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.
Here, the authors introduce a set of quantities to diagnose symmetry-protected topological phases of fermions protected by antiunitary symmetries. These quantities, which can be written as expectation values of nonlocal operators, effectively simulate the partition function on nonorientable spacetime manifolds in the operator formalism. The important observation is that the proposed quantities are complex valued for topological states, where the complex phase is the many-body topological invariant. Numerous symmetry classes and dimensionalities are studied in detail. The key ingredient of all quantities is the ``fermionic partial transpose''. For example, the
Z2 topological invariant for time-reversal symmetric topological insulators in two spatial dimensions is given.
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.
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.
Abstract We review a recipe to produce a lattice construction of fermionic phases of matter in the presence of time reversal symmetry by extending the fermionization and bosonization known in (1+1) dimensions to various setups including higher spacetime dimensions in the presence of global symmetries. As an application, we provide a state sum lattice path integral for a (1+1)-dimensional topological superconductor with time reversal symmetry generating the
Z8 classification of the symmetry-protected topological phase. We also illustrate a state sum path integral for a (3+1)-dimensional topological superconductor with time reversal symmetry that generates the
Z16 classification.
We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the ‘collapse’ of the non-interacting classification, where an integer classification (‘’) at non-interacting level reduces to or in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.
We give a brief overview on interacting symmetry-protected topological (SPT) phases. In particular, we will focus on various cases where the classification of non-interacting fermionic systems breaks down in the presence of interactions. The examples include topological (and trivial) superconductors in one, two and three spatial dimensions, protected by various symmetries such as time-reversal, spin parity, and reflection symmetries. Common to these examples is the ‘collapse’ of the non-interacting classification, where an integer classification (‘ ’) at non-interacting level reduces to or in the presence of interactions. We make use of these examples to illustrate various approaches to interacting SPT phases.