Emergent topological excitations in a two-dimensional quantum spin system

Abstract

We study the mechanism of decay of a topological (winding-number) excitation due to finitesize effects in a two-dimensional valence-bond solid state, realized in an S = 1/2 spin model (J-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 life time 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 "quasi-eigenvalue" 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 quasi-eigenvalues. Including Heisenberg exchange interactions which brings the system to a quantum-critical point separating the valencebond 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.

Quantum criticality with two length scales

“…The previous results in Ref. [40] were likely affected in the same way by remaining scaling corrections, and ν/ν ≈ 0.72 should hold universally for different variants of the J-Q model and for different ways of generating domain walls.…”

“…[40] we employed a different approach to studying domain walls in periodic systems, by restricting the trial state used in projector QMC simulations in the valence-bond basis to a topological (winding number) sector corresponding to the presence of a given number of domain walls. We found anomalous scaling for κ, but with a somewhat larger exponent ratio ν/ν = 0.80(1), for a different variant of the J-Q model with products of three singlet projectors (the J-Q 3 model) instead of the two projectors used in the model (3) (the J-Q 2 model).…”

“…The multi-canonical approach employed in the previous section, developed to circumvent the difficulties of MC calculations of individual free energies at T > 0, are therefore neither useful nor needed. We use the projector QMC approach with e −βH applied to a valence-bond trial state of the amplitude-product type [39,40], choosing the "projection time" β sufficiently large, up to β = 4L, to converge the ground-state energy. Domain walls are introduced by boundary conditions in two different ways, schematically illustrated in Fig.…”

Quantum Monte Carlo studies of spinons in one-dimensional spin systems

“…However, in the simulations there are no explicit references to sublattices and in effect the system is then translationally invariant. Then, under the further assumption that no bonds with length as large as N/4 are present (such configurations having ill-defined signs), 36 the momentum is k = 0 or π, for N of the form 4n + 1 and 4n + 3, respectively.…”

Round table: What can we learn about confinement and anoma-lous effects in QCD using analog systems?

“…An ideal model in which to test this numerically is the so-called J-Q model. The J-Q model on a cubic lattice is well known to be in the VBS phase for a certain range of its parameters, and it has been extensively used for numerical studies of the VBS phase and quantum critical phenomena in quantum magnets [42][43][44][45] (see [46] for a review). Indeed as Shao, Sandvik, Ünsal and TS demonstrated in [40] that while the usual VBS typically has exponential correlation functions, in the presence of the domain wall, the Z 2 VBS shows algebraic correlations along the wall and is in perfect agreement with the 1D spin chain, which is well known to host spin 1/2 excitations (spinons) [47][48][49].…”