Continuous matrix product states with periodic boundary conditions and an application to atomtronics

We introduce a time evolution algorithm for one-dimensional quantum field theories with periodic boundary conditions. This is done by applying the Dirac-Frenkel time-dependent variational principle to the set of translational invariant continuous matrix product states with periodic boundary conditions. Moreover, the ansatz is accompanied with additional boundary degrees of freedom to study quantum impurity problems. The algorithm allows for a cutoff in the spectrum of the transfer matrix and thus has an efficient computational scaling. In particular we study the prototypical example of an atomtronic system—an interacting Bose gas rotating in a ring shaped trap in the presence of a localized barrier potential.

Figure 1

Left: Persistent current in units of $I_0=2\pi /m{L}^2$ for $\lambda =9.5$ (blue, $D=10$) and $\lambda =38.2$ (red, $D=10$) both at $\gamma =2.33$. Right: Energy density difference at $\lambda =38.2$ and $\gamma =33$ at $D=12$. The other parameters in both panels are $L=120$ and $N=18$.

Figure 2

Density profiles vs angular coordinate $\mathrm{\Theta }=\frac{2\pi }{L}x$ at $L=120$ and $\mathrm{\Omega }=0$ for different values of the interaction $\gamma$ and the barrier height $\lambda$. (a) Weak interaction at $\gamma =0.03, N=18$, and $D=4$. Inset: GP solution (black dots) and cMPS result (red) at $\lambda =19.1$. (b) Intermediate interaction at $\gamma =2.33, N=18$, and $D=10$. (c) Strong interaction at $\gamma =33, N=18$, and $D=12$. Inset: DMRG data (black dots) compared to cMPS results (red) at $\lambda =19.1$. (d) Density profile at fixed $\gamma =33, \lambda =19.1$, and $L=120$ but for different expectation value of the particle number $N=80(D=18),N=60(D=18),N=40(D=16)$, and $N=18(D=12)$. Inset: loglog plot of the width $\sigma$ (blue dots) of the central depletion around $\mathrm{\Theta }=0$ as a function of $N$. The black line is a linear fit indicating an algebraic decay of $\sigma \left(N\right)=A{N}^{\eta }$ with $A$ a constant and the exponent $\eta \approx -1.3$.

Figure 3

Persistent current amplitude in units of $I_0=2\pi /m{L}^2$ as a function of $\gamma$ for several values of $\lambda$ at $L=120$ and $N=18$. Three different approaches were used: cMPS (dots), DMRG (squares) at $D_{\mathrm{MPS}}=20$ (and unitary lattice spacing $a=1$), and GP (black dashed line). The bond dimension increases from $D=4$ at $\gamma =0.03$ up to $D=14$ for $\gamma \ge 5$.

Figure 4

Energy density for different system sizes. The TTN simulation (red triangles) was performed at $D_{\mathrm{TTN}}=96$ while the largest value for the cMPS simulations (blue circles) was $D=20$ at $L=1024$. Inset: Linear fit of the cMPS data. All simulations were done at $\mathrm{\Omega }=0, \rho =0.125, \gamma =80$, and $\mathrm{\Lambda }=4$.

Figure 5

Left: Inverse ground state energy density of the bulk as a function of the cutoff $\epsilon$. Right: Ground state expectation value of the boundary terms in Hamiltonian (10) for several values of $\epsilon$. The other parameters in both panels are $D=16, L=256, N=32, \gamma =80, \mathrm{\Lambda }=4$, and $\mathrm{\Omega }=0$.

Figure 6

Density profile for three different values of $D$ at $L=128, N=16, \gamma =80, \mathrm{\Lambda }=4$, and $\mathrm{\Omega }=0$. The inset shows a zoom around the bulk value of the density.