Time-dependent variational Monte Carlo

The time-dependent variational Monte Carlo (t-VMC) method is a quantum Monte Carlo approach to study the dynamics of closed, non-relativistic quantum systems in the context of the quantum many-body problem. It is an extension of the variational Monte Carlo method, in which a time-dependent pure quantum state is encoded by some variational wave function, generally parametrized as

$\Psi (X,t)=\exp \left(\sum _{k}a_{k}(t)O_{k}(X)\right)$

where the complex-valued $a_{k}(t)$ are time-dependent variational parameters, $X$ denotes a many-body configuration and $O_{k}(X)$ are time-independent operators that define the specific ansatz. The time evolution of the parameters $a_{k}(t)$ can be found upon imposing a variational principle to the wave function. In particular one can show that the optimal parameters for the evolution satisfy at each time the equation of motion

$i\sum _{k^{\prime }}\langle O_{k}O_{k^{\prime }}\rangle _{t}^{c}{\dot {a}}_{k^{\prime }}=\langle O_{k}{\mathcal {H}}\rangle _{t}^{c},$

where ${\mathcal {H}}$ is the Hamiltonian of the system, $\langle AB\rangle _{t}^{c}=\langle AB\rangle _{t}-\langle A\rangle _{t}\langle B\rangle _{t}$ are connected averages, and the quantum expectation values are taken over the time-dependent variational wave function, i.e., $\langle \cdots \rangle _{t}\equiv \langle \Psi (t)|\cdots |\Psi (t)\rangle$.

In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret ${\frac {|\Psi (X,t)|^{2}}{\int |\Psi (X,t)|^{2}\,dX}}$ as a probability distribution function over the multi-dimensional space spanned by the many-body configurations $X$. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time $t$, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories $a(t)$ of the variational parameters are then found upon numerical integration of the associated differential equation.

References

• G. Carleo; F. Becca; M. Schiró & M. Fabrizio (2012). "Localization and glassy dynamics of many-body quantum systems". Sci. Rep. 2: 243. arXiv:1109.2516. Bibcode:2012NatSR...2E.243C. doi:10.1038/srep00243. PMC 3272662. PMID 22355756.

• G. Carleo; F. Becca; L. Sanchez-Palencia; S. Sorella & M. Fabrizio (2014). "Light-cone effect and supersonic correlations in one- and two-dimensional bosonic superfluids". Phys. Rev. A. 89 (3): 031602(R). arXiv:1310.2246. Bibcode:2014PhRvA..89c1602C. doi:10.1103/PhysRevA.89.031602. S2CID 45660254.

• G. Carleo (2011). Spectral and dynamical properties of strongly correlated systems (PhD Thesis). pp. 107–128. hdl:20.500.11767/4289.