slides

Strict Conditions for Confinement in the extended Bose-Hubbard model

April 2025

Tanausú Hernández-Yanes

Extended Bose-Hubbard Hamiltonian

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

Initial assumptions about the model

\(J \ll U, V\)
\(U \to \infty\)

Formation of inter-site bound clusters

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L. Barbiero et al, Out-of-equilibrium states and quasi-many-body localization in polar lattice gases, Phys. Rev. B 92, 180406(R) (2015)

When is \(U \to \infty\) reasonable?

A. J. Daley et al, Repulsively Bound Atom Pairs: Overview, Simulations and Links, AIP Conf. Proc. 869, 212–218 (2006)

M. Valiente and D. Petrosyan, Scattering resonances and two-particle bound states of the extended Hubbard model, Journal of Physics B, Vol. 42 (2009)

When is \(U \to \infty\) reasonable?

Resonant states

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

Resonant states

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

Second-order processes

From cluster to another state in two hops

\[ \hat{H}_{\mathrm{eff},i\neq j}^{(2)} = \frac{1}{2}\sum_l \left( \frac{\hat{V}_{il}\hat{V}_{lj}}{E_i - E_l} + \frac{\hat{V}_{jl}\hat{V}_{li}}{E_j - E_l} \right) \]

Second-order processes

From cluster to another state in two hops

Center of Mass Displacement

Examples of CoM conserving Fock states after two hops \[\ket{\dots, 1, 0, 3, 0, 1, \dots}\] \[\ket{2, 0, 1, \dots, 1, 0, 2}\] \[\ket{1, 0, 1, \dots, 1, 0, 2}\]

Center of Mass Displacement

Expansion for different U/V

\(V=10J, N = 8, M = 20\)

Is this applicable to other results?

Quasi Many-Body Localization

\(V = 50J\)

W. Li et al, Disorderless Quasi-localization of Polar Gases in One-Dimensional Lattices, PRL 124, 010404 (2020)

Quasi Many-Body Localization

\(V = 50J\)

Dimerization resuts

L. Barbiero et al, Out-of-equilibrium states and quasi-many-body localization in polar lattice gases, Phys. Rev. B 92, 180406(R) (2015)

Inhomogeneity and IPR

L. Barbiero et al, Out-of-equilibrium states and quasi-many-body localization in polar lattice gases, Phys. Rev. B 92, 180406(R) (2015)

Inhomogeneity and IPR

Hilbert Space Shattering

Wei-Han Li et al, Hilbert Space Shattering and Disorder-Free Localization in Polar Lattice Gases, PRL 127, 260601 (2021)

TBD

What happens to the expansion when we increase the number of particles?

First passage test

First passage

Approximations with MPS

\(N = 5, V = 10J, U = 17J, dt = 0.1/J\)

Truncation error

First passage (with MPS)

Expansion of 2x2 cluster

Resonant states

Expansion can be anisotropic, depending on dipole polarization

Isotropic Expansion

\(J = 1 Hz, V = 10 J, \theta= 0, \phi = 0\)

Ballistic expansion

\(J = 1 Hz, V = 31 J, \sin\theta= \sqrt{2/3}, \phi = \pi/4\)

Anisotropic Expansion

\(J = 1 Hz, V = 50 J, \theta= 0.1982\pi, \phi = 0.7953\pi\)

\(U < \infty\)?

Isotropic dipolar interactions

Conclusions

We require \(U > 4V\) to see confinement-related phenomena at the second-order time scales
Results like dimerization, quasi-localization, and Hilbert space shattering are affected
2D lattice can show anisotropic expansion rate at \(U\to\infty\)
Results also dependent on finite U in a similar fashion