Finding suitable nonlocal order parameters that distinguish various symmetry-protected topological (SPT) phases is an important subject in view of experimental and numerical detection of SPT phases. By ``simulating'' the generating manifold of cobordism group in the operator formalism, the authors here propose nonlocal operations as diagnoses for SPT phases protected by point group symmetries. The nonlocal operations involve ``partial point group transformations'', which are obtained by point group transformations restricted to a spatial subregion on a given ground-state wave function. Through analytical and numerical calculations, the authors show that the complex
U(1) phase of the ground state expectation value of such partial point group transformations may serve as an order parameter for those SPT phases. The examples in the paper include the
Z8 and
Z16 invariants of topological superconductors protected by inversion symmetry in
(1+1) and
(3+1) dimensions, respectively, as well as the lens space topological invariants in
(2+1)-dimensional fermionic topological phases.
We define and compute many-body topological invariants of interacting fermionic symmetry-protected topological phases, protected by an orientation-reversing symmetry, such as time-reversal or reflection symmetry. The topological invariants are given by partition functions obtained by a path integral on unoriented spacetime which, as we show, can be computed for a given ground state wave function by considering a nonlocal operation, "partial" reflection or transpose. As an application of our scheme, we study the Z_{8} and Z_{16} classification of topological superconductors in one and three dimensions.
Although symmetry-protected topological phases (SPTs) can be distinguished by their entanglement properties, it has been unclear how to extract this information directly from expectation values beyond the 1D case. Here, we close this gap and propose a pair of nonlocal order parameters that can detect and distinguish all bosonic SPTs in 2D protected by internal discrete, Abelian unitary symmetries. The desired topological invariants are extracted by these quantities by effectively simulating the SPT path integral on topologically non-trivial spacetime manifolds. Our order parameters are defined in terms of expectation values of partial symmetry and permutation operations acting on fixed numbers of replicas of the system in finite spatial regions. These expectation values correspond to symmetry-twisted versions of multipartite entanglement quantities known as multi-entropies. We show explicitly that our two order parameters detect symmetry-protected four-party and six-party entanglement, respectively, and we constrain possible "spurious" contributions. We analytically test our proposal in fixed-point lattice models. Our results suggest multipartite entanglement to be a defining feature of SPTs; indeed, we expect our methods to generalize to fermionic and higher-dimensional systems.
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.
Quantum anomalies, breakdown of classical symmetries by quantum effects, provide a sharp definition of symmetry protected topological phases. In particular, they can diagnose interaction effects on the noninteracting classification of fermionic symmetry protected topological phases. In this paper, we identify quantum anomalies in two kinds of (3+1)d fermionic symmetry protected topological phases: (i) topological insulators protected by CP (charge conjugation $\ifmmode\times\else\texttimes\fi{}$ reflection) and electromagnetic
U(1) symmetries, and (ii) topological superconductors protected by reflection symmetry. For the first example, which is related to, by CPT-theorem, time-reversal symmetric topological insulators, we show that the CP-projected partition function of the surface theory is not invariant under large
U(1) gauge transformations, but picks up an anomalous sign, signaling a
Z2 topological classification. Similarly, for the second example, which is related to, by CPT-theorem, class DIII topological superconductors, we discuss the invariance/noninvariance of the partition function of the surface theory, defined on the three-torus and its descendants generated by the orientifold projection, under large diffeomorphisms (coordinate transformations). The connection to the collapse of the noninteracting classification by an integer (
Z) to
Z16, in the presence of interactions, is discussed.
We explore one-dimensional fermionic symmetry-protected topological (SPT) phases related by the crystalline equivalence principle. In particular, we study charge-conserving many-body topological phases of fermions protected respectively by chiral and reflection symmetries. While the classifications of the two crystalline-equivalent SPT phases are identical, their topological properties and phase structures can be very different, depending on the microscopic details. Specifically, we consider certain extensions of the Su-Schrieffer-Heeger model, with and without interactions, that preserve both chiral and reflection symmetries, and explicitly compute the many-body topological invariants based on the systems' ground states. The phase structures determined by these topological invariants align perfectly with the many-body spectra of deformations among the models. As expected, gapped deformations exist only when all the topological invariants remain unchanged. Moreover, we show that decomposable systems -- those that can be decomposed into local and decoupled subsystems -- can be topologically characterized by real-space quantum numbers directly associated with the symmetries. For reflection-symmetric systems, these quantum numbers are related to the many-body topological invariants via a bulk-center correspondence, which can be justified using the Atiyah-Hirzebruch spectral sequence in generalized homology theory. Finally, we discuss the role of transition symmetry in the many-body topologies of these SPT phases.