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arXiv:1810.00325 · Physical Review Letters

Exactly Solvable Points and Symmetry Protected Topological Phases of Quantum Spins on a Zig-Zag Lattice

Haiyuan Zou, Erhai Zhao, Xi-Wen Guan, W. Vincent Liu

Published May 10, 2019• 24 citations
arXivHTMLPDF

Abstract

A large number of symmetry-protected topological (SPT) phases have been hypothesized for strongly interacting spin-1/2 systems in one dimension. Realizing these SPT phases, however, often demands fine-tunings hard to reach experimentally. And the lack of analytical solutions hinders the understanding of their many-body wave functions. Here we show that two kinds of SPT phases naturally arise for ultracold polar molecules confined in a zigzag optical lattice. This system, motivated by recent experiments, is described by a spin model whose exchange couplings can be tuned by an external field to reach parameter regions not studied before for spin chains or ladders. Within the enlarged parameter space, we find the ground state wave function can be obtained exactly along a line and at a special point, for these two phases, respectively. These exact solutions provide a clear physical picture for the SPT phases and their edge excitations. We further obtain the phase diagram by using infinite time-evolving block decimation and discuss the phase transitions between the two SPT phases and their experimental signatures.

DOI
10.1103/physrevlett.122.180401

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A large number of symmetry-protected topological (SPT) phases have been hypothesized for strongly interacting spin-1/2 systems in one dimension. Realizing these SPT phases, however, often demands fine-tunings hard to reach experimentally. And the lack of analytical solutions hinders the understanding of their many-body wave functions. Here we show that two kinds of SPT phases naturally arise for ultracold polar molecules confined in a zigzag optical lattice. This system, motivated by recent experiments, is described by a spin model whose exchange couplings can be tuned by an external field to reach parameter regions not studied before for spin chains or ladders. Within the enlarged parameter space, we find the ground state wave function can be obtained exactly along a line and at a special point, for these two phases, respectively. These exact solutions provide a clear physical picture for the SPT phases and their edge excitations. We further obtain the phase diagram by using infinite time-evolving block decimation and discuss the phase transitions between the two SPT phases and their experimental signatures.
Symmetry fractionalization, generating a large amount of symmetry-protected topological phases, provides scenarios for continuous phase transitions different from spontaneous symmetry breaking. However, it is hard to detect these symmetry-protected topological phase transitions experimentally. Motivated by the recent development of highly tunable ultracold polar molecules, we show that the setup in a zigzag optical lattice of this system provides the perfect platform to realize symmetry-protected topological phase transitions. By using infinite time-evolving block decimation, we obtain the phase diagram in a large parameter region and find another scheme to realize the long-sought vector chiral phase, which is robust from quantum fluctuations. We discuss the existence of the chiral phase by an effective field analysis.
Symmetry-protected topological (SPT) phases of matter have been the focus of many recent theoretical investigations, but controlled mechanisms for engineering them have so far been elusive. In this work, we demonstrate that by driving interacting spin systems periodically in time and tuning the available parameters, one can realize lattice models for bosonic SPT phases in the limit where the driving frequency is large. We provide concrete examples of this construction in one and two dimensions, and discuss signatures of these phases in stroboscopic measurements of local observables.
We theoretically demonstrate that interacting symmetry-protected topological (SPT) phases can be realized with ultracold spinful bosonic atoms loaded on the lattices which have a flat band at the bottom of the band structure.Ground states of such systems are not conventional Mott insulators in the sense that the ground states possess not only spin fluctuations but also non-negligible charge fluctuations.The SPT phases in such systems are determined by both spin and charge fluctuations at zero temperature.We find that the many-body ground states of such systems can be exactly obtained in some special cases, and these exact ground states turn out to serve as representative states of the SPT phases.As a concrete example, we demonstrate that spin-1 bosons on a sawtooth chain can be in an SPT phase protected by Z 2 × Z 2 spin rotation symmetry or time-reversal symmetry, and this SPT phase is a result of spin fluctuations.We also show that spin-3 bosons on a kagome lattice can be in an SPT phase protected by D 2 point group symmetry, but this SPT phase is, however, a result of charge fluctuations.
We propose that ultracold alkaline-earth-like atoms confined in one-dimensional optical lattice can realize a Kondo lattice model which hosts a symmetry-protected topological (SPT) phase and an associated quantum phase transition in a controllable manner. The symmetry protection of the phase transition is discussed from two different viewpoints: topological properties related to spatial patterns of Kondo singlets and symmetry eigenvalues of the spin states. We uncover the role of various symmetries in the phase diagram of this system by combining a weak-coupling approach by Abelian bosonization and strong-coupling pictures of ground states. Furthermore, we show that the bosonization approach correctly describes a crossover from a fermionic SPT phase to a bosonic SPT phase and an associated change of protecting symmetries as the charge degrees of freedom are frozen by the Hubbard repulsion.
Here we study the emergence of different symmetry-protected topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $({\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2})+T$ symmetry group, where Z2{\mathbb{Z}}_{2}Z2​ is some $\ensuremath{\pi}$ rotation in the spin internal space and TTT is time reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an SO(5)SO(5)SO(5)-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous and can be described by a c=2c=2c=2 conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using matrix product states via the infinite time-evolving block decimation method, as well as by effective field theory considerations. Our findings show that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.
Symmetry-protected trivial (SPt) phases of matter are the product-state analog of symmetry-protected topological (SPT) phases. This means, SPt phases can be adiabatically connected to a product state by some path that preserves the protecting symmetry. Moreover, SPt and SPT phases can be adiabatically connected to each other when interaction terms that break the symmetries protecting the SPT order are added in the Hamiltonian. It is also known that spin-1 SPT phases in quantum spin chains can emerge as effective intermediate phases of spin-2 Hamiltonians. In this paper we show that a similar scenario is also valid for SPt phases. More precisely, we show that for a given spin-2 quantum chain, effective intermediate spin-1 SPt phases emerge in some regions of the phase diagram, these also being adiabatically connected to nontrivial intermediate SPT phases. We characterize the phase diagram of our model by studying quantities such as the entanglement entropy, symmetry-related order parameters, and 1-site fidelities. Our numerical analysis uses matrix product states and the infinite time evolving block decimation method to approximate ground states of the system in the thermodynamic limit. Moreover, we provide a field theory description of the possible quantum phase transitions between the SPt phases. Together with the numerical results, such a description shows that the transitions may be described by conformal field theories with central charge c=1c=1c=1. Our results are in agreement with, and further generalize, those of Y. Fuji, F. Pollmann, and M. Oshikawa [Phys. Rev. Lett. 114, 177204 (2015)].