Schedule for: 21w5502 - Topology and Entanglement in Many-Body Systems (Online)

A brief introduction to BIRS with important logistical information, technology instruction, and opportunity for participants to ask questions.

In this talk, we discuss a recent result on a bulk gap for a truncated Haldane pseudopotential with maximal half filling, which describes a strongly correlated system of spinless bosons in a cylinder geometry. For this Hamiltonian with either open or periodic boundary conditions, we prove a spectral gap above the highly degenerate ground-state space which is uniform in the volume and particle number. Our proofs rely on identifying invariant subspaces to which we apply gap-estimate methods previously developed only for quantum spin Hamiltonians. In the case of open boundary conditions, the lower bound on the spectral gap accurately reflects the presence of edge states, which do not persist into the bulk. Customizing the gap technique to the invariant subspace, we avoid the edge states and establish a more precise estimate on the bulk gap in the case of periodic boundary conditions. The same approach can also be applied to prove a bulk gap for the analogously truncated 1/3-filled Haldane pseudopotential for the fractional quantum Hall effect.

We will discuss an area law for ground states of locally gapped frustration-free 2D lattice spin systems. We first generalize the optimal approximation of the boolean AND function to a non-commuting setting, showing that the ground state projector of a locally gapped frustration-free 1D spin system can be optimally approximated in a similar manner. For 2D spin systems we then construct an approximate ground state projector (AGSP) that employs the optimal 1D approximation along the boundary of the bipartition of interest. If time permits, we will also discuss the challenges in extending the proof to 3D systems. Joint work with Itai Arad and David Gosset (https://arxiv.org/abs/2103.02492).

I will talk about our trial on classification of SPT phases in quantum statistical mechanics.

Quantum phases (without symmetry) in two-dimensional (2D) gapped systems are expected to be classified into different topologically ordered phases, which cannot be described by the conventional theory of symmetry-breaking. A vast amount of work has been dedicated to classify/characterize these new phases, and nowadays, it is widely believed that all topologically ordered phases can be classified by topological quantum field theory (TQFT). Although the existing analytical studies as well as numerical studies are consistent with the prediction of this theory, we still lack a general, concrete proof that directly shows that any 2D gapped quantum phases should be classified in this framework. In this talk, we propose the entanglement bootstrap program to make progress on this important problem. In this program, we posit that local reduced density matrices of such systems obey a set of entropic identities followed by the area law of entanglement. We show that the axioms of the fusion rules of anyon are derived from the entropic identities only. We then discuss recent applications of this approach to S-matrix and central charges. We also discuss an obstacle of the universality of this approach, spurious topological entropy.

I discuss the stability of spectral gaps for quantum lattice systems subject to small perturbations of the interactions defining the Hamiltonian and its connection to topological and symmetry-protected topological phases. I will review the current status of the problem and then discuss the application of recent results to the stability of a dimerized phases in a family of quantum spin chains with O(n) symmetry and certain features of their phase diagram, including in the neighborhood of a non-frustration-free point. (Includes joint work with Robert Sims and Amanda Young and with Jakob Bjoernberg, Peter Muehlbacher, and Daniel Ueltschi).

Some gapped quantum many-body systems have the interesting property that they have multiple nearly degenerate ground states with an energy splitting that is exponentially small in the system size. Furthermore, this nearly exact degeneracy is a robust phenomenon in the sense that it does not require fine tuning parameters in the Hamiltonian. While this kind of nearly exact ground state degeneracy is a well-established phenomenon for systems with short-range interactions, much less is known about long-range interactions (e.g., power-law decaying interactions). In this talk, I will present a rigorous result showing that a robust, exponentially small ground state splitting also occurs in some models with long-range interactions.

Tensor Networks constitute a very rich family of states that is known to capture the low energy sector of locally interacting quantum systems. One of their most interesting features is the existence of a rigorous bulk-boundary correspondence: interesting (and usually complex) bulk properties have an easier characterization at the boundary. Examples of such bulk properties include the existence of a spectral gap or the presence of topological order. Both can be combined to shed new light in the problem of the existence of self-correcting quantum memories in 2D.

A QCA is a locality preserving automorphism of operator algebra on lattices. Such an automorphism is characterized by a subalgebra on a boundary. The existence of a maximal commutative set of local operators within the subalgebra restricts feasible QCAs. From this approach, we classify Clifford QCA in 3d, the ones that map every Pauli operator to another.

Unitary evolutions of quantum lattice systems that preserve locality are called "quantum cellular automata", or QCAs. QCAs in one dimension have been completely classified by an index theory. However, physical systems often only preserve locality approximately. For example, Hamiltonian evolutions on the lattice satisfy Lieb-Robinson bounds rather than strict locality. In this talk, we will discuss QCAs that approximately preserve locality. As we will see, the index theory of 1D QCAs is robust even when considering these more general evolutions. As a consequence, we also obtain a converse to the Lieb-Robinson bounds in one dimension. Based on work with Daniel Ranard and Freek Witteveen in arXiv:2012.00741.

I will talk about how to define invariants for families of d-dimensional gapped ground states on the lattice known as Higher Berry classes. As an application I will show how to construct the invariant of 2d gapped ground states with semisimple Lie group symmetry G taking values in H^3(G,R) and show it’s quantization for invertible states.