• Source: 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




Ψ
(
X
,
t
)
=
exp
⁡

(


∑

k



a

k


(
t
)

O

k


(
X
)

)



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


where the complex-valued




a

k


(
t
)


{\displaystyle a_{k}(t)}

are time-dependent variational parameters,



X


{\displaystyle X}

denotes a many-body configuration and




O

k


(
X
)


{\displaystyle O_{k}(X)}

are time-independent operators that define the specific ansatz. The time evolution of the parameters




a

k


(
t
)


{\displaystyle 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

∑


k

′




⟨

O

k



O


k

′





⟩

t


c






a
˙





k

′




=
⟨

O

k




H



⟩

t


c


,


{\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},}


where





H




{\displaystyle {\mathcal {H}}}

is the Hamiltonian of the system,



⟨
A
B

⟩

t


c


=
⟨
A
B

⟩

t


−
⟨
A

⟩

t


⟨
B

⟩

t




{\displaystyle \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.,



⟨
⋯

⟩

t


≡
⟨
Ψ
(
t
)

|

⋯

|

Ψ
(
t
)
⟩


{\displaystyle \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







|

Ψ
(
X
,
t
)


|


2




∫

|

Ψ
(
X
,
t
)


|


2



d
X





{\displaystyle {\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


{\displaystyle X}

. The Metropolis–Hastings algorithm is then used to sample exactly from this probability distribution and, at each time



t


{\displaystyle t}

, the quantities entering the equation of motion are evaluated as statistical averages over the sampled configurations. The trajectories



a
(
t
)


{\displaystyle 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.

Kata Kunci Pencarian:

• Time-dependent variational Monte Carlo
• Variational Monte Carlo
• Quantum Monte Carlo
• Giuseppe Carleo
• Particle filter
• Time-dependent density functional theory
• Quantum master equation
• Positron annihilation spectroscopy
• Hartree–Fock method
• Resampling (statistics)