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Optimal time-dependent lattice models for nonequilibrium dynamics

Lattice models are the workhorses of condensed matter physics. They provide a simple interpretation of the underlying many-body dynamics in terms of particles hopping from one lattice site to another. At the heart of a lattice model is the idea of lattice site localized orbitals which are commonly referred to as Wannier functions. Among the most popular lattice models are the fermionic and the bosonic version of the Hubbard model. In our recent work [1] we show that conventional lattice models, e.g. the Bose-Hubbard model, can be greatly improved at little extra cost. In fact the improvement is maximal and hence the resulting lattice model optimal. The idea is simple and relies entirely on the principle of least action.

An open-source code for the optimal time-dependent Bose-Hubbard model is currently in preparation. Please contact us if you are interested.

The idea in a nutshell

We illustrate the idea for bosons. In a conventional lattice model the ansatz for the many-boson wave function is
\begin{displaymath}
\vert\Psi(t)\rangle=\sum_{\vec{n}}C_{\vec{n}}(t)\left\vert\vec n\right>,
\end{displaymath} (1)

where the sum is over all permanents $\left\vert\vec n \right>=\vert n_1, n_2,\dots,n_M\rangle$ of $N$ bosons residing in $M$ Wannier functions $w_k(x), k=1,\dots,M$. With this ansatz, Eq. (1), the action functional of the many-body Schrödinger equation $S[\{C_{\vec{n}}\}]=\int dt \langle \Psi \vert H -i\partial_t\vert\Psi\rangle$ depends only on the coefficients $\{C_{\vec{n}}(t)\}$. The equations of motion are then obtained by requiring stationarity of the action functional with respect to variations of the coefficients $\{C_{\vec{n}}(t)\}$.

Is it possible to include more variational parameters to improve the lattice model? The answer is Yes!

In our work [1] we start from a more general ansatz for the many-boson wave function. We let not only the coefficients $\{C_{\vec{n}}\}$ depend on time, but also the Wannier functions themselves:
\begin{displaymath}
\vert\Psi(t)\rangle=\sum_{\vec{n}}C_{\vec{n}}(t)\left\vert\vec n{\color{red};t}\right>,
\end{displaymath} (2)

where the sum is over all permanents $\left\vert\vec n {\color{red} ;t} \right>=\vert n_1, n_2,\dots,n_M {\color{red};t}\rangle$ of $N$ bosons residing in $M$ Wannier functions $w_k(x{\color{red}, t}), k=1,\dots,M$ which are allowed to depend on time. With this ansatz the action functional $S[\{C_{\vec{n}}\}, \{w_k\}]=\int dt \langle \Psi \vert H -i\partial_t\vert\Psi\rangle$ of the many-body Schrödinger equation depends not only on the coefficients $\{C_{\vec{n}}(t)\}$, but also on the Wannier functions $\{w_k(x,t)\}$ themselves. This is a whole new class of variational parameters! As we show in our work [1] it is then again possible to derive equations of motion by requiring stationarity of the action functional. Since all parameters in the ansatz wave function, Eq. (2), are determined by the variational principle, the lattice model is optimal. The variational principle ensures that the results can only improve on the respective standard lattice model with time-independent Wannier functions.

Results in brief

We illustrate the idea of optimal time-dependent lattice models for a quantum quench and the Bose-Hubard model. By comparison with exact results of the full many-body Schrödinger equation we show that it is crucial to use optimal time-dependent lattice models as opposed to standard lattice models in order to properly describe the physics. In particular, the well-known parameters J and U of the Bose-Hubbard model become time-dependent and oscillate at a high frequency. The respective standard Bose-Hubbard model cannot describe this dynamics.

Bibliography

[1] Optimal time-dependent lattice models for nonequilibrium dynamics
K. Sakmann, A. I. Streltsov, O. E. Alon and L. S. Cederbaum, New J. Phys. 13 043003 (2011).


[Uni Heidelberg] [Institute of Physical Chemistry] [Theoretical Chemistry]