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).
The large-N limit of a nearest-neighbor SU(N) antiferromagnet on a bipartite lattice exhibits in dimensions d\ensuremath{\ge}2 a zero-temperature phase transition between a N\'eel ordered state and a resonating-valence-bond state. Here it is shown in d=1,2 that topological effects produce spin-Peierls or valence-bond-solid order in the non-N\'eel phase with a ground-state degeneracy which varies periodically with ``spin'' for fixed N with periodicity given by the coordination number of the lattice. Thus a non-N\'eel phase of the spin-(1/2 Heisenberg model on a square lattice would be a spin-Peierls state with a fourfold degeneracy due to broken lattice rotational symmetry.
We examine large-N limits of the nearest-neighbor SU(N) quantum antiferromagnets on bipartite lattices in d=1,2. In d=2 the model displays a transition from a N\'eel to a disordered phase. The properties of the disordered phase close to the phase boundary are crucially dependent upon the nature of ``hedgehog''-like instanton tunneling events. We calculate the Berry phases of the instantons and show that, at scales larger than the spin-correlation length, the system can be described by a Coulomb plasma of instantons with complex fugacities. The properties of the Coulomb plasma vary periodically with the ``spin''
nc of the states at each site, with periodicity given by the coordination number Z of the lattice [
nc=2S for SU(2)]. For
nc\ensuremath{\ne}0 (mod Z) the disordered phase has a broken lattice symmetry with spin-Peierls order, while for
nc=0 (mod Z), the ground state is a valence-bond solid state with no broken symmetry. Related topological effects for the d=1 chain lead to spin-Peierls order for odd
nc. These results are for a class of models which have, at sites of the A sublattice, representations of SU(N) described by a Young tableau with a single row, and the conjugate on the B sublattice. Similar results are also obtained for representations with m rows, using U(m) gauge theory.
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 introduce an SU(N) symmetric two-dimensional quantum spin model, the X-Q model, which hosts a ground state transition between Néel antiferromagnetic and spontaneously dimerized states. The Q terms are products of two adjacent singlet projectors on nearest-neighbor sites, as in the often studied J-Q model (where J is the Heisenberg exchange), while the X terms are products of two permutation operators on second-neighbor sites. Quantum Monte Carlo simulations reveal close proximity to a deconfined quantum critical point for N=2, as in the J-Q model. However, for N>2 the transition becomes strongly first order, contrary to conventional expectations that increasing N should weaken discontinuities. We attribute this behavior to the inability of the X term to induce U(1) fluctuations of the dimer pattern, while those from the Q term are suppressed by 1/N. These results provide insights into the interactions that support deconfined criticality.
We consider an exchange model describing two isotropic spin-1/2 Heisenberg antiferromagnets coupled by a quartic term on the square lattice. The model is relevant for systems with orbital degeneracy and strong electron-vibron coupling in the large Hubbard repulsion limit, and is known to show a spin-Peierls-like dimerization in one dimension. In two dimensions we calculate energy gaps, susceptibilities, and correlation functions with a Green's function Monte Carlo method. We find a finite triplet and singlet gap, with no evidence of order of any kind. We conclude that the ground state is, most likely, a spin liquid of resonating valence bonds.