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arXiv:1502.01085 · Physical Review B

Emergent topological excitations in a two-dimensional quantum spin system

Hui Shao, Wenan Guo, Anders W. Sandvik

Published Mar 24, 2015• 11 citations
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Abstract

We study the mechanism of decay of a topological (winding-number) excitation due to finite-size effects in a two-dimensional valence-bond solid state, realized in an $S=1/2$ spin model $(J\ensuremath{-}Q$ model) with six-spin interactions and studied using projector Monte Carlo simulations in the valence bond basis. A topological excitation with winding number $|W|>0$ contains domain walls, which are unstable due to the emergence of long valence bonds in the wave function, unlike in effective descriptions with the quantum dimer model (which by construction includes only short bonds). We find that the lifetime of the winding number in imaginary time, which is directly accessible in the simulations, diverges as a power of the system length $L$. The energy can be computed within this time (i.e., it converges toward a ``quasieigenvalue'' before the winding number decays) and agrees for large $L$ with the domain-wall energy computed in an open lattice with boundary modifications enforcing a domain wall. Constructing a simplified two-state model which can be solved in real and imaginary time, and using the imaginary-time behavior from the simulations as input, we find that the real-time decay rate out of the initial winding sector is exponentially small in $L$. Thus, the winding number rapidly becomes a well-defined conserved quantum number for large systems, supporting the conclusions reached by computing the energy quasieigenvalues. Including Heisenberg exchange interactions which bring the system to a quantum-critical point separating the valence-bond solid from an antiferromagnetic ground state (the putative ``deconfined'' quantum-critical point), we can also converge the domain-wall energy here and find that it decays as a power law of the system size. Thus, the winding number is an emergent quantum number also at the critical point, with all winding number sectors becoming degenerate in the thermodynamic limit. This supports the description of the critical point in terms of a U(1) gauge-field theory.

DOI
10.1103/physrevb.91.094426

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We study the mechanism of decay of a topological (winding-number) excitation due to finite-size effects in a two-dimensional valence-bond solid state, realized in an S=1/2S=1/2S=1/2 spin model (J\ensuremath−Q(J\ensuremath{-}Q(J\ensuremath−Q model) with six-spin interactions and studied using projector Monte Carlo simulations in the valence bond basis. A topological excitation with winding number ∣W∣>0|W|>0∣W∣>0 contains domain walls, which are unstable due to the emergence of long valence bonds in the wave function, unlike in effective descriptions with the quantum dimer model (which by construction includes only short bonds). We find that the lifetime of the winding number in imaginary time, which is directly accessible in the simulations, diverges as a power of the system length LLL. The energy can be computed within this time (i.e., it converges toward a quasieigenvalue'' before the winding number decays) and agrees for large $L$ with the domain-wall energy computed in an open lattice with boundary modifications enforcing a domain wall. Constructing a simplified two-state model which can be solved in real and imaginary time, and using the imaginary-time behavior from the simulations as input, we find that the real-time decay rate out of the initial winding sector is exponentially small in $L$. Thus, the winding number rapidly becomes a well-defined conserved quantum number for large systems, supporting the conclusions reached by computing the energy quasieigenvalues. Including Heisenberg exchange interactions which bring the system to a quantum-critical point separating the valence-bond solid from an antiferromagnetic ground state (the putative deconfined'' quantum-critical point), we can also converge the domain-wall energy here and find that it decays as a power law of the system size. Thus, the winding number is an emergent quantum number also at the critical point, with all winding number sectors becoming degenerate in the thermodynamic limit. This supports the description of the critical point in terms of a U(1) gauge-field theory.
Dirac fermions in (2 + 1) dimensions with dynamically generated anticommuting SO(3) antiferromagnetic and Z2{\mathrm{Z}}_{2}Z2​ Kekul\'e valence-bond solid (KVBS) masses map onto a field theory with a topological $\ensuremath{\theta}$ term. This term provides a mechanism for continuous phase transitions between different symmetry-broken states: topological defects of one phase carry the charge of the other and proliferate at the transition. The $\ensuremath{\theta}$ term implies that a domain wall of the Z2{\mathrm{Z}}_{2}Z2​ KVBS order parameter harbors a spin-1/21/21/2 Heisenberg chain, as described by a (1 + 1)-dimensional SO(3) nonlinear sigma model with $\ensuremath{\theta}$ term at $\ensuremath{\theta}=\ensuremath{\pi}$. Using pinning fields to stabilize the domain wall, we show that our auxiliary-field quantum Monte Carlo simulations indeed support the emergence of a spin-1/21/21/2 chain at the Z2{\mathrm{Z}}_{2}Z2​ topological defect. Surprisingly, the consequences of the topological term are seen far from the critical point such that the physics of apparently unrelated model systems are naturally understood by invoking them. This concept can be generalized to higher dimensions where (2 + 1)-dimensional SO(4) or SO(5) theories with topological terms are realized at a domain wall.
Two-dimensional quantum antiferromagnets are believed to host phases of matter whose excitations are more fundamental than those of the ordered phases. When combining two such spin systems in a bilayer, strong interaction between the emergent excitations can produce phases not realized in either of its subsystems. We show that the critical fluctuations of a two-dimensional spin-liquid state can induce a deconfined quantum critical point in a proximate antiferromagnet. The most relevant coupling between the associated effective field theories is given by a mixed Chern-Simons term of the emergent gauge fields in each layer. This describes a topological current-current interaction. In contrast to the local spin-spin interaction, it strongly modifies the renormalization group flow of the theory describing the N\'eel--valence-bond-solid transition of the antiferromagnet. In particular, the protected coupling constant associated with it implies nontrivial quantum critical scaling characterized by a nonuniversal power-law divergence of the correlation length in the critical domain and Berezinskii-Kosterlitz-Thouless divergence approaching it.
We study spin-1/21/21/2 chains with long-range power-law decaying unfrustrated (bipartite) Heisenberg exchange Jr∝r−αJ_r \propto r^{-α}Jr​∝r−α and multi-spin interactions QQQ favoring a valence-bond solid (VBS) ground state. Employing quantum Monte Carlo techniques and Lanczos diagonalization, we analyze order parameters and excited-state level crossings to characterize quantum states and phase transitions in the (α,Q)(α,Q)(α,Q) plane. For weak QQQ and sufficiently slowly decaying Heisenberg interactions (small ααα), the system has a long-range-ordered antiferromagnetic (AFM) ground state, and upon increasing ααα there is a continuous transition into a quasi long-range ordered (QLRO) critical state of the type in the standard Heisenberg chain. For rapidly decaying long-range interactions, there is transition between QLRO and VBS ground states of the same kind as in the frustrated J1J_1J1​-J2J_2J2​ Heisenberg chain. Our most important finding is a direct continuous quantum phase transition between the AFM and VBS states - a close analogy to the 2D deconfined quantum-critical point. In previous 1D analogies the ordered phases both have gapped fractional excitations, and the critical point is a conventional Luttinger Liquid. In our model the excitations fractionalize upon transitioning from the AFM state, changing from spin waves to deconfined spinons. We extract critical exponents at the AFM-VBS transition and use order-parameter distributions to study emergent symmetries. We find emergent O(444) symmetry of the O(333) AFM and scalar VBS order parameters. Thus, the order parameter fluctuations exhibit the covariance of a uniaxially deformed O(444) sphere (an "elliptical" symmetry). This unusual quantum phase transition does not yet have any known field theory description, and our detailed results can serve to guide its construction. We discuss possible experimental realizations.
Using ground-state projector quantum Monte Carlo simulations in the valence-bond basis, it is demonstrated that nonfrustrating four-spin interactions can destroy the Néel order of the two-dimensional S=1/2 Heisenberg antiferromagnet and drive it into a valence-bond solid (VBS) phase. Results for spin and dimer correlations are consistent with a single continuous transition, and all data exhibit finite-size scaling with a single set of exponents, z=1, nu=0.78+/-0.03, and eta=0.26+/-0.03. The unusually large eta and an emergent U(1) symmetry, detected using VBS order parameter histograms, provide strong evidence for a deconfined quantum critical point.
Observing constituent particles with fractional quantum numbers in confined and deconfined states is an interesting and challenging problem in quantum many-body physics. Here, we further explore a computational scheme [Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011)] based on valence-bond quantum Monte Carlo simulations of quantum spin systems. Using several different one-dimensional models, we characterize S=12S=\frac{1}{2}S=21​ spinon excitations using the intrinsic spinon size $\ensuremath{\lambda}$ and confinement length $\mathrm{\ensuremath{\Lambda}}$ (the size of a bound state). The spinons have finite size in valence-bond-solid states, infinite size in the critical region (with overlaps characterized by power laws), and become ill defined (completely unlocalizable) in the N\'eel state (which we stabilize in one dimension by introducing long-range interactions). We also verify that pairs of spinons are deconfined in uniform spin chains but become confined upon introducing a pattern of alternating coupling strengths (dimerization) or coupling two chains (forming a ladder). In the dimerized system, an individual spinon can be small when the confinement length is large; this is the case when the imposed dimerization is weak but the ground state of the corresponding uniform chain is a spontaneously formed valence-bond solid (where the spinons are deconfined). Based on our numerical results, we argue that a system with $\ensuremath{\lambda}\ensuremath{\ll}\mathrm{\ensuremath{\Lambda}}$ is associated with weak repulsive short-range spinon-spinon interactions. In principle, both the length scales $\ensuremath{\lambda}$ and $\mathrm{\ensuremath{\Lambda}}$ can still be individually tuned from small to infinite (with $\ensuremath{\lambda}\ensuremath{\le}\mathrm{\ensuremath{\Lambda}}$) by varying model parameters. In contrast, in the ladder system the two lengths are always similar, and this is the case also in the weakly dimerized systems when the corresponding uniform chain is in the critical phase. In these systems, the effective spinon-spinon interactions are purely attractive and there is only a single large length scale close to criticality, which is reflected in the standard spin correlations as well as in the spinon characteristics.