Entanglement spectrum of one-dimensional extended Bose-Hubbard models (original) (raw)
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Spectral and Entanglement Properties of the Bosonic Haldane Insulator
Physical Review Letters, 2014
We discuss the existence of a nontrivial topological phase in one-dimensional interacting systems described by the extended Bose-Hubbard model with a mean filling of one boson per site. Performing large-scale density-matrix renormalization group calculations we show that the presence of nearest-neighbor repulsion enriches the ground-state phase diagram of the paradigmatic Bose-Hubbard model by stabilizing a novel gapped insulating state, the so-called Haldane insulator, which, embedded into superfluid, Mott insulator and density wave phases, is protected by the lattice inversion symmetry. The quantum phase transitions between the different insulating phases were determined from the central charge via the von Neumann entropy. The Haldane phase reveals a characteristic four-fold degeneracy of the entanglement spectrum. We finally demonstrate that the intensity maximum of the dynamical charge structure factor, accessible by Bragg spectroscopy, features the gapped dispersion known from the spin-1 Heisenberg chain.
Bosonization and entanglement spectrum for one-dimensional polar bosons on disordered lattices
New Journal of Physics, 2013
Ultra cold polar bosons in a disordered lattice potential, described by the extended Bose-Hubbard model, display a rich phase diagram. In the case of uniform random disorder one finds two insulating quantum phases-the Mott-insulator and the Haldane insulator-in addition to a superfluid and a Bose glass phase. In the case of a quasiperiodic potential, further phases are found, e.g. the incommensurate density wave, adiabatically connected to the Haldane insulator. For the case of weak random disorder we determine the phase boundaries using a perturbative bosonization approach. We then calculate the entanglement spectrum for both types of disorder, showing that it provides a good indication of the various phases.
Physical Review B, 2015
We reexamine the one-dimensional spin-1 XXZ model with on-site uniaxial single-ion anisotropy as to the appearance and characterization of the symmetry-protected topological Haldane phase. By means of large-scale density-matrix renormalization group (DMRG) calculations the central charge can be determined numerically via the von Neumann entropy, from which the ground-sate phase diagram of the model can be derived with high precision. The nontrivial gapped Haldane phase shows up in between the trivial gapped even Haldane and Néel phases, appearing at large single-ion and spin-exchange interaction anisotropies, respectively. We furthermore carve out a characteristic degeneracy of the lowest entanglement level in the topological Haldane phase, which is determined using a conventional finite-system DMRG technique with both periodic and open boundary conditions. Defining the spin and neutral gaps in analogy to the single-particle and neutral gaps in the intimately connected extended Bose-Hubbard model, we show that the excitation gaps in the spin model qualitatively behave just as for the bosonic system. We finally compute the dynamical spin structure factor in the three different gapped phases and find significant differences in the intensity maximum which might be used to distinguish these phases experimentally.
Quantum phase transitions in the dimerized extended Bose-Hubbard model
Physical Review A
We present an unbiased numerical density-matrix renormalization group study of the onedimensional Bose-Hubbard model supplemented by nearest-neighbor Coulomb interaction and bond dimerization. It places the emphasis on the determination of the ground-state phase diagram and shows that, besides dimerized Mott and density-wave insulating phases, an intermediate symmetryprotected topological Haldane insulator emerges at weak Coulomb interactions for filling factor one, which disappears, however, when the dimerization becomes too large. Analyzing the critical behavior of the model, we prove that the phase boundaries of the Haldane phase to Mott insulator and density-wave states belong to the Gaussian and Ising universality classes with central charges c = 1 and c = 1/2, respectively, and merge in a tricritical point. Interestingly we can demonstrate a direct Ising quantum phase transition between the dimerized Mott and density-wave phases above the tricritical point. The corresponding transition line terminates at a critical end point that belongs to the universality class of the dilute Ising model with c = 7/10. At even stronger Coulomb interactions the transition becomes first order.
Bulk-edge correspondence in the Haldane phase of the bilinear-biquadratic spin-1 Hamiltonian
Journal of Statistical Mechanics: Theory and Experiment, 2021
The Haldane phase is the prototype of symmetry protected topological (SPT) phases of spin chain systems. It can be protected by several symmetries having in common the degeneracy of the entanglement spectrum (ES). Here we explore in depth this degeneracy for the spin-1 Affleck–Kennedy–Lieb–Tasaki and bilinear-biquadratic Hamiltonians and show the emergence of a bulk-edge correspondence that relates the low energy properties of the entanglement Hamiltonian of a periodic chain and that of the physical Hamiltonian of an open chain. We find that the ES can be described in terms of two spins-1/2 behaving as the effective spins at the end of the open chain. In the case of non-contiguous partitions, we find that the entanglement Hamiltonian is given by the spin-1/2 Heisenberg Hamiltonian, which suggests a relationship between SPT phases and conformal field theory. We finally investigate the string order parameter and the relation with the bulk-edge correspondence.
Advances in pure optical trapping techniques now allow the creation of degenerate Bose gases with internal degrees of freedom. Systems such as 87 Rb, 39 K or 23 Na in the F = 1 hyperfine state offer an ideal platform for studying the interplay of superfluidity and quantum magnetism. Motivated by the experimental developments, we study ground state phases of a two-component Bose gas loaded on an optical lattice. The system is described effectively by the Bose-Hubbard Hamiltonian with onsite and near neighbor spin-spin interactions. An important feature of our investigation is the inclusion of interconversion (spin flip) terms between the two species. Using mean-field theory and quantum Monte Carlo simulations, we map out the phase diagram of the system. A rich variety of phases is identified, including antiferromagnetic (AF) Mott insulators, ferromagnetic and AF superfluids.
Physical Review B, 2011
We investigate the nature of the Mott-insulating phases of half-filled 2N -component fermionic cold atoms loaded into a one-dimensional optical lattice. By means of conformal field theory techniques and large-scale DMRG calculations, we show that the phase diagram strongly depends on the parity of N . First, we single out charged, spin-singlet, degrees of freedom, that carry a pseudo-spin S = N/2 allowing to formulate a Haldane conjecture: for attractive interactions, we establish the emergence of Haldane insulating phases when N is even, whereas a metallic behavior is found when N is odd. We point out that the N = 1, 2 cases do not have the generic properties of each family. The metallic phase for N odd and larger than 1 has a quasi-long range singlet pairing ordering with an interesting edge-state structure. Moreover, the properties of the Haldane insulating phases with even N further depend on the parity of N/2. In this respect, within the low-energy approach, we argue that the Haldane phases with N/2 even are not topologically protected but equivalent to a topologically trivial insulating phase and thus confirm the recent conjecture put forward by Pollmann et al. [Pollmann et al., arXiv:0909.4059 (2009)]. PACS numbers: 71.10.Pm, 71.10.Fd, 03.75.Mn arXiv:1107.0171v1 [cond-mat.quant-gas] 1 Jul 2011
Interplay of local order and topology in the extended Haldane-Hubbard model
Physical Review B
We investigate the ground-state phase diagram of the spinful extended Haldane-Hubbard model on the honeycomb lattice using an exact-diagonalization, mean-field variational approach, and further complement it with the infinite density matrix renormalization group, applied to an infinite honeycomb cylinder. This model, governed by both on-site and nearest-neighbor interactions, can result in two types of insulators with finite local order parameters, either with spin or charge ordering. Moreover, a third one, a topologically nontrivial insulator with nonlocal order, is also manifest. We test expectations of previous analyses in spinless versions asserting that once a local order parameter is formed, the topological characteristics of the ground state, associated with a finite Chern number, are no longer present, resulting in a topologically trivial wave function. Our study confirms this overall picture, and highlights how finite-size effects may result in misleading conclusions on the coexistence of finite local order parameters and nontrivial topology in this model.
Physical Review B, 2005
We use the finite-size density-matrix-renormalization-group (FSDMRG) method to obtain the phase diagram of the one-dimensional (d = 1) extended Bose-Hubbard model for density ρ = 1 in the U − V plane, where U and V are, respectively, onsite and nearest-neighbor interactions. The phase diagram comprises three phases: Superfluid (SF), Mott Insulator (MI) and Mass Density Wave (MDW). For small values of U and V , we get a reentrant SF-MI-SF phase transition. For intermediate values of interactions the SF phase is sandwiched between MI and MDW phases with continuous SF-MI and SF-MDW transitions. We show, by a detailed finite-size scaling analysis, that the MI-SF transition is of Kosterlitz-Thouless (KT) type whereas the MDW-SF transition has both KT and two-dimensional-Ising characters. For large values of U and V we get a direct, first-order, MI-MDW transition. The MI-SF, MDW-SF and MI-MDW phase boundaries join at a bicritical point at (U, V) = (8.5 ± 0.05, 4.75 ± 0.05).