We use stochastic series expansion (SSE) quantum Monte Carlo (QMC) methods to study the phases and transitions displayed by a class of sign-free designer Hamiltonians for
SU(N) analogs of spin
S=1 quantum antiferromagnets on the square lattice. The
SU(N) spins are generators of the single-row two-column representation (complex conjugate of single-row two-column representation) on
A (
B) sublattices of the square lattice, and the Hamiltonian is designed to explore the competition between the nearest-neighbor antiferromagnetic exchange couplings
J and four-spin interactions
Q that favor a plaquette-ordered valence bond solid (p-VBS) ground state. We find that this state is indeed established at large
Q/J for all
N>3. For $3<N\ensuremath{\le}9$, the ground state exhibits a direct first-order quantum phase transition from a small-
Q/J N\'eel ordered antiferromagnetic state to this large-
Q/J p-VBS state. The ground state at
Q/J=0 for $N\ensuremath{\ge}10$ has been previously reported to be a valence bond nematic state, dubbed the Haldane nematic in recent literature. For small nonzero
Q/J and $N\ensuremath{\ge}10$, we additionally find an unusual intermediate state in which the four-site plaquette energy has a Bragg peak at wave vector $(\ensuremath{\pi},\ensuremath{\pi})$ with no accompanying Bragg peaks at wave vectors $(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi})$. This $(\ensuremath{\pi},\ensuremath{\pi})$ state also appears to be metastable at
Q/J=0, as evidenced by low-temperature histograms of the Haldane nematic and $(\ensuremath{\pi},\ensuremath{\pi})$ order parameters. Deep in the p-VBS phase of the ground state phase diagram, we find the temperature-driven melting of the p-VBS order is in the Ashkin-Teller universality class. In this regime, we identify an interesting signature of Ashkin-Teller criticality in four-site plaquette singlet correlations at wave vector $(\ensuremath{\pi},\ensuremath{\pi})$; this is in addition to the expected critical fluctuations of the conventional p-VBS order parameter at wave vectors $(\ensuremath{\pi},0)$ and $(0,\ensuremath{\pi})$.
We study the N\'eel to fourfold columnar valence bond solid (cVBS) quantum phase transition in a sign-free
S=1 square-lattice model. This is the same kind of transition that for
S=1/2 has been argued to realize the prototypical deconfined critical point. Extensive numerical simulations of the square-lattice
S=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=1/2 case, in our quantum Monte Carlo simulations for the
S=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=1/2 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 the phase diagram of the most general SU(4)-symmetric two-site Hamiltonian for a system of two fermions per site (i.e., self-conjugate
6 representation) on the square lattice. It is known that this model hosts magnetic phases breaking SU(4) symmetry and quantum disordered dimerlike phases breaking lattice translation symmetry. Motivated by a previous work [O. Gauth\'e, S. Capponi, and D. Poilblanc, Phys. Rev. B 99, 241112(R) (2019)], we investigate the possibility of the existence of SU(4) quantum spin liquid phases in this model, using SU(4)-symmetric projected entangled pair states (PEPS) of small bond dimensions, which can be classified according to point group and charge (C) symmetries. Among several (disconnected) families of SU(4)-symmetric PEPS, breaking or not C-symmetry, we identify critical or topological spin liquids which may be stable in some regions of the phase diagram. These results are confronted to exact diagonalization (ED) and density matrix renormalization group (DMRG) calculations.
SU ( N ) quantum spin systems may be realized in a variety of physical systems including ultracold atoms in optical lattices. The study of such models also leads to insights into possible novel quantum phases and phase transitions of SU (2) spin models. Here we use Gutzwiller projected fermionic variational wavefunctions to explore the phase diagram and correlation functions of SU ( N ) quantum spin models in the self-conjugate representation. In one dimension, the ground state of the SU (4) spin chain with Heisenberg bilinear and biquadratic interactions is studied by examining instabilities of the Gutzwiller projected free fermion ground state to various broken symmetries. The variational phase diagram so obtained agrees well with exact results. The spin–spin and dimer–dimer correlation functions of the Gutzwiller projected free fermion state with N flavours of fermions are in good agreement with exact and 1/ N calculations for the critical points of SU ( N ) spin chains. In two dimensions, the phase diagram of the antiferromagnetic Heisenberg model on the square lattice is obtained by finding instabilities of the Gutzwiller projected π-flux state. In the absence of biquadratic interactions, the model exhibits long-range Néel order for N = 2 and 4, and spin Peierls (columnar dimer) order for N >4. Upon including biquadratic interactions in the SU (4) model (with a sign appropriate to a fermionic Hubbard model), the Néel order diminishes and eventually disappears, giving way to an extended valence bond crystal. In the case of the SU (6) model, the dimerized ground state melts at sufficiently large biquadratic interaction, yielding a projected π-flux spin liquid phase which in turn undergoes a transition into an extended valence bond crystal at even larger biquadratic interaction. The spin correlations of the projected π-flux state at N = 4 are in good agreement with 1/ N calculations. We find that the state shows strongly enhanced dimer correlations, in qualitative agreement with recent theoretical predictions. We also compare our results with a recent quantum Monte Carlo study of the SU (4) Heisenberg model.
We discuss zero‐temperature phase transititions from an antiferromagnetic ground state into a valence‐bond‐solid (VBS) in two different square‐lattice quantum spin models. In one limit these models reduce to the standard S = 1/2 XY and Heisenberg models, respectively, which have long‐range antiferromagnetic order at T = 0. Introducing particular types of four‐spin interactions, amenable to quantum Monte Carlo (QMC) simulation without negative‐sign problems, drives phase transitions into VBS states with broken Z4 symmetry. It has recently been argued that continuous quantum phase transitions of this nature represent a new class of deconfined quantum‐critical points [Senthil et al., Science 303, 1490 (2004)], which are not captured within the standard Ginzburg‐Landau framework of phase transitions. Here we discuss a QMC study aimed at accurately characterizing the transition in the XY model. We find some evidence that the transition is actually weakly first‐order, but other scenarios, such as a continuous transition violating hyperscaling, also cannot be completely ruled out. For the Heisenberg model with four‐spin interactions we have confirmed that a VBS state exists. Preliminary finite‐size scaling results show consistency with a continuous transition into the Néel state, with dynamic exponent z = 1. This model is thus a candidate for deconfined quantum criticality.
We investigate the ground state phase diagram of an SU(
N)-symmetric antiferromagnetic spin model on a square lattice where each site hosts an irreducible representation of SU(
N) described by a square Young tableau of
N/2 rows and
2S columns. We show that negative sign free fermion Monte Carlo simulations can be carried out for this class of quantum magnets at any
S and even values of
N. In the large-
N limit, the saddle point approximation favors a four-fold degenerate valence bond solid phase. In the large
S-limit, the semi-classical approximation points to Néel state. On a line set by
N=8S+2 in the
S versus
N phase diagram, we observe a variety of phases proximate to the Néel state. At
S=1/2 and
3/2 we observe the aforementioned four fold degenerate valence bond solid state. At
S=1 a two fold degenerate spin nematic state in which the C
4 lattice symmetry is broken down to C
2 emerges. Finally at
S=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=1/2 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).