Session V26: Focus Session: Computational Frontiers in Quantum Spin Systems III

A dynamical Marshall sign convention

Monte Carlo sampling of quantum spin models is only practical when it is possible to gauge away simultaneously all negative signs in the coefficients of the ground state wavefunction. The existence of such a transformation is related to the possibility of establishing a bipartite pattern of magnetic order on the lattice and to the choice of a so-called Marshall sign convention. In practice, identifying the correct Marshall sign convention is the responsibility of the QMC practitioner, and the convention itself is generally hard coded. It turns out, however, that a locally optimal sign convention can be determined dynamically within the simulation---meaning that for nonfrustrated systems the simulation quickly establishes a Marshall sign convention that leads to sign-problem-free sampling and that for frustrated systems the Marshall sign convention continually evolves in Monte Carlo time so as to minimize the severity of the sign problem. For concreteness, we focus on a worm algorithm formulated in the basis of singlet product states.   

Monte Carlo Simulations of Quantum Spin Systems in the Valence Bond Basis

We propose a quantum Monte Carlo method for frustrated spin systems that partially alleviates the sign problem --- thereby extending the range of frustrated couplings over which the system can be reliably sampled. The scheme is projective and takes advantage of the overcompleteness and nonorthogonality of the valence bond basis. It provides a framework for further semi-controlled approximations that are fully sign-problem-free in which the transition weights between bond configurations take on effective, renormalized values. We present results for the frustrated (J1-J2), spin-half Heisenberg model on the square lattice in the vicinity of its phase transition at $J2/J1 \approx 0.4$.   

Observing spinon excitations in quantum spin models

We develop a technique to directly study spinons (emergent spin $S=1/2$ particles) in quantum spin models in any number of dimensions [1]. Two characteristic lengths---the size of a spinon wave packet and the size of a bound pair (a triplon)---are defined in terms of wave-function overlaps that can be evaluated by quantum Monte Carlo simulations. We find that these two lengths are well distinguishable in one-dimensional models with valence-bond-solid (VBS) ground states and explicitly dimerized models, yet hardly separable in 2-leg ladder systems. We provide some physics insights for these phenomena. We also study spinons in two-dimensional resonating-valence-bond states and models with N\'eel-VBS transitions.\\[4pt] [1] Y. Tang and A. W. Sandvik, Phys. Rev. Lett. 107, 157201 (2011).   

Lightcone renormalization and quantum quenches in one-dimensional Hubbard models

The Lieb-Robinson bound implies that the unitary time evolution of an operator can be restricted to an effective light cone for any Hamiltonian with short-range interactions. Here we present a very efficient new renormalization group algorithm based on this light cone structure to study the time evolution of prepared initial states in the thermodynamic limit in one-dimensional quantum systems. The algorithm does not require translational invariance and allows for an easy implementation of local conservation laws. We use the algorithm to investigate the relaxation dynamics of a doublon lattice in fermionic Hubbard models as well as a possible thermalization. Furthermore, we present results for a doublon impurity in a N\'eel background. We find that the excess charge and spin spread at different velocities, providing an example of spin-charge separation in a highly excited state.   

Time Evolution within a Comoving Window

We present a modification of Matrix Product State time evolution to simulate the propagation of a signal front on an infinite system. The time evolution is calculated within a finite window that moves along with the signal front, in such a way that boundary effects do not occur. Signal fronts can then be studied unperturbed for much longer times than on truly finite systems, where boundary perturbations and reflections would interfere. The entanglement within in the comoving window remains small. Our approach avoids the large entanglement which develops around the location of the signal source and therefore requires significantly lower computational effort. We verify our approach against exact results and show examples of propagating signals for the XXZ model and the transverse Ising model.   

Emergence of prominent bound states in the spin-1/2 Heisenberg XXZ chain after a local quantum quench

We calculate the non-equilibrium evolution in the spin-1/2 XXZ Heisenberg chain for fixed magnetization after a \emph{local quantum quench}. Initially an infinite magnetic field is applied to $n$ consecutive sites in the center of a large chain, and the ground state is determined. Then the field is switched off and the time evolution of observables such as the z-component of spin is computed using the Time Evolving Block Decimation (TEBD) algorithm. We find that the observables exhibit strong signatures of propagating spinon as well as bound state excitations. These persist even when integrability-breaking perturbations are included. Since bound states (``strings'') are notoriously difficult to observe using conventional probes such as inelastic neutron scattering we conclude that local quantum quenches are an ideal setting for studying their properties. We comment on implications of our results for cold atom experiments.   

Non-local order parameters in 1D symmetry protected topological phases

A topological phase is a phase of matter which cannot be characterized by a local order parameter. It has been shown that gapped phases in 1D systems can be completely characterized using tools related to projective representations of the symmetry groups. An example of a symmetry protected topological phase is the Haldane phase of S = 1 chains. Here the phase is protected by any of the following symmetries: dihedral group of $\pi$-rotations about two orthogonal axes, time-reversal symmetry, or bond centered inversion symmetry. We introduce non-local order parameters for each case which can be simply calculated using numerical methods such as Density-Matrix Renormalization Group (DMRG). These non-local order parameters provide a practical tool for numerically detecting these non-trivial phases.   

Projected Density of Transitions for Heisenberg Models

The projected density of transitions (PDoT) is the interacting analogue of the projected or local density of states. The PDoT is proportional to the probability, averaged over all states of the system, that some disturbance (the projection) induces a transition with a specific energy. It is calculated in the same way as the density of states, but using Heisenberg's equation instead of Schr\"{o}dinger's equation. As an example we have applied it to the Heisenberg model for spin interactions of electrons on linear, square, and cubic lattices. One surprise in these calculations is what seems to be a consequence of the spin's rotational symmetry.   

Criticality in one-dimensional supersymmetric lattice fermions

A supersymmetric model for lattice fermions has been seen to host a plethora of interesting phenomena. On one-dimensional and quasi-one-dimensional lattices, the model naturally becomes critical and it has been conjectured that it is described by superconformal field theory. While this relation has been confirmed for the chain, establishing this link and exploring the phases adjacent to the critical point for the case of the square ladder has turned out to be a challenging problem for numerical simulations. In our work, we collect evidence in support of the conjecture and obtain insights into the adjacent phases using a variety of numerical techniques, including the density-matrix renormalization group and the multi-scale entanglement renormalization ansatz.   

Study of Kitaev-Heisenberg model with second-neighbor Heisenberg coupling by DMRG simulations and slave-particle theories

We study the effect of second-neighbor Heisenberg coupling $J_2$ to the first neighbor Kitaev-Heisenberg model on the honeycomb lattice by doing DMRG simulations and slave-particle theories. In the Kitaev limit, we find that the gapless spin liquid phase at $J_2=0$ survives up to a finite critical value $J_{2c}$. In an intermediate range, namely $J_{2c} [Preview Abstract]

Thermodynamics and phase transitions of the pinwheel-distorted Kagome lattice Heisenberg model

We study the Heisenberg model on the pinwheel-distorted Kagome lattice as observed in the material $Rb_2Cu_3SnF_{12}$. Experimentally relevant thermodynamic properties at finite temperatures are computed utilizing numerical linked-cluster expansions [1]. We introduce a Lanczos-based zero-temperature numerical linked-cluster expansion and study the approach of the pinwheel distorted lattice to the uniform Kagome lattice Heisenberg model. We find strong evidence for a phase transition before the uniform limit is reached, implying that the ground state of the Kagome lattice Heisenberg model is likely not pinwheel dimerized and is stable to finite pinwheel dimerizing perturbations [2]. \\[4pt] [1] M. Rigol and R. R. P. Singh, Phys. Rev. Lett. 98, 207204 (2007); Phys. Rev. B 76, 184403 (2007). \\[0pt] [2] E. Khatami, R. R. P. Singh, M. Rigol, preprint: arXiv:1105.4147   

Numerical Schwinger boson approach to the Bethe lattice antiferromagnet at percolation

What are the lowest energy excitations of a spin-1/2 Heisenberg antiferromagnet on a critical percolation cluster? On the square lattice, Wang et al.,\footnote{Phys. Rev. B 81, 054417 (2010)} discovered anomalously low S=1 excitations, with energy scaling as $\Delta \sim 1/N^2$. Normally the ``Anderson tower'' would be the lowest, having energy $S(S+1)/2\chi N \sim 1/N$ ($\chi$= transverse susceptibility). These anomalous excitations were attributed to ``dangling spins'' appearing in parts of the cluster where there was an imbalance of even and odd sites. Here, we look at the diluted z=3 Bethe lattice at the percolation threshold. Previous work confirmed the existence of emergent spin-1/2 degrees of freedom with an effective Heisenberg Hamiltonian but did not explain their origin. New results from DMRG (Density Matrix Renormalization Group) show strong dimerization tendency in the (singlet) ground state, yet there is long-range Neel order on the percolation cluster.\footnote{A. Sandvik, Phys. Rev. B 66,024418 (2002)} To harmonize these results, we set up a numerical Schwinger Boson mean field calculation (with site dependent parameters); we find lowering of mean field energy and spin correlations which agree well with ED and DMRG.   

Spin Mott Glass Phase in the Disordered Spin Systems

We use quantum Monte Carlo simulations to study a glassy ground state of S=1/2 quantum spins by using a dimerized J1-J2-J3 Heisenberg model on the square lattice. J1 corresponds to weak bonds, and J2 and J3 are stronger bonds which are randomly distributed on columnar rungs forming coupled 2-leg ladders. By tuning the average value of J2 and J3, the system undergoes Neel-glass-paramagnetic quantum phase transition. The size of the glass region is affected by the value of the disorder strength. In the glass phase, we find that the uniform susceptibility decreases with T according to exp(-b/$T^a$) with $a<1$; thus the state is incompressible at T=0 and classified as a Mott glass (MG). At the Neel-MG transition, the susceptibility behaves as $T^{2/z-1}$. The dynamical exponent z is found to be larger than 1.   

Tuning the disorder in a bosonic superglass

We study the phase transitions associated with tuning the disorder in a Bose-Hubbard model exhibiting a superglass phase. By shifting the distribution of a nearest neighbor $\pm J$ interaction between hard-core bosons on a three-dimensional lattice, we can produce anti-ferromagnetic, glassy, and ferromagnetic (diagonal) types of ordering, each of which interacts uniquely with the superfluid (off-diagonal) ordering. Quantum Monte Carlo simulation results using the worm algorithm show that the existence of superfluidity guards against the formation of large ferromagnetic clusters, presumably due to a lowering of the cost of interfaces. This leads to a strongly-temperature-dependent phase boundary between the glassy and ferromagnetic regions.   

Impact of spin-orbit coupling on the Holstein polaron

We utilize an exact variational numerical procedure to calculate the ground state properties of a polaron in the presence of a Rashba-like spin orbit interaction. Our results corroborate with previous work performed with the Momentum Average approximation and with weak coupling perturbation theory. We find that spin orbit coupling increases the effective mass in the regime with weak electron phonon coupling, and decreases the effective mass in the intermediate and strong electron phonon coupling regime. Analytical strong coupling perturbation theory results confirm our numerical results in the small polaron regime. A large amount of spin orbit coupling can lead to a significant lowering of the polaron effective mass