Geometric entanglement and quantum phase transitions in two-dimensional quantum lattice models

Geometric entanglement (GE), as a measure of multipartite entanglement, has been investigated as a universal tool to detect phase transitions in quantum many-body lattice models. In this paper we outline a systematic method to compute GE for two-dimensional (2D) quantum many-body lattice models based on the translational invariant structure of infinite projected entangled pair state (iPEPS) representations. By employing this method, the $q$-state quantum Potts model on the square lattice with $q\in \{2,3,4,5\}$ is investigated as a prototypical example. Further, we have explored three 2D Heisenberg models: the antiferromagnetic spin-1/2 $XXX$ and anisotropic $XYX$ models in an external magnetic field, and the antiferromagnetic spin-1 $XXZ$ model. We find that continuous GE does not guarantee a continuous phase transition across a phase transition point. We observe and thus classify three different types of continuous GE across a phase transition point: (i) GE is continuous with maximum value at the transition point and the phase transition is continuous, (ii) GE is continuous with maximum value at the transition point but the phase transition is discontinuous, and (iii) GE is continuous with nonmaximum value at the transition point and the phase transition is continuous. For the models under consideration, we find that the second and the third types are related to a point of dual symmetry and a fully polarized phase, respectively.

Figure 1

Numerical values for (a) the local order parameter $\langle S_x\rangle$ and (b) the GE per site ${\mathcal{E}}_{\infty }\left(\lambda \right)$ for the 2D quantum Ising model as a function of the transverse field strength $\lambda$ for the indicated values of the iPEPS truncation dimension $D$.

Figure 2

Numerical values for the local order parameter $\langle M_{x_1}\rangle$ and the GE per site $\mathcal{E}\left(\lambda \right)$ for the 2D quantum $q$-state Potts model for (a) $q=3$, (b) $q=4$, and (c) $q=5$ as a function of the transverse field strength $\lambda$ for the indicated values of the iPEPS truncation dimension $D$. For each case the curves for both the local order parameter and the GE show a discontinuous behavior, indicative of a discontinuous phase transition.

Figure 3

Plots for the 2D spin-$\frac{1}{2} XXX$ model with varying magnetic field $h$. (a) Local magnetization $m$ for sublattices A and B, obtained with iPEPS truncation dimension $D=2$. (b) Amplitude of the local order parameters $M$. (c) GE per site $\mathcal{E}\left(h\right)$.

Figure 4

Plots for the spin-$\frac{1}{2} XYX$ model with fixed magnetic field $h=0.25$ and varying anisotropy parameter ${\mathrm{\Delta }}_y$. (a) Components of the local magnetization $m$ for sublattices A and B for truncation dimension $D=6$. (b) Components of the staggered magnetizations $M$ for truncation dimension $D=6$. (c) GE per site $\mathcal{E}\left(\lambda \right)$ with truncation dimension $D$.

Figure 5

Single-copy entanglement (a) $S_1\left(\mathrm{\Delta }\right)$ and (b) $S_2\left(\mathrm{\Delta }\right)$ as a function of the anisotropy parameter $\mathrm{\Delta }$ for the 2D spin-1 $XXZ$ model with increasing iPEPS truncation dimension $D$. Here the single-copy entanglement is calculated based on the one-site and two-site reduced density matrix, obtained by using the iTEBD method.

Figure 6

(a) Sublattice components of the local magnetization $m$, (b) amplitude of the local order parameters $M_{xy}$ and $M_z$, (c) total antiferromagnetic order parameter $M_t$, and (d) GE per lattice site $\mathcal{E}\left(\mathrm{\Delta }\right)$ as a function of anisotropy parameter $\mathrm{\Delta }$ for the 2D spin-1 $XXZ$ model. Values of the iPEPS truncation dimension $D$ are as indicated, with $D=9$ for (a).

Figure 7

(i) A five-index tensor $A_{lrud}^s$ labeled by one physical index $s$ and four bond indices $l, r, u$, and $d$. (ii) The iPEPS representation of a wave function on the square lattice. Copies of the tensors $A_{lrud}^s$ and $B_{lrud}^s$ are connected through four types of bonds. (iii) A one-index tensor ${\tilde{A}}^s$ labeled by one physical index $s$. (iv) The iPEPS representation of a separable state in the square lattice. (v) A reduced four-index tensor $a_{lrud}$ from a five-index tensor $A_{lrud}^s$ and a one-index ${\tilde{A}}^{s*}$. (vi) The tensor network representation for the fidelity between a quantum wave function (described by $A_{lrud}^s$ and $B_{lrud}^s$) and a separable state (described by ${\tilde{A}}^s$ and ${\tilde{B}}^s$), consisting of the reduced tensors $a_{lrud}$ and $b_{lrud}$.

Figure 8

Key ingredients for obtaining the gradient of the fidelity between a given ground-state wave function $|\psi \rangle$ and a separable state $|\varphi \rangle$ in the iPEPS representation. (i) Zero-dimensional transfer matrix $E_0$ and its dominant eigenvectors $V_L$ and $V_R$. Here the infinite matrix product state representation of the dominant eigenvectors for the one-dimensional transfer matrix $E_1$ follows from Ref. [25], and $V_L$ and $V_R$ may be evaluated using the Lanczos method. The contraction of the entire tensor network is the dominant eigenvalue ${\eta }_{\langle \varphi |\psi \rangle }$ of the zero-dimensional transfer matrix $E_0$ for $\langle \varphi |\psi \rangle$. (ii) A half-filled square denotes $a_{-}$, the derivative of the four-index tensor $a_{lrud}$ with respect to ${\tilde{A}}^{s*}$, which is nothing but the five-index tensor $A_{lrud}^s$. Similarly, we may define $b_{-}$, the derivative of the four-index tensor $b_{lrud}$ with respect to ${\tilde{B}}^{s*}$. (iii) and (iv) Pictorial representation of the contributions to the derivative of ${\eta }_{\langle \varphi |\psi \rangle }$ with respect to ${\tilde{A}}^{s*}$, with different relative positions between filled circles and half-filled squares.

Figure 9

Key ingredients in the diagonal contraction scheme for obtaining the gradient of the fidelity between a given ground-state wave function $|\psi \rangle$ and a separable state $|\varphi \rangle$ in the iPEPS representation. (i) Tensor network representation for the fidelity between a quantum wave function $|\psi \rangle$ and a separable state $|\varphi \rangle$. (ii) Tensor network representation for a zero-dimensional transfer matrix $E_0$ and its dominant eigenvectors $V_L$ and $V_R$. Here the infinite matrix product state representation of the dominant eigenvectors for the one-dimensional transfer matrix $E_1$ follows from Ref. [25], and $V_L$ and $V_R$ are evaluated using the Lanczos method. The contraction of the entire tensor network is the dominant eigenvalue ${\eta }_{\langle \varphi |\psi \rangle }$ of the zero-dimensional transfer matrix $E_0$ for $\langle \varphi |\psi \rangle$. (iii) The pictorial representation of the contributions to the derivative of ${\eta }_{\langle \varphi |\psi \rangle }$ with respect to ${\tilde{A}}^{s*}$, with different relative positions between filled circles and half-filled squares.