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arXiv:1406.4083 · Physical Review B

Exact diagonalization study of the anisotropic triangular lattice Heisenberg model using twisted boundary conditions

Mischa Thesberg, Erik Schwartz Sørensen

Published Sep 8, 2014• 21 citations
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Abstract

The spin-$1/2$ anisotropic triangular model, which is believed to describe the materials ${\mathrm{Cs}}_{2}{\mathrm{CuCl}}_{4}$ and ${\mathrm{Cs}}_{2}{\mathrm{CuBr}}_{4}$, among others, is dominated by incommensurate spiral physics and is thus extremely resistant to numerical analysis on small system sizes. In this paper, we use twisted boundary conditions and exact diagonalization techniques to study the ground state of this model as a function of exchange parameter strength. With these boundary conditions we are able to extract the inter- and intrachain ordering $q$ vectors for the $\frac{{J}^{\ensuremath{'}}}{J}<1$ region finding very close agreement with recent DMRG results on much larger systems. Our results suggest a phase transition between a long-range incommensurate spiral ordered phase, and a more subtle phase with short-range spiral correlations with the $q$ vector describing the incommensurate correlations varying smoothly through the transition. In the latter phase correlations between next-nearest chains exhibit an extremely close competition between predominantly antiferromagnetic and ferromagnetic correlations. Further analysis suggests that the antiferromagnetic next-nearest chain correlations may be slightly stronger than the ferromagnetic ones. This difference is found to be slight but in line with previous renormalization group predictions of a collinear antiferromagnetic ordering in this region.

DOI
10.1103/physrevb.90.115117

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Ground- and excited-state quantum fidelities in combination with generalized quantum fidelity susceptibilites, obtained from exact diagonalizations, are used to explore the phase diagram of the anisotropic next-nearest-neighbour triangular Heisenberg model. Specifically, the J'-J2 plane of this model, which connects the J1-J2 chain and the anisotropic triangular lattice Heisenberg model, is explored using these quantities. Through the use of a quantum fidelity associated with the first excited-state, in addition to the conventional ground-state fidelity, the BKT-type transition and Majumdar-Ghosh point of the J1-J2 chain (J'=0) are found to extend into the J'-J2 plane and connect with points on the J2=0 axis thereby forming bounded regions in the phase diagram. These bounded regions are then explored through the generalized quantum fidelity susceptibilities χρ, χ₁₂₀°, χD and χCAF which are associated with the spin stiffness, 120° spiral order parameter, dimer order parameter and collinear antiferromagnetic order parameter respectively. These quantities are believed to be extremely sensitive to the underlying phase and are thus well suited for finite-size studies. Analysis of the fidelity susceptibilities suggests that the J', J2≪J phase of the anisotropic triangular model is either a collinear antiferromagnet or possibly a gapless disordered phase that is directly connected to the Luttinger phase of the J1-J2 chain. Furthermore, the outer region is dominated by incommensurate spiral physics as well as dimer order.
We study the anisotropic spin-1/2 antiferromagnetic triangular Heisenberg lattice in two dimensions, seen as a set of chains with couplings JJJ (${J}^{\ensuremath{'}}$) along (in-between) chains, respectively. Our focus is on the incommensurate correlation that emerges in this system in a wide parameter range due to the intrinsic frustration of the spins. We study this system with traditional density matrix renormalization group using cylindrical boundary conditions to least constrain possible incommensurate order. Despite that the limit of essentially decoupled chains ${J}^{\ensuremath{'}}/J\ensuremath{\lesssim}0.5$ is not very accessible numerically, it appears that the spin-spin correlations remain incommensurate for any finite $0<{J}^{\ensuremath{'}}<{J}_{{}^{{}_{\mathrm{C}}}}^{\ensuremath{'}}$, where ${J}_{{}^{{}_{\mathrm{C}}}}^{\ensuremath{'}}/J>1$. The incommensurate wave vector qJ{q}_{J}qJ​, however, approaches the commensurate value corresponding to the antiferromagnetic correlation of a single chain very rapidly with decreasing ${J}^{\ensuremath{'}}/J$, roughly as ${q}_{J}\ensuremath{\sim}\ensuremath{\pi}\ensuremath{-}{c}_{1}{({J}^{\ensuremath{'}}/J)}^{n}{e}^{\ensuremath{-}{c}_{2}J/{J}^{\ensuremath{'}}}$.
We study two Heisenberg spin-1/2 chains coupled by a frustrating ``zigzag'' interaction. We are particularly interested in the regime of weak interchain coupling, which is difficult to analyze by either numerical or analytical methods. Previous density matrix renormalization-group studies of the isotropic model with open boundary conditions and sizable interchain coupling have established the presence of incommensurate correlations and of a spectral gap. By using twisted boundary conditions with an arbitrary twist angle, we are able to determine the incommensurabilities both in the isotropic case and in the presence of an exchange anisotropy by means of exact diagonalization of relatively short finite chains of up to 242424 sites. Using twisted boundary conditions results in a very smooth dependence of the incommensurabilities on system size, which makes the extrapolation to infinite systems significantly easier than for open or periodic chains.
We study two Heisenberg spin-1/2 chains coupled by a frustrating ''zigzag'' interaction. We are particularly interested in the regime of weak interchain coupling, which is difficult to analyze by either numerical or analytical methods. Previous density matrix renormalization-group studies of the isotropic model with open boundary conditions and sizable interchain coupling have established the presence of incommensurate correlations and of a spectral gap. By using twisted boundary conditions with an arbitrary twist angle, we are able to determine the incommensurabilities both in the isotropic case and in the presence of an exchange anisotropy by means of exact diagonalization of relatively short finite chains of up to 24 sites. Using twisted boundary conditions results in a very smooth dependence of the incommensurabilities on system size, which makes the extrapolation to infinite systems significantly easier than for open or periodic chains.
The ground state of an S=1/2S=1/2S=1/2 antiferromagnetic Heisenberg model on a spatially anisotropic triangular lattice, which is an effective model of Mott insulators on a triangular layer of organic charge transfer salts or Cs2{}_{2}2​CuCl4{}_{4}4​, is numerically studied. We apply a numerical variational method by using a tensor network with entanglement renormalization, which improves the capability of describing a quantum state. Magnetic ground states are identified for $0.7\ensuremath{\le}{J}_{2}/{J}_{1}\ensuremath{\le}1$ in the thermodynamic limit, where J1{J}_{1}J1​ and J2{J}_{2}J2​ denote the innerchain and interchain coupling constants, respectively. Except for the isotropic case (J1=J2{J}_{1}={J}_{2}J1​=J2​), the magnetic structure is spiral with an incommensurate wave vector that is different from the classical one. The quantum fluctuation weakens the effective coupling between chains, but the magnetic order remains in the thermodynamic limit. In addition, the incommensurate wave number is in good agreement with that of the series expansion method.
We investigate the effect of dimensional crossover in the ground state of the antiferromagnetic spin-1 Heisenberg model on the anisotropic triangular lattice that interpolates between the regime of weakly coupled Haldane chains $({J}^{\ensuremath{'}}\ensuremath{\ll}J)$ and the isotropic triangular lattice $({J}^{\ensuremath{'}}=J)$. We use the density-matrix renormalization group (DMRG) and Schwinger boson theory performed at the Gaussian correction level above the saddle-point solution. Our DMRG results show an abrupt transition between decoupled spin chains and the spirally ordered regime at ${({J}^{\ensuremath{'}}/J)}_{c}\ensuremath{\sim}0.42$, signaled by the sudden closing of the spin gap. Coming from the magnetically ordered side, the computation of the spin stiffness within Schwinger boson theory predicts the instability of the spiral magnetic order toward a magnetically disordered phase with one-dimensional features at ${({J}^{\ensuremath{'}}/J)}_{c}\ensuremath{\sim}0.43$. The agreement of these complementary methods, along with the strong difference found between the intra- and the interchain DMRG short spin-spin correlations for sufficiently large values of the interchain coupling, suggests that the interplay between the quantum fluctuations and the dimensional crossover effects gives rise to the one-dimensionalization phenomenon in this frustrated spin-1 Hamiltonian.