Numerical Studies of Various Néel-VBS Transitions in SU(<em>N</em>) Anti-Ferromagnets

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) 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 Néel 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. 1. Overview The study of ground states of lattice models of quantum spins has become a major field in condensed matter physics [1]. Despite their simplicity these models can be extremely hard to study theoretically, due in part to the rich variety of ground states that they can host. Recent years have seen a dramatic increase in the use of numerical methods to study the ground states of quantum spin models. Of particular interest are “unbiased” numerical methods, which solve for physical properties of model systems with numerical errors that can be controlled and estimated in a reliable fashion. Quantum Monte Carlo occupies a special place among the unbiased methods because when applicable, it is the only technique that is able to access the large systems sizes required for reliable extrapolation to the thermodynamic limit, especially in the proximity of critical phenomena [2]. The simplest ground state that can arise in a quantum spin system is a magnetic ground state, where the expectation value of the spin on a site is finite, causing the spin to effectively “point” in a certain direction, thus spontaneously breaking the global symmetry associated with the rotation of spins. Such magnetic states are well known to arise in the low-energy space of classical spin models and their appearance in quantum models can be understood from semiclassical arguments. The focus of recent numerical studies of spin models has been in large part on the nature of the non-magnetic phases that arise at T = 0 due to quantum fluctuations and the quantum phase transitions separating magnetic and non-magnetic phases. The non-magnetic phases may either break another symmetry, most often a lattice translational symmetry in which case they are called “solids” or be completely symmetric in both the lattice and spin symmetries XXVI IUPAP Conference on Computational Physics (CCP2014) IOP Publishing Journal of Physics: Conference Series 640 (2015) 012041 doi:10.1088/1742-6596/640/1/012041 Content from this work may be used under the terms of the Creative Commons Attribution 3.0 licence. Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. Published under licence by IOP Publishing Ltd 1 in which case they are called “liquids.” The central questions of interest to numerical studies are: What is the precise characterization of the non-magnetic phases, both of the “solid” and “liquid” type that are found in simple microscopic models? How can the phase transitions between the magnetic and non-magnetic phases observed in numerical simulations be understood in terms of long wavelength quantum field theories? While these general questions have received copious attention from a multitude of complementary approaches in different contexts over the last three decades [2, 3, 4, 5], in this review we will outline how new “designer” Hamiltonians with SU(N) symmetry have contributed to answers concerning phase transitions between SU(N) symmetry breaking magnetic phases and lattice translational symmetry breaking valence-bond solid states, for which an exotic direct continuous transition – “deconfined criticality” – has been proposed [6, 7, 8]. For a fuller appreciation of this short review, a familiarity with the deconfined theory is recommended. In the interest of space this is not provided here, the interested reader is encouraged to refer to the original literature. In Section 2 we briefly review the discovery of a large family of “designer” SU(N) models that do not suffer from the sign problem. In Sec 3 we describe the phase diagrams obtained for the Hamiltonians that have been studied so far and the nature of the critical points that arise between different phases. Finally, in Section 4 we conclude with an outlook and directions for future work. 2. SU(N) Hamiltonians, loop models and the sign problem In order to use unbiased quantum Monte-Carlo techniques efficiently, one needs to identify models that do not suffer from the sign problem. It is well known that Hamiltonians that satisfy Marshall’s sign condition are also sign-problem free. So a basic questions of central importance to the numerical simulations of quantum spin models is: What is the family of spin models that satisfies Marshall’s sign criteria? In order to make this question concrete, we specialize our considerations to bipartite models with a specific representation of SU(N) symmetry, originally introduced to condensed matter physics by Affleck [9]. This realization of SU(N) symmetry requires spins on one sub-lattice to transform in the fundamental representation of SU(N) and spins on the other sub-lattice to transform in the conjugate to fundamental representation. We note here for N = 2, since the fundamental and conjugate to fundamental representations are identical, this realization of SU(2) symmetry gives rise to the familiar Heisenberg-like models. The standard quantum-classical mapping allows us to rewrite the quantum statistical mechanics of d-dimensional Hamiltonians as a classical statistical mechanics problem in d + 1dimensions, where the extra dimension is of extent β = 1/T . The stochastic series expansion (SSE) is an elegant and well-documented method to execute this step [10]. When carried out and as discussed in detail [11], the Affleck SU(N) spin models on bipartite lattices can be mapped to oriented tightly packed loop models with N colors in one higher dimension. In order to carry out Monte Carlo sampling we require these configurations to have positive weights. In the language of loop models it is straightforward to systematically write down all possible interactions that keep the weights of the loop configurations positive. When the quantum-classical mapping is run backwards, the interactions in the classical loop model correspond to terms in quantum Hamiltonians that are Marshall positive. This picture allows one to systematically write down a large class of SU(N) spin Hamiltonians that are Marshall positive. We note parenthetically that the Marshall positivity of these “designer” Hamiltonians is not obvious when viewed directly in the spin language, and that the “designer” models include all previously known Marshall positive spin models as particular cases. As a concrete example of the results, let us discuss the familiar N = 2 case. Here the spins ~ S on either sub-lattice can be written in terms of Pauli matrices in the usual way. Previously XXVI IUPAP Conference on Computational Physics (CCP2014) IOP Publishing Journal of Physics: Conference Series 640 (2015) 012041 doi:10.1088/1742-6596/640/1/012041