Variational study of hard-core bosons in a two-dimensional optical lattice using projected entangled pair states

We have studied the system of hard-core bosons on a two-dimensional optical lattice using a variational algorithm based on projected entangled-pair states. We have investigated the ground-state properties of the system as well as the responses of the system to sudden changes in the parameters. We have compared our results to mean-field results based on a Gutzwiller ansatz.

Figure 1

Structure of the coefficient related to the state $\mid k_{11},..,k_{44}⟩$ in the PEPS $\mid {\Psi }_A⟩$. The bonds represent the indices of the tensors ${\left[A_i\right]}^k$ that are contracted.

Figure 2

Structure of the contractions in $⟨{\Psi }_A\mid {\Psi }_A⟩$. In this scheme, the first and last rows can be interpreted as MPS $\mid U_1⟩$ and $⟨U_4\mid$ and the rows in between as MPO $U_2$ and $U_3$. The contraction of all tensors is then equal to $⟨U_4\mid U_3{U}_2\mid U_1⟩$.

Figure 3

Energy as a function of time for the imaginary time evolution of the system of hard-core bosons on a $4\times 4$ lattice. The evolutions are performed sequentially with PEPS of virtual dimension $D=2$, $D=3$, $D=4$, and $D=5$. The times at which $D$ is increased are indicated by vertical lines. For comparison, the exact ground-state energy, the exact imaginary time evolution, and the energy of the optimal Gutzwiller ansatz are included.

Figure 4

Energy as a function of time for the imaginary time evolution of the system of hard-core bosons on a $11\times 11$ lattice. The evolutions are performed sequentially with PEPS of virtual dimension $D=2$, $D=3$, $D=4$, and $D=5$. The times at which $D$ is increased are indicated by vertical lines. For comparison, the energy of the optimal Gutzwiller ansatz is included.

Figure 5

(Quasi)momentum distribution of the particles in the ground state of a $11\times 11$ lattice. Plotted are results of the variational calculations with PEPS of dimension $D=5$ and with the Gutzwiller ansatz. From the inset, the density of the particles can be gathered.

Figure 6

Time evolution of the condensate density after a sudden change of the magnitude of the trapping potential from $V_0∕J=100$ to $V_0∕J=64$. As a starting state, we use the Gutzwiller approximation of the ground state. The evolution is performed on the basis of the Gutzwiller ansatz and PEPS with $D=2$, $D=3$, and $D=4$. From the inset, the overlap between the PEPS with $D=2$ and $D=3$ (solid line) and the PEPS with $D=3$ and $D=4$ (dashed line) can be gathered.

Figure 7

Time evolution of the condensate density after a sudden shift of the center of the trap by one site in $x$ and $y$ direction. The starting state is the Gutzwiller approximation of the ground state. The evolution is performed using the Gutzwiller ansatz and PEPS with $D=2$, $D=3$, and $D=4$. The inset shows the overlap between the PEPS with $D=2$ and $D=3$ (solid line) and $D=3$ and $D=4$ (dashed line).

Figure 8

Time evolution of the condensate density starting from a Mott distribution with 14 particles arranged in the center of the trap. The magnitude of the trapping potential is $V_0∕J=100$. For the evolution, the Gutzwiller ansatz and PEPS with $D=2$, $D=3$, and $D=4$ are used. The inset shows the overlap between the $D=2$ and $D=3$ PEPS (solid line) and the $D=3$ and $D=4$ PEPS (dashed line).

Figure 9

Distance $K$ between the time-evolved state and the state with reduced virtual dimension. The virtual dimensions $D=2$, $D=3$, and $D=4$ are included. The distance is plotted for the evolution of a Mott distribution with $N=14$, as explained in Fig. 8. From the inset, the deviation of the particle number from the value 14 can be gathered.