Haldane phase on the sawtooth lattice: Edge states, entanglement spectrum, and the flat band

Grémaud, Benoît; Batrouni, G. G.

doi:10.1103/physrevb.95.165131

Cited by 22 publications

(13 citation statements)

References 39 publications

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“…They appear for a sufficiently strong repulsive interaction as dynamically stable excited states. More recently, pair formation was proposed in the ground state of the bosonic Hubbard model in some special onedimensional lattice structures [29,32,25,27,10,6]. The pair formation occurring here is a collective effect and is caused by the interplay between the repulsive interaction and the movement of the particles in these lattice structures.…”

Section: Introductionmentioning

confidence: 93%

Pair Formation of Hard Core Bosons in Flat Band Systems

Mielke

2018

J Stat Phys

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Hard core bosons in a large class of one or two dimensional flat band systems have an upper critical density, below which the ground states can be described completely. At the critical density, the ground states are Wigner crystals. If one adds a particle to the system at the critical density, the ground state and the low lying multi particle states of the system can be described as a Wigner crystal with an additional pair of particles. The energy band for the pair is separated from the rest of the multi-particle spectrum. The proofs use a Gerschgorin type of argument for block diagonally dominant matrices. In certain one-dimensional or tree-like structures one can show that the pair is localised, for example in the chequerboard chain. For this one-dimensional system with periodic boundary condition the energy band for the pair is flat, the pair is localised.

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Section: Introductionmentioning

confidence: 93%

Pair Formation of Hard Core Bosons in Flat Band Systems

Mielke

2018

J Stat Phys

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“…For certain systems like the spin-1 XXZ model, the topological order is also captured by an even degeneracy of the entanglement spectrum (ES) [28]. From the other side, in different models, like for instance the AKLT [29] or exotic bosonic Hamiltonians [30], the ES does not present even degeneracy but the topological order is assured by the presence of edge modes and finite SOPs. We checked that this happens also in our case where the ES of t < 1 deformed Motzkin chains does not display any degeneracy.…”

Section: Pacs Numbersmentioning

confidence: 99%

Haldane topological orders in Motzkin spin chains

et al. 2017

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Motzkin spin chains are frustration-free models whose ground-state is a combination of Motzkin paths. The weight of such path contributions can be controlled by a deformation parameter t. As a function of the latter these models, beside the formation of domain wall structures, exhibit gapped Haldane topological orders with constant decay of the string order parameters for t < 1. A behavior compatible with a Berezinskii-KosterlitzThouless phase transition at t = 1 is also presented. By means of numerical calculations we show that the topological properties of the Haldane phases depend on the spin value. This allows to classify different kinds of hidden antiferromagnetic Haldane gapped regimes associated to nontrivial features like symmetry-protected topological order. Our results from one side allow to clarify the physical properties of Motzkin frustration-free chains and from the other suggest them as a new interesting and paradigmatic class of local spin Hamiltonians. [6] has been discovered for spin-1 XXZ Heisenberg chains. This has driven significant efforts to look for new kinds of models whose topological order can be described in terms of a SOP [7] motivating the discovery of the celebrated AffleckKennedy-Lieb-Tasaki (AKLT) model [8]. Although the argument of Haldane is given for integer spin chains, only integer spin XXZ-like and AKLT-like chains own topological HP and is therefore non-trivial to find and study new classes of Hamiltonians where HP emerges. Thanks to the strongest quantum "resource", namely the entanglement, spin models have also a fundamental role in the simulation of quantum logical gates for quantum computation [4]. For this reason finding and studying Hamiltonians with highly entangled spins is currently one of the most challenging and intriguing fields in quantum physics. In this direction local integer frustration-free spin Hamiltonians whose ground-state can be expressed as a combination of Motzkin paths [9] have been recently introduced [10,11]. Among others interesting aspects, their importance is given by the fact that they own a level of entanglement entropy which strongly exceeds the one exhibited by other previously known local models. Relevantly, also for half-integer spins, a similar class of Hamiltonians, the Fredkin spin chains, exhibiting the same features [12,13] has been introduced. In addition to their entanglement properties Motzkin chains own further very peculiar properties. Indeed, even if they are purely local models, for high spin values s (i.e., s ≥ 2) they behave as de facto long range Hamiltonians being able to violate cluster decomposition properties (CDP) and area law (AL) scaling DMRG local magnetization for a system of length 2n = 60 at different t deformation values S z i for c) s = 1 and d) s = 2. Lower panels: Thermodynamic limit of the gap ∆ = E1 − E0 as a function of t for e) s = 1 (red circles) and f) s = 2 (blue squares). The continuos lines are fitted with the form ∼ exp(−b/ √ tc − t) with tc = 1 and b a fitting parameter. The thermodynamic limit ...

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“…In the sawtooth lattice, it was found that doping above the critical density leads to a phase where the peak of the momentum distribution is at nonzero momentum [13], while doping the kagome lattice leads to a supersolid phase [13]. Adding longer range interactions between the bosons on the sawtooth lattice uncovers topological effects such as the Haldane insulator phase and edge states in open systems [19]. Bosons were also studied on the flat band Creutz lattice ( Fig.…”

Section: Introductionmentioning

confidence: 99%

Pairing and superconductivity in the flat band: Creutz lattice

2018

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We use unbiased numerical methods to study the onset of pair superfluidity in a system that displays flat bands in the noninteracting regime. This is achieved by using a known example of flat band systems, namely the Creutz lattice, where we investigate the role of local attractive interactions in the U < 0 Hubbard model. Going beyond the standard approach used in these systems where weak interactions are considered, we map the superfluid behavior for a wide range of interaction strengths and exhibit a crossover between BCS and tightly bound bosonic fermion pairs. We further contrast these results with a standard two-leg fermionic ladder, showing that the pair correlations, although displaying algebraic decay in both cases, are longer ranged in the Creutz lattice, signifying the robustness of pairing in this system.

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Haldane phase on the sawtooth lattice: Edge states, entanglement spectrum, and the flat band

Cited by 22 publications

References 39 publications

Pair Formation of Hard Core Bosons in Flat Band Systems

Pair Formation of Hard Core Bosons in Flat Band Systems

Haldane topological orders in Motzkin spin chains

Pairing and superconductivity in the flat band: Creutz lattice

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