Innsbruck Big Meeting

# Big Meeting
### July 2024
#### Tanausú Hernández Yanes

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## Expansion of dipolar gas in 1D and 2D lattices

Main points of research:
- Experimentally revelant interaction strengths
- Expansion rates
- Anisotropy (in 2D)

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## Extended Bose-Hubbard Hamiltonian

`
$$
\begin{align}
\hat{H} =& -J \sum_j (\hat{b}_{j+1}^\dagger \hat{b}_j +  \mathrm{h.c.}) \\
&+ \frac{U}{2} \sum_j \hat{n}_j(\hat{n}_j - 1) \\
&+ \sum_{i<j}^\mathrm{cutoff} \frac{V_{ij}}{|i-j|^3}\hat{n}_i \hat{n}_j,
\end{align}
$$
`

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## Initial assumptions about the model

- $J \ll U, V$
- $U \to \infty$

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## Brief Reminder of Perturbation Theory

![perturbation theory diagram](img/sketch_t-J.svg)

Note: We are used to perturbation theory as a method to correct eigenenergies only. But it also allows to project a given Hamiltonian in a reduced Hilbert space depending on the perturbation strength. Example of t-J model.

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## Isotropic Expansion of 2x2 cluster

$J = 1 Hz, V = 10 J, \theta= 0, \phi = 0$

</br>
<img src='img/expansion_iso.png' height='350'/>

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## $U < \infty$?

![sc](img/expansion_soft_core_iso.png)

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## Resonant states in 1D

Can we find states of equal energy to that of a cluster?

<img height="150px" src='img/resonant_states_1D.svg' style='-webkit-filter: invert(1);'>

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## Resonant states

Values of $U$ where, at least, one resonant state exists at a given perturbation order (number of hops):

![resonances](img/resonant_U_1D.png)

Notes: In reality, states that are not strictly resonant will also be populated depending on the perturbation strength. Moreover, long-range tail should increase number and variety of resonances.

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## Expansion for different U/V

![expansion comparison](img/expansion_1D_comparison.png)

$J=1, V=10J, N =4, L = 20$

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## Short time scales, Larger V

![expansion comparison](img/1D_ED_soft_core_N=4_J=1_V=50_cutoff=None.png)

$J=1, V=50J, N = 4, L = 12$
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## What happens to the expansion when we increase the number of particles?

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## First passage test

![resonances](img/test_first_passage.png)

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## First passage

![resonances](img/first_passage_ED.png)

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## First passage (with MPS)

![resonances](img/first_passage_MPS.png)

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## Is this applicable to other results?

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## Quasi Many-Body Localization

DOI: 10.1103/PhysRevLett.124.010404

![](img/fig1.png)

$V = 50J$

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## Quasi Many-Body Localization

$V = 50J$

![](img/mbl_paper_comparison.png)

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## Dimerization resuts

DOI: 10.1103/PhysRevB.92.180406

![dimers](img/dimers.png)

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## Inhomogeneity and IPR

DOI: 10.1103/PhysRevB.92.180406

![resonances](img/dN.png)

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## Inhomogeneity and IPR

![resonances](img/inhomogeneity_test.png)

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## Next steps

- Increase $N$
- Increase $V$
- Hilbert space shattering
- Write up