This work focuses on numerical studies of quantum spin systems. These simple models are known to exhibit a variety of phases, some of which have no classical counterpart. Phase transitions between them are driven by quantum fluctuations and the unconventional nature of some such transitions make them a fascinating avenue of study. Quantum Monte Carlo (QMC) is an indispensable tool in the study of these phases and phase transitions in two and higher dimensions. Nevertheless, we are limited by our inability to simulate models that suffer from the infamous sign problem. While the case of S=1/2 has been studied extensively due to availability of sign problem free models, not much progress has been made for higher S. In a first part of this dissertation, a systematic procedure to write down a large family of ``designer hamiltonians'', i.e. models constructed to be free from the sign problem (``de-signed''), for arbitrary spin-S is given. Three applications of this procedure are also presented. As a first application, a S=1 interaction is constructed on the square lattice realizing a novel ``Haldane Nematic (HN)'' phase that breaks the lattice rotational symmetry while preserving lattice translational symmetry and spin rotational symmetry. By supplementing our model with a two-spin Heisenberg interaction, a study of the transition between the antiferromagnetic and HN phase is presented, which we find to be of first order. In a second application, the antiferromagnetic to four-fold columnar valence bond solid (cVBS) phase transition in a sign free S=1 square lattice model is studied. Our simulations provide unambiguous evidence for a direct conventional first-order quantum phase transition demonstrating a sharp contrast with the S=1/2 case, where this transition is a prototypical example of an unconventional continuous transition. In our third application, all possible sign-free two-site spin-S interactions are constructed and the phases that are realized by these new nearest neighbour models are investigated on the square lattice. In a second part, the superfluid-VBS quantum phase transition is studied in a spin model in presence of easy plane anisotropy, i.e. spins preferentially align in a plane. The model studied is an interpolation of two models: (a) a rotationally symmetric model that appears to host a continuous transition even on the biggest lattices studied (b) the other is an easy plane version of the aforementioned model that clearly shows a first order transition even on relatively small lattices. In our simulations, the nature of the transition was found to be first order in the presence of an easy plane anisotropy, indicating the superfluid-VBS transition in the two-component easy plane model is generically discontinuous. In a third part, we study the SU(N) generalization of the Heisenberg antiferromagnet on the BCC lattice. Our numerical studies of the model show that magnetic order present for N=2 is destroyed for N>15 and valence bond solid order is observed for N>16. The nature of the phase at N=16 and the nature of the phase transition between different phases is investigated.
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 $(0,\ensuremath{\pi})$.
Infinite projected entangled pair states simulations of the
S=1 bilinear-biquadratic Heisenberg model on the square lattice reveal an emergent Haldane phase in between the previously predicted antiferromagnetic and three-sublattice ${120}^{\ensuremath{\circ}}$ magnetically ordered phases. This intermediate phase preserves SU(2) spin and translational symmetry but breaks lattice rotational symmetry, and it can be adiabatically connected to the Haldane phase of decoupled
S=1 chains. Our results contradict previous studies which found a direct transition between the two magnetically ordered states.
Frustration in quantum spin systems promote a variety of novel quantum phases. An important example is the frustrated spin-1 model on the square lattice with the nearest-neighbor bilinear (
J1) and biquadratic (
K1) interactions. We provide strong evidence for a nematic spin liquid phase in a range of
K1/J1 near the SU(3)-symmetric point (
J1=K1), based on the linear flavor-wave theory and extensive density matrix renormalization group calculation. This phase displays no spin dipolar or quadrupolar order, preserves translational symmetry but spontaneously breaks
C4 lattice rotational symmetry, and possesses fluctuations peaked at the wave vector $(\ensuremath{\pi},2\ensuremath{\pi}/3)$. The spin excitation gap drops rapidly with system size and appears to be gapless, and the nematic order is attributed to the dominant $(\ensuremath{\pi},2\ensuremath{\pi}/3)$ fluctuations. Our results provide a novel mechanism for electronic nematic order and, more generally, open up a new avenue to explore frustration-induced exotic ground states.
Spontaneous discrete symmetry breaking can be described in the framework of projected entangled pair states (PEPS) by linearly superposing local tensors belonging to two (or more) symmetry classes of tensors. This is illustrated in the case of a frustrated spin-1 Heisenberg model on the square lattice, which hosts a nematic spin liquid spontaneously breaking lattice $\ensuremath{\pi}/2$-rotation symmetry. A superposition of SU(2)-symmetric PEPS tensors belonging to two irreducible representations of the lattice point group is shown to capture accurately the properties of the nematic phase, as shown from a comparison to exact diagonalizations and density matrix renormalization group results.
The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.