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arXiv:2309.12262 · Physical review. B./Physical review. B

Competition between Neel, Haldane nematic, plaquette valence bond solid, and ( π , π ) valence bond solid phases in SU ( N ) analogs of S = 1 square-lattice antiferromagnets

Souvik Kundu, Nisheeta Desai, Kedar Damle

Published May 13, 2024• 2 citations
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Abstract

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 $\mathrm{SU}(N)$ analogs of spin $S=1$ quantum antiferromagnets on the square lattice. The $\mathrm{SU}(N)$ 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 $(0,\ensuremath{\pi})$.

DOI
10.1103/physrevb.109.195146

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Phase diagram of the SU(NNN) antiferromagnet of spin SSS on a square lattice
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We study the N\'eel to fourfold columnar valence bond solid (cVBS) quantum phase transition in a sign-free S=1S=1S=1 square-lattice model. This is the same kind of transition that for S=1/2S=1/2S=1/2 has been argued to realize the prototypical deconfined critical point. Extensive numerical simulations of the square-lattice S=1/2S=1/2S=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=1/2S=1/2S=1/2 case, in our quantum Monte Carlo simulations for the S=1S=1S=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=1/2S=1/2S=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.
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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.
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