As is well-known in the context of topological insulators and superconductors, short-range-correlated fermionic pure Gaussian states with fundamental symmetries are systematically classified by the periodic table. We revisit this topic from a quantum-information-inspired operational perspective without referring to any Hamiltonians, and apply the formalism to bosonic Gaussian states as well as (both fermionic and bosonic) locality-preserving unitary Gaussian operations. We find that while bosonic Gaussian states are all trivial, there exist nontrivial bosonic Gaussian operations that cannot be continuously deformed into the identity under the locality and symmetry constraint. Moreover, we unveil unexpectedly complicated relations between fermionic Gaussian states and operations, pointing especially out that some of the former can be disentangled by the latter under the same symmetry constraint, while some cannot. In turn, we find that some topological operations are genuinely dynamical, in the sense that they cannot create any topological states from a trivial one, yet they are not connected to the identity. The notions of disentanglability and genuinely dynamical topology can be unified in a general picture of unitary-to-state homomorphism, and apply equally to interacting topological phases and quantum cellular automata.
As is well-known in the context of topological insulators and superconductors, short-range-correlated fermionic pure Gaussian states with fundamental symmetries are systematically classified by the periodic table. We revisit this topic from a quantum-information-inspired operational perspective without referring to any Hamiltonians, and apply the formalism to bosonic Gaussian states as well as (both fermionic and bosonic) locality-preserving unitary Gaussian operations. We find that while bosonic Gaussian states are all trivial, there exist nontrivial bosonic Gaussian operations that cannot be continuously deformed into the identity under the locality and symmetry constraint. Moreover, we unveil unexpectedly complicated relations between fermionic Gaussian states and operations, pointing especially out that some of the former can be disentangled by the latter under the same symmetry constraint, while some cannot. In turn, we find that some topological operations are genuinely dynamical, in the sense that they cannot create any topological states from a trivial one, yet they are not connected to the identity. The notions of disentanglability and genuinely dynamical topology can be unified in a general picture of unitary-to-state homomorphism, and apply equally to interacting topological phases and quantum cellular automata.
This dissertation brings together ideas from quantum information theory and condensed matter physics to study many-particle systems of fermions. We revisit the structure of the Hilbert space and operator algebras in Fermi systems which provides a natural platform to discuss the notion of entanglement and local operations. We generalize the partial transpose, which is a well-known operation to diagnose entanglement in density matrices of qubits and bosonic systems, to fermionic systems. Our idea was inspired by the observation that time-reversal acts as transposition on density matrices; hence, we looked for a way to apply partial time-reversal to density matrices of fermions. We present a comprehensive set of benchmarks on our proposed definition of fermionic partial transpose and explain the fundamental differences between the bosonic and fermionic partial transpose. We use this new framework and the associated entanglement measure, logarithmic negativity, to study the entanglement content of free fermions with an arbitrary shape of Femi surface in all dimensions as well as topological insulators and superconductors. In particular, we show how thermal fluctuations destroy the quantum coherence of the ground state as temperature is increased. Furthermore, we report the discovery of a surprising connection between topological invariants and the partial transpose in time-reversal symmetric topological insulators and superconductors. In short, we find that partition functions on non-orientable spacetime manifolds such as the Klein bottle or real-projective plane can be obtained by combining untransposed and partially transposed density matrix. This relation turns out to be general and can be used to study various other topological phases protected by anti-unitary symmetries. Similar to the partition functions, the quantities we introduce are complex numbers and their complex phase is a topological invariant. These topological invariants can be regarded as order parameters of topological phases in the sense that they admit discrete values and can only change by jumping from one value to another as we transition from one topological phase to another.
In a recent paper [Bardyn et al., Phys. Rev. X 8, 011035 (2018)], it was shown that the generalization of many-body polarization to mixed states can be used to construct a topological invariant that is also applicable to finite-temperature and nonequilibrium Gaussian states of lattice fermions. The many-body polarization defines an ensemble geometric phase that is identical to the Zak phase of a fictitious Hamiltonian, whose symmetries determine the topological classification. Here we show that in the case of Gaussian states of bosons, the corresponding topological invariant is always trivial. This also applies to finite-temperature states of bosons in lattices with a topologically nontrivial band structure. As a consequence, there is no quantized topological charge pumping for translationally invariant bulk states of noninteracting bosons.
Supersymmetry (SUSY), originally proposed in particle physics, refers to a dual relation that connects fermionic and bosonic degrees of freedom in a system. Recently, there has been considerable interest in applying the idea of SUSY to topological phases, motivated by the attempt to gain insights from the fermion side into the boson side and vice versa. We present a systematic study of this construction when applied to band topology in noninteracting systems. First, on top of the conventional tenfold way, we find that topological insulators and superconductors are divided into three classes depending on whether the supercharge can be local and symmetric, must break a symmetry to preserve locality, or needs to break locality. Second, we resolve the apparent paradox between the nontriviality of free fermions and the triviality of free bosons by noting that the topological information is encoded in the identification map. We also discuss how to understand a recently revealed SUSY entanglement duality in this context. These findings are illustrated by prototypical examples. In this paper, we shed light on band topology from the perspective of SUSY.
Topological insulators and superconductors are subsets of (almost) noninteracting symmetry protected topological (SPT) phases. More generally, SPT phases represent equivalence classes of (equilibrium) gapped quantum states with short-range entanglement that are protected by certain symmetries. Two quantum states are “equivalent” if they can be adiabatically deformed into one another without closing the bulk gap. Thus, quantum states belonging to different SPT phases cannot be smoothly deformed into each other without a phase transition, as long as the underlying symmetry is preserved. However, upon removing the symmetry all SPT states can be adiabatically deformed to a trivial product state (e.g., an atomic insulator) without undergoing a phase transition. In this chapter, we generalize our discussion of (almost) noninteracting topological states by removing some of the explicit and implicit constraints underlying the classification and properties of topological insulators and superconductors discussed in Chapter 3 . In Section 4.1 , we enlarge the type of symmetries that protect the topological phases by supplementing internal symmetries (e.g., time-reversal and particle-hole symmetries) with crystalline symmetries (e.g., mirror and rotation symmetries) and discuss topological crystalline insulators and superconductors , a subsets of SPT phases protected by internal and crystalline symmetries. In Section 4.2 , we remove the requirement of having a finite (bulk) gap and discuss Weyl semimetals – crystalline systems characterized by topologically protected bulk band degeneracy points separated in momentum space, which support low energy excitations that are massless chiral fermions, i.e., Weyl fermions. Finally, in Section 4.3 , we remove the condition of thermodynamic equilibrium and consider topological quantum states in periodically driven systems, focusing on the so-called Floquet topological insulators , which exhibit a richer topological structure than their static counterparts. We note that aspects regarding the effects of particle-particle interactions, including the emergence of topological order in systems with long-range entanglement, will be addressed in Chapter 5 . We also point out that the areas summarized in this chapter have grown tremendously in recent years and continue to grow; even a brief mentioning of all recent developments would be impossible within the constraints of a book chapter. Instead, we point out the key elements that determine the nontrivial topology of the quantum systems mentioned above, discuss specific manifestations of their topological properties (e.g., the presence of protected gapless states), and provide representative examples.
Recently, it was realized that quantum states of matter can be classified as long-range entangled states (i.e., with nontrivial topological order) and short-range entangled states (i.e., with trivial topological order). We can use group cohomology class Hd(SG,R/Z) to systematically describe the SRE states with a symmetry SG [referred as symmetry-protected trivial (SPT) or symmetry-protected topological (SPT) states] in d-dimensional space-time. In this paper, we study the physical properties of those SPT states, such as the fractionalization of the quantum numbers of the global symmetry on some designed point defects and the appearance of fractionalized SPT states on some designed defect lines/membranes. Those physical properties are SPT invariants of the SPT states which allow us to experimentally or numerically detect those SPT states, i.e., to measure the elements in Hd(G,R/Z) that label different SPT states. For example, 2+1-dimensional bosonic SPT states with Zn symmetry are classified by a Zn integer $m\ensuremath{\in}{\mathcal{H}}^{3}({Z}_{n},\mathbb{R}/\mathbb{Z})={\mathbb{Z}}_{n}$. We find that n identical monodromy defects, in a Zn SPT state labeled by m, carry a total Zn charge 2m (which is not a multiple of n in general).
This thesis is an attempt to understand the physics of short-range entangled phases of fermions through several related approaches. The first angle is topological quantum field theory. We discuss the classification of interacting fermionic short-range entangled phases by spin cobordism and give an algebraic characterization of unoriented equivariant bosonic topological quantum field theories in one spatial dimension. A second tool is tensor network representation. We develop the formalism of fermionic matrix product states and use it to derive the stacking group law for one dimensional symmetry-enriched fermionic short-range entangled phases. We also study its relationship with state sum constructions of topological quantum field theories and develop a state sum construction for pin-minus theories in one spatial dimension. The third approach is topological band theory. We classify free fermionic phases enriched by a unitary symmetry in any dimension and determine the map into the interacting classification.