Session T26: Focus Session: Computational Frontiers in Quantum Spin Systems II

Cubic Interactions and Quantum Criticality in Dimerized Antiferromagnets

Dimerized quantum antiferromagnets can be driven through a quantum phase transition from a dimerized phase into a magnetically ordered state upon tuning the exchange parameters. In recent years, the critical properties in such dimerized antiferromagnets were examined in detail, based on large-scale quantum Monte Carlo simulations, which reported deviations from O(3) universality for specific two-dimensional geometries, in particular for the staggered-dimer antiferromagnet. Symmetry arguments and microscopic calculations exhibit that a nontrivial cubic term arises in the relevant order-parameter quantum field theory, related to three-particle interactions among the triplet excitations within the paramagnetic phase of this model. The consequences of such cubic terms are explored using a combination of analytical and numerical methods. Complemented by finite-temperature quantum Monte Carlo simulations, these results lead to the conclusion that critical exponents in dimerized antiferromagnets are identical to that of the standard O(3) universality class, but with anomalously large corrections to scaling for these specific dimerization geometries.   

Computational Studies of $T=0$ Neel-Valence bond solid transitions in two dimensional quantum antiferromagnets

We use Quantum Monte Carlo techniques to study a direct quantum phase transition in two dimensional quantum antiferromagnets between a collinear Neel ordered state and a valence bond solid ordered singlet state. We contrast the strongly first order behavior of the transition in cases where the valence bond solid order is of a ``staggered'' type with the deconfined critical behavior seen in cases where the valence-bond solid order is of a columnar type. In the deconfined case, we find evidence for weak, apparently logarithmic violations of scaling. [References: preprint; Phys. Rev. B 83, 235111 (2011); Phys. Rev. B 83, 134419 (2011); Phys. Rev. B 82, 155139 (2010)]   

Velocities of Goldstone and critical modes in SU(2) symmetric quantum spin systems

The low-energy excitations of many interesting quantum spin systems are gapless and linearly dispersing. Examples include Goldstone modes in the N\'eel phase and critical modes at a z=1 quantum critical point. We calculate the velocities of such modes for a variety of SU(2) symmetric S=1/2 systems using quantum Monte Carlo (QMC) methods. We use two complementary approaches:a)The lowest triplet gap from the singlet ground state is calculated using T=0 projector QMC by measuring appropriate imaginary-time correlation functions. The velocity is obtained from extracted momentum dependent gaps.b)We use a method based on tuning the system to the cubic regime by varying its temperature to equate the variance of spatial and temporal winding numbers, which was recently used by Jiang [1] for a system with Goldstone modes. We find that this method can also be applied to a z=1 critical point (the critical point of an S=1/2 Heisenberg bilayer) and to the 1D Heisenberg spin chain, where there are no Goldstone modes. We also extract the velocity of the critical modes of the J-Q model. It agrees very well with the velocity obtained from a phenomenological approach [2] based on a spinon gas picture. [1] Jiang, Phys. Rev B 83, 024419 (2011) [2] Sandvik et al., Phys. Rev. Lett. 106, 207203 (2011)   

Unconventional versus conventional destruction of square lattice SU(N) magnetism

Recently we have found SU($N$) symmetric square lattice spin models of quantum anti-ferromagnets, which have quantum phase transitions between magnetic and non-magnetic phases for arbitrary $N$, and which are nonetheless free of the ``sign-problem'' of quantum Monte Carlo. The absence of the notorious sign-problem allows detailed unbiased numerical simulations of two-dimensional magnetic quantum phase transitions on lattices containing in excess of 10$^4$ spins. Depending on the absence or presence of uncompensated Berry phases in our microscopic models we find evidence for both conventional first order phase transitions and unconventional continuous quantum phase transitions at which both N\'eel- and valence bond solid-order (VBS) are {\em simultaneously} quantum critical. A detailed quantitative study of the N\'eel and VBS scaling dimensions as a function of $N$ provides compelling evidence that the long-wavelength description of these quantum critical points may be found in the CP$^{N-1}$ gauge theory, as predicted by the deconfined quantum criticality scenario. R. K. Kaul and A. W. Sandvik, http://arxiv.org/abs/1110.4130. R. K. Kaul (forthcoming).   

Neel to valence-bond solid transitions in generalized amplitude-product states with correlated weights

An amplitude-product state is a superposition of valence-bond states with the expansion coefficients being products of individual bond amplitudes that depend only on the bond shape. In two dimensions, these states have Neel order or are spin liquids, but they never have any valence-bond solid order. We construct generalized amplitude-product states on the square lattice which include correlated weights for short-range bonds. Using these states, we explore phase transitions between Neel phase, valence-bond solid phases, and spin liquid. We also study the Neel-VBS transition in the standard amplitude-product states in one dimension.   

Emergent U(1) Symmetry in Square Lattice Quantum Dimer Models

We report an exact diagonalization study of Rokhsar-Kivelson quantum dimer models on square lattice. Using a finite-size scaling analysis of excited energy levels, we are able to identify a regime of length scales where the the quantum dimer model exhibits a $U(1)$ symmetry. Dimer order parameter histograms confirm this remarkable symmetry. Beyond this crossover length, columnar dimer order emerges at least for $v/t \la 0.6$. Our interpretation is supplemented with field-theory analysis as well as large-scale quantum Monte-Carlo simulations.   

A round-trip from spin to quantum dimer models

Short-range valence bonds wave-functions are often used as a paradigm for non-magnetic states (such as spin liquids or valence bond crystals). Recently, two local S=1/2 spin Hamiltonians which admits nearest neighbor valence bond wave-function(s) as ground-state(s) on the square lattice have been proposed by Cano and Fendley [Phys. Rev. Lett. {\bf 105}, 067205 (2010)]. We present a numerical study, by means of exact diagonalizations and diagonalizations restricted in the subspace spanned by nearest neighbor valence bond states, of the ground state and excitations of these models. We show that it corresponds to a new type of spin liquid state, with gapped spin but gapless non-magnetic excitations. Mixing the two models is shown to stabilize valence bond crystal ground states that are very reminiscent of the phases present in the Rokhsar-Kivelson quantum dimer model phase diagram. Using a generic mapping scheme [Phys. Rev. B {\bf 81}, 214413 (2010)] of spin hamiltonians to generalized quantum dimer models we show how this correspondence between the phase diagram of a local S=1/2 spin hamiltonian and the Rokhsar-Kivelson quantum dimer model can be understood.   

Phase Diagram of a Frustrated Quantum Antiferromagnet on the Honeycomb Lattice: Magnetic Order versus Valence-Bond Crystal Formation

We present a comprehensive computational study of the phase diagram of the frustrated $S=1/2$ Heisenberg antiferromagnet on the honeycomb lattice, with second-nearest $(J_2)$ and third-neighbor $(J_3)$ couplings. Using a combination of exact diagonalizations of the original spin model, of the Hamiltonian projected into the nearest neighbor short range valence bond basis, and of an effective quantum dimer model, we determine the boundaries of several magnetically ordered phases in the region $J_2,J_3\in [0,1]$, and find a sizable magnetically disordered region in between. We characterize part of this magnetically disordered phase as a {\em plaquette} valence bond crystal phase.   

Plaquette order and deconfined quantum critical point in the spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice

We have precisely determined the ground state phase diagram of the quantum spin-1 bilinear-biquadratic Heisenberg model on the honeycomb lattice using the tensor renormalization group method. We find that the ferromagnetic, ferroquadrupolar, and a large part of the antiferromagnetic phases are stable against quantum fluctuations. However, around the phase where the ground state is antiferro-quadrupolar ordered in the classical limit, quantum fluctuations suppress completely all magnetic orders, leading to a plaquette order phase which breaks the lattice symmetry but preserves the spin SU(2) symmetry. The quantum phase transition between the antiferromagnetic phase and the plaquette phase is found to be a direct second order transition, being the first candidate of the deconfined quantum critical point for the spin-1 quantum systems.   

Three-sublattice order in the SU(3) Heisenberg model on the square and triangular lattice

We present a numerical study of the SU(3) Heisenberg model on the triangular and square lattice by means of the density-matrix renormalization group (DMRG) and infinite projected entangled-pair states (iPEPS). For the triangular lattice we confirm that the ground state has a three-sublattice order with a finite ordered moment which is compatible with the result from linear flavor wave theory (LFWT). The same type of order has recently been predicted also for the square lattice. However, for this case the ordered moment cannot be computed with LFWT due to divergent fluctuations. Our numerical study clearly supports this three-sublattice order, with an ordered moment of m=0.2-0.4 in the thermodynamic limit.   

Zero- vs. finite-field transition in S=1 Heisenberg antiferromagnet with single-ion anisotropy

We use large scale quantum Monte Carlo simulation on finite size lattices to study the ground state and finite temperature transitions in a $S=1$ Heisenberg antiferromagnet (HAFM) with single-ion anisotropy ($D$) in the presence of an external magnetic field ($h_z$). The ground state phase diagram (in the $h_z-D$ plane) is characterized by a quantum paramegnetic phase (QPM) at large $D$ and small $h_z$ and an XY-AFM phase at small $D$ and/or large $h_z$ separated by a continuous transition. We show that the QPM to XY-AFM transition belongs to the XY universality class at $h_z=0$ (driven by varying $D$) whereas it belongs to the BEC universality class at $h_z\neq 0$ (driven by either varying $D$ at constant $h_z$ or varying $h_z$ at constant $D$). This has important implication in the behavior of the specific heat at the transition point which can be verified experimentally. Finally, we discuss the experimental relevance of our results in the case of DTN.   

Ferronematic order in a spin-1 Heisenberg antiferromagnet

We study the field-induced ground-state phase transition of a spin-1 Heisenberg antiferromagnet with large easy-axis single-ion anisotropy $D$. Direct spin-wave treatment predicts a single first-order phase transition from an antiferromagnetic N\'{e}el phase at low magnetic fields to a fully polarized state at high magnetic fields. Mean field arguments, based on an effective spin-1/2 model that is exact in the $D\rightarrow\infty$ limit, show that this transition is preempted by an intermediate phase with double-spin-flip correlations. We call this phase the {\em ferronematic} phase, as the effective spin model for large (negative) $D$ is a spin-1/2 XXZ model with {\em ferromagnetic} transverse exchange. Using exact diagonalization and quantum Monte Carlo, we confirm the presence of the ferronematic phase. Long range order is observed in the equal-time Green's function $\langle S_{i}^{+}S_{i}^{+}S_{j}^{-}S_{j}^{-}+H.c.\rangle$, which is the correlation function for ferronematic order. We also show the rapid convergence to the effective model for large values of $D$.   

Monte Carlo study of a $U(1)\times U(1)$ system with $\pi$-statistical interaction

We study a $U(1)\times U(1)$ system with two species of loops with mutual $\pi$-statistics in (2+1) dimensions. We are able to reformulate the model in a way that can be studied by Monte Carlo and we determine the phase diagram. In addition to a phase with no loops, we find two phases with only one species of loop proliferated. The model has a self-dual line, a segment of which separates these two phases. Everywhere on the segment, we find the transition to be first-order, signifying that the two loop systems behave as immiscible fluids when they are both trying to condense. Moving further along the self-dual line, we find a phase where both loops proliferate, but they are only of even strength, and therefore avoid the statistical interactions. We study another model which does not have this phase, and also find first-order behavior on the self-dual segment.