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arXiv:1709.08631 · Physical review. B./Physical review. B

String order parameters for one-dimensional Floquet symmetry protected topological phases

Ajesh Kumar, Philipp T. Dumitrescu, Andrew C. Potter

Published Jun 4, 2018• 17 citations
arXivHTMLPDF

Abstract

Floquet symmetry protected topological (FSPT) phases are nonequilibrium topological phases enabled by time-periodic driving. FSPT phases of one-dimensional (1D) chains of bosons, spins, or qubits host dynamically protected edge states that can store quantum information without decoherence, making them promising for use as quantum memories. While FSPT order cannot be detected by any local measurement, here we construct nonlocal string order parameters that directly measure general 1D FSPT order. We propose a superconducting-qubit array based realization of the simplest Ising FSPT phase, which can be implemented with existing quantum computing hardware. We devise an interferometric scheme to directly measure the nonlocal string order using only simple one- and two-qubit operations and single-qubit measurements.

DOI
10.1103/physrevb.97.224302

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Floquet symmetry protected topological (FSPT) phases are nonequilibrium topological phases enabled by time-periodic driving. FSPT phases of one-dimensional (1D) chains of bosons, spins, or qubits host dynamically protected edge states that can store quantum information without decoherence, making them promising for use as quantum memories. While FSPT order cannot be detected by any local measurement, here we construct nonlocal string order parameters that directly measure general 1D FSPT order. We propose a superconducting-qubit array based realization of the simplest Ising FSPT phase, which can be implemented with existing quantum computing hardware. We devise an interferometric scheme to directly measure the nonlocal string order using only simple one- and two-qubit operations and single-qubit measurements.
Topological phases which host Majorana fermions can not be identified via local order parameters. We give simple nonlocal order parameters to distinguish quasi-one-dimensional (1D) topological superconductors of spinless fermions, for any interacting model in the absence of time reversal symmetry. These string or ``brane'' order parameters are natural for measurements in cold atom systems using quantum gas microscopy. We propose them as a way to identify symmetry-protected topological phases of Majorana fermions in cold atom experiments via bulk rather than edge degrees of freedom. Subsequently, we study two-dimensional (2D) topological superconductors via the quasi-1D limit of coupling NNN identical chains on the cylinder. We classify the symmetric, interacting topological phases protected by the additional ZN{\mathbb{Z}}_{N}ZN​ translation symmetry. The phases include quasi-1D analogs of (i) the p+ipp+ipp+ip chiral topological superconductor, which can be distinguished up to the 2D Chern number mod 2, and (ii) the 2D weak topological superconductor. We devise general rules for constructing nonlocal order parameters which distinguish the phases. These rules encode the signature of the fermionic topological phase in the symmetry properties of the terminating operators of the nonlocal string or brane. The nonlocal order parameters for some of these phases simply involve a product of the string order parameters for the individual chains. Finally, we give a physical picture of one of the topological phases as a condensate of certain defects, which motivates the form of the nonlocal order parameter and is reminiscent of higher dimensional constructions of topological phases.
A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped symmetric phases in one-dimensional (1D) systems can be completely characterized using tools related to projective representations of the symmetry groups. We explain two ways to detect these symmetry protected topological phases in 1D. First, we give a numerical approach for directly extracting the projective representations from a matrix-product state representation. Second, we derive nonlocal order parameters for time-reversal and inversion symmetry, and discuss a generalized string order for local symmetries for which the regular string-order parameter cannot be applied. We furthermore point out that the nonlocal order parameter for these topological phases is actually related to topological surfaces.
Floquet topological matter has emerged as one exciting platform to explore rich physics and game-changing applications of topological phases. As one remarkable and recently discovered feature of Floquet symmetry protected topological (SPT) phases, in principle, a simple periodically driven system can host an arbitrary number of topological protected zero edge modes and $\ensuremath{\pi}$ edge modes, with Majorana zero modes and Majorana $\ensuremath{\pi}$ modes as examples protected by the particle-hole symmetry. This work advocates a new route to holonomic quantum computation by exploiting the co-existence of many Floquet SPT edge modes, all of which have trivial dynamical phases during a computation protocol. As compelling evidence supporting this ambitious goal, three pairs of Majorana edge modes, hosted by a periodically driven one-dimensional (1D) superconducting superlattice, are shown to suffice to encode two logical qubits, realize quantum gate operations, and execute two simple quantum algorithms through adiabatic lattice deformation. When compared with early studies on quantum computation based on Majorana zero modes of topological quantum wires, significant resource saving is now made possible by use of Floquet SPT phases.
The periodically driven quantum Ising chain has recently attracted a large attention in the context of Floquet engineering. In addition to the common paramagnet and ferromagnet, this driven model can give rise to new topological phases. In this work, we systematically explore its quantum phase diagram by examining the properties of its Floquet ground state. We specifically focus on driving protocols with time-reversal invariant points, and demonstrate the existence of an infinite number of distinct phases. These phases are separated by second-order quantum phase transitions, accompanied by continuous changes of local and string order parameters, as well as sudden changes of a topological winding number and of the number of protected edge states. When one of these phase transitions is adiabatically crossed, the correlator associated to the order parameter is nonvanishing over a length scale which shows a Kibble-Zurek scaling. In some phases, the Floquet ground state spontaneously breaks the discrete time-translation symmetry of the Hamiltonian. Our findings provide a better understanding of topological phases in periodically driven clean integrable models.
We propose an order parameter for the symmetry-protected topological (SPT) phases which are protected by Abelian on-site symmetries. This order parameter, called the SPT entanglement, is defined as the entanglement between AAA and BBB, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region CCC is measured and known, where CCC is a connected region surrounded by A,BA, BA,B, and the boundaries of the system. In the case of one-dimensional systems we prove that in the limit where AAA and BBB are large and far from each other compared to the correlation length, the SPT entanglement remains constant throughout a SPT phase, and furthermore, it is zero for the trivial phase while it is nonzero for all the nontrivial phases. Moreover, we show that the SPT entanglement is invariant under the low-depth quantum circuits which respect the symmetry, and hence it remains constant throughout a SPT phase in the higher dimensions as well. Also, we show that there is an intriguing connection between SPT entanglement and the Fourier transform of the string order parameters, which are the traditional tool for detecting SPT phases. This leads to an algorithm for extracting the relevant information about the SPT phase of the system from the string order parameters. Finally, we discuss implications of our results in the context of measurement-based quantum computation.
The topological orders in amorphous systems that lack crystalline symmetry have gained considerable attention recently. Here we propose the Floquet amorphous topological matter, among which the topological orders are explored in experimentally accessible one-dimensional array of randomly pointed Rydberg atoms with periodic driving. The topological properties are comprehensively characterized, considering both the single-particle and many-body perspectives. It is found that the periodic driving leads to rich topological phases of matter. At the single-particle level, we evaluate the real space winding numbers and polarization, revealing robust amorphous topological phases with 0-type and π-type edge modes. We show a structural disorder induced topological phase transition associated with localization transition in the nonequilibrium system. Remarkably, in the many-body case it is discovered that the amorphous topological order exists in the chain of hardcore bosons, captured by the topological entanglement entropy and the string order. Moreover, feasible experimental probe protocols are also elaborated. The topological phase in amorphous systems that lack long-range order brings a fresh perspective to the topological states of matter. Here, the authors propose the concept of Floquet amorphous topological order and consider its realization with periodically-driven Rydberg atoms.