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arXiv:cond-mat/0212589 · Physical review. B, Condensed matter

XY ring-exchange model on the triangular lattice

Leon Balents, Arun Paramekanti

Published Apr 28, 2003• 21 citations
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Abstract

We study ring-exchange models for bosons or $\mathrm{XY}$ spins on a triangular lattice. A four-spin exchange leads to a manifold of ground states with gapless excitations and critical power-law correlations. With a nearest-neighbor exchange, fluctuations select a four-fold ferrimagnetically ordered ground state with a small spin (superfluid) stiffness which breaks the global U(1) and translational symmetry. We explore consequences for phase transitions at finite temperature and in an in-plane magnetic field.

DOI
10.1103/physrevb.67.134427

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We study ring-exchange models for bosons or XY\mathrm{XY}XY spins on a triangular lattice. A four-spin exchange leads to a manifold of ground states with gapless excitations and critical power-law correlations. With a nearest-neighbor exchange, fluctuations select a four-fold ferrimagnetically ordered ground state with a small spin (superfluid) stiffness which breaks the global U(1) and translational symmetry. We explore consequences for phase transitions at finite temperature and in an in-plane magnetic field.
We present the linear spin wave theory calculation of the superfluid phase of a hard-core boson J-K model with nearest neighbour exchange J and four-particle ring-exchange K at half filling on the triangular lattice, as well as the phase diagrams of the system at zero and finite temperatures. A similar analysis has been done on a square lattice (Schaffer et al. Phys. Rev. B, 80, 014503 (2009)). We find similar behaviour to that of a square lattice but with different spin wave values of the thermodynamic quantities. We also find that the pure J model (XY model), which has a well-known uniform superfluid phase with an ordered parameter [Formula: see text] at zero temperature is quickly destroyed by the inclusion of negative-K ring-exchange interactions, favouring a state with a (4π/3, 0) ordering wavevector. We further study the behaviour of the finite-temperature Kosterlitz–Thouless phase transition (TKT) in the uniform superfluid phase, by forcing the universal quantum jump condition on the finite-temperature spin wave superfluid density. We find that for K < 0, the phase boundary monotonically decreases to T = 0 at K/J = −4/3, where a phase transition is expected and TKT decreases rapidly, while for positive K, TKT reaches a maximum at some K ≠ 0. It has been shown on a square lattice using quantum Monte Carlo (QMC) simulations that for small K > 0 away from the XY point, the zero-temperature spin stiffness value of the XY model is decreased (Melko and Sandvik. Ann. Phys. 321, 1651 (2006)). Our result seems to agree with this trend found in QMC simulations for two-dimensional systems.
We examine the superfluid phase of a hard-core boson model with nearest-neighbor exchange JJJ and four-particle ring-exchange KKK at half filling on the square lattice. At zero temperature we find that the uniform superfluid in the pure-JJJ model is quickly destroyed by the inclusion of negative-KKK ring-exchange interactions, favoring a state with a $(\ensuremath{\pi},\ensuremath{\pi})$ ordering wave vector. Minimization of the mean-field energy suggests that another type of superfluid state, characterized by nonzero bond chirality, is formed. We also study the behavior of the finite-TTT Kosterlitz-Thouless phase transition in the uniform superfluid phase, by forcing the Nelson-Kosterlitz universal jump condition on the finite-TTT spin-wave superfluid density. Away from the pure JJJ point, TKT{T}_{KT}TKT​ decreases rapidly for negative KKK, while for positive KKK, TKT{T}_{KT}TKT​ reaches a maximum at some $K\ensuremath{\ne}0$ in agreement with recent quantum Monte Carlo simulations.
Using Lanczos exact diagonalization (ED), we study the spin-1/2 ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model with the four-spin ring-exchange interaction Jr{J}_{r}Jr​ on a triangular lattice. We mainly use the level spectroscopic technique of two 36-site tori to investigate the ground-state phase diagram, and further characterize phases by spin, dimer, and chiral correlation functions. The ground state has rich phases including several magnetic-ordered phases like the zigzag phase and tetrahedral phase, as well as several novel nonmagnetic phases, some of which exhibit valence bond solid behavior in their dimer correlation functions. However, we do not find direct evidence of a quantum spin liquid phase with spinon Fermi surface in this model. Our results can give a better understanding of the ground-state properties of the triangular Heisenberg model with ring-exchange interaction and help to understand the relevant triangular materials.
A spin-1 Heisenberg model on trimerized kagome lattice is studied by doing a low-energy bosonic theory in terms of plaquette triplons defined on its triangular unit cells. The model considered has an intratriangle antiferromagnetic exchange interaction JJJ (set to 1) and two intertriangle couplings ${J}^{\ensuremath{'}}>0$ (nearest neighbor) and ${J}^{\ensuremath{''}}$ (next nearest neighbor; of both signs). The triplon analysis performed on this model investigates the stability of the trimerized singlet ground state (which is exact in the absence of intertriangle couplings) in the ${J}^{\ensuremath{'}}\ensuremath{-}{J}^{\ensuremath{''}}$ plane. It gives a quantum phase diagram that has two gapless antiferromagnetically ordered phases separated by the spin-gapped trimerized singlet phase. The trimerized singlet ground state is found to be stable on ${J}^{\ensuremath{''}}=0$ line (the nearest-neighbor case), and on both sides of it for ${J}^{\ensuremath{''}}\ensuremath{\ne}0$, in an extended region bounded by the critical lines of transition to the gapless antiferromagnetic phases. The gapless phase in the negative ${J}^{\ensuremath{''}}$ region has a coplanar ${120}^{\ensuremath{\circ}}$ antiferromagnetic order with $\sqrt{3}\ifmmode\times\else\texttimes\fi{}\sqrt{3}$ structure. In this phase, all the magnetic moments are of equal length, and the angle between any two of them on a triangle is exactly ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this case has a unit cell consisting of three triangles. The other gapless phase, in the positive ${J}^{\ensuremath{''}}$ region, is found to exhibit a different coplanar antiferromagnetic order with ordering wave vector q=(0,0)\mathbf{q}=(0,0)q=(0,0). Here, two magnetic moments in a triangle are of the same magnitude, but shorter than the third. While the angle between two short moments is ${120}^{\ensuremath{\circ}}\ensuremath{-}2\ensuremath{\delta}$, it is ${120}^{\ensuremath{\circ}}+\ensuremath{\delta}$ between a short and the long one. Only when ${J}^{\ensuremath{''}}={J}^{\ensuremath{'}}$, their magnitudes become equal and the relative angles ${120}^{\ensuremath{\circ}}$. The magnetic lattice in this q=(0,0)\mathbf{q}=(0,0)q=(0,0) phase has the translational symmetry of the kagome lattice with triangular unit cells of reduced (isosceles) symmetry. This reduction in the point-group symmetry is found to show up as a difference in the intensities of certain Bragg peaks, whose ratio ${I}_{(1,0)}/{I}_{(0,1)}=4{sin}^{2}(\frac{\ensuremath{\pi}}{6}+\ensuremath{\delta})$ presents an experimental measure of the deviation $\ensuremath{\delta}$ from the ${120}^{\ensuremath{\circ}}$ order.
In our previous work [Phys. Rev. Lett. 121, 046401 (2018)PRLTAO0031-900710.1103/PhysRevLett.121.046401], we found a quantum spin liquid phase with a spinon Fermi surface in the two dimensional spin-1/2 Heisenberg model with four-spin ring exchange on a triangular lattice. In this work we dope the spinon Fermi surface phase by studying the t-J model with four-spin ring exchange. We perform density matrix renormalization group calculations on four-leg cylinders of a triangular lattice and find that the dominant pair correlation function is that of a pair density wave; i.e., it is oscillatory while decaying with distance with a power law. The doping dependence of the period is studied. This is the first example where a pair density wave is the dominant pairing in a generic strongly interacting system where the pair density wave cannot be explained as a composite order and no special symmetry is required.