Time-dependent variational principle for quantum lattices

TL;DR

A new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real- and imaginary-time dynamics for infinite one-dimensional quantum lattices is developed.

Abstract

We develop a new algorithm based on the time-dependent variational principle applied to matrix product states to efficiently simulate the real- and imaginary time dynamics for infinite one-dimensional quantum lattice systems. This procedure: (1) is argued to be optimal; (2) does not rely on the Trotter decomposition and thus has no Trotter error; (3) explicitly preserves all symmetries and conservation laws; and (4) has low computational complexity. The algorithm is illustrated using both imaginary time and real-time examples.

Figures (2)

• Figure 1: An illustration of our construction: the wireframe surface represents the variational manifold $\mathcal{M}=\mathcal{M}_{\text{uMPS}}$ embedded in state space, with the black dot a point representing a uMPS $\ket{\psi(A)}$. The rotated gray square represents the tangent plane $T_{A}\mathcal{M}$ to $\mathcal{M}$ in $\ket{\psi(A)}$, with two generally non-orthogonal coordinate axes $\ket{\partial_{1}\psi(A)}$ and $\ket{\partial_{2}(A)}$ displayed as dotted lines. The arrow with solid head is the direction $\mathrm{i}\hat{H}\ket{\psi(A)}$ of time evolution, and the arrow with open head represents the vector that best approximates $\mathrm{i}\hat{H}\ket{\psi(A)}$ within the tangent plane. The gray curve is the optimal path $\ket{\psi(A(t))}$ which follows the vector field generated by these vectors with open head throughout $\mathcal{M}$.
• Figure 2: Comparison of real-time simulation results at $D=128$ with time step $dt=5\times 10^{-3}$ for conserved quantities $e$ (energy density), $\braket{\hat{S}^{x}}$ and $\braket{\hat{S}^{z}}$ with TDVP (dashed lines) and TEBD (dotted lines).