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arXiv:1403.5678 · Physical Review B

Marshall-positive SU(N) quantum spin systems and classical loop models: A practical strategy to design sign-problem-free spin Hamiltonians

Ribhu K. Kaul

Published Feb 19, 2015• 19 citations
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Abstract

We consider bipartite $\mathrm{SU}(N)$ 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 $\mathrm{SU}(N)$ symmetric spin Hamiltonians that satisfy Marshall's sign condition. This family includes all previously known sign-free $\mathrm{SU}(N)$ 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 $Q$ term, there is an independent nontrivial four-spin $R$ term that is sign free. Using numerical simulations, we show how the $R$ term provides a new route to the study of quantum criticality of N\'eel order.

DOI
10.1103/physrevb.91.054413

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arXiv:2304.07329•Apr 14, 2023•0 citations
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.
In this manuscript we review recent developments in the numerical simulations of bipartite SU( N ) spin models by quantum Monte Carlo (QMC) methods. We provide an account of a large family of newly discovered sign-problem free spin models which can be simulated in their ground states on large lattices, containing O (10 5 ) spins, using the stochastic series expansion method with efficient loop algorithms. One of the most important applications so far of these Hamiltonians are to unbiased studies of quantum criticality between Neel and valence bond phases in two dimensions - a summary of this body of work is provided. The article concludes with an overview of the current status of and outlook for future studies of the "designer" Hamiltonians.
This work focuses on numerical studies of quantum spin systems. These simple models are known to exhibit a variety of phases, some of which have no classical counterpart. Phase transitions between them are driven by quantum fluctuations and the unconventional nature of some such transitions make them a fascinating avenue of study. Quantum Monte Carlo (QMC) is an indispensable tool in the study of these phases and phase transitions in two and higher dimensions. Nevertheless, we are limited by our inability to simulate models that suffer from the infamous sign problem. While the case of S=1/2 has been studied extensively due to availability of sign problem free models, not much progress has been made for higher S. In a first part of this dissertation, a systematic procedure to write down a large family of ``designer hamiltonians'', i.e. models constructed to be free from the sign problem (``de-signed''), for arbitrary spin-S is given. Three applications of this procedure are also presented. As a first application, a S=1 interaction is constructed on the square lattice realizing a novel ``Haldane Nematic (HN)'' phase that breaks the lattice rotational symmetry while preserving lattice translational symmetry and spin rotational symmetry. By supplementing our model with a two-spin Heisenberg interaction, a study of the transition between the antiferromagnetic and HN phase is presented, which we find to be of first order. In a second application, the antiferromagnetic to four-fold columnar valence bond solid (cVBS) phase transition in a sign free S=1 square lattice model is studied. Our simulations provide unambiguous evidence for a direct conventional first-order quantum phase transition demonstrating a sharp contrast with the S=1/2 case, where this transition is a prototypical example of an unconventional continuous transition. In our third application, all possible sign-free two-site spin-S interactions are constructed and the phases that are realized by these new nearest neighbour models are investigated on the square lattice. In a second part, the superfluid-VBS quantum phase transition is studied in a spin model in presence of easy plane anisotropy, i.e. spins preferentially align in a plane. The model studied is an interpolation of two models: (a) a rotationally symmetric model that appears to host a continuous transition even on the biggest lattices studied (b) the other is an easy plane version of the aforementioned model that clearly shows a first order transition even on relatively small lattices. In our simulations, the nature of the transition was found to be first order in the presence of an easy plane anisotropy, indicating the superfluid-VBS transition in the two-component easy plane model is generically discontinuous. In a third part, we study the SU(N) generalization of the Heisenberg antiferromagnet on the BCC lattice. Our numerical studies of the model show that magnetic order present for N=2 is destroyed for N>15 and valence bond solid order is observed for N>16. The nature of the phase at N=16 and the nature of the phase transition between different phases is investigated.
We consider the easy-plane limit of bipartite SU(N\mathrm{SU}(NSU(N) Heisenberg Hamiltonians, which have a fundamental representation on one sublattice and the conjugate to fundamental on the other sublattice. For N=2N=2N=2 the easy plane limit of the SU(2) Heisenberg model is the well-known quantum XYXYXY model of a lattice superfluid. We introduce a logical method to generalize the quantum XYXYXY model to arbitrary NNN, 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 NNN 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>5N>5N>5. By introducing SU(N\mathrm{SU}(NSU(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=2N=2N=2 and N=5N=5N=5, suggesting that easy-plane deconfined criticality runs away generically to a first-order transition for small NNN.
Spin systems are a fundamental concept in condensed matter and statistical physics. Here, the authors provide the very first computation of the spin-SSS SU(N) antiferromagnet in a representation amenable to negative sign free fermion Monte Carlo simulations. They offer a numerical proof of the seminal works of Read and Sachdev. It highlights the close connection between topology and the melting of the quantum antiferromagnet, with the Berry phase explaining the various states observed in the vicinity of the quantum antiferromagnet.
For spin rotational symmetric models with a positive-definite high-temperature expansion of the partition function, a stochastic sampling of the series expansion upon partial resummation becomes logically equivalent to sampling an uncolored closely packed loop-gas model in one higher dimension. Based on this, we devise quantum Monte Carlo updates that importance-sample loop configurations for general SU(N)SU(N)SU(N) in fundamental and higher-symmetric representations. The algorithmic performance systematically improves with an increase in (continuous) NNN, allowing the efficient simulation of quantum paramagnets. The underlying reason for the increased efficacy is the correspondence of quantum paramagnetic phases such as valence bond solids to short-loop phases on the loop-gas side rather than the particular value of NNN. This also gives a connection between Sandvik's JQJQJQ model class and classical loop-gas models in the deconfined universality class.
Phase diagram of the SU(NNN) antiferromagnet of spin SSS on a square lattice
We investigate the ground state phase diagram of an SU(NNN)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU(NNN) described by a square Young tableau of N/2N/2N/2 rows and 2S2S2S columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any SSS and even values of NNN. In the large-NNN limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large SSS-limit, the semi-classical approximation points to Néel state. On a line set by N=8S+2N=8S + 2N=8S+2 in the SSS versus NNN phase diagram, we observe a variety of phases proximate to the Néel state. At S=1/2S = 1/2S=1/2 and 3/23/23/2 we observe the aforementioned four fold degenerate valence bond solid state. At S=1S=1S=1 a two fold degenerate spin nematic state in which the C4_44​ lattice symmetry is broken down to C2_22​ emerges. Finally at S=2S=2S=2 we observe a unique ground state that pertains to a two-dimensional version of the Affleck-Kennedy-Lieb-Tasaki state. For our specific realization, this symmetry protected topological state is characterized by an SU(18), S=1/2S=1/2S=1/2 boundary state, that has a dimerized ground state. These phases that are proximate to the Néel state are consistent with the notion of monopole condensation of the antiferromagnetic order parameter. In particular one expects spin disordered states with degeneracy set by mod(4,2S)\text{mod}(4,2S)mod(4,2S).