Random sampling neural network for quantum many-body problems

The eigenvalue problem of quantum many-body systems is a fundamental and challenging subject in condensed-matter physics because the dimension of the Hilbert space (and hence the required computational memory and time) grows exponentially as the system size increases. A few numerical methods have been developed for some specific systems but may not be applicable in others. Here we propose a general numerical method, random sampling neural networks (RSNNs), to utilize the pattern recognition technique for the random sampling matrix elements of an interacting many-body system via a self-supervised learning approach. Several exactly solvable one-dimensional models, including the Ising model with a transverse field, the Fermi-Hubbard model, and the spin-$\frac{1}{2}XXZ$ model, are used to test the applicability of RSNN. Pretty high accuracy ($>96\%$) of energy spectra, magnetization, critical exponents, etc. can be obtained within the strongly correlated regime or near the quantum phase-transition point, even the corresponding RSNN models are trained in the weakly interacting regime. After including data augmentation and transfer learning methods, RSNN can be further applied to systems of larger sizes with much less training data if only pretrained in a smaller system. Our results demonstrate the possibility to combine the existing numerical methods and the RSNN to explore quantum many-body problems in a much wider parameter space.

Figure 1

A typical flowchart of a RSNN model. The input data are $M\times M$ matrices ${\mathbf{H}}_S^{\left(m\right)}$, randomly sampled from the original full system Hamiltonian (see text). After a standard self-supervised learning by a CNN model with known results in the training regime ($R_{\text{train}}$), the obtained RSNN model can be used to predict physical results in the test regime ($R_{\text{test}}$). In most physical problems, the training and test regimes can be defined by a continuous parameter $\lambda$.

Figure 2

(a) Prediction result of the lowest three eigenenergies of 1D IMTF by RSNN (solid lines) for $N=12$. The training regime, $R_{\text{train}}=(0,0.3)\cup (0.7,1.0)$, is shown by the colored background, and the test regime, $R_{\text{test}}=(0.3,0.7)$ is shown by the white background. The blue-gray area near the RSNN result indicates the uncertainty, resulting from five independent calculations. The results obtained by the exact diagonalization (dashed lines) are shown together for comparison. The inset shows the predicted magnetization (solid line) for $N=30$ compared with the results obtained by MPS (dashed line). Here we use $N_{\text{train}}=10000, N_S=200$, and $M=10$ for the training process (see text). Note that we have scaled the energy in unit of $J+h$ in order provide a more balanced expression of eigenstate energies in the training regime. Panel (b) shows the average computation time (solid line) to generate a test data by RSNN for $N_{\text{test}}=100$ data points (see text). The computation time by MPS (dashed lines) as a function of system sizes are also shown together for comparison. The accuracy of the magnetization of the RSNN method is also shown together by open squares. The inset shows the finite-size scaling of the phase-transition point, which is defined when the separation between the lowest two eigenenergies is larger than their uncertainties. The horizontal dashed line is the quantum critical point (${\lambda }_c=0.5$) in the thermodynamic limit. The other parameters are the same as in panel (a).

Figure 3

(a) Accuracy and average computation time of RSNN ($t_{\text{RSNN}}$) for the lowest three eigenenergies of 1D IMTF as a function of the number $N_s$ of sampling at each $\lambda$. Here $N=28, N_S=200$, and $M=10$. Panel (b) shows the same calculation but for a different size of sampling matrix, $M$, for $N_S=50$ and $N=28$. The average computation time stops increasing due to the early stop mechanism when the training loss is saturated. Panel (c) shows the same calculation for different value of training regime, ${\lambda }_0$ (see the text). The total number of $\lambda \mathrm{s}$ for different training regimes are still the same ($N_{\text{train}}=10000$). Here $N_S=200, M=10$, and $N=28$.

Figure 4

Panels (a) and (b) show the predicted holon excitation spectrum (blue dots) of 1D FHM by RSNN for $\lambda \equiv U/(t+U)=0.76$ and 0.93, respectively. The training regime is $R_{\text{train}}=(0.23,0.73)$ with $L=30, N=10$, and $N_{\downarrow }=5$. Exact results calculated by using the Bethe ansatz (BA, red crosses) are also shown together for comparison. Panel (c) shows the associate density of states (DOS) obtained by RSNN. Panel (d) shows the accuracy of the energy spectrum with an uncertainty by RSNN in the test regime, $R_{\text{test}}=(0.73,0.96)$ (white background). Results for two different system sizes are shown together for comparison.

Figure 5

(a) Spinon excitation spectrum of 1D $XXZ$ model for $N=28$ and $\lambda =-1.1$. The spin excitation gap $\mathrm{\Delta }$ is defined by the excitation energy at $p=\pi$. (b) The spin excitation gap $\mathrm{\Delta }$ predicted by RSNN for several system sizes up to $N=400$ with the corresponding uncertainties. The training and test regime is indicated by the colored and white background. (c) RSNN-predicted $\mathrm{\Delta }$ as a function of $1/N$ (on a log-log plot) for different values of $\lambda$. The predicted phase-transition point, ${\lambda }_c^{\text{RSNN}}=-1.07\pm 0.02$, is defined as the fixed point of phenomenological renormalization group (see text). The dashed line is the slope given by ${\lambda }_c^{\text{RSNN}}$. The inset shows the $\mathrm{\Phi }\left(\lambda \right)$ obtained (see text) for several system sizes. The position of its minimum value gives an estimate of critical quantum phase-transition point. (d) Predicted $\mathrm{\Delta }$ as a function of $|\lambda -{\lambda }_c^{\text{RSNN}}|$ (on a log-log plot) for $M=300$ and 400. The solid lines are the fitting results for different values of dynamical exponent. Inset shows how such critical exponent ($z\nu$) changes as a function of the system size. $z\nu =2.16$ in the thermodynamic limit, as indicated by the horizontal dashed line.

Figure 6

Comparison of the accuracy for the energy spectrum of 1D FHM in different extended RSNN models. All the training data are taken from the weak-interaction regime (pink background, $U/t<3$) with $L=18$ and $N=6$, while the test data are calculated in the medium- and strong-interaction regimes ($3<U/t<30$). Green circles, blue crosses, and red boxes are respectively the results obtained by a naive linear regression method, by data augmentation (DA) during the training process, and by the transfer learning (TL) model for a larger system size with $L=72$ and $N=24$. See the text for more details of these calculations.