Quantum phase transitions in three-leg spin tubes

We investigate the properties of a three-leg quantum spin tube using several techniques such as the density-matrix renormalization-group (DMRG) method, strong-coupling approaches and the nonlinear sigma model. For integer $S$, the model proves to exhibit a particularly rich phase diagram consisting of an ensemble of $2S$ phase transitions. They can be accurately identified by the behavior of a nonlocal string order parameter associated to the breaking of a hidden symmetry in the Hamiltonian. The nature of these transitions is further elucidated within the different approaches. We carry a detailed DMRG analysis in the specific cases $S=1$. The numerical data confirm the existence of two Haldane phases with broken hidden symmetry separated by a trivial singlet state. The study of the gap and of the von Neumann entropy suggest a first-order phase transition but at the close proximity of a tricritical point separating a gapless and a first-order transition line in the phase diagram of the quantum spin tube.

Figure 1

(Color online) The three different parameters defining the coupling of the three-leg spin tube.

Figure 2

(Color online) Top: the collinear configuration which minimizes the energy for $0\le \alpha \le 0.5$. Bottom: the coplanar configuration which minimizes the energy for $0.5\le \alpha$.

Figure 3

(Color online) Spectrum of a single triangle for $S=2$ as a function of $\alpha$ as given by the two sequences (4) and (5). The higher energy levels are not represented.

Figure 4

(Color online) Numerical phase diagram for the spin-1 tube obtained with DMRG (see Sec. 6B). Both phases can be distinguished according to the string order parameters ${\mathcal{O}}^z$ [see Eq. (9) or Eq. (16) and Sec. 6B1]. The label $S_b$ for the different phases stands for the value of the spin of the boundary state for OBC. The transition between the $S_b=0$ and $S_b=1/2$ regime is analyzed in more detail in this paper and turns out to be first order. Phase boundaries obtained from an effective model (see Sec. 6A) are also shown with dashed red lines and agree quite well with numerical results.

Figure 5

(Color online) Sketch of the phase diagram for the $S=2$ spin tube. Again, the label $S_b$ for the different phases stands for the value of the spin of the boundary state. The $S_b=1/2$ phases (light red) must be separated from the others with real phase transitions (full lines). On the other hand, there is no reason to expect a real phase transition between the regions $S_b=0$ and $S_b=1$ so we have plotted these boundaries with dashed-dotted lines.

Figure 6

(Color online) The low-energy modes deduced from the linear spin-wave theory are connected to the three rotations of the initial triad around the axes $\widehat{x},\widehat{y},\widehat{z}$. When propagating to the tube, they underpin slow twists of the original structure which asymptotically cost no energy at long wavelengths.

Figure 7

(Color online) Top: renormalization flow at $\alpha =0.55$ for (a) the three velocities $c_1$, $c_2$ and $c_3$, (b) the anisotropy parameters ${\alpha }_2$ and ${\alpha }_3$, and (c) the effective coupling constant $g$. Bottom: same plots for $\alpha =0.95$. The curves were obtained by integrating numerically the RG equations of Mouhanna (Ref. 30).

Figure 8

(Color online) Evolution of the spin tube from the two different topological sectors of $SO\left(3\right)$. Each triad is represented by two orthonormal vectors ${\mathbf{e}}_1\left(x,t\right)$ and ${\mathbf{e}}_2\left(x,t\right)$ that one can connect with the classical configuration of spins on each triangle $\left({\mathbf{S}}_{i,1},{\mathbf{S}}_{i,2},{\mathbf{S}}_{i,3}\right)$ (Ref. 16). In the continuum, ${\mathbf{e}}_1$ and ${\mathbf{e}}_2$ also stand for the first two vector columns of the rotation matrix $\mathcal{R}\left(x,t\right)$. Starting from a Néel configuration, the system tunnels to a nontrivial configuration via an instanton event $(+)$. The system returns to the trivial configuration via an anti-instanton $(-)$.

Figure 9

(Color online) Two space-time configurations differing by the shift of one instanton. The two crosses represent the positions of the singularities. A straight line represents a trivial path in $SO\left(3\right)$ while a loop is a nontrivial path. The Berry phase associated with each loop is $\pm 2\pi S$ depending on the sublattice.

Figure 10

(Color online) String order parameter along $z$ of the spin-1 tube as a function of the anisotropy parameter for $J_{\parallel }=0.1J_{\perp }$ and various lengths $L$. The extrapolated spin gap (see Fig. 14 below) is also shown, as well as the nonlocal order parameter of the spin-1 Haldane chain.

Figure 11

(Color online) String order parameter along $z$ of the spin-1 tube as a function of the anisotropy parameter for $J_{\parallel }=J_{\perp }$ and various lengths $L$. A linear extrapolation with the largest sizes is also plotted and is finite for all $\alpha$.

Figure 12

(Color online) String order parameter along $z$ of the spin-1 tube as a function of $J_{\parallel }/J_{\perp }$ for a fixed anisotropy parameter $\alpha =0.75$ and various sizes.

Figure 13

(Color online) Various string order parameters of the spin-1 ladder $\left(\alpha =0\right)$ as a function of $J_{\perp }/J_{\parallel }$ for various sizes. Usual (respectively, product) string order along $z$ is shown with open (respectively filled) symbols. The dashed line indicates the known value for the spin-1 Haldane chain (Ref. 44).

Figure 14

(Color online) Spin gap of the spin-1 tube as a function of the anisotropy parameter. $J_{\parallel }=0.1J_{\perp }$ and both OBC/PBC are considered.

Figure 15

(Color online) Spin gap of the spin-1 tube as a function of the anisotropy parameter (open symbols) or of the effective two-leg spin-1/2 ladder (filled symbols). $J_{\parallel }=0.1J_{\perp }$ and open boundary conditions are used. The infinite-size extrapolation was performed with the two largest ladder clusters.

Figure 16

(Color online) Plot of the von Neumann entropy as a function of $d\left(\ell \right)$ for different values of the anisotropy parameter and fixed $J_{\parallel }=0.1J_{\perp }$.

Figure 17

(Color online) Various spin gaps ${\Delta }_{ab}$ for the spin-2 tube as a function of $\alpha$ for $J_{\parallel }/J_{\perp }=0.1$. Data are extrapolated to the thermodynamic limit from simulations with systems of length $L=8$, 16, and 32.

Figure 18

(Color online) String order for the spin-2 tube as a function of $\alpha$ for various lengths $L$ with (a) $J_{\parallel }/J_{\perp }=0.1$ or (b) $J_{\parallel }/J_{\perp }=1$.

Figure 19

(Color online) Numerical phase diagram for the spin-2 tube obtained from the spin gap and the string order parameter.