Efficient neural-network based variational Monte Carlo scheme for direct optimization of excited energy states in frustrated quantum systems

We examine applicability of the valence bond basis correlator product state ansatz, equivalent to the restricted Boltzmann machine quantum artificial neural-network ansatz, and variational Monte Carlo method for direct optimization of excited energy states to study properties of strongly correlated and frustrated quantum systems. The energy eigenstates are found by stochastic minimization of the variational function for the energy eigenstates, which allows direct optimization of particular energy state without knowledge of the lower energy states. This approach combined with numerous tensor network or artificial neural-network ansatz wave functions then allows further insight into quantum phases and phase transitions in various strongly correlated models by considering properties of these systems beyond the ground-state properties. Also, the method is in general applicable to any dimension and has no sign instability. An example that we consider is the square lattice $J_1-J_2$ antiferromagnetic Heisenberg model. The model is one of the most studied models in frustrated quantum magnetism since it is closely related to the disappearance of the antiferromagnetic order in the high-Tc superconducting materials and there is still no agreement about the properties of the system in the highly frustrated regime near $J_2/J_1=0.5$. For the $J_1-J_2$ model, we write the variational ansatz in terms of the two site correlators and in the valence bond basis and calculate the lowest energy eigenstates in the highly frustrated regime near $J_2/J_1=0.5$ where the system has a paramagnetic phase. We find that our results are in good agreement with previously obtained results, which confirms applicability of the method to study frustrated spin systems.

Figure 1

Transposition graph (c) $\langle \alpha |\beta \rangle$ of two valence bond states (a) $|\alpha \rangle$ and (b) $\langle \beta |$. The valence bond basis (5) is an overcomplete basis and the overlap between two valence bond states is $\langle \alpha |\beta \rangle =\pm 2^{N_l-N/2}$, where $N_l$ is the number of loops in the transposition graph and $N$ is the number of lattice sites. Here $N=16$ and $N_l=2$.

Figure 2

Local two-bond updates. First site $i_1$ is randomly chosen, then one of four diagonal neighbor sites $i_2$ is chosen randomly, and the ends of bonds are exchanged: $(i_1,j_1)(i_2,j_2)\to (i_1,j_2)(i_2,j_1)$. The reconfiguration is accepted with probability (19).

Figure 3

Nonlocal bond-loop updates. The first step in a bond-loop update discussed in the text creates two defects (one site with two bonds and one empty site). Since such defects cannot be present in a valence bond state, one of these defects is further moved by subsequent bond moves until it annihilates with the second defect and the loop closes.

Figure 4

$\mathrm{\Omega }$ as a function of the energy shift parameter $\omega$ for the ground state and the first excited energy state ($M_z={\sum }_{i=1}^N{S}_i^z=0$ sector) for $J_2/J_1=0.5$ ($J_1$ is set to $J_1=1$) and the system sizes $N=L\times L$ with (a) $L=6$, (b) $L=8,$ and with periodic boundary conditions. Here, $|{\psi }^{\mathrm{ansatz}}\rangle$ is optimized using a stochastic optimization method to minimize $\mathrm{\Omega }({\psi }^{\mathrm{ansatz}},\omega )$ for each value of $\omega$. Red lines denote minimal value of $\mathrm{\Omega }\left(\omega \right), {\mathrm{\Omega }}_1^{\min }$, for $\omega$ within the range $E_0\le \omega \le E_1$. Energy of the first excited energy state obtained by optimization procedure is calculated using the optimized ansatz that minimized $\mathrm{\Omega }$ at ${\omega }_1^{\min }$, where $\mathrm{\Omega }\left({\omega }_1^{\min }\right)={\mathrm{\Omega }}_1^{\min }$.

Figure 5

Energy $E$ as a function of the energy shift parameter $\omega$ calculated by taking the optimized ansatz that minimizes $\mathrm{\Omega }({\psi }^{\mathrm{ansatz}},\omega )$ for each value of $\omega$. The results are for the system sizes $N=L\times L$ with (a) $L=6$, (b) $L=8,$ and periodic boundary conditions. Here, $J_2/J_1=0.5$ and $J_1$ is set to $J_1=1$. Blue lines denote the ground-state energies obtained by exact diagonalization and the cluster update algorithm for tensor product states [35].

Figure 6

Variance ${\sigma }^2$ as a function of the energy shift parameter $\omega$ for $J_2/J_1=0.5$ ($J_1=1$) and the system sizes $N=L\times L$ with (a) $L=6$, (b) $L=8,$ and periodic boundary conditions. Gray lines denote values of $\omega$ for which the energy eigenvalues for the ground and first excited states are calculated.