Phase diagram of the anisotropic spin-2 XXZ model: Infinite-system density matrix renormalization group study

We study the ground-state phase diagram of the quantum spin-2 XXZ chain in the presence of on-site anisotropy using a matrix-product state based infinite-system density matrix renormalization group (iDMRG) algorithm. One of the interests in this system is in connecting the highly quantum-mechanical spin-1 phase diagram with the classical $S=\infty$ phase diagram. Several of the recent advances within DMRG make it possible to perform a detailed analysis of the whole phase diagram. We consider different types of on-site anisotropies, which allows us to establish the validity of the following statements: (1) the spin-2 model can be tuned into a phase, which is equivalent to the “topologically nontrivial” spin-1 Haldane phase, and (2) the spin-2 Haldane phase at the isotropic Heisenberg point is adiabatically connected to the “trivial” large-$D$ phase, with a continuous change of the Hamiltonian parameters. Furthermore, we study the spin-3 XXZ chain to help explain the development of the classical phase diagram. We present details on how to use the iDMRG method to map out the phase diagram and include an extensive discussion of the numerical methods.

Figure 1

The phases present in a spin-$S$ XXZ chain, for $S=\frac{1}{2},1,2$. The Haldane phases appear only for integer spins and their width decreases rapidly with increasing $S$. At the same time, the phase transitions into the Haldane phases approach $\Delta =1$, where the direct XY-AFM phase transition occurs for all half-integer $S$.

Figure 2

A sketch of the spin-1 phase diagram from Ref. 43 (with our labeling of the phases). (Inset) Phase diagram in the semiclassical limit $S\to \infty$.

Figure 3

The spin-2 phase diagram at $D_4=0$ for $\Delta \ge 0$ and $D_2\ge 0$ as obtained with iDMRG. Examples of data used to obtain the phase boundaries are presented in Sec. 5. In particular, data corresponding to the vertical line are given in Figs. 12 and 13 and the tilted line in Figs. 13 and 14.

Figure 4

The spin-2 phase diagram at $\Delta =1.0$ as obtained with iDMRG. The OH phase is easily reached by including the $D_4$ on-site anisotropy term. The dots indicate the values at which the simulations were performed. To resolve the narrow EH phase, simulations on a finer grid were performed between the AFM and XY phases (points not shown).

Figure 5

Diagrammatic representation of (a) the tensors $\Gamma$ and $\Lambda$. The horizontal lines represent the bond indices $\alpha \in \{1,\cdots ,\chi \}$ and the vertical lines the physical indices $j\in \{1,\cdots ,d\}$. (b) An MPS formed by the tensors $\Gamma$ and $\Lambda$. Connected lines between tensors (or within a single tensor) denote summation over the corresponding indices. (c) Definition of the right and left (Schmidt) basis states with respect to a partition on a bond with index $\alpha$. (d) Condition for the MPS to be in the canonical form. The transfer matrix $T$ of Eq. (10) has been shaded. The upside-down triangles are the complex conjugates of the $\Gamma$ tensors. (e) If the state is in canonical form, then the dominant right eigenvector of $T$ is the “identity matrix” with eigenvalue equal to 1. A similar condition applies for the left transfer matrix $\tilde{T}$. (f) The correlation function defined in Eq. (13). The squares correspond to the operators $F\left(0\right)$ and $G\left(r\right)$.

Figure 6

In iTEBD each time step $\delta t$ of a time evolution is approximated using a Trotter-Suzuki decomposition, i.e., the time evolution operator is expressed as a product of unitary two-site operators.

Figure 7

The iTEBD update scheme for a unitary two-site transformation of a two-site unit cell MPS in canonical form (see Sec. 4b for details).

Figure 8

(a) An operator $O$ acting on an entire chain expressed as a matrix-product operator. (b) A matrix-product operator acting on a matrix-product state $O|\psi \rangle$. (c) The expectation value $\langle \psi \left|O\right|\psi \rangle$ expressed in an MPO form.

Figure 9

Pictorial representation of a contraction of the left and right environments. The boundaries are initiated by the tensors $R_{\alpha \overline{\alpha },a}^0={\delta }_{\alpha \overline{\alpha }}{\vec{v}}_{\mathrm{right};a}$ and $L_{\alpha \overline{\alpha },a}^0={\delta }_{\alpha \overline{\alpha }}{\vec{v}}_{\mathrm{left};a}$.

Figure 10

A pictorial representation of an iDRMG iteration step update. Refer to Sec. 4d for details.

Figure 11

A schematic plot illustrating the idea of finite-entanglement scaling. The black solid line shows the exact correlation length ${\xi }_{\mathrm{phys}}$ of the Hamiltonian, which diverges at the critical point. The dotted lines show the correlation lengths $\xi$ of the optimized iMPS at a finite ${\chi }_2>{\chi }_1$. The horizontal dashed lines are the correlation lengths ${\xi }_{{\chi }_1\left({\chi }_2\right)}$ from Eq. (36), which are induced by the finite-entanglement cut-off at the critical point. The color shaded background indicates the two different regimes: blue is the regime in which the iMPS is converged to the exact ground state using bond dimensions ${\chi }_1,{\chi }_2$ and red is the scaling regime, which shrinks with increasing $\chi$ (see main text for further details).

Figure 12

AFM $\leftrightarrow$ EH phase transitions. (a)–(c) Example of a second-order phase transition. Data away from the Heisenberg point as a function of $D_2$ $(\Delta =1,D_4=0)$, see Fig. 3. (a) The magnetic order parameter $|\langle S_n^z\rangle |$ of the AFM phase. (b) The entanglement entropy $S_E$. (c) The ${\xi }_0$ correlation length. (d) Example of a first-order phase transition. Data along the line $D_2=3.95-D_4$ at $\Delta =4.5$, see Fig. 14a. The magnetic order parameter $|\langle S_n^z\rangle |$ (black symbols) and the ground-state energy $E_{\mathrm{GS}}+4$ (red symbols) are plotted.

Figure 13

BKT phase transition. The data are obtained from finite entanglement scaling averaged over the $\chi$ values given in the insets [for (b) the same $\chi$ values as in (a) were used]. (a)–(c) Critical quantities away from the Heisenberg point as a function of $D_2$ $(\Delta =1,D_4=0)$, see Fig. 3. (a) and (c) show the central charge, while (b) and (d) the Luttinger parameter. (d) Luttinger parameter along the line $D_2=0.85\Delta -0.055$ at $D_4=0$, see Fig. 3.

Figure 14

(a) Phase diagram at $\Delta =4.5$. (b) Central charge $c$ ($\circ$ and $\times$) and string order SO ($▿$) across the EH-OH phase transition along the line $D_2=3.95-D_4$ shown in (a). (c) Projected inversion symmetry along the line $D_2=2.155-D_4$ at $\Delta =2.6$. (d) Projected inversion symmetry ${\mathcal{O}}_{\mathcal{I}}$ along the line $D_2=0.85·\Delta -0.055$ at $D_4=0$, see Fig. 3 [the same $\chi$ values as in (c) were used].

Figure 15

Spin-3 XXZ chain. (a) The magnetic order parameter vanishes $|\langle S_n^z\rangle |\to 0$ at different ${\Delta }^{*}$ as a function of $\chi$. (b) The correlation length ${\xi }_0$ for various $\chi$ across the XY $\leftrightarrow$ OH and the OH $\leftrightarrow$ AFM phase transitions. (c) The central charge, fitted to data at three different $\chi$, across the XY $\leftrightarrow$ OH phase transition. (d) String order at the Heisenberg point ($\Delta =1.0$) for various $\chi$ and scaled to the infinite bond dimension limit (red dashed line).