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arXiv:2112.04616 · Physical review. B./Physical review. B

Persistent corner spin mode at the quantum critical point of a plaquette Heisenberg model

Yining Xu, Chen Peng, Zijian Xiong, Long Zhang

Published Dec 8, 2022• 3 citations
arXivHTMLPDF

Abstract

Gapless edge states are the hallmark of a large class of topological states of matter. Recently, intensive research has been devoted to understanding the physical properties of the edge states at the quantum phase transitions of the bulk topological states. A higher-order symmetry-protected topological state is realized in a plaquette Heisenberg model on the square lattice. In its disordered phase, the lattice with an open boundary hosts either dangling corner states with spin-$1/2$ degeneracy characterizing the topological phase or nondangling corner states without degeneracy, which depends on the bond configuration near the corners. In this work, we study the critical behavior of these corner states at the quantum critical point (QCP) and find that the spin-$1/2$ corner state induces a new universality class of the corner critical behavior, which is distinct from the ordinary transition of the nondangling corners. In particular, we find that the dangling spin-$1/2$ corner state persists at the QCP despite its coupling to the critical spin fluctuations in the bulk. This shows the robustness of the corner state of the higher-order topological state.

DOI
10.1103/physrevb.106.214409

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Gapless edge states are the hallmark of a large class of topological states of matter. Recently, intensive research has been devoted to understanding the physical properties of the edge states at the quantum phase transitions of the bulk topological states. A higher-order symmetry-protected topological state is realized in a plaquette Heisenberg model on the square lattice. In its disordered phase, the lattice with an open boundary hosts either dangling corner states with spin-1/21/21/2 degeneracy characterizing the topological phase or nondangling corner states without degeneracy, which depends on the bond configuration near the corners. In this work, we study the critical behavior of these corner states at the quantum critical point (QCP) and find that the spin-1/21/21/2 corner state induces a new universality class of the corner critical behavior, which is distinct from the ordinary transition of the nondangling corners. In particular, we find that the dangling spin-1/21/21/2 corner state persists at the QCP despite its coupling to the critical spin fluctuations in the bulk. This shows the robustness of the corner state of the higher-order topological state.
A higher-order (HO) symmetry-protected topological (SPT) state can be realized in a plaquette-modulated square lattice antiferromagnet, which hosts a spin-1/21/21/2 degenerate mode on each corner of the lattice with open boundaries. In this work, we show with the field-theoretic analysis and quantum Monte Carlo simulations that the plaquette modulation can drive a direct topological quantum phase transition from the HOSPT to a trivial disordered phase across the deconfined quantum critical point (DQCP) between the antiferromagnetic (AF) order and the valence bond solid (VBS) order, thus the DQCP is a multicritical point bridging both the AF-VBS transition and the topological transition of the HOSPT phase. Our work thus reveals the ubiquitous duality between topological transitions of SPT phases and DQCPs.
We study the interplay between two nontrivial boundary effects: (1) the two-dimensional (2d2d2d) edge states of three-dimensional (3d3d3d) strongly interacting bosonic symmetry-protected topological states, and (2) the boundary fluctuations of 3d3d3d bulk disorder-to-order phase transitions. We then generalize our study to 2d2d2d gapless states localized at an interface embedded in a 3d3d3d bulk, when the bulk undergoes a quantum phase transition. Our study is based on generic long-wavelength descriptions of these systems and controlled analytic calculations. Our results are summarized as follows: (i) The edge state of a prototype bosonic symmetry-protected state can be driven to a new fixed point by coupling to the boundary fluctuations of a bulk quantum phase transition; (ii) the states localized at a 2d2d2d interface of a 3dSU(N)3d\phantom{\rule{4pt}{0ex}}\mathrm{SU}(N)3dSU(N) quantum antiferromagnet may be driven to a new fixed point by coupling to the bulk quantum critical modes. Properties of the new fixed points identified are also studied.
Quantum paramagnets are strongly correlated phases of matter where competing interactions frustrate magnetic order down to zero temperature. In certain cases, quantum fluctuations instead induce topological order, supporting fractionalized quasiparticles. In this Letter, we investigate paradigmatic spin models and show how magnetic frustration can also give rise to higher-order topological properties. We first study the frustrated Heisenberg model in a square lattice, where a plaquette valence bond solid appears through the spontaneous breaking of translational invariance. Despite the amount of effort that has been devoted to study this phase, its topological nature has so far been overlooked. By means of tensor network simulations, we establish how such a state belongs to a higher-order symmetry-protected topological phase, where long-range plaquette order and nontrivial topology coexist. Through this interplay, we uncover excitations that would be absent otherwise, such as cornerlike bulk states bound to dynamical topological defects. Finally, we demonstrate how this higher-order topological quantum paramagnet is also induced by dipolar interactions, indicating the possibility to directly observe this phase using atomic quantum simulators.
We study the higher-order topological spin phases based on a spin analogue of Benalcazar-Bernevig-Hughes model in two dimensions using large-scale quantum Monte Carlo simulations. A continuous Néel-valence bond solid quantum phase transition is revealed by tuning the ratio between dimerized spin couplings, namely, the weak and strong exchange couplings. Through the finite-size scaling analysis, we identify the phase critical points, and consequently, map out the full phase diagrams in related parameter spaces. Particularly, we find that the valence bond solid phase can be a higher-order topological spin phase, which has a gap for spin excitations in the bulk while demonstrates characteristic gapless spin modes at corners of open lattices. We further discuss the connection between the higher-order topological spin phases and the electronic correlated higher-order phases, and find both of them possess gapless spin corner modes that are protected by higher-order topology. Our result exemplifies higher-order physics in the correlated spin systems and will contribute to further understandings of the many-body higher-order topological phenomena.
The nonordinary surface transitions at the (2+12+12+1)-dimensional quantum critical point precluded in the corresponding classical critical point have been found recently. The mechanism of such behavior that is only found in dimerized Heisenberg models to date is still under debate. To illuminate the role of symmetry-protected topological (SPT) phases in inducing such nonordinary behaviors, we study a system on a two-dimensional square lattice consisting of interacting spin-1 Haldane chains, which has a genuine SPT phase---the Haldane phase---at weak interchain interactions and a quantum critical point, belonging to the classical three-dimensional (3D) O(3) universality class, to the N\'eel phase. Different from previously studied models, there is no dimerization in the current model. Cutting the system along the chain direction or perpendicular to the chain direction exposes two different surfaces. Using unbiased quantum Monte Carlo simulations, we find that the two different types of surfaces show completely different surface critical behaviors at the bulk critical point, resulting from different surface states in the SPT phase. For the system with surfaces along the chain direction, the surface critical behavior is of ordinary type of the bulk 3D O(3) critical point, while for the surfaces perpendicular to the chain direction, the surface critical behavior is nonordinary, consistent with nonordinary transitions found in dimerized Heisenberg models. Our numerical results demonstrate that the gapless surface state in the gapped SPT phase together with the gapless mode of the critical point is a pure quantum scenario that leads to the nonordinary transition.
Based on large-scale quantum Monte Carlo simulations, we examine the correlations along the edges of two-dimensional semi-infinite quantum critical Heisenberg spin-1/21/21/2 systems. In particular, we consider coupled quantum spin-dimer systems at their bulk quantum critical points, including the columnar-dimer model and the plaquette-square lattice. The alignment of the edge spins strongly affects these correlations and the corresponding scaling exponents, with remarkably similar values obtained for various quantum spin-dimer systems. We furthermore observe subtle effects on the scaling behavior from perturbing the edge spins that exhibit the genuine quantum nature of these edge states. Our observations furthermore challenge recent attempts that relate the edge spin criticality to the presence of symmetry-protected topological phases in such quantum spin systems.