Fermionic symmetry protected topological phases induced by iterations (original) (raw)

Fermionic symmetry-protected topological phase induced by interactions

Physical Review B, 2015

Strong interactions can give rise to new fermionic symmetry protected topological phases which have no analogs in free fermion systems. As an example, we have systematically studied a spinless fermion model with U (1) charge conservation and time reversal symmetry on a three-leg ladder using density-matrix renormalization group. In the non-interacting limit, there are no topological phases. Turning on interactions, we found two gapped phases. One is trivial and is adiabatically connected to a band insulator, while another one is a nontrivial symmetry protected topological phase resulting from strong interactions.

Realizing a symmetry protected topological phase through dimerized interactions

Physical Review B

We show that in a system of one dimensional spinless fermions a topological phase and phase transition can emerge only through interaction. By allowing a dimerized or bond-alternating nearest neighbour interaction we show that the system exhibits a symmetry protected topological phase while its non-interacting limit is a gapless state. The non-trivial topological character appears due to the onset of two degenerate bond-order phases as a function of dimerized interaction which are found to be topologically distinct from each other. As a result a topological phase transition occurs between these bond order phases through a gap closing point. However, in the limit of strong interaction, the bond order phases are connected through a gapped charge density wave phase possessing local antiferromagnetic order. The topological nature is characterized by the edge states, Berry phase and non-local string order parameter. At the end we provide possible experimental signatures of the emergent symmetry protected topological phase transition in terms of Thouless charge pumping and density-density correlation.

Symmetry-protected topological phases in spin ladders with two-body interactions

Spin-1/2 two-legged ladders respecting interleg exchange symmetry σ and spin rotation symmetry D 2 have new symmetry-protected topological (SPT) phases which are different from the Haldane phase. Three of the new SPT phases are t x ,t y ,t z , which all have symmetry-protected twofold degenerate edge states on each end of an open chain. However, the edge states in different phases have different responses to magnetic field. For example, the edge states in the t z phase will be split by the magnetic field along the z direction, but not by the fields in the x and y directions. We give the Hamiltonian that realizes each SPT phase and demonstrate a proof-of-principle quantum simulation scheme for Hamiltonians of the t 0 and t z phases based on the coupled-QED-cavity ladder.

Topological Insulators and Nematic Phases from Spontaneous Symmetry Breaking in 2D Fermi Systems with a Quadratic Band Crossing

Physical Review Letters, 2009

We investigate the stability of a quadratic band-crossing point (QBCP) in 2D fermionic systems. At the non-interacting level, we show that a QBCP exists and is topologically stable for a Berry flux ±2π, if the point symmetry group has either fourfold or sixfold rotational symmetries. This putative topologically stable free-fermion QBCP is marginally unstable to arbitrarily weak shortrange repulsive interactions. We consider both spinless and spin-1/2 fermions. Four possible ordered states result: a quantum anomalous Hall phase, a quantum spin Hall phase, a nematic phase, and a nematic-spin-nematic phase.

Symmetry-protected Topological Phases in Lattice Gauge Theories: Topological QED

Phys. Rev. D 99, 014503, 2019

The interplay of symmetry, topology, and many-body effects in the classification of phases of matter poses a formidable challenge in condensed-matter physics. Such many-body effects are typically induced by inter-particle interactions involving an action at a distance, such as the Coulomb interaction between electrons in a symmetry-protected topological (SPT) phase. In this work we show that similar phenomena also occur in certain relativistic theories with interactions mediated by gauge bosons, and constrained by gauge symmetry. In particular, we introduce a variant of the Schwinger model or quantum electrodynamics (QED) in 1+1 dimensions on an interval, which displays dynamical edge states localized on the boundary. We show that the system hosts SPT phases with a dynamical contribution to the vacuum θ-angle from edge states, leading to a new type of topological QED in 1+1 dimensions. The resulting system displays an SPT phase which can be viewed as a correlated version of the Su-Schrieffer-Heeger topological insulator for polyacetylene due to non-zero gauge couplings. We use bosonization and density-matrix renormalization group techniques to reveal the detailed phase diagram, which can further be explored in experiments of ultra-cold atoms in optical lattices. Global and local symmetries play a crucial role in our understanding of Nature at very different energy scales [1, 2]. At high energies, they govern the behavior of fundamental particles [3], their spectrum and interactions [4, 5]. At low energies [6], spontaneous symmetry breaking and local order parameters characterize a wide range of phases of matter [7] and a rich variety of collective phenomena [8]. There are, however, fundamental physical phenomena that can only be characterized by non-local order parameters, such as the Wil-son loops distinguishing confined and deconfined phases in gauge theories [9], or hidden order parameters distinguishing topological phases in solids [10]. The former, requiring a non-perturbative approach to quantum field theory (e.g. lattice gauge theories (LGTs)), and the latter, demanding the introduction of mathematical tools of topology in condensed matter (e.g. topological invariants), lie at the forefront of research in both high-energy and condensed-matter physics. The interplay of symmetry and topology can lead to a very rich, and yet partially-uncharted, territory. For instance, different phases of matter can arise without any symmetry breaking: symmetry-protected topological (SPT) phases. Beyond the celebrated integer quantum Hall effect [11-14], a variety of SPT phases have already been identified [15-17] and realized [18]. Let us note that some representative models of these SPT phases [19] can be understood as lower-dimensional versions of the so-called domain-wall fermions [20], introduced in the context of chiral symmetry in lattice field theories [21]. A current problem of considerable interest is to understand strong-correlation effects in SPT phases as interactions are included [22], which may, for instance, lead to exotic fractional excitations [23, 24]. So far, the typical interactions considered involve an action at a distance (e.g. screened Coulomb or Hubbard-like nearest or next-to-nearest neighbor interactions). To the best of our knowledge, and with the recent exception [25], the study of correlated SPT phases with mediated interactions remains a largely-unexplored subject. In this work, we initiate a systematic study of SPT phases with interactions dictated by gauge symmetries focusing on a 2a x c 2n+1 c † 2n (1 − )U 2n U 2n−1 c † 2n−1 c 2n a b U n c n+1 +m s −m s c † n c n n+ + +1 +m m s −m m m m s fermion fields gauge fields FIG. 1. Discretizations for standard and topological QED 2 : (a) Staggered-fermion approach to the massive Schwinger model. The relativistic Dirac field is discretized into spinless lattice fermions subjected to a staggered on-site energy ±m s , represented by filled/empty circles in a 1D chain with alternating heights. The gauge field is discretized into rotor-angle operators that reside on the links, depicted as shaded ellipses with various levels representing the electric flux eigenbasis. The gauge-invariant term c † n U n c n+1 involves the tunneling of neighboring fermions, dressed by a local excitation of the gauge field in the electric-flux basis U n | = | + 1, represented by the zigzag grey arrow joining two neighboring fermion sites, via an excitation of the link electric-flux level. (b) Dimerized-tunneling approach to the topological Schwinger model. The previous staggered mass is substituted by a gauge-invariant tunneling with alternating strengths (1 − δ n)c † n U n c n+1 , where δ n = 0, ∆ for even/odd sites. This dimerization of the tunneling matrix elements is represented by alternating big/small ellipses at the odd/even links. the lattice Schwinger model, an Abelian LGT that regularizes quantum electrodynamics in 1+1 dimensions (QED 2) [26]. We show that a discretization alternative to the standard lattice approach [27] leads to a topological Schwinger model, and derive its continuum limit referred to as topological QED 2. This continuum quantum field theory is used to predict a phase diagram that includes SPT, confined, and fermion-condensate phases, which are then discussed in the context of the afore-mentioned domain-wall fermions in LGTs.

Gapped symmetric edges of symmetry protected topological phases

Symmetry protected topological (SPT) phases are gapped quantum phases which host symmetry-protected gapless edge excitations. On the other hand, the edge states can be gapped by spontaneously breaking symmetry. We show that topological defects on the symmetry-broken edge cannot proliferate due to their fractional statistics. A gapped symmetric boundary, however, can be achieved between an SPT phase and certain fractionalized phases by condensing the bound state of a topo-logical defect and an anyon. We demonstrate this by two examples in two dimensions: an exactly solvable model for the boundary between topological Ising paramagnet and double semion model, and a fermionic example about the quantum spin Hall edge. Such a hybrid structure containing both SPT phase and fractionalized phase generally support ground state degeneracy on torus.

Topological phases of quasi-one-dimensional fermionic atoms with a synthetic gauge field

New Journal of Physics, 2013

We theoretically investigate the effect of intertube tunneling in topological superfluid phases of a quasi-one-dimensional Fermi gas with a Rashba-type spin-orbit interaction. It is shown that the effective Hamiltonian is analogous to that of a nanowire topological superconductor with multibands. Using a hidden mirror symmetry in the system, we introduce a new topological number that ensures the existence of non-Abelian Majorana zero modes even in the presence of intertube tunneling. It is demonstrated from the full numerical calculation of self-consistent equations that some of the Majorana modes survive against the intertube tunneling, when the number of one-dimensional tubes is odd in the y-direction. We also discuss a generalization of our consideration to nanowire topological superconductors. Gesellschaft opened an exciting new chapter in condensed matter physics. These original works predicted that the mysterious fermions exist as zero-energy quasi-particles bound at vortices and edges of a spinless p-wave superconductor. Subsequently, tremendous progress has succeeded in extending the platform for realizing Majorana fermions to some categories of the so-called topological superconductors . The remarkable consequence of the self-charge conjugate property of the Majorana fermion is non-Abelian braiding statistics, where a pair of Majorana zero modes are created or annihilated by braiding their host vortices . Hence, Majorana fermions possessing non-Abelian braiding statistics can provide a promising platform for faulttolerant topological quantum computation . Moreover, it has been recently unveiled that zeroenergy quasi-particles exhibit multifaceted properties, not only as Majorana fermions but also as odd-frequency Cooper pair correlations .

Journal of Physics B: Atomic, Molecular and Optical Physics, 2013

Motivated by the recent progress in engineering artificial non-Abelian gauge fields for ultracold fermions in optical lattices, we investigate the time-reversal-invariant Hofstadter-Hubbard model. We include an additional staggered lattice potential and an artificial Rashba-type spin-orbit coupling term available in experiment. Without interactions, the system can be either a (semi)-metal, a normal or a topological insulator, and we present the non-Abelian generalization of the Hofstadter butterfly. Using a combination of real-space dynamical mean-field theory (RDMFT), analytical arguments, and Monte-Carlo simulations we study the effect of strong on-site interactions. We determine the interacting phase diagram, and discuss a scenario of an interactioninduced transition from normal to topological insulator. At half-filling and large interactions, the system is described by a quantum spin Hamiltonian, which exhibits exotic magnetic order due to the interplay of Rashba-type spinorbit coupling and the artificial time-reversal-invariant magnetic field term. We determine the magnetic phase diagram: both for the itinerant model using RDMFT and for the corresponding spin model in the classical limit using Monte-Carlo simulations.

Unwinding fermionic symmetry-protected topological phases: Supersymmetry extension

Physical Review B, 2021

We show how 1+1-dimensional fermionic symmetry-protected topological states (SPTs, i.e. nontrivial short-range entangled gapped phases of quantum matter whose boundary exhibits 't Hooft anomaly and whose bulk cannot be deformed into a trivial tensor product state under finite-depth local unitary transformations only in the presence of global symmetries), indeed can be unwound to a trivial state by enlarging the Hilbert space via adding extra degrees of freedom and suitably extending the global symmetries. The extended projective global symmetry on the boundary can become supersymmetric in a specific sense, i.e., it contains group elements that do not commute with the fermion number parity (−1) F , while the anti-unitary time-reversal symmetry becomes fractionalized. This also means we can uplift and remove certain exotic fermionic anomalies (e.g., "parity" anomaly in time-reversal or reflection symmetry) via appropriate supersymmetry extensions in terms of group extensions. We work out explicit examples for multi-layers of 1+1d Majorana fermion chains, then comment on models with Sachdev-Ye-Kitaev (SYK) interactions, intrinsic fermionic gapless SPTs protected by supersymmetry, and generalizations to higher spacetime dimensions via a cobordism theory.

Symmetry-protected topological phases in lattice gauge theories: Topological QED2

Physical Review D, 2019

Fermionic symmetry protected topological phases induced by iterations (original) (raw)

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