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arXiv:1512.05082 · Physical review. B./Physical review. B

First-order superfluid to valence-bond solid phase transitions in easy-plane SU(N) magnets for small N

Jonathan D’Emidio, Ribhu K. Kaul

Published Feb 3, 2016• 21 citations
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Abstract

We consider the easy-plane limit of bipartite $\mathrm{SU}(N$) Heisenberg Hamiltonians, which have a fundamental representation on one sublattice and the conjugate to fundamental on the other sublattice. For $N=2$ the easy plane limit of the SU(2) Heisenberg model is the well-known quantum $XY$ model of a lattice superfluid. We introduce a logical method to generalize the quantum $XY$ model to arbitrary $N$, which keeps the Hamiltonian sign-free. We show that these quantum Hamiltonians have a world-line representation as the statistical mechanics of certain tightly packed loop models of $N$ colors in which neighboring loops are disallowed from having the same color. In this loop representation we design an efficient Monte Carlo cluster algorithm for our model. We present extensive numerical results for these models on the two dimensional square lattice, where we find the nearest neighbor model has superfluid order for $N\ensuremath{\le}5$ and valence-bond order for $N>5$. By introducing $\mathrm{SU}(N$) easy-plane symmetric four-spin couplings we are able to tune across the superfluid-VBS phase boundary for all $N\ensuremath{\le}5$. We present clear evidence that this quantum phase transition is first order for $N=2$ and $N=5$, suggesting that easy-plane deconfined criticality runs away generically to a first-order transition for small $N$.

DOI
10.1103/physrevb.93.054406

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We study the quantum phase transition between the superfluid and valence bond solid in easy-plane J-Q models on the square lattice. The Hamiltonian we study is a linear combination of two model Hamiltonians: (1) an SU(2) symmetric model, which is the well known J-Q model that does not show any direct signs of a discontinuous transition on the largest lattices and is presumed continuous and (2) an easy-plane version of the J-Q model, which shows clear evidence for a first-order transition even on rather small lattices of size $L\ensuremath{\approx}16$. A parameter $0\ensuremath{\le}\ensuremath{\lambda}\ensuremath{\le}1$ [$\ensuremath{\lambda}=0$ being the easy-plane model and $\ensuremath{\lambda}=1$ being the SU(2) symmetric J-Q model] allows us to smoothly interpolate between these two limiting models. We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) to investigate the nature of this transition as $\ensuremath{\lambda}$ is varied---here we present studies for $\ensuremath{\lambda}=0,0.5,0.75,0.85,0.95$, and 1. While we find that the first-order transition weakens as $\ensuremath{\lambda}$ is increased from 0 to 1, we find no evidence that the transition becomes continuous until the SU(2) symmetric point, $\ensuremath{\lambda}=1$. We thus conclude that the square-lattice superfluid-VBS transition in the two-component easy-plane model is generically first order.
To examine the validity of the scenario of the deconfined critical phenomena, we carry out a quantum Monte Carlo simulation for the SU(NNN) generalization of the Heisenberg model with four-body and six-body interactions. The quantum phase transition between the SU(NNN) N\'eel and valence-bond solid phases is characterized for N=2N=2N=2, 3, and 4 on the square and honeycomb lattices. While finite-size scaling analysis works well up to the maximum lattice size (L=256L=256L=256) and indicates the continuous nature of the phase transition, a clear systematic change towards the first-order transition is observed in the estimates of the critical exponent $y\ensuremath{\equiv}1/\ensuremath{\nu}$ as the system size increases. We also confirm the relevance of a squared valence-bond solid field ${\ensuremath{\Psi}}^{2}$ for the SU(3) model.
We generalize the SU(N=2) S=1/2 square-lattice quantum magnet with nearest-neighbor antiferromagnetic coupling (J(1)) and next-nearest-neighbor ferromagnetic coupling (J(2)) to arbitrary N. For all N>4, the ground state has valence-bond-solid order for J(2)=0 and Néel order for J(2)/J(1)≫1, allowing us access to the transition between these types of states for large N. Using quantum Monte Carlo simulations, we show that both order parameters vanish at a single quantum-critical point, whose universal exponents for large enough N (here up to N=12) approach the values obtained in a 1/N expansion of the noncompact CP(N-1) field theory. These results lend strong support to the deconfined quantum-criticality theory of the Néel-valence-bond-solid transition.
We study the N\'eel to fourfold columnar valence bond solid (cVBS) quantum phase transition in a sign-free S=1S=1S=1 square-lattice model. This is the same kind of transition that for S=1/2S=1/2S=1/2 has been argued to realize the prototypical deconfined critical point. Extensive numerical simulations of the square-lattice S=1/2S=1/2S=1/2 N\'eel-VBS transition have found consistency with the deconfined critical point scenario with no direct evidence for first-order behavior. In contrast to the S=1/2S=1/2S=1/2 case, in our quantum Monte Carlo simulations for the S=1S=1S=1 model, we present unambiguous evidence for a direct conventional first-order quantum phase transition. Classic signs of a first-order transition demonstrating coexistence including double-peaked histograms and switching behavior are observed. The sharp contrast from the S=1/2S=1/2S=1/2 case is remarkable; we hypothesize that this is a striking demonstration of the role of the size of the quantum spin in the phase diagram of two-dimensional lattice models.
We consider a model Hamiltonian with two SU(4)\mathrm{SU}(4)SU(4) fermions per site on a square lattice, showing a competition between bilinear and biquadratic interactions. This model has generated interest due to possible realizations in ultra-cold-atom experiments and existence of spin-liquid ground states. Using a basis transformation, we show that part of the phase diagram is amenable to quantum Monte Carlo simulations without a sign problem. We find evidence for spin nematic and valence-bond crystalline phases, which are separated by a weak first-order phase transition. A U(1\mathrm{U}(1U(1) symmetry is found to emerge in the valence-bond crystal histograms, suggesting proximity to a deconfined quantum critical point. Our results are obtained with the help of a loop algorithm which allows large-scale simulations of bilinear-biquadratic SO(N\mathrm{SO}(NSO(N) models on arbitrary lattices in a certain parameter regime.
We consider bipartite SU(N)\mathrm{SU}(N)SU(N) spin Hamiltonians with a fundamental representation on one sublattice and a conjugate to fundamental on the other sublattice. By mapping these antiferromagnets to certain classical loop models in one higher dimension, we provide a practical strategy to write down a large family of SU(N)\mathrm{SU}(N)SU(N) symmetric spin Hamiltonians that satisfy Marshall's sign condition. This family includes all previously known sign-free SU(N)\mathrm{SU}(N)SU(N) spin models in this representation and in addition provides a large set of new models that are Marshall positive and can hence be studied efficiently with quantum Monte Carlo methods. As an application of our idea to the square lattice, we show that in addition to Sandvik's QQQ term, there is an independent nontrivial four-spin RRR term that is sign free. Using numerical simulations, we show how the RRR term provides a new route to the study of quantum criticality of N\'eel order.