Visualizing quantum phases and identifying quantum phase transitions by nonlinear dimensional reduction

Identifying quantum phases and phase transitions is key to understanding complex phenomena in statistical physics. In this work, we propose an unconventional strategy to access quantum phases and phase transitions by visualization based on the distribution of ground states in Hilbert space. By mapping the quantum states in Hilbert space onto a two-dimensional feature space using an unsupervised machine learning method, distinct phases can be directly specified and quantum phase transitions can be well identified. Our proposal is benchmarked on gapped, critical, and topological phases in several strongly correlated spin systems. As this proposal directly learns quantum phases and phase transitions from the distributions of the quantum states, it does not require priori knowledge of order parameters of physical systems, which thus indicates a perceptual route to identify quantum phases and phase transitions particularly in complex systems by visualization through learning.

Figure 1

Visualizing quantum states via an unsupervised nonlinear dimensionality reduction: An illustration. The ground states in the Hilbert space $\mathcal{H}$ of exponentially large dimensionality can be mapped onto a two-dimensional feature space ${\mathcal{R}}^2$ via $t$-SNE, taking negative logarithmic fidelity to measure the distance between two quantum states. Different points in the feature space represent the ground states at different physical parameters in the Hamiltonian. The quantum phases and phase transitions can be clearly specified through the visualized distribution of the states in ${\mathcal{R}}^2$.

Figure 2

The visualizations of distinct quantum phases of the 1D transverse field Ising model in two-dimensional feature space ${\mathcal{R}}^2$ for three different measuring distances between quantum states. In panel (a), the map is implemented by $t$-SNE where the distances of states are measured by the Euclidean distances $D_{\alpha ,{\alpha }^{\prime }}^{\mathcal{M}}=\parallel {\mathbf{v}}^{\alpha }-{\mathbf{v}}^{{\alpha }^{\prime }}\parallel$ of the MPS parameters. In panel (b), we use PCA for mapping and the entanglement spectra [16] for measuring the distances of states. Panel (c) shows the results of our proposal, where the map is implemented by $t$-SNE with negative logarithmic fidelity (NLF) $D_{\alpha ,{\alpha }^{\prime }}^{\mathcal{H}}=-log\left(\right|\langle {\psi }^{\alpha }|{\psi }^{{\alpha }^{\prime }}\rangle \left|\right)$ as the distance between the ground state. The antiferromagnetic (AFM) and polarized ferromagnetic (FM) phases are clearly specified. Each state is represented by a point on a plane of two components $(y_1,y_2)$ in space ${\mathcal{R}}^2$, and the color bar indicates the magnitudes of $h_x$. We calculate 50 GS's at different $h_x$ by DMRG, where the values of $h_x$ are taken uniformly from 0 to 1. For DMRG, we take the size of the system $L=80$ and the dimension cutoff $\chi =30$. For $t$-SNE, we take the number of iteration steps $n_{\text{iter}}=5000$ and the perplexity $\mathcal{P}=24$.

Figure 3

(a) The three-dimensional (3D) visualization of distinct phases in the 1D anisotropic XXZ antiferromagnetic (AFM) Heisenberg model with anisotropy $\mathrm{\Delta }$. In addition to the two dimensions of features $y_1$ and $y_2$ of ${\mathcal{R}}^2, \mathrm{\Delta }$ is also plotted as the third dimension for a better visualization. Two expected transition points at $\mathrm{\Delta }=-1$ and 1 are indicated by two semitransparent planes to assist visualization. (b) The visualization of panel (a) in feature space ${\mathcal{R}}^2$. Three phases (AFM, XY and FM) are clearly visualized with different patterns in the 3D or 2D space. There are 400 ground states calculated by discretizing $\mathrm{\Delta }$ with an interval 0.01. We take the system size $L=120$, the dimension cutoff of DMRG $\chi =160$, iteration steps in $t$-SNE $n_{\text{iter}}=5000$ with the perplexity $\mathcal{P}=24$.

Figure 4

The visualization of quantum phases in the spin-1 Heisenberg chain (a) in different magnetic field $h_z$, and (b) the visualization of those of the spin-1 Heisenberg antiferromagnetic zigzag chain with different strength of the next-nearest neighboring (NNN) couplings $J_2/J_1$ (the nearest neighboring coupling is fixed at 1). In panel (a), the Haldane phase Luttinger liquid (LL) phase and fully polarized (FP) phase are clearly specified, where it gives the critical field $h_{z_1}=0.42\pm 0.02$. and $h_{z_2}=4\pm 0.02$ In panel (b), the two distinct Haldane phases are obviously separated, which shows the critical NNN coupling $(J_2/J_1)=0.745\pm 0.005$. The distance of quantum states is measured by $D_{\alpha ,{\alpha }^{\prime }}^{\mathcal{H}}=-log\left(\right|\langle {\psi }^{\alpha }|{\psi }^{{\alpha }^{\prime }}\rangle \left|\right)$. We take 250 values of $h_z$ and 40 values of $J_2/J_1$ with the interval $\delta h=0.02$ and $\delta J_2/J_1=0.005$ for the two models, respectively. We take the system size $L=128$ and 100, the dimension cutoff of DMRG $\chi =128$ and 60 for panels (a) and (b), respectively. Iterative steps in $t$-SNE $n_{\text{iter}}=5000$ with the perplexity $\mathcal{P}=20$ for both.

Figure 5

Panels (a)-(e) demonstrate the evolution of samples classified into different clusters (marked by red or blue) and the centers of clusters (indicated by the two stars) after different iteration time in $k$-means. The numbers in panels (a)-(e) denote the magnitudes of the applied magnetic fields. f shows the difference of the centers before and after the $n_{it}\mathrm{th}$ iteration [see Eq. (10)], showing that the iteration converges only after $n_{it}=4$ steps.

Figure 6

The $SC$ [Eq. (11a)] and $CHI$ [Eq. (12a)] versus $K$ for the spin $S=1$ antiferromagnetic Heisenberg uniform chain (a) and zigzag chain (b).

Figure 7

Distribution of quantum states in 2D feature space varies with iteration time of $t$-SNE. The distribution in ${\mathcal{R}}^2$ of the grounds states of TFIM by $t$-SNE with the iteration time $n_{it}=250$, 300, 350 400, 500 in panels (a)-(e), respectively. Panel (f) shows the KL divergence versus $n_{it}$. Here we take the perplexity $\mathcal{P}=24$, the system size $L=80$, and dimension cutoff in DMRG $\chi =30$. The red and blue numbers represent the applied magnetic fields, and the green number 0.5 denotes the critical field.

Figure 8

Distribution of quantum states in 2D feature space varies with perplexities of $t$-SNE. Visualization of the grounds states of TFIM by $t$-SNE with different perplexities $\mathcal{P}=18$, 20, 22, 26, 28, and 30. We take the total iteration times in $t$-SNE $n_{\text{it}}=5000$, the system size $L=80$, and dimension cutoff in DMRG $\chi =30$.

Figure 9

Distribution of quantum states in 2D feature space varies with noises. The visualization of the ground states of TFIM under small noise by $t$-SNE with different strengths of noise $\delta =0$, 0.01, 0.03, 0.05, 0.07, and 0.1 in panels (a)-(f), respectively. We take the perplexity $\mathcal{P}=24$, the total iteration time $n_{\text{it}}=1000$, the system size $L=80$, and dimension cutoff in DMRG $\chi =30$.

Figure 10

Visualization of classical datasets by $t$-SNE with NLF and Euclidean distance. The visualization of the MNIST dataset and the fashion-MNIST dataset by $t$-SNE with NLF or the Euclidean distance. In panels (a) and (c), the images are mapped onto the Hilbert space, where the distance of two different images are measured by NLF. In panels (b) and (d), the distance of two different images are measured by the Euclidean distance in the original feature space. For both MNIST and Fashion-MNIST datasets, we take 1000 images (100 images from each class) as the input of $t$-SNE. We take $\mathcal{P}=16$ and total iteration times $n_{\text{it}}=5000$.