mathbb Z}_2$ symmetry on a $3+1$ dimensional boundary cannot be described by a quantum circuit and instead is a nontrivial quantum cellular automaton (QCA). A related property is that a codimension-two defect (for example, the termination of a ${mathbb Z}_2$ domain wall at a trivial boundary) will carry nontrivial chiral central charge $4$ mod $8$. We also construct a gapped symmetric topologically-ordered boundary state for our model, which constitutes an anomalous symmetry enriched topological phase outside of the classification of arXiv:1602.00187, and define a corresponding anomaly indicator.
We present a 2D bosonization duality using the language of tensor networks. Specifically, we construct a tensor network operator (TNO) that implements an exact 2D bosonization duality. The primary benefit of the TNO is that it allows for bosonization
at the level of quantum states. Thus, we use the TNO to provide an explicit algorithm for bosonizing fermionic projected entangled pair states (fPEPs). A key step in the algorithm is to account for a choice of spin-structure, encoded in a set of bonds of the bosonized fPEPS. This enables our tensor network approach to bosonization to be applied to systems on arbitrary triangulations of orientable 2D manifolds.
We analyze the class of Generalized Double Semion (GDS) models in arbitrary dimensions from the point of view of lattice Hamiltonians. We show that on a $d$-dimensional spatial manifold $M$ the dual of the GDS is equivalent, up to constant depth loca
l quantum circuits, to a group cohomology theory tensored with lower dimensional cohomology models that depend on the manifold $M$. We comment on the space-time topological quantum field theory (TQFT) interpretation of this result. We also investigate the GDS in the presence of time reversal symmetry, showing that it forms a non-trivial symmetry enriched toric code phase in odd spatial dimensions.
We construct a three-dimensional quantum cellular automaton (QCA), an automorphism of the local operator algebra on a lattice of qubits, which disentangles the ground state of the Walker-Wang three fermion model. We show that if this QCA can be reali
zed by a quantum circuit of constant depth, then there exists a two-dimensional commuting projector Hamiltonian which realizes the three fermion topological order which is widely believed not to be possible. We conjecture in accordance with this belief that this QCA is not a quantum circuit of constant depth, and we provide two further pieces of evidence to support the conjecture. We show that this QCA maps every local Pauli operator to a local Pauli operator, but is not a Clifford circuit of constant depth. Further, we show that if the three-dimensional QCA can be realized by a quantum circuit of constant depth, then there exists a two-dimensional QCA acting on fermionic degrees of freedom which cannot be realized by a quantum circuit of constant depth; i.e., we prove the existence of a nontrivial QCA in either three or two dimensions. The square of our three-dimensional QCA can be realized by a quantum circuit of constant depth, and this suggests the existence of a $mathbb{Z}_2$ invariant of a QCA in higher dimensions, totally distinct from the classification by positive rationals (i.e., by one integer index for each prime) in one dimension. In an appendix, unrelated to the main body of this paper, we give a fermionic generalization of a result of Bravyi and Vyalyi on ground states of 2-local commuting Hamiltonians.
We prove that neither Integer nor Fractional Quantum Hall Effects with nonzero Hall conductivity are possible in gapped systems described by Local Commuting Projector Hamiltonians.
We construct fixed point lattice models for group supercohomology symmetry protected topological (SPT) phases of fermions in 2+1D. A key feature of our approach is to construct finite depth circuits of local unitaries that explicitly build the ground
states from a tensor product state. We then recover the classification of fermionic SPT phases, including the group structure under stacking, from the algebraic composition rules of these circuits. Furthermore, we show that the circuits are symmetric, implying that the group supercohomology phases can be many body localized. Our strategy involves first building an auxiliary bosonic model, and then fermionizing it using the duality of Chen, Kapustin, and Radicevic. One benefit of this approach is that it clearly disentangles the role of the algebraic group supercohomology data, which is used to build the auxiliary bosonic model, from that of the spin structure, which is combinatorially encoded in the lattice and enters only in the fermionization step. In particular this allows us to study our models on 2d spatial manifolds of any topology and to define a lattice-level procedure for ungauging fermion parity.
Symmetry protected topological (SPT) phases are well understood in the context of free fermions and in the context of interacting but essentially bosonic models. Recently it has been realized that intrinsically fermionic SPTs exist which only appear
in interacting models. Here we show that the 3+1 dimensional realizations of these phases have surface states characterized by a new t Hooft anomaly, captured by a $H^3(G, Z_2)$ class. This is encoded in the anomalous action of symmetry on the surface states with topological order, which must necessarily permute the anyons. We discuss in detail an example with symmetry group $G = Z_2 times Z_4$. Using a network model of the surface we derive a candidate surface topological order given by a $Z_4$ gauge theory. We relate our findings to anomalies valued in $H^3$ with various coefficients introduced previously in both bosonic and fermionic settings, and describe a general framework that unifies these various anomalies.
We study the eigenstate phases of disordered spin chains with on-site finite non-Abelian symmetry. We develop a general formalism based on standard group theory to construct local spin Hamiltonians invariant under any on-site symmetry. We then specia
lize to the case of the simplest non-Abelian group, $S_3$, and numerically study a particular two parameter spin-1 Hamiltonian. We observe a thermal phase and a many-body localized phase with a spontaneous symmetry breaking (SSB) from $S_3$ to $mathbb{Z}_3$ in our model Hamiltonian. We diagnose these phases using full entanglement distributions and level statistics. We also use a spin-glass diagnostic specialized to detect spontaneous breaking of the $S_3$ symmetry down to $mathbb{Z}_3$. Our observed phases are consistent with the possibilities outlined by Potter and Vasseur [Phys. Rev. B 94, 224206 (2016)], namely thermal/ ergodic and spin-glass many-body localized (MBL) phases. We also speculate about the nature of an intermediate region between the thermal and MBL+SSB regions where full $S_3$ symmetry exists.
We construct exactly solvable models for a wide class of symmetry enriched topological (SET) phases. Our construction applies to 2D bosonic SET phases with finite unitary onsite symmetry group $G$ and we conjecture that our models realize every phase
in this class that can be described by a commuting projector Hamiltonian. Our models are designed so that they have a special property: if we couple them to a dynamical lattice gauge field with gauge group $G$, the resulting gauge theories are equivalent to modified string-net models. This property is what allows us to analyze our models in generality. As an example, we present a model for a phase with the same anyon excitations as the toric code and with a $mathbb{Z}_2$ symmetry which exchanges the $e$ and $m$ type anyons. We further illustrate our construction with a number of additional examples.
