We introduce a simple model of SO($N$) spins with two-site interactions which is amenable to quantum Monte-Carlo studies without a sign problem on non-bipartite lattices. We present numerical results for this model on the two-dimensional triangular lattice where we find evidence for a spin nematic at small $N$, a valence-bond solid (VBS) at large $N$ and a quantum spin liquid at intermediate $N$. By the introduction of a sign-free four-site interaction we uncover a rich phase diagram with evidence for both first-order and exotic continuous phase transitions.
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.
We present a study of the scaling behavior of the R{e}nyi entanglement entropy (REE) in SU($N$) spin chain Hamiltonians, in which all the spins transform under the fundamental representation. These SU($N$) spin chains are known to be quantum critical and described by a well known Wess-Zumino-Witten (WZW) non-linear sigma model in the continuum limit. Numerical results from our lattice Hamiltonian are obtained using stochastic series expansion (SSE) quantum Monte Carlo for both closed and open boundary conditions. As expected for this 1D critical system, the REE shows a logarithmic dependence on the subsystem size with a prefector given by the central charge of the SU($N$) WZW model. We study in detail the sub-leading oscillatory terms in the REE under both periodic and open boundaries. Each oscillatory term is associated with a WZW field and decays as a power law with an exponent proportional to the scaling dimension of the corresponding field. We find that the use of periodic boundaries (where oscillations are less prominent) allows for a better estimate of the central charge, while using open boundaries allows for a better estimate of the scaling dimensions. For completeness we also present numerical data on the thermal R{e}nyi entropy which equally allows for extraction of the central charge.
We consider bipartite 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($N$) symmetric spin Hamiltonians that satisfy Marshalls sign condition. This family includes all previously known sign-free 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 Sandviks $Q$-term, there is an independent non-trivial 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 Neel order.
We present an extensive quantum Monte Carlo study of the Neel-valence bond solid (VBS) phase transition on rectangular and honeycomb lattice SU($N$) antiferromagnets in sign problem free models. We find that in contrast to the honeycomb lattice and previously studied square lattice systems, on the rectangular lattice for small $N$ a first order Neel-VBS transition is realized. On increasing $Ngeq 4$, we observe that the transition becomes continuous and with the {em same} universal exponents as found on the honeycomb and square lattices (studied here for $N=5,7,10$), providing strong support for a deconfined quantum critical point. Combining our new results with previous numerical and analytical studies we present a general phase diagram of the stability of $mathbb{CP}^{N-1}$ fixed points with $q$-monopoles.
We present results for the phase diagram of an SU($N$) generalization of the Heisenberg antiferromagnet on a bipartite three-dimensional anisotropic cubic (tetragonal) lattice as a function of $N$ and the lattice anisotropy $gamma$. In the isotropic $gamma=1$ cubic limit, we find a transition from N{e}el to valence bond solid (VBS) between N=9 and N=10. We follow the N{e}el-VBS transition to the limiting cases of $gamma ll 1 $ (weakly coupled layers) and $gamma gg 1$ (weakly coupled chains). Throughout the phase diagram we find a direct first-order transition from N{e}el at small-$N$ to VBS at large-$N$. In the three-dimensional models studied here, we find no evidence for either an intervening spin-liquid photon phase or a continuous transition, even close to the limit $gamma ll 1$ where the isolated layers undergo continuous N{e}el-VBS transitions.
We establish compelling evidence for the existence of new quasi-one-dimensional descendants of the d-wave Bose liquid (DBL), an exotic two-dimensional quantum phase of uncondensed itinerant bosons characterized by surfaces of gapless excitations in momentum space [O. I. Motrunich and M. P. A. Fisher, Phys. Rev. B {bf 75}, 235116 (2007)]. In particular, motivated by a strong-coupling analysis of the gauge theory for the DBL, we study a model of hard-core bosons moving on the $N$-leg square ladder with frustrating four-site ring exchange. Here, we focus on four- and three-leg systems where we have identified two novel phases: a compressible gapless Bose metal on the four-leg ladder and an incompressible gapless Mott insulator on the three-leg ladder. The former is conducting along the ladder and has five gapless modes, one more than the number of legs. This represents a significant step forward in establishing the potential stability of the DBL in two dimensions. The latter, on the other hand, is a fundamentally quasi-one-dimensional phase that is insulating along the ladder but has two gapless modes and incommensurate power law transverse density-density correlations. In both cases, we can understand the nature of the phase using slave-particle-inspired variational wave functions consisting of a product of two distinct Slater determinants, the properties of which compare impressively well to a density matrix renormalization group solution of the model Hamiltonian. Stability arguments are made in favor of both quantum phases by accessing the universal low-energy physics with a bosonization analysis of the appropriate quasi-1D gauge theory. We will briefly discuss the potential relevance of these findings to high-temperature superconductors, cold atomic gases, and frustrated quantum magnets.
We present evidence for an exotic gapless insulating phase of hard-core bosons on multi-leg ladders with a density commensurate with the number of legs. In particular, we study in detail a model of bosons moving with direct hopping and frustrating ring exchange on a 3-leg ladder at $ u=1/3$ filling. For sufficiently large ring exchange, the system is insulating along the ladder but has two gapless modes and power law transverse density correlations at incommensurate wave vectors. We propose a determinantal wave function for this phase and find excellent comparison between variational Monte Carlo and density matrix renormalization group calculations on the model Hamiltonian, thus providing strong evidence for the existence of this exotic phase. Finally, we discuss extensions of our results to other $N$-leg systems and to $N$-layer two-dimensional structures.
High temperature superconductivity emerges in the cuprate compounds upon changing the electron density of an insulator in which the electron spins are antiferromagnetically ordered. A key characteristic of the superconductor is that electrons can be extracted from them at zero energy only if their momenta take one of four specific values (the `nodal points). A central enigma has been the evolution of the zero energy electrons in the metallic state between the antiferromagnet and the superconductor, and recent experiments yield apparently contradictory results. The oscillation of the resistance in this metal as a function of magnetic field indicate that the zero energy electrons carry momenta which lie on elliptical `Fermi pockets, while ejection of electrons by high intensity light indicates that the zero energy electrons have momenta only along arc-like regions. We present a theory of new states of matter, which we call `algebraic charge liquids, which arise naturally between the antiferromagnet and the superconductor, and reconcile these observations. Our theory also explains a puzzling dependence of the density of superconducting electrons on the total electron density, and makes a number of unique predictions for future experiments.
