Numerical study of boson mixtures with multi-component continuous matrix product states

Abstract

The continuous matrix product state (cMPS) ansatz is a promising numerical tool for studying quantum many-body systems in continuous space. Although it provides a clean framework that allows one to directly simulate continuous systems, the optimization of cMPS is known to be a very challenging task, especially in the case of multi-component systems. In this work, we have developed an improved optimization scheme for multi-component cMPS that enables simulations of bosonic quantum mixtures with substantially larger bond dimensions than previous works. We benchmark our method on the two-component Lieb-Liniger model, obtaining numerical results that agree well with analytical predictions. Our work paves the way for further numerical studies of quantum mixture systems using the cMPS ansatz.

Figures (5)

• Figure 1: Half-infinite entanglement entropy in the cMPS ground state approximation, plotted with respect to the logarithm of the cMPS correlation length $\xi$ (as naturally obtained for different bond dimensions), for $c_{12} = -0.6, -0.3, 0.0, 0.3, 0.6$ (shown in different colors). The bond dimensions of the cMPS data are $\chi = 8, 16, 32, 64$. The entanglement entropy data for different values of $c_{12}$ are vertically shifted by a constant so that the data points are collapsed onto a single line. The dashed line represents the scaling $S_{\mathrm{EE}} = (C/6) \log(\xi) + S_0$, where $C=2$ is the central charge and $S_0$ is a constant.
• Figure 2: The cMPS prediction for the Luttinger parameter $K_\pm$ and the sound velocity $v_\pm$ in the charge and spin sectors, respectively. All the results are obtained with bond dimension $\chi = 32$. The dashed lines represent the analytical results obtained from the weak-coupling expansion results \ref{['eq:weak-coupling-approximation-sound-velocity']} and \ref{['eq:weak-coupling-approximation-luttinger-parameter']}.
• Figure 3: The cMPS results for the correlation functions $C_+(x)$ and $C_-(x)$. For each choice of $c_{12}$, the correlation functions are plotted with respect to the ratio between $x$ and the correlation length $\xi$ of the system. The cMPS results are obtained with bond dimension $\chi=32$. The dashed lines represent the scaling predicted by the Luttinger liquid theory, where the values of $K_+$ and $K_-$ are taken from the numerical results obtained previously.
• Figure 4: The ground-state particle density $\rho$ as a function of the inter-species interaction strength $c_{12}$. We plot the results with respect to different functions of $c_{12}$, in regard to the TG limit and the weak-coupling limit, respectively. The cMPS results are obtained with bond dimension $\chi=32$. The dashed lines represent the analytical predictions in the two different limits.
• Figure 5: A comparison of the optimization performance of our optimization scheme with and without preparation steps. As a comparison, we also plot the energy curve of the ordinary L-BFGS optimization scheme in the diagonal gauge. The bond dimension of the cMPS is $\chi=8$, and the Hamiltonian parameters are $c = 10.0$, $c_{12} = -7.0$, and $\mu = 0.0$.