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arXiv:1901.03595 · Physical Review A

Bosonic fractional quantum Hall states on a finite cylinder

Paolo Rosson, Michael Lubasch, Martin Kiffner, Dieter Jaksch

Published Mar 7, 2019• 34 citations
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Abstract

We investigate the ground-state properties of a bosonic Harper-Hofstadter model with local interactions on a finite cylindrical lattice with filling fraction $\ensuremath{\nu}=1/2$. We find that our system supports topologically ordered states by calculating the topological entanglement entropy, and its value is in good agreement with the theoretical value for the $\ensuremath{\nu}=1/2$ Laughlin state. By exploring the behavior of the density profiles, edge currents, and single-particle correlation functions, we find that the ground state on the cylinder shows all signatures of a fractional quantum Hall state even for large values of the magnetic flux density. Furthermore, we determine the dependence of the correlation functions and edge currents on the interaction strength. We find that depending on the magnetic flux density, the transition toward Laughlin-like behavior can be either smooth or it can happen abruptly for some critical interaction strength.

DOI
10.1103/physreva.99.033603

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We investigate the ground-state properties of a bosonic Harper-Hofstadter model with local interactions on a finite cylindrical lattice with filling fraction \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2. We find that our system supports topologically ordered states by calculating the topological entanglement entropy, and its value is in good agreement with the theoretical value for the \ensuremathν=1/2\ensuremath{\nu}=1/2\ensuremathν=1/2 Laughlin state. By exploring the behavior of the density profiles, edge currents, and single-particle correlation functions, we find that the ground state on the cylinder shows all signatures of a fractional quantum Hall state even for large values of the magnetic flux density. Furthermore, we determine the dependence of the correlation functions and edge currents on the interaction strength. We find that depending on the magnetic flux density, the transition toward Laughlin-like behavior can be either smooth or it can happen abruptly for some critical interaction strength.
We numerically study the behavior of spin-1/21/21/2 fermions on a two-dimensional square lattice subject to a uniform magnetic field, where opposite spins interact via an on-site attractive interaction. Starting from the noninteracting case where each spin population is prepared in a quantum Hall state with unity filling, we follow the evolution of the system as the interaction strength is increased. Above a critical value and for sufficiently low flux density, we observe the emergence of a twofold quasidegeneracy accompanied by the opening of an energy gap to the third level. Analysis of the entanglement spectra shows that the gapped ground state is the bosonic 1/21/21/2 Laughlin state. Our work therefore provides compelling evidence of a topological phase transition from the fermionic quantum Hall state at unity filling to the bosonic Laughlin state at a critical attraction strength of the order of the one-body spectrum linewidth.
We show via tensor network methods that the Harper-Hofstadter Hamiltonian for hard-core bosons on a square geometry supports a topological phase realizing the $\ensuremath{\nu}=\frac{1}{2}$ fractional quantum Hall (FQH) effect on the lattice. We address the robustness of the ground-state degeneracy and of the energy gap, measure the many-body Chern number, and characterize the system using Green functions, showing that they decay algebraically at the edges of open geometries, indicating the presence of gapless edge modes. Moreover, we estimate the topological entanglement entropy by taking a combination of lattice bipartitions that reproduces the topological structure of the original proposals by Kitaev and Preskill [Phys. Rev. Lett. 96, 110404 (2006)] and Levin and Wen [Phys. Rev. Lett. 96, 110405 (2006)]. The numerical results show that the topological contribution is compatible with the expected value $\ensuremath{\gamma}=\frac{1}{2}$. Our results provide extensive evidence that FQH states are within reach of state-of-the-art cold-atom experiments.
The Bose-Hubbard model subjected to an effective magnetic field hosts a plethora of phases with different topological orders when tuning the chemical potential. Using the density matrix renormalization group method, we identify several gapped phases near the first Mott lobe at strong interactions. They are connected by a particle-hole symmetry to a variety of quantum Hall states stabilized at low fillings. We characterize phases of both particle and hole type and identify signatures compatible with Laughlin, Moore-Read, and bosonic integer quantum Hall states by calculating the quantized Hall conductance and by extracting the topological entanglement entropy. Furthermore, we analyze the entanglement spectrum of Laughlin states of bosonic particles and holes for a range of interaction strengths, as well as the entanglement spectrum of a Moore-Read state. These results further corroborate the existence of topological states at high fillings, close to the first Mott lobe, as hole analogs of the respective low-filling states.
We present numerical evidence of an interaction-driven quantum Hall plateau transition between a ∣C∣>1|C|>1∣C∣>1 Chern Insulator (CI) and a ν=1/3ν= 1/3ν=1/3 Laughlin state in the Harper-Hofstadter model. We study the model at flux densities p/qp/qp/q, where the lowest Landau level (LLL) manifold comprises ppp magnetic sub-bands. For weak interactions, the model realises integer CIs corresponding to filled sub-bands, while strongly interacting candidate states include fractional quantum Hall (FQH) states at LLL filling fractions ν=r/tν=r/tν=r/t. These phases may compete at the same particle density when p=tp=tp=t. As a concrete example, we numerically explore the physics at flux density nϕ=3/11n_ϕ = 3/11nϕ​=3/11, where we show evidence that a direct transition occurs between a CI and a ν=1/3ν= 1/3ν=1/3 Laughlin state, which we characterise in terms of its critical, topological and entanglement properties. We also show that strong interactions generically stabilise a ν=1/3ν= 1/3ν=1/3 Laughlin state even when the LLL is split into multiple bands, and introduce a powerful methodology to extract its topological entanglement entropy by exploiting the scaling of magnetic length with nϕn_ϕnϕ​.
Determining the statistics of elementary excitations supported by fractional quantum Hall states is crucial to understanding their properties and potential applications. In this paper, we use the topological entanglement entropy as an indicator of Abelian statistics to investigate the single-component $\ensuremath{\nu}=2/5$ and 3/73/73/7 states for the Hofstadter model in the band mixing regime. We perform many-body simulations using the infinite cylinder density matrix renormalization group and present an efficient algorithm to construct the area law of entanglement, which accounts for both numerical and statistical errors. Using this algorithm, we show that the $\ensuremath{\nu}=2/5$ and 3/73/73/7 states exhibit Abelian topological order in the case of two-body nearest-neighbor interactions. Moreover, we discuss the sensitivity of the proposed method and fractional quantum Hall states with respect to interaction range and strength.
Topological states of matter, such as fractional quantum Hall states, are an active field of research due to their exotic excitations. In particular, ultracold atoms in optical lattices provide a highly controllable and adaptable platform to study such new types of quantum matter. However, finding a clear route to realize non-Abelian quantum Hall states in these systems remains challenging. Here we use the density-matrix renormalization-group (DMRG) method to study the Hofstadter-Bose-Hubbard model at filling factor $\ensuremath{\nu}=1$ and find strong indications that at $\ensuremath{\alpha}=1/6$ magnetic flux quanta per plaquette the ground state is a lattice analog of the continuum non-Abelian Pfaffian. We study the on-site correlations of the ground state, which indicate its paired nature at $\ensuremath{\nu}=1$, and find an incompressible state characterized by a charge gap in the bulk. We argue that the emergence of a charge density wave on thin cylinders and the behavior of the two- and three-particle correlation functions at short distances provide evidence for the state being closely related to the continuum Pfaffian. The signatures discussed in this letter are accessible in current cold atom experiments and we show that the Pfaffian-like state is readily realizable in few-body systems using adiabatic preparation schemes.