Two spin liquid phases in the spatially anisotropic triangular Heisenberg model

Seiji Yunoki and Sandro Sorella

Phys. Rev. B 74, 014408 – Published 11 July 2006

Abstract

The quantum spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on a two dimensional triangular lattice geometry with spatial anisotropy is relevant to describe materials such as ${Cs}_{2} Cu {Cl}_{4}$ and organic compounds such as $κ − {(ET)}_{2} {Cu}_{2} {(C N)}_{3}$ . The strength of the spatial anisotropy can increase quantum fluctuations and can destabilize the magnetically ordered state leading to nonconventional spin liquid phases. In order to understand these intriguing phenomena, quantum Monte Carlo methods are used to study this model system as a function of the anisotropic strength, represented by the ratio $J^{'} ∕ J$ between the intrachain nearest neighbor coupling $J$ and the interchain one $J^{'}$ . We have found evidence of two spin liquid regions. The first one is stable for small values of the coupling $J^{'} ∕ J ≲ 0.65$ , and appears gapless and fractionalized, whereas the second one is a more conventional spin liquid with a small spin gap and is energetically favored in the region $0.65 ≲ J^{'} ∕ J ≲ 0.8$ . We have also shown that in both spin liquid phases there is no evidence of broken translation symmetry with dimer or spin-Peirls order or any broken spatial reflection symmetry of the lattice. The various phases are in good agreement with the experimental findings, thus supporting the existence of spin liquid phases in two dimensional quantum spin- $1 ∕ 2$ systems.

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Received 7 February 2006

DOI:https://doi.org/10.1103/PhysRevB.74.014408

Authors & Affiliations

Seiji Yunoki and Sandro Sorella

Istituto Nazionale di Fisica della Materia (INFM)-Democritos, National Simulation Centre, and Scuola Internazionale Superiore di Studi Avanzati (SISSA), I-34014 Trieste, Italy

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Vol. 74, Iss. 1 — 1 July 2006

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Figure 1
(Color online) Antiferromagnetic Heisenberg models on the triangular lattice studied. A spin- $1 ∕ 2$ is located at each dot. ${\vec{τ}}_{1} = (1, 0)$ and ${\vec{τ}}_{2} = (\frac{1}{2}, \frac{\sqrt{3}}{2})$ denote the primitive translational vectors. A lattice constant is set to be 1.Reuse & Permissions
Figure 2
(Color online) Monte Carlo evolution of (a) variational parameters ( $Δ_{1}$ and $Δ_{3}$ ) and (b) energy as a function of stochastic reconfiguration (SR) iterations for the 1D spin- $1 ∕ 2$ antiferromagnetic Heisenberg model ${\hat{H}}_{1 D}$ on a $L = 22$ ring. Here the SR method with $δ t = 0.2 ∕ J$ [Eq. (23)] is used. In (a), $Δ_{1}$ and $Δ_{3}$ are the two variational parameters of the wave function (for details, see Sec. 3B), which are both initialized to the value $Δ_{1} = Δ_{3} = 1$ . In (b), at each iteration, the energy is computed by a small VMC simulation for the wave function with the fixed variational parameters given at the corresponding iteration in (a).Reuse & Permissions
Figure 3
(Color online) (a) Imaginary time $τ$ evolution of $S (k, τ)$ at $k = π$ for the 1D spin- $1 ∕ 2$ antiferromagnetic Heisenberg model ${\hat{H}}_{1 D}$ on a $L = 22$ ring. (b) Excitation energy $E (k) = E_{k}^{S = 1} - E_{0}^{eff}$ extracted from $S (k, τ)$ with large $τ$ shown in (a) (squares). The error bars are smaller than the size of the symbols. For comparison, the exact values are also presented by solid circles.Reuse & Permissions
Figure 4
(Color online) Lowest spin- $1 ∕ 2$ excitation energy $E (k)$ and $E (k^{(3)})$ (lower panel) and the variance $σ^{2} (∣ k ⟩)$ and $σ^{2} (∣ k^{(3)} ⟩)$ (upper panel) of the projected BCS states with a single BCS mode $∣ k ⟩$ and three BCS modes $∣ k^{(3)} ⟩$ (see text) as a function of momentum $k$ . The model studied is the 1D spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on a ring with $L = 31$ . For comparison, in the lower panel, the exact lowest excitation energy is plotted by crosses, and the BCS spectrum $E_{k}$ denoted by dashed line is scaled by a factor to match the exact bandwidth. The remaining lines are guides to the eye.Reuse & Permissions
Figure 5
(Color online) Loci of $k$ points where $ξ_{k} = 0$ (dashed lines) and $Δ_{k} = 0$ (solid lines) determined by using the optimized variational parameters for $J^{'} ∕ J = 0.33$ . The boundary of the first Brillouin zone (BZ) for the triangular lattice is also denoted by long dashed lines.Reuse & Permissions
Figure 6
(Color online) The same as in Fig. 5 but for $J^{'} ∕ J = 0.5$ .Reuse & Permissions
Figure 7
(Color online) Spin-spin correlation functions $C (\vec{r})$ with $\vec{r} = l_{1} {\vec{τ}}_{1} + l_{2} {\vec{τ}}_{2}$ for $J^{'} ∕ J = 0.33$ [(a) and (b)] and $J^{'} ∕ J = 0.5$ [(c) and (d)] calculated using both variational Monte Carlo (VMC) and effective Hamiltonian (FNE) methods. The cluster sizes used are $30 \times 30$ [(a) and (b)] and $18 \times 18$ [(c) and (d)]. $C_{1} (l)$ and $C_{2} (l)$ indicate $C (l {\vec{τ}}_{1})$ and $C (l {\vec{τ}}_{2})$ , respectively.Reuse & Permissions
Figure 8
(Color online) System size dependence of (a) spin-spin correlation functions $C (\vec{l})$ at the largest distance in the ${\vec{τ}}_{1}$ direction $(P_{s})$ and (b) the dimer order parameter squared $P_{d}^{2}$ (see the text) for clusters of $L = l \times l$ . The coupling studied is $J^{'} ∕ J = 0.33$ . The variational Monte Carlo and effective Hamiltonian (FNE) results are denoted by squares and crosses, respectively. The straight dashed line in (a) is a guide to the eye.Reuse & Permissions
Figure 9
(Color online) Dimer-dimer correlation functions $D (\vec{r})$ with $\vec{r} = l {\vec{τ}}_{1}$ for $J^{'} ∕ J = 0.33$ calculated using both variational Monte Carlo (squares) and effective Hamiltonian (crosses) methods.Reuse & Permissions
Figure 10
(Color online) Lowest triplet excitation energy as a function of momentum for $J^{'} ∕ J = 0.33$ and different cluster sizes $L$ (indicated in the figures), calculated using the method introduced in Sec. 2D. For comparison, the experimental values (open circles) measured by inelastic neutron scattering on ${Cs}_{2} Cu {Cl}_{4}$ (Ref. 11) are also plotted. Here $G_{2} = 4 π ∕ \sqrt{3}$ , and the experimentally estimated value of $J = 0.37 meV$ is used (Ref. 11).Reuse & Permissions
Figure 11
(Color online) Nearest neighbor gap functions $Δ ({\vec{τ}}_{1}) = Δ ({\vec{τ}}_{2}) = Δ ({\vec{τ}}_{2} - {\vec{τ}}_{1}) = Δ$ [Eq. (55)], which are consistent with the sign convention for the short range RVB state explained in Appendix . The dashed (solid) bonds represent a positive (negative) sign [Eqs. (D8, D9)]. Notice that the unit cell is $(2 \times 1)$ . The origin of the lattice (0,0) [Eq. (55)] is denoted by O.Reuse & Permissions
Figure 12
(Color online) Variational energy for the isotropic triangular lattice with $J^{'} ∕ J = 1.0$ calculated using the short-range RVB state $∣ Ψ_{RVB} ⟩$ (circles), the short-range RVB state with $μ = 0$ (squares), the classical Néel state $∣ Ψ_{Néel} ⟩$ (diamonds) (Ref. 4), and the present projected BCS state $J_{s} ∣ p − BCS ⟩$ with spin Jastrow factor $J_{s}$ (triangles).Reuse & Permissions
Figure 13
(Color online) System size dependence of the nearest neighbor spin-spin correlation functions $C (\vec{r})$ [Eq. (49)] for different cluster sizes $L = l \times l$ up to $30 \times 30$ sites and for the three different directions of the triangular lattice ( $\vec{r} = {\vec{τ}}_{1}$ , ${\vec{τ}}_{2}$ , and ${\vec{τ}}_{2} - {\vec{τ}}_{1}$ ). Here $∣ ψ ⟩ = ∣ p − BCS ⟩$ with $Δ ({\vec{τ}}_{1}) = Δ ({\vec{τ}}_{2}) = Δ ({\vec{τ}}_{2} - {\vec{τ}}_{1}) = 1$ [see in Fig. 11 and Eq. (55)], $t_{i, j} = 0$ , and $μ = - 1$ . This state is a good but not optimal variational wave function for the spatially isotropic model $(J = J^{'})$ , and it is used only to show that wave functions of this type recover all spatial symmetries of the lattice in the thermodynamic limit.Reuse & Permissions
Figure 14
(Color online) System size dependence of spin structure factor $S (\vec{q})$ at $\vec{q} = Q^{*} = (4 π ∕ 3, 0)$ for the optimized variational state $J_{s} ∣ p − BCS ⟩$ (see the text) for the spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on the isotropic triangular lattice of size $L$ . Note that $Q^{*}$ corresponds to the momentum at which $S (Q^{*}) ∕ L$ is finite for $L \to \infty$ when the state is classical Néel ordered with relative angle of 120° between the nearest neighbor spins on the different sublattices.Reuse & Permissions
Figure 15
(Color online) Spin-spin correlation functions $C (l {\vec{τ}}_{1})$ in the ${\vec{τ}}_{1}$ direction for the spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on the isotropic triangular lattice with $L = 18 \times 18$ calculated using both variational Monte Carlo (VMC) and effective Hamiltonian (FNE) methods. The variational wave function considered here is the projected BCS state $J_{s} ∣ p − BCS ⟩$ with spin Jastrow factor $J_{s}$ (see the text), and this wave function is used as the guiding function $∣ ψ_{G} ⟩$ for the FNE calculation. The spin-spin correlation functions in the ${\vec{τ}}_{2}$ and ${\vec{τ}}_{2} - {\vec{τ}}_{1}$ directions are essentially the same as the ones presented here.Reuse & Permissions
Figure 16
(Color online) Forward walking time evolution $τ$ (unit $J^{- 1}$ ) [Eq. (57)] of the spin structure factor $S (\vec{q})$ at $\vec{q} = Q^{*} = (4 π ∕ 3, 0)$ for $J^{'} ∕ J = 1.0$ . Here our best variational wave function $∣ p − BCS ⟩$ with spin Jastrow factor $J_{s}$ (see the text) is used as the guiding function $∣ ψ_{G} ⟩$ . For clarity, for each cluster, the FN results $S {(Q^{*})}_{FN}$ are referenced to the mixed-average estimate $S {(Q^{*})}_{mix}$ , which is set to zero.Reuse & Permissions
Figure 17
(Color online) System size dependence of the spin structure factor $S (\vec{q})$ at $\vec{q} = Q^{*} = (4 π ∕ 3, 0)$ for the spin- $1 ∕ 2$ isotropic triangular antiferromagnet with $J^{'} ∕ J = 1.0$ calculated using FN and FNE methods. Here our best variational wave function $J_{s} ∣ p − BCS ⟩$ with spin Jastrow factor $J_{s}$ (see the text) is used as the guiding function $∣ ψ_{G} ⟩$ . In order to reduce finite size effects, for each cluster, the variational result $S {(Q^{*})}_{VMC} ∕ L$ for the spin liquid state $J_{s} ∣ p − BCS ⟩$ (which is zero for $L \to \infty$ ) is subtracted from the FN and FNE results $S {(Q^{*})}_{FN ∕ FNE} ∕ L$ .Reuse & Permissions
Figure 18
(Color online) System size dependence of the spin structure factor $S (\vec{q})$ at $\vec{q} = Q^{*} = (q^{*}, 0)$ for $J^{'} ∕ J = 0.7$ calculated using the FNE method. The incommensurate momenta $q^{*}$ as well as the corresponding value for the classical limit (Ref. 5) are plotted in the upper panel. For clarity, the variational estimates of the structure factor $S {(Q^{*})}_{VMC}$ are subtracted from the FNE results $S {(Q^{*})}_{FNE}$ . The optimized projected BCS state $∣ p − BCS ⟩$ without spin Jastrow factor $J_{s}$ (see the text) is used for the variational calculations. The same wave function $∣ p − BCS ⟩$ is used as the guiding function $∣ ψ_{G} ⟩$ for the FNE calculations. The system sizes used are indicated in the figure.Reuse & Permissions
Figure 19
(Color online) Monte Carlo evolution of the variational parameters defined in Eq. (58) as a function of SR iterations for the spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on the anisotropic triangular lattice with $J^{'} ∕ J = 0.7$ and $L = 18 \times 18$ . Here the SR minimization method explained in Sec. 2B is used with $δ t = 0.25 ∕ J$ [Eq. (23)]. Inset: the corresponding energy evolution as a function of SR iterations. At each SR iteration, the energy is computed for the wave function with the fixed variational parameters given at the corresponding iteration in the main figure.Reuse & Permissions
Figure 20
(Color online) Dimer-dimer correlation functions $D (\vec{r})$ with $\vec{r} = l {\vec{τ}}_{1}$ for the spin- $1 ∕ 2$ triangular antiferromagnet with $J^{'} ∕ J = 0.7$ and $L = 30 \times 30$ calculated using both variational Monte Carlo (VMC) and FNE methods. The variational ansatz state considered here is the projected BCS state $∣ p − BCS ⟩$ (see text), and this state is used as the guiding function $∣ ψ_{G} ⟩$ for the FNE calculations.Reuse & Permissions
Figure 21
(Color online) Energy per site for the spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on the spatially anisotropic triangular lattice calculated using the variational Monte Carlo (VMC) and the effective Hamiltonian (FNE) methods. Here two types of spin liquid wave functions are considered, the one described in Sec. 3C (denoted by “1D type”) and the other described in Secs. 3D and 3E (denoted by “2D type”). These wave functions are used as the guiding functions for the FNE calculations.Reuse & Permissions
Figure 22
(Color online) Lowest triplet spin excitations as a function of momentum for the spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on the triangular lattice with $J^{'} ∕ J = 0.8$ and $L = 18 \times 18$ , calculated using the method introduced in Sec. 2D. Here $G_{2} = 4 π ∕ \sqrt{3} .$ Reuse & Permissions
Figure 23
(Color online) A schematic phase diagram of the spin- $1 ∕ 2$ antiferromagnetic Heisenberg model on the triangular lattice. $J$ $(J^{'})$ is the nearest neighbor antiferromagnetic coupling between the spins in the chain (in different chains) (see Fig. 1). SL and AFMLO stand for spin liquid and antiferromagnetic long range order (incommensurate spin order for $J^{'} ∕ J$ away from the isotropic case), respectively. Materials possibly described by this class of models are also indicated.Reuse & Permissions
Figure 24
(Color online) (a) Region of $(m_{1}, m_{2})$ (denoted by solid circles) considered for the sum over ${\vec{t}}_{m} = m_{1} {\vec{τ}}_{1} + m_{2} {\vec{τ}}_{2}$ . Each circle corresponds to a pair of integers $(m_{1}, m_{2})$ . (b) The first Brillouin zone (hexagon drown by dashed lines) for the triangular lattice shown in Fig. 1. The corresponding reciprocal lattice vectors are ${\vec{g}}_{1} = 2 π (1, - \frac{1}{\sqrt{3}})$ and ${\vec{g}}_{2} = 2 π (0, \frac{2}{\sqrt{3}})$ . The reduced Brillouin zone (tilted rectangle) is also given by solid lines. Several symmetric points are $Γ$ : (0,0), P: $(\frac{4}{3} π, 0)$ , Q: $(π, \frac{π}{\sqrt{3}})$ , $P^{'}$ : $(\frac{2}{3} π, \frac{2}{\sqrt{3}} π)$ , and $Q^{'}$ : $(0, \frac{2}{\sqrt{3}} π)$ .Reuse & Permissions

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