Many-Body Topological Invariants for Fermionic Symmetry-Protected Topological Phases

Abstract

We define and compute many-body topological invariants of interacting fermionic symmetryprotected 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 non-local operation, "partial" reflection or transpose. As an application of our scheme, we study the Z8 and Z16 classification of topological superconductors in one and three dimensions.The Thouless-Kohmoto-Nightingale-den Nijs (TKNN) formula [1, 2] is the prototype for topological characterization of phases of matter. It relates the quantized Hall conductance to the (first) Chern number defined for Bloch wave functions. At the level of many-body physics, the quantized Hall conductance can also be formulated in terms of ground state wave functions in the presence of twisted boundary conditions ("the many-body Chern number") [3]. In contrast to local order parameters, the TKNN integer distinguishes different quantum phases of matter by focusing on their global topological properties.More recently, the discovery of topological insulators and superconductors [4, 5] has led to a new research frontier, generally referred to as symmetry protected topological (SPT) phases. These phases are adiabatically connected to topologically trivial states, i.e., atomic insulators which can be represented as simple product states without any entanglement. Nevertheless, they are topologically distinct once a symmetry condition, e.g., timereversal symmetry, is imposed. A complete classification of the noninteracting fermionic SPT phases protected by non-spatial discrete symmetries [6][7][8], as well as crystalline SPT phases protected by spatial symmetries [9][10][11][12], were achieved.However, it was later discovered that the noninteracting topological classification is not the full story, and can be dramatically altered once interaction effects are taken into account [13]. Since then, there have been several works which discuss the breakdown of the noninteracting classification in the presence of interactions [14][15][16][17][18][19][20][21][22][23][24][25].There are various topological invariants for noninteracting fermionic SPT phases using single-particle states (e.g., Bloch wave functions). For example, the Z 2 -valued topological invariants have been introduced for topological insulators both in two and three spatial dimensions [26][27][28][29]. For topological superconductors protected by time-reversal symmetry, the integer-valued topological invariants ("the winding number") have been introduced [6]. However, the discovered breakdown of non-interacting classification clearly indicates that the situation at the interacting level is more intricate, and a general framework to distinguish interacting fermionic SPT phases is lacking. This should be contrasted from the quantized Hall conductance, which can be formulate...

Pin TQFT and Grassmann integral

“…(1. 2) In contrast, it is sometimes useful to consider a fermionic topological phase on an unoriented manifold [11][12][13][14][15], when the system has a symmetry that reverses the orientation of spacetime. In such a situation, the corresponding theory requires a pin structure, which encodes the orientation reversing symmetry.…”

Quantum information dynamics in multipartite integrable systems

“…In the following, we denote as E (f ) A1:A2 this fermionic negativity (see Ref. [92] and below for its definition), and E (b) A1:A2 the one in (1). The quasiparticle picture for the entanglement spreading [4] is illustrated in Fig.…”

Measuring fermionic entanglement: Entropy, negativity, and spin structure

“…The above calculation of negativities is just a special case of a general fermionic system with nontrivial spin structure. Similar quantities appear in, e.g., string theory [100] or the classification of interacting topological systems [98,99]. We use the connection of such manifolds to entanglement measures of fermionic systems [96] in order to show that they are not mere theoretical constructs, but could actually be measured with ultracold atoms.…”