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arXiv:1711.11580

Symmetry protected bosonic topological phase transitions: Quantum Anomalous Hall system of weakly interacting spinor bosons in a square lattice

Fadi Sun, Wang, Junsen, Jinwu Ye, Shaui Chen, Youjin Deng

Published Nov 30, 2017• 0 citations
arXivHTMLPDF

Abstract

We study possible many body phenomena in the Quantum Anomalous Hall system of weakly interacting spinor bosons in a square lattice. There are various novel spin-bond correlated superfluids (SF) and quantum or topological phase transitions among these SF phases. One transition is a first order one driven by roton droppings ( but with non-zero gaps $ Δ_R $ ) tuned by the Zeeman field $ h $. Another is a second order bosonic Lifshitz transition with the dynamic exponents $ z_x=z_y=2 $ and an accompanying $ [C_4 \times C_4]_D $ symmetry breaking. It is driven by the softening of the superfluid Goldstone mode tuned by the ratio of spin-orbit coupled (SOC) strength over the hopping strength. The two phase boundaries meet at a topological tri-critical (TT) point which separates the $ h=0 $ line into two SF phases with $ N=2 $ and $ N=4 $ condensation momenta respectively. At the $ h=0 $ line where the system has an anti-unitary $ Z_2 $ Reflection symmetry, there are infinite number of classically degenerate family of states on both sides. We perform a systematic order from quantum disorder analysis to find the quantum ground states, also calculate the roton gaps $ Δ_R $ generated by the order from disorder mechanism on both sides of the TT point. The $ N=2 $ and $ N=4 $ SF phases have the same spin-orbital XY-AFM spin structure, respect the anti-unitary symmetry and break the $ [C_4 \times C_4]_D $ symmetry, so they be distinguished only by the different topology of the BEC condensation momenta instead of by any differences in the symmetry breaking patterns. All these novel quantum or topological phenomena can be probed in the recent experimentally realized weakly interacting Quantum Anomalous Hall (QAH) model of $ ^87 Rb $ by Wu, {\sl et.al}, Science 354, 83-88 (2016).

DOI
10.48550/arxiv.1711.11580

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We study possible many body phenomena in the Quantum Anomalous Hall system of weakly interacting spinor bosons in a square lattice. There are various novel spin-bond correlated superfluids (SF) and quantum or topological phase transitions among these SF phases. One transition is a first order one driven by roton droppings ( but with non-zero gaps ΔRΔ_RΔR​ ) tuned by the Zeeman field hhh. Another is a second order bosonic Lifshitz transition with the dynamic exponents zx=zy=2z_x=z_y=2zx​=zy​=2 and an accompanying [C4×C4]D[C_4 \times C_4]_D[C4​×C4​]D​ symmetry breaking. It is driven by the softening of the superfluid Goldstone mode tuned by the ratio of spin-orbit coupled (SOC) strength over the hopping strength. The two phase boundaries meet at a topological tri-critical (TT) point which separates the h=0h=0h=0 line into two SF phases with N=2N=2N=2 and N=4N=4N=4 condensation momenta respectively. At the h=0h=0h=0 line where the system has an anti-unitary Z2Z_2Z2​ Reflection symmetry, there are infinite number of classically degenerate family of states on both sides. We perform a systematic order from quantum disorder analysis to find the quantum ground states, also calculate the roton gaps ΔRΔ_RΔR​ generated by the order from disorder mechanism on both sides of the TT point. The N=2N=2N=2 and N=4N=4N=4 SF phases have the same spin-orbital XY-AFM spin structure, respect the anti-unitary symmetry and break the [C4×C4]D[C_4 \times C_4]_D[C4​×C4​]D​ symmetry, so they be distinguished only by the different topology of the BEC condensation momenta instead of by any differences in the symmetry breaking patterns. All these novel quantum or topological phenomena can be probed in the recent experimentally realized weakly interacting Quantum Anomalous Hall (QAH) model of 87Rb^87 Rb87Rb by Wu, {\sl et.al}, Science 354, 83-88 (2016).
The study of continuous phase transitions triggered by spontaneous symmetry breaking has brought revolutionary ideas to physics. Recently, through the discovery of symmetry protected topological phases, it is realized that continuous quantum phase transition can also occur between states with the same symmetry but different topology. The subject of this dissertation is the phase transition between these symmetry protected topological states (SPTs). There are two main parts in this dissertation.In the first part we consider spatial dimension ddd and symmetry group GGG so that the cohomology group, Hd+1(G,U(1))H^{d+1}(G,U(1))Hd+1(G,U(1)), contains at least one Z2nZ_{2n}Z2n​ or ZZZ factor. We show that the phase transition between the trivial SPT and the root states that generate the $ Z_{2n} $ or ZZZ groups can be induced on the boundary of a d+1 dimensional GtimesZ2TG\\times Z_2^TGtimesZ2T​-symmetric SPT by a Z2TZ_2^TZ2T​ symmetry breaking field. Moreover we show these boundary phase transitions can be ``transplanted'' to d dimensions and realized in lattice models as a function of a tuning parameter. The price one pays is for the critical value of the tuning parameter there is an extra non-local (duality-like) symmetry. In the case where the phase transition is continuous, our theory predicts the presence of unusual (sometimes fractionalized) excitations corresponding to delocalized boundary excitations of the non-trivial SPT on one side of the transition. This theory also predicts other phase transition scenarios including first order transition and transition via an intermediate symmetry breaking phase. In the second part, we study the phase transition between bosonic topological phases protected by ZntimesZnZ_n\\times Z_nZn​timesZn​ in 1+1 dimensions. We find a direct transition occurs when $ n=2,3,4 $ and in all cases the critical point possesses two gap opening relevant operators: one leads to a Landau-forbidden symmetry breaking phase transition and the other to the topological phase transition. We also obtained a constraint($ c\\geq1 $) on the central charge for general phase transitions between symmetry protected bosonic topological phases in 1+1D.
Topological Mott Insulators and Discontinuous U(1)U(1)U(1) θθθ-Terms
We introduce a lattice field theory that describes the transition between a superfluid (SF) and a bosonic topological Mott Insulator (tMI) -- a U(1)U(1)U(1) symmetry protected topological phase labeled by an integer level kkk and possessing an even integer 2ke2h2k\frac{e^2}{h}2khe2​ quantized Hall conductance. Our model differs from the usual 2+12+12+1d XY model by a topological term that vanishes on closed manifolds and in the absence of an applied gauge field, which implies that the critical exponents of the SF-tMI transition are identical to those of the well-studied 2+12+12+1d XY transition. Our formalism predicts a "level-shift" symmetry: in the absence of an applied gauge field, the bulk correlation functions of all local operators are identical for models differing by the topological term. %, for example, near the SF-MI and SF-tMI transitions, and hence the extremely well-studied critical exponents of the 2+12+12+1d XY model apply to the SF-tMI transition. In the presence of a background gauge field, the topological term leads to a quantized Hall response in the tMI phase, and we argue that this quantized Hall effect persists in the vicinity of the phase transition into the SF phase. Our formalism paves the way for other exact lattice descriptions of symmetry-protected-topological (SPT) phases, and mappings of critical exponents between transitions from symmetry-breaking to trivial states and transitions from symmetry-breaking to SPT states. A similar "level shift" symmetry should appear between all group cohomology SPT states protected by the same symmetry.
In this thesis, we study the topological phases of quantum spin systems. One project is to investigate a class of anti-ferromagnetic SU(N) Heisenberg models, describing Mott insulators of fermionic ultra-cold alkaline earth atoms on the three-dimensional simple cubic lattice. Our large-N analysis maps a rich phase diagram. One particularly striking state we found spontaneously breaks lattice rotation symmetry, where the cubic lattice breaks up into bilayers, each of which forms a two-dimensional chiral spin liquid state. In the other projects, we study the phenomenon of symmetry fractionalization on anyons as a tool to characterize two-dimensional symmetry enriched topological phases. In particular, we focus on how crystalline symmetries may fractionalize in gapped Z2 spin liquids. If the system has the symmetry of the square lattice, then there are 2080 symmetry fractionalization patterns possible. With exactly solvable models, we realize 487 of these in strictly two-dimensional systems. In addition, we succeed to understand why the remaining patterns cannot be found in the family of models we construct. Some can only appear on the surface of three-dimensional systems with non-trivial point group symmetry protected topological (pgSPT) order, whose boundary degrees of freedom transform non-locally under the symmetries. We construct a simple toy model to show this anomalous crystalline symmetry fractionalization phenomenon associated with a reflection. Moreover, our approach establishes the connection between the pgSPT phases and the topological phases with on-site symmetries in lower dimensions. This insight is very useful for classification of pgSPT orders in general.
Topological phase transitions represent a new paradigm beyond conventional Landau-Ginzburg symmetry breaking. Even in the absence of symmetry breaking, the symmetries of a Hamiltonian play an important role in stabilizing topological signatures. In this thesis, we will study the interplay between symmetry and topology in many-body systems, with three distinct physical setups in mind. First, we investigate a two-dimensional system of ultracold bosons which is condensed in the p-band of a triangular lattice. Certain smooth interaction profiles result in non-Abelian symmetry generators of the order parameter manifold, leading to anomalous physical consequences. These include a lack of Berezinskii-Kosterlitz-Thouless transition due to unstable vortex configurations, additional Goldstone modes, and a marginally divergent normal fluid density which implies lack of superfluidity in the thermodynamic limit. Our results suggest possible connections to solid state experiments on 4He films. We then show that disorder which keeps a Hamiltonian in the same Altland-Zirnbauer symmetry class is capable of inducing a topological transition. This is achieved by considering the normalizability of Majorana edge modes in a topological superconductor, and corroborated with entanglement metrics. While degeneracies of the many-body spectrum are robust with respect to weak perturbations, strong disorder can both promote and destroy a topological index. Lastly, we study topological edge modes and degeneracies in atomic, molecular, and optical systems which evolve non-unitarily in time or space due to non-equilibrium effects, including: condensate instabilities after a quench, decoherence channels from an external environment, and dissipation into a medium. Mathematically, the non-unitary evolution is generated by effective Hamiltonians which break Hermiticity. These systems can possess uniquely non-Hermitian symmetries which are absent from equilibrium counterparts. We present a symmetry-based classification, and investigate several systems which can be solved exactly.
We study the system of attractively interacting fermions hopping in a square lattice with any linear combinations of Rashba or Dresselhaus spin-orbit coupling in a normal Zeeman field. By imposing self-consistence equations at half filling with zero chemical potential, we find that there are three phases: band insulator (BI), superfluid (SF), and topological superfluid (TSF) with a Chern number C=2C=2C=2. The C=2C=2C=2 TSF happens in small Zeeman fields and weak interactions which is in the experimentally most easily accessible regime. The transition from the BI to the SF is a first-order one due to the multiminima structure of the ground state energy landscape. There is a class of topological phase transitions (TPTs) from the SF to the C=2C=2C=2 TSF at the low critical field hc1{h}_{c1}hc1​, then another one from the C=2C=2C=2 TSF to the BI at the upper critical field hc2{h}_{c2}hc2​. We derive effective actions to describe these two classes of topological phase transitions, then use them to study the Majorana edge modes and the zero modes inside the vortex core of the C=2C=2C=2 TSF near both hc1{h}_{c1}hc1​ and hc2{h}_{c2}hc2​, especially exploring their spatial and spin structures. We find that the edge modes decay into the bulk with oscillating behaviors and determine both the decay and oscillating lengths. We compute the bulk spectra and map out the Berry curvature distribution in momentum space near both hc1{h}_{c1}hc1​ and hc2{h}_{c2}hc2​. We elaborate some intriguing bulk-Berry curvature-edge-vortex correspondences. We also discuss the competitions between SFs and charge density wave states in more general cases. A possible classification scheme of all the TPTs in a square lattice is outlined. Comparisons with previous works on related systems are discussed. Possible experimental implications in cold atoms in an optical lattice are given.
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