Bulletin of the American Physical Society
APS March Meeting 2014
Volume 59, Number 1
Monday–Friday, March 3–7, 2014; Denver, Colorado
Session B43: Symmetry Protected Topological Phases |
Hide Abstracts |
Sponsoring Units: DCMP Chair: Maxim Dzero, Kent State University Room: Mile High Ballroom 4B |
Monday, March 3, 2014 11:15AM - 11:27AM |
B43.00001: Detection of Symmetry Enriched Topological Phases Frank Pollmann, Ching-Yu Huang, Xie Chen Topologically ordered systems in the presence of symmetries can exhibit new structures which are referred to as symmetry enriched topological (SET) phases. We introduce simple methods to detect the SET order directly from a complete set of topologically degenerate ground state wave functions. In particular, we first show how to directly determine the characteristic symmetry fractionalization of the quasiparticles from the reduced density matrix of the minimally entangled states. Second, we show how a simple generalization of a string order parameter can be measured to detect SETs. The usefulness of the proposed approached is demonstrated by examining two concrete model states which exhibit SET: (i) a spin-1 model on the honeycomb lattice and (ii) the resonating valence bond state on a kagome lattice. [Preview Abstract] |
Monday, March 3, 2014 11:27AM - 11:39AM |
B43.00002: Detecting two dimensional symmetry protected topological order in a ground state wave function Michael Zaletel Symmetry protected topological states cannot be deformed to a trivial state so long as the symmetry is preserved, yet there is no local order parameter that can distinguish them from a trivial state. We demonstrate how to detect whether a two dimensional ground state has symmetry protected topological order; the measurement play a similar role as the topological entanglement entropy does for detecting anyons. For finite abelian symmetries the measurement gives a complete characterization of the 3rd cohomology class that describes the order. The proposed measurement is validated numerically for a model with $Z_2$ symmetry protected order. [Preview Abstract] |
Monday, March 3, 2014 11:39AM - 11:51AM |
B43.00003: Symmetry-Protected Topological Entanglement Iman Marvian We propose an order parameter for the Symmetry-Protected Topological (SPT) phases which are protected under an Abelian on-site symmetry. This order parameter, called the SPT entanglement, is defined as the entanglement between A and B, two distant regions of the system, given that the total charge (associated with the symmetry) in a third region C is measured and known, where C is a connected region surrounded by A and B and the boundaries of the system. In the case of 1-dimensional systems we prove that at the limit where A and B 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 non- trivial phases. Moreover, we show that the SPT entanglement is invariant under the low-depth local quantum circuits which respect the symmetry, suggesting that the SPT entanglement remains constant throughout a SPT phase in the higher dimensions as well. Finally, based on the concept of SPT-Ent, we propose a new interpretation of string order parameters and also an algorithm for extracting the relevant information about the SPT phase from them. [Preview Abstract] |
Monday, March 3, 2014 11:51AM - 12:03PM |
B43.00004: Symmetry protected topological phases in two dimensions: Generalized Laughlin's argument and quantum pumps Chang-Tse Hsieh, Olabode Mayodele Sule, Shinsei Ryu, Rob Leigh We generalize Laughlin's flux insertion argument in a way that it is applicable to interacting topological phases protected by unitary symmetries -- either on-site or non-on-site -- in two spatial dimensions. Large gauge invariance of the symmetry projected partition function of the one-dimensional edge theory can be used to argue the (non)conservation of the quantum number (corresponding to the projected unitary symmetry) under the large gauge transformation. If the edge does not conserve such quantum number, there is a flux-driven ``quantum pump'' between edges of the two dimensional system, which can be diagnosed as the nontrivial symmetry protected topological phase. This also gives the criteria of stability/gappability of the edge states that respect the symmetry. For non-on-site symmetry such as parity symmetry, the one dimensional edge theory is considered as the conformal field theory on an unoriented surface, such as Klein bottle, which arise naturally from a parity symmetry projection operation. [Preview Abstract] |
Monday, March 3, 2014 12:03PM - 12:15PM |
B43.00005: Diagnosing gapless edge theory of symmetry protected topological phases via twist operators Gil Cho, Shinsei Ryu A symmetry protected topological (SPT) phase is a new phase of matter which has been actively studied recently. The bulk of a SPT phase is gapped and disordered, and thus it is featureless and difficult to be distinguished from a trivially disordered phase. Remarkably there are gapless edge modes emerging at the boundary between the vacuum and the SPT phase. The gapless edge state is protected by the symmetries of the SPT phase and is a only measurable signature of the SPT phase, and thus we can learn about the SPT phase by studying only its edge modes. One can write down a conformal field theory describing the edge modes, and we consider twist operators of the theory to diagnose the stability of the conformal field theory against symmetry-respecting perturbations. When acted on a state, a twist operator changes the boundary condition for the quantum fields in the conformal field theories. It manifests in the mode expansion of the fields and changes only the behavior of the ``zero'' mode of the fields. At the edge of the SPT phase, we consider only the twist operators consistent with symmetries. Then we investigate the algebra between the twist operators and their symmetry quantum numbers in various methods to study the stability of the edge theory. [Preview Abstract] |
Monday, March 3, 2014 12:15PM - 12:27PM |
B43.00006: Detecting symmetry protected topological states by generalized correlation Yizhuang You, Alex Rasmussen, Zhen Bi, Cenke Xu The symmetry protected topological (SPT) states has attracted much research attention recently. They classify a large family of disordered gapped quantum states that have non-trivial topological twist in their wave functions. Examples include topological insulators and the Haldane spin-1 chain. To better understand the physical properties of the SPT states, we focus on their many-body wave functions. We propose a simply way to distinguish the SPT state from the trivial state by studying the behavior of a generalized static bulk correlation function. We show that for 2D SPT states, the generalized correlation function will exhibit a long-range or quasi-long-range behavior, distinct from the short-range behavior for trivial states. This quasi-long-range behavior in the bulk is closely related to the symmetry protected gapless edge modes on the boundary of SPT state. The effective theory for the gapless edge can be described by the conformal field theory (CFT), whose central charge may be extracted from the scaling behavior of the entanglement entropy, which can be given by the wave function overlap on a double torus. We demonstrate our proposal with lattice models for both the fermion and the boson SPT states. [Preview Abstract] |
Monday, March 3, 2014 12:27PM - 12:39PM |
B43.00007: Topological Response Theory of Abelian Symmetry-Protected Topological Phases in Two Dimensions Meng Cheng, Zheng-Cheng Gu Symmetry-protected topological (SPT) phases in two-dimensions can be largely described by Chern-Simons topological field theories. We propose a topological response theory to uniquely identify the SPT orders, which allows us to obtain a systematic scheme to classify bosonic SPT phases with any finite Abelian symmetry group. We also apply the theory to fermionic SPT phases with $Z_m$ symmetry and find the classification of SPT phases depends on the parity of $m$: for even $m$ there are $2m$ classes, $m$ out of which is intrinsically fermionic SPT phases and can not be realized in any bosonic system. We outline the general classification scheme for fermionic SPT phases in two dimensions. [Preview Abstract] |
Monday, March 3, 2014 12:39PM - 12:51PM |
B43.00008: Detecting topological phase transitions of insulators in the complex crystal momentum space Xugang He, Wei Ku We present an intuitive picture of topological phase transitions in insulators via topological properties of band dispersion in the {\it complex} crystal momentum space. Specifically, the dispersion, when analytically contiuned to the complex crystal momentum space, has doubly degenerate ``branch point'' where two bands can meet, and the topological property of the branch point contains clear signature of the phase transition. In addition, the residue of the branch point in the reduced Berry curvature is shown to give the change of topological invariants across the phase transitions, thus providing a convenient way to detect topological phase transition. We demonstrate the general idea using the generic Bernevig-Hughes-Zhang (BHZ) model originated in the quantum spin Hall effect on a square lattice. [Preview Abstract] |
Monday, March 3, 2014 12:51PM - 1:03PM |
B43.00009: Phase diagram of the isotropic spin-3/2 model on the z=3 Bethe lattice Stefan Depenbrock, Frank Pollmann We study an $SU(2)$ symmetric spin-3/2 model on the $z=3$ Bethe lattice using the infinite Time Evolving Block Decimation (iTEBD) method. This model is shown to exhibit a rich phase diagram. We compute the expectation values of several order parameters which allow us to identify a ferromagnetic, a ferrimagnetic, a anti-ferromagnetic as well as a dimerized phase. We calculate the entanglement spectra from which we conclude the existence of a symmetry protected topological phase that is characterized by $S=1/2$ edge spins. [Preview Abstract] |
Monday, March 3, 2014 1:03PM - 1:15PM |
B43.00010: Quantum Distance and the Classification of Topological States of Matter Jiahua Gu, Kai Sun In this talk, we provide a generic geometrical classification for topological states of matter, which is applicable for all topologically nontrivial band insulators (with or without symmetry protections), as well as certain strongly-correlated topological states (e.g. the fractional quantum Hall effect and the fractional Chern insulators). We prove that generically, quantum distance measurement contains direct information about the topology of a quantum wavefunction. Specific examples will be provided to demonstrate this principle. The experimental implications will also be discussed. [Preview Abstract] |
Monday, March 3, 2014 1:15PM - 1:27PM |
B43.00011: Integer characterization of 2D topological insulators at finite temperature Zhoushen Huang, Daniel Arovas 2D band topological insulators (TI) are characterized by the TKNN number and its variants. However, this only works for zero temperature as the TKNN number is no longer quantized for $T>0$. We show that using Uhlmann's parallel transport for density matrices, TI at finite temperature can still be characterized by an integer, which (1) reduces to the corresponding TKNN number at $T = 0$, and (2) exhibits a phase transition, i.e. drops to zero, at a critical temperature. Prototypical models such as Haldane's honeycomb lattice model and the Bernevig-Hughs-Zhang model will be discussed. [Preview Abstract] |
Monday, March 3, 2014 1:27PM - 1:39PM |
B43.00012: Topological field theory for 2+1 TRI TSC Yingfei Gu, Xiaoliang Qi Time-reversal invariant topological superconductors (TRI TSC) are gapped TRI superconductors with topologically robust gapless modes on the boundary. In the work by X. L. Qi et al, [PRB, 87, 134519(2013)], a topological field theory description was proposed for 3+1-dimensional TRI TSC, which contains an axionic coupling between superconducting phase and electromagnetic field. In my talk, I will describe a generalization of this theory to the 2+1 dimensional TRI TSC. The 2+1d topological field theory describes a topological coupling between electromagnetic field, superconducting phase fluctuation and magneto-electric polarization. I will also talk about the corresponding physical consequences. [Preview Abstract] |
Monday, March 3, 2014 1:39PM - 1:51PM |
B43.00013: Unitary engineering of two- and three-band Chern insulators Soo-Yong Lee, Jin-Hong Park, Gyungchun Go, Jung Hoon Han In this talk, we discuss how to engineer the topological number and ordering in some two- and three-band Chern insulators. First, we investigate a way to extend the unit Chern number of a two-band lattice model such as Haldane model and Bernevig-Hughes-Zhang model to the one in possession of higher Chern numbers, relying crucially on the monopole number-changing unitary transformations. The scheme is generalized to a class of three-band model Hamiltonian where the a pair of monopole charges can be introduced to manipulate the Chern numbers of each band. [Preview Abstract] |
Monday, March 3, 2014 1:51PM - 2:03PM |
B43.00014: Analytical approach to the edge state of the Kane-Mele model Hyeonjin Doh, Gun Sang Jeon, Hyoung Joon Choi We investigate the edge state of a two-dimensional topological insulator based on the Kane- Mele model. We consider the two semi-infinite honeycomb lattices with a zig-zag and an armchair boundary, respectively. We construct the effective Hamiltonians for the edge states assuming exponentially decaying wave functions. With the boundary conditions for the both types of the boundaries, we derive the self-consistent equations for the energies and the decaying factors of the edge states. The numerical solutions of the self-consistent equations exhibit intriguing spatial behaviors of the edge states with respect to the spin-orbit coupling and the sub-lattice potential. We found the bifurcation behavior of the edge state width with respect to the sub-lattice potential in zigzag boundary. The bifurcation behavior discriminates the boundary dependencies of the edge state properties. We also discuss the relation between the sample size and the interaction parameters in the phase transition from normal insulator to topological insulator. [Preview Abstract] |
Monday, March 3, 2014 2:03PM - 2:15PM |
B43.00015: Edge states and the quantized Berry phase of general massless Dirac fermions Toshikaze Kariyado, Yasuhiro Hatsugai Topological properties of massless Dirac fermion systems are investigated in terms of the quantized (Z$_2$) Berry phase. Although the Berry phase is gauge dependent and can take any value in modulo $2\pi$, it is quantized with symmetry protection. For this protection, the chiral symmetry is often employed. Here, we show that this symmetry protection is effective in much more general situation, namely, the inversion combined with the time reversal symmetry or the spatial reflection is sufficient for the quantization. Then, the topological stability of the massless Dirac fermions in two dimension is discussed in relation to the quantized Berry phase. We also demonstrate the bulk-edge correspondence of the generic massless Dirac fermions, that is, giving topological reasoning for the existence of edge states, using a model containing the massless Dirac fermion, but having no chiral symmetry [T. Kariyado and Y. Hatsugai, arXiv:1307.7926]. Further generic applications of the symmetry protection for the bulk-edge correspondence will be discussed as well. [Preview Abstract] |
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