We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)]. Whereas the AKLT states are created by application of bond singlet operators for SU(2) spins, the simplex solid construction is based on N-simplex singlet operators for SU(N) spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a CP^{N-1} coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of SU(4) edge- and SU(8) face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the SU(3) models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to `order by disorder'.
We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)]. Whereas the AKLT states are created by application of bond singlet operators for SU(2) spins, the simplex solid construction is based on N-simplex singlet operators for SU(N) spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a CP^{N-1} coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of SU(4) edge- and SU(8) face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the SU(3) models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to `order by disorder'.
Abstract. We study the structure of quantum ground states of simplex solid models, which are generalizations of the valence bond construction for quantum antiferromagnets originally proposed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) (1987 Phys. Rev. Lett . 59 799). Whereas the AKLT states are created by application of bond singlet operators for spins, the simplex solid construction is based on N -simplex singlet operators for spins. In both cases, a discrete one-parameter family of translationally-invariant models with exactly solvable ground states is defined on any regular lattice, and the equal time ground state correlations are given by the finite temperature correlations of an associated classical model on the same lattice, owing to the product form of the wave functions when expressed in a coherent state representation. We study these classical companion models via a mix of Monte Carlo simulations, mean-field arguments, and low-temperature effective field theories. Our analysis reveals that the ground states of edge- and face-sharing cubic lattice simplex solid models are long range ordered for sufficiently large values of the discrete parameter, whereas the ground states of the models on the kagome (2D) and hyperkagome (3D) lattices are always quantum disordered. The kagome simplex solid exhibits strong local order absent in its three-dimensional hyperkagome counterpart, a contrast that we rationalize with arguments similar to those leading to ‘order by disorder’.
I define a set of wave functions for SU(N) lattice antiferromagnets, analogous to the valence bond solid states of Affleck et al. [Phys. Rev. Lett. 59, 799 (1987); Commun. Math. Phys. 115, 477 (1988)], in which the singlets are extended over N-site simplixes. As with the valence bond solids, the new simplex solid (SS) states are extinguished by certain local projection operators, allowing one to construct Hamiltonians with local interactions which render the SS states exact ground states. Using a coherent state representation, I show that the quantum correlations in each SS state are calculable as the finite temperature correlations of an associated classical model with N-spin interactions on the same lattice. In three and higher dimensions, the SS states can spontaneously break SU(N) and exhibit N-sublattice long-ranged order as a function of a discrete parameter which fixes the local representation of SU(N). I analyze this transition using a classical mean field approach. For N>2, the ordered state is selected via an ``order by disorder'' mechanism. As in the Affleck-Kennedy-Lieb-Tasaki case, the bulk representations fractionalize at an edge, and the ground state entropy is proportional to the volume of the boundary.
The models constructed by Affleck, Kennedy, Lieb, and Tasaki (AKLT) [Phys. Rev. Lett. 59, 799 (1987)] describe a family of quantum antiferromagnets on arbitrary lattices, where the local spin S is an integer multiple M of half the lattice coordination number. The equal-time quantum correlations in their ground states may be computed as finite temperature correlations of a classical O(3) model on the same lattice, where the temperature is given by T=1/M. In dimensions d=1 and 2 this mapping implies that all AKLT states are quantum disordered. We consider AKLT states in d=3 where the nature of the AKLT states is now a question of detail depending upon the choice of lattice and spin; for sufficiently large S some form of N\'eel order is almost inevitable. On the unfrustrated cubic lattice, we find that all AKLT states are ordered, while for the unfrustrated diamond lattice the minimal S=2 state is disordered while all other states are ordered. On the frustrated pyrochlore lattice, we find (conservatively) that several states starting with the minimal S=3 state are disordered. The disordered AKLT models we report here are a significant addition to the catalog of magnetic Hamiltonians in d=3 with ground states known to lack order on account of strong quantum fluctuations.
We present a numerical study of the SU(N) Heisenberg model with the fundamental representation at each site for the kagome lattice (for N=3) and the checkerboard lattice (for N=4), which are the line graphs of the honeycomb and square lattices and thus belong to the class of bisimplex lattices. Using infinite projected entangled-pair states and exact diagonalizations, we show that in both cases the ground state is a simplex solid state with a twofold degeneracy, in which the N spins belonging to a simplex (i.e., a complete graph) form a singlet. Theses states can be seen as generalizations of valence bond solid states known to be stabilized in certain SU(2) spin models.
Phase diagram of the SU(N) antiferromagnet of spin S on a square lattice
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 C4 lattice symmetry is broken down to C2 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 study of various types of orders and phases of matter in magnetic systems is extremely popular theme in condensed matter physics. Not only frequently applicable in real world, quantum magnetism provides a fruitful and at the same time frequently feasible framework to search for new phases of matter. In this work we study quantum and associated classical phase transitions of spin systems by combinations of numerical and analytical methods.In Chapter 1 we study quantum paramagnets that are a generalization of antiferromagnetic AKLT [2] construction for the spin group SU(N). We map the so called simplex solid state ansatz wavefunction into a partition function of the classical system on the same lattice using coherent state representation. Then we study phase transitions of that classical models via classical Monte Carlo methods to determine properties of the original quantum simplex solid state [3].In Chapter 2 we are interested in the O(2) model in (2+1)-dimensional system with the dilution, i.e. when some sites are removed from the system. This relativistic O(2) model originates from the description of a transition between superfluid and Mott insulator phases. We study critical properties of the transition using a recently invented [4] worm algorithm.In Chapter 3 we are studying large-N expansion for the symplectic spin group Sp(N). When the original SU(2) spin model proves to be too difficult to be solved, for example at low temperatures, one of the alternative ways is to consider a system based on a more general spin group, for example SU(N) or Sp(N). Results often remain relevant for the original SU(2) group. We are using the mean-field theory approach to find phase transitions associated with the cubic Sp(N) system at large N.Another fundamental question of interest is how different systems behave when they are periodically driven by some external field. Such systems are called Floquet systems, and we will study them in Chapter 4. We will consider interacting and non-interacting systems, that might have a non-conventional type of ordering, topological order.