Physics: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

[math]\displaystyle{ \Psi(X,t) = \exp \left ( \sum_k a_k(t) O_k(X) \right ) }[/math]

where the complex-valued [math]\displaystyle{ a_k(t) }[/math] are time-dependent variational parameters, [math]\displaystyle{ X }[/math] denotes a many-body configuration and [math]\displaystyle{ O_k(X) }[/math] are time-independent operators that define the specific ansatz. The time evolution of the parameters [math]\displaystyle{ a_k(t) }[/math] 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

[math]\displaystyle{ 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, }[/math]

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

In analogy with the Variational Monte Carlo approach and following the Monte Carlo method for evaluating integrals, we can interpret [math]\displaystyle{ \frac{ | \Psi(X,t) | ^2 } { \int | \Psi(X,t) | ^2 \, dX } }[/math] as a probability distribution function over the multi-dimensional space spanned by the many-body configurations [math]\displaystyle{ X }[/math]. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time [math]\displaystyle{ t }[/math], the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories [math]\displaystyle{ a(t) }[/math] 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. doi:10.1038/srep00243. PMID 22355756. Bibcode: 2012NatSR...2E.243C. 

• 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). doi:10.1103/PhysRevA.89.031602. Bibcode: 2014PhRvA..89c1602C. 

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