Phase identification in many-body systems by virtual configuration binarization

Artificial intelligence provides an unprecedented perspective for studying phases of matter in condensed-matter systems. Image segmentation is a basic technique of computer vision that belongs to a branch of artificial intelligence. Inspired by the image segmentation techniques, in this work, we propose a scheme named virtual configuration binarization (VCB) to unveil quantum phases and quantum phase transitions in many-body systems. By encoding the information of renormalized quantum states into a color image and binarize the color image through the VCB, the renormalized quantum states can be visualized, from which quantum phase transitions can be revealed and the corresponding critical points can be identified. Our scheme is benchmarked on several strongly correlated spin systems, which does not depend on the priori knowledge of order parameters of quantum phases. This demonstrates the potential to disclose the underlying structure of quantum phases by the techniques of computer vision.

Figure 1

(a) A three-index tensor related to matrix product states (MPS), where one unit of MPS is shown. $d$ is the physical index with dimension $\sigma$, which represents the local physical space of particles. $m$ and $n$ are the virtual indexes with dimension ${\chi }_1$ and ${\chi }_2$, which are related to the entanglement between two subsystems. (b) The RGB color image with random pixels, which consists of three color channels where R, G, and B correspond to red, green, and blue colors. Each color channel is related to a pixel matrix. So, the RGB color image can also be represented by a three-index tensor, where $d_c$ is the color space, and $d{p}_1$ and $d{p}_2$ are the pixel space.

Figure 2

Graphical illustration of the image binarization scheme VCB. (a) The color image $T^{\mathcal{G}}\left[\mathrm{cat}\right]$, which is the visualization of three-index tensor $T\left[\mathrm{cat}\right]$ by mapping $\mathcal{G}$, where $\mathcal{G}=\mathcal{I}$ is the identical transformation. (b) The gray images $T^{\mathcal{G},d}\left[\mathrm{cat}\right]$ with $d=1$, 2, and 3 are corresponding to three color channels $R$ (red), $G$ (green), and $B$ (blue). (c) The binary images $T^{\mathcal{B},d}\left[\mathrm{cat}\right]$ corresponding to three color channels $R, G$, and $B$ are obtained by mapping $\mathcal{B}$. (d) The binary images $T^{\mathcal{AD},\delta }$ are obtained by mapping $\mathcal{AD}$ according to Eq. (2).

Figure 3

The color image ${\mathrm{\Phi }}^{\mathcal{G}}$, binary images ${\mathrm{\Phi }}^{\mathcal{B}}$, and binary images ${\mathrm{\Phi }}^{\mathcal{AD},{\delta }_{1,2}}$ of the center tensor $\mathrm{\Phi }$ of the ground states of the transverse field Ising model (TFIM) with different transverse fields $h_x=0.2$, 0.5, and 0.8 are in (a), (b), and (c), respectively. (a) The distinguished details of ${\mathrm{\Phi }}^{\mathcal{G}}$ with different $h_x$ are marked by red circles. (b) The binary images ${\mathrm{\Phi }}^{\mathcal{B},d=1\left(2\right)}$ with different $h_x$, and its textures are always nonregular. (c) The antiferromagnetic (AFM) and paramagnetic (PM) phases are clearly specified by binary images ${\mathrm{\Phi }}^{\mathcal{AD},{\delta }_{1,2}}$. The partial regular texture of ${\mathrm{\Phi }}^{\mathcal{AD},{\delta }_{1,2}}$ when the system at QCP ($h_x=0.5$) is marked by the red triangular. (d) The histogram that counts the occurring times of the tensor elements of $\mathrm{\Phi }$ with $h_x=0.2$, 0.5, and 0.8. (e) More binary images ${\mathrm{\Phi }}^{\mathcal{AD},{\delta }_{1,2}}$ with different transverse fields $h_x=0.1$, 0.3, 0.4, 0.6, 0.7, and 0.9. For DMRG, we take the size of the system $L$ = 80 and the dimension cutoff $\chi$ = 30. The size of binary images ${\mathrm{\Phi }}^{\mathcal{AD},{\delta }_{1,2}}$ is $\chi \times \chi$.

Figure 4

The binary images ${\mathrm{\Theta }}^{\mathcal{AD},{\delta }_{1,2}}$ of the ground states in the 1D anisotropic XXZ antiferromagnetic Heisenberg model (XXZHM) with anisotropy $\mathrm{\Delta }$. (a) ${\mathrm{\Theta }}^{\mathcal{AD},{\delta }_{1,2}}$ in the ferromagnetic (FM) phase with $\mathrm{\Delta }=-1.8$ and $-1.4$. The textures are in inhomogeneous patterns. (b) ${\mathrm{\Theta }}^{\mathcal{AD},{\delta }_{1,2}}$ of the quantum critical point (QCP) $\mathrm{\Delta }=-1$, where FM and XY phases are separated. The texture shows a distorted pattern. (c) ${\mathrm{\Theta }}^{\mathcal{AD},{\delta }_{1,2}}$ in the ferromagnetic (XY) phase with $\mathrm{\Delta }=-0.6$, 0, and 0.6. The textures show homogeneous patterns like the regular brick wall and display approximate spatial uniformity and consistency (global homogeneity). (d) ${\mathrm{\Theta }}^{\mathcal{AD},{\delta }_{1,2}}$ of the QCP $\mathrm{\Delta }=1$, where XY and antiferromagnetic (AFM) phases are separated. The texture is a mixture of homogeneous patterns (the bottom of the image) and inhomogeneous patterns (the upper part of the image). (e) ${\mathrm{\Theta }}^{\mathcal{AD},{\delta }_{1,2}}$ in the AFM phase with $\mathrm{\Delta }=1.4$ and 1.8. The textures are in inhomogeneous patterns, where the distribution of the pixels is nonregular. For DMRG, we take the size of the system $L=80$ and the dimension cutoff $\chi =30$.

Figure 5

The binary images ${\mathrm{\Omega }}^{\mathcal{AD},{\delta }_{1,3}}$ of ground states in the spin-1 Heisenberg antiferromagnetic model with different magnetic field $h_z$. (a) ${\mathrm{\Omega }}^{\mathcal{AD},{\delta }_{1,3}}$ in Haldane phase with $h_z=0$ and 0.1. The textures are homogeneous and show the regularity of the lattice window with diamond holes. (b) ${\mathrm{\Omega }}^{\mathcal{AD},{\delta }_{1,3}}$ of quantum critical point (QCP) $h_z=0.414$. The texture is like a weak-distorted brick wall. (c) ${\mathrm{\Omega }}^{\mathcal{AD},{\delta }_{1,3}}$ in topological trivial magnetic phase (TTM) with $h_z=0.6$. The textures are inhomogeneous. For DMRG, we take the size of the system $L=120$ and the dimension cutoff $\chi =40$.

Figure 6

Cumulative Shannon entropies $S_{{\mathrm{\Phi }}^{\mathcal{AD}}}, S_{{\mathrm{\Theta }}^{\mathcal{AD}}}$, and $S_{{\mathrm{\Omega }}^{\mathcal{AD}}}$ of binary images ${\mathrm{\Phi }}^{\mathcal{AD}}, {\mathrm{\Theta }}^{\mathcal{AD}}$, and ${\mathrm{\Omega }}^{\mathcal{AD}}$ are shown in (a), (b), and (c), respectively, and the corresponding virtual correlations $E_{{\mathrm{\Phi }}^{\mathcal{W}}}, E_{{\mathrm{\Theta }}^{\mathcal{W}}}$, and $E_{{\mathrm{\Omega }}^{\mathcal{W}}}$ are shown in (d), (e), and (f). The image sizes of $S_{{\mathrm{\Phi }}^{\mathcal{AD}}}, S_{{\mathrm{\Theta }}^{\mathcal{AD}}}$, and $S_{{\mathrm{\Omega }}^{\mathcal{AD}}}$ are $\chi \times \chi =30\times 30, 30\times 30$, and $40\times 40$. The windows for getting the cumulative Shannon entropies $S_{{\mathrm{\Phi }}^{\mathcal{AD}}}, S_{{\mathrm{\Theta }}^{\mathcal{AD}}}$, and $S_{{\mathrm{\Omega }}^{\mathcal{AD}}}$ are $W_{10,10}^{x_1,y_1}, W_{10,10}^{x_2,y_2}$, and $W_{10,10}^{x_3,y_3}$, where $1\le x_1(y_1,x_2,y_2)\le 21$ and $1\le x_3\left(y_3\right)\le 31$. The windows for getting the virtual correlations $E_{{\mathrm{\Phi }}^{\mathcal{W}}}, E_{{\mathrm{\Theta }}^{\mathcal{W}}}, E_{{\mathrm{\Omega }}^{\mathcal{W}}}$ are $W_{30,30}^{1,1}, W_{9,9}^{1,1}$, and $W_{40,40}^{1,1}$, respectively. In DMRG, the system sizes for the phases of TFIM, XXZHM, and spin-1 HM are taken as $L$=80, 80, and 200, respectively.