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arXiv:1606.05822 · Physical review. B./Physical review. B

Visualizing a bosonic symmetry protected topological phase in an interacting fermion model

Han-Qing Wu, Yuan-Yao He, Yi‐Zhuang You, Tsuneya Yoshida, Norio Kawakami, Cenke Xu, Zi Yang Meng, Zhong-Yi Lu

Published Oct 11, 2016• 16 citations
arXivHTMLPDF

Abstract

Symmetry protected topological (SPT) phases in free fermion and interacting bosonic systems have been classified, but the physical phenomena of interacting fermionic SPT phases have not been fully explored. Here, employing large-scale quantum Monte Carlo simulation, we investigate the edge physics of a bilayer Kane-Mele-Hubbard model with zigzag ribbon geometry. Our unbiased numerical results show that the fermion edge modes are gapped out by interaction, while the bosonic edge modes remain gapless at the $(1+1)d$ boundary, before the bulk quantum phase transition to a topologically trivial phase. Therefore, finite fermion gaps both in the bulk and on the edge, together with the robust gapless bosonic edge modes, prove that our system becomes an emergent bosonic SPT phase at low energy, which is directly observed in an interacting fermion lattice model.

DOI
10.1103/physrevb.94.165121

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Symmetry protected topological (SPT) phases in free fermion and interacting bosonic systems have been classified, but the physical phenomena of interacting fermionic SPT phases have not been fully explored. Here, employing large-scale quantum Monte Carlo simulation, we investigate the edge physics of a bilayer Kane-Mele-Hubbard model with zigzag ribbon geometry. Our unbiased numerical results show that the fermion edge modes are gapped out by interaction, while the bosonic edge modes remain gapless at the (1+1)d(1+1)d(1+1)d boundary, before the bulk quantum phase transition to a topologically trivial phase. Therefore, finite fermion gaps both in the bulk and on the edge, together with the robust gapless bosonic edge modes, prove that our system becomes an emergent bosonic SPT phase at low energy, which is directly observed in an interacting fermion lattice model.
We propose a series of simple two-dimensional (2D) lattice interacting fermion models that we demonstrate at low energy describe bosonic symmetry-protected topological (SPT) states and quantum phase transitions between them. This is because due to interaction, the fermions are gapped both at the boundary of the SPT states and at the bulk quantum phase transition, thus these models at low energy can be described completely by bosonic degrees of freedom. We show that the bulk of these models is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a topological $\mathrm{\ensuremath{\Theta}}$ term, whose boundary is described by a Sp(N\mathrm{Sp}(NSp(N) principal chiral model with a Wess-Zumino-Witten term at level 1. The quantum phase transition between SPT states in the bulk is tuned by a particular interaction term, which corresponds to tuning $\mathrm{\ensuremath{\Theta}}$ in the field theory, and the phase transition occurs at $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$. The simplest version of these models with N=1N=1N=1 is equivalent to the familiar O(4) nonlinear sigma model (NLSM) with a topological term, whose boundary is a (1+1)D(1+1)\text{D}(1+1)D conformal field theory with central charge c=1c=1c=1. After breaking the O(4) symmetry to its subgroups, this model can be viewed as bosonic SPT states with U(1), or Z2{Z}_{2}Z2​ symmetries, etc. All of these fermion models, including the bulk quantum phase transitions, can be simulated with the determinant quantum Monte Carlo method without the sign problem. Recent numerical results strongly suggest that the quantum disordered phase of the O(4) NLSM with precisely $\mathrm{\ensuremath{\Theta}}=\ensuremath{\pi}$ is a stable (2+1)D(2+1)\text{D}(2+1)D conformal field theory with gapless bosonic modes.
Symmetry-protected topological (SPT) phases are bulk-gapped quantum phases with symmetries, which have gapless or degenerate boundary states as long as the symmetries are not broken. The SPT phases in free fermion systems, such as topological insulators, can be classified; however, it is not known what SPT phases exist in general interacting systems. We present a systematic way to construct SPT phases in interacting bosonic systems. Just as group theory allows us to construct 230 crystal structures in three-dimensional space, we use group cohomology theory to systematically construct different interacting bosonic SPT phases in any dimension and with any symmetry, leading to the discovery of bosonic topological insulators and superconductors.
Symmetry protected topological (SPT) states are short-range entangled states with symmetry. The boundary of a SPT phases has either gapless excitations or degenerate ground states, around a gapped bulk. Recently, we proposed a systematic construction of SPT phases in interacting bosonic systems, however it is not very clear what is the form of the low-energy excitations on the gapless edge. In this paper, we answer this question for two-dimensional (2D) bosonic SPT phases with ZN{\mathbb{Z}}_{N}ZN​ and U(1)\text{U}(1)U(1) symmetry. We find that while the low-energy modes of the gapless edges are nonchiral, symmetry acts on them in a ``chiral'' way, i.e., acts on the right movers and the left movers differently. This special realization of symmetry protects the gaplessness of the otherwise unstable edge states by prohibiting a direct scattering between the left and right movers. Moreover, understanding of the low-energy effective theory leads to experimental predictions about the SPT phases. In particular, we find that all the 2D U(1)\text{U}(1)U(1) SPT phases have even integer quantized Hall conductance.
The low energy behavior of a huge variety of one-dimensional interacting spinful fermionic systems exhibits spin-charge separation, described in the continuum limit by two sine-Gordon models decoupled in the charge and spin channels. Interaction is known to induce, besides the gapless Luttinger liquid phase, eight possible gapped phases, among which are the Mott, Haldane, charge-/spin-density, and bond-ordered wave insulators, and the Luther Emery liquid. Here we prove that some of these physically distinct phases have nontrivial topological properties, notably the presence of degenerate protected edge modes with fractionalized charge/spin. Moreover, we show that the eight gapped phases are in one-to-one correspondence with the symmetry-protected topological (SPT) phases classified by group cohomology theory in the presence of particle-hole symmetry P. The latter result is also exploited to characterize SPT phases by measurable nonlocal order parameters which follow the system evolution to the quantum phase transition. The implications on the appearance of exotic orders in the class of microscopic Hubbard Hamiltonians, possibly without P symmetry at higher energies, are discussed.
Symmetry-protected topological (SPT) phases are gapped short-range-entangled quantum phases with a symmetry GGG, which can all be smoothly connected to the trivial product states if we break the symmetry. It has been shown that a large class of interacting bosonic SPT phases can be systematically described by group cohomology theory. In this paper, we introduce a (special) group supercohomology theory which is a generalization of the standard group cohomology theory. We show that a large class of short-range interacting fermionic SPT phases can be described by the group supercohomology theory. Using the data of supercocycles, we can obtain the ideal ground state wave function for the corresponding fermionic SPT phase. We can also obtain the bulk Hamiltonian that realizes the SPT phase, as well as the anomalous (i.e., non-onsite) symmetry for the boundary effective Hamiltonian. The anomalous symmetry on the boundary implies that the symmetric boundary must be gapless for (1+1)-dimensional [(1+1)D] boundary, and must be gapless or topologically ordered beyond (1+1)D. As an application of this general result, we construct a new SPT phase in three dimensions, for interacting fermionic superconductors with coplanar spin order (which have T2=1{T}^{2}=1T2=1 time-reversal Z2T{Z}_{2}^{T}Z2T​ and fermion-number-parity Z2f{Z}_{2}^{f}Z2f​ symmetries described by a full symmetry group ${Z}_{2}^{T}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$). Such a fermionic SPT state can neither be realized by free fermions nor by interacting bosons (formed by fermion pairs), and thus are not included in the KKK-theory classification for free fermions or group cohomology description for interacting bosons. We also construct three interacting fermionic SPT phases in two dimensions (2D) with a full symmetry group ${Z}_{2}\ifmmode\times\else\texttimes\fi{}{Z}_{2}^{f}$. Those 2D fermionic SPT phases all have central-charge c=1c=1c=1 gapless edge excitations, if the symmetry is not broken.
Recently, it has been found that there exist symmetry-protected topological phases of fermions, which have no realizations in non-interacting fermionic systems or bosonic models. We study the edge states of such an intrinsically interacting fermionic SPT phase in two spatial dimensions, protected by \mathbb{Z}_4\times\mathbb{Z}_2^\mathsf{T} ℤ 4 × ℤ 2 𝖳 symmetry. We model the edge Hilbert space by replacing the internal \mathbb{Z}_4 ℤ 4 symmetry with a spatial translation symmetry, and design an exactly solvable Hamiltonian for the edge model. We show that at low-energy the edge can be described by a two-component Luttinger liquid, with nontrivial symmetry transformations that can only be realized in strongly interacting systems. We further demonstrate the symmetry-protected gaplessness under various perturbations, and the bulk-edge correspondence in the theory.