Phases of the disordered Bose-Hubbard model with attractive interactions

Letter

Phases of the disordered Bose-Hubbard model with attractive interactions

Olli Mansikkamäki, Sami Laine, and Matti Silveri

Phys. Rev. B 103, L220202 – Published 23 June 2021

Abstract

We study the quantum ground-state phases of the one-dimensional disordered Bose-Hubbard model with attractive interactions, realized by a chain of superconducting transmon qubits or cold atoms. We map the phase diagram using perturbation theory and exact diagonalization. Compared to the repulsive Bose-Hubbard model, the quantum ground-state behavior is dramatically different. At strong disorder of the on-site energies, all the bosons localize into the vicinity of a single site, contrary to the Bose glass behavior of the repulsive model. At weak disorder, depending on hopping, the ground state is either superfluid or a $W$ state, which is a multisite and multiparticle entangled superposition of states where all the bosons occupy a single site. We show that the robustness of the $W$ phase against disorder diminishes as the total number of bosons increases.

Received 15 January 2021
Revised 26 May 2021
Accepted 8 June 2021

DOI:https://doi.org/10.1103/PhysRevB.103.L220202

Physics Subject Headings (PhySH)

Localization Phase diagrams Quantum simulation

Disordered systems Superconducting qubits

Bose-Hubbard model Exact diagonalization Perturbation theory

Quantum Information, Science & TechnologyCondensed Matter, Materials & Applied Physics

Authors & Affiliations

Olli Mansikkamäki , Sami Laine , and Matti Silveri

Nano and Molecular Systems Research Unit, University of Oulu, P.O. Box 3000, FI-90014 Oulu, Finland

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Issue

Vol. 103, Iss. 22 — 1 June 2021

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Images

Figure 1
A schematic illustration and the local occupation density $p_{n ℓ} = 〈 n | {\hat{ρ}}_{ℓ} | n 〉$ as a function of the transmon site $ℓ$ and the local Fock state $| n 〉$ for (a) the localized state, (b) the $W$ state, and (c) the superfluid state at the scaled hopping frequency $τ = 0.05$ , 0.15, and 1, respectively. A single disorder realization at the scaled disorder strength $δ = 0.33 \times 10^{- 3}$ was used. The local density matrix ${\hat{ρ}}_{ℓ}$ is calculated by tracing over all the other sites: ${\hat{ρ}}_{ℓ} = {Tr}_{{i \neq ℓ}} (| ψ 〉 〈 ψ |)$ , where $| ψ 〉$ is the ground state.
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Figure 2
(a), (b) In white, the inverse participation ratios $P_{s}$ (solid) and $P_{r}$ (dashed), defined in Eq. (11), of the quantum ground state of the disordered attractive Bose-Hubbard model (1) as a function of the scaled hopping frequency $τ$ at two values of the scaled disorder strength, (a) $δ = 0.001$ and (b) $δ = 0.036$ . In black, the corresponding critical scaled hopping frequencies $τ_{c}^{s / r}$ defined in Eq. (12). (c) The two inverse participation ratios $P_{s / r}$ , represented as a single colormap, as a function of the scaled hopping frequency $τ$ and the disorder strength $δ$ . The amount of blue (red) indicates the value of $P_{s}$ ( $P_{r}$ ): The bluer (redder) the pixel, the more delocalized the ground state is spatially (in the reciprocal space). When both of the inverse participation ratios are nonzero, the color of the corresponding pixel is purple, indicating the $W$ phase. The overlaid lines indicate the phase boundaries estimated from the analytical results: localized-to- $W$ at $δ = 2 {(α τ)}^{N}$ (white dashed line), superfluid-to-localized at $ɛ_{SF} = ɛ_{loc}$ (gray dash-dotted line), and $W$ -to-superfluid at $ɛ_{W} = ɛ_{SF}$ (yellow dotted line). The arrows indicate the locations of the horizontal cutoffs shown in (a) and (b). The ground states were numerically computed for an open chain of length $L = 8$ with the total number of bosons $N = 4$ , and averaged over 1000 disorder realizations. See Ref. [41] for the results for corresponding periodic chains. Note that $τ \geq$ 0.05 in this and all the following figures.
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Figure 3
The critical scaled hopping frequencies (a) $τ_{c}^{r}$ (colored filled triangles) and (b) $τ_{c}^{s}$ (colored filled circles), defined in Eq. (12), as a function of the total boson number $N = 4, ..., 9$ and the scaled disorder strength $δ$ . The chain length is $L = 8$ and the results are averaged over 2000 disorder realizations. The analytically predicted phase boundary between the localized phase and the $W$ phase is shown as colored lines at $δ = 2 {[α (N) τ_{c}^{s}]}^{N}$ in (b).
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