Based on large-scale quantum Monte Carlo simulations, we examine the correlations along the edges of two-dimensional semi-infinite quantum critical Heisenberg spin-

1/2

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.

DOI: 10.1103/physrevb.98.140403

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arXiv:2006.14152Physical review. B./Physical review. BAug 31, 202018 citations

Based on large-scale quantum Monte Carlo simulations, we examine the correlations along the edges of two-dimensional semi-infinite quantum critical Heisenberg spin-

1/2

Based on large-scale quantum Monte Carlo simulations, we examine the correlations along the edges of two-dimensional semi-infinite quantum critical Heisenberg spin-

1/2

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.

The nonordinary surface transitions at the (

2+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.

Using quantum Monte Carlo simulations, we study effects of disorder on the

S=1/2

Heisenberg model with exchange constant

J

on the square lattice supplemented by multispin interactions

Q

. It was found recently [L. Liu et al., Phys. Rev. X 8, 041040 (2018)] that the ground state of this $J\ensuremath{-}Q$ model with random couplings undergoes a quantum phase transition from the N\'eel antiferromagnetic state into a randomness-induced spin-liquid-like state that is a close analog to the well known random-singlet (RS) state of the Heisenberg chain with random couplings. This 2D RS state arises from a spontaneously symmetry-broken fourfold degenerate columnar valence-bond solid that is broken up by the disorder into finite domains, with spinons localized at topological defects. The interacting spinons form a critical collective many-body state without magnetic long range order but with the mean spin-spin correlations decaying with distance

r

as ${r}^{\ensuremath{-}2}$, as in the one-dimensional RS state. The dynamic exponent $z\ensuremath{\ge}2$, varying continuously with the model parameters. In this work, we further investigate the properties of the RS state in the $J\ensuremath{-}Q$ model with random

Q

couplings. We study the temperature dependence of the specific heat and various susceptibilities for large enough systems to reach the thermodynamic limit. We also analyze the size dependence of the critical magnetic order parameter and its susceptibility in the ground state. For all these quantities, we find consistency with the conventional quantum-critical scaling laws when the condition implied by the ${r}^{\ensuremath{-}2}$ form of the spin correlations is imposed. In particular, all the different quantities can be explained by the same value of the dynamic exponent

z

at fixed model parameters. We argue that the RS state identified in the $J\ensuremath{-}Q$ model corresponds to a generic renormalization group fixed point that can be reached in many quantum magnets with random couplings and that it has already been observed experimentally.