We investigate the nature of the phase transition between the period-three charge-density wave and the disordered phase of a hard-boson model proposed in the context of cold-atom experiments. Building on a density-matrix renormalization group algorithm that takes full advantage of the hardboson constraints, we study systems with up to 9'000 sites and calculate the correlation length and the wave-vector of the incommensurate short-range correlations with unprecedented accuracy. We provide strong numerical evidence that there is an intermediate floating phase far enough from the integrable Potts point, while in its vicinity, our numerical data are consistent with a unique transition in the Huse-Fisher chiral universality class. arXiv:1808.08990v2 [cond-mat.str-el] 1 Nov 2018

Accurate and predictive computations of the quantum-mechanical behavior of many interacting electrons in realistic atomic environments are critical for the theoretical design of materials with desired properties, and they require solving the grand-challenge problem of the many-electron Schrödinger equation. An infinite chain of equispaced hydrogen atoms is perhaps the simplest realistic model for a bulk material, embodying several central themes of modern condensed-matter physics and chemistry while retaining a connection to the paradigmatic Hubbard model. Here, we report a combined application of cutting-edge computational methods to determine the properties of the hydrogen chain in its quantummechanical ground state. Varying the separation between the nuclei leads to a rich phase diagram, including a Mott phase with quasi-long-range antiferromagnetic order, electron density dimerization with power-law correlations, an insulator-to-metal transition, and an intricate set of intertwined magnetic orders.

We study spontaneous dimerization transitions in a Heisenberg spin-1 chain with additional nextnearest neighbor (NNN) and 3-site interactions using extensive numerical simulations and a conformal field theory analysis. We show that the transition can be second order in the WZW SU(2)2 or Ising universality class, or first-order. We argue that these features are generic because of a marginal operator in the WZW SU(2)2 model, and because of two topologically distinct non-dimerized phases with or without edge states. We also provide explicit numerical evidence of conformal towers of singlets inside the spin gap at the Ising transition. Implications for other models are briefly discussed.Topological matter is currently attracting a lot of attention. One of the first examples is the spin-1 Heisenberg chain, which has a bulk gap 1 but spin-1/2 edge states.2,3 Spin-1 chains with more general interactions have been extensively studied over the years, and they have in particular been shown to undergo a spontaneous dimerization in the presence of a negative biquadratic interaction at an integrable critical point.4,5 The universality class of this critical point is SU(2) 2 Wess-ZuminoWitten (WZW) with central charge c = 3/2.6-8 It has been identified in other models exhibiting spontaneous dimerization, 9 and it is usually considered to describe the generic behaviour of spin-1 chains at the transition to a spontaneously dimerized phases.In this Letter, we identify two other generic possibilities, Ising and first order, and we show that these alternatives are natural consequences of general properties: i) the presence of topological and non-topological phases with and without edge states respectively; ii) the existence of a marginal operator in the WZW SU(2) 2 model. We also show that combining density matrix renormalization group (DMRG) simulations with conformal field theory (CFT) predictions for open systems gives access to the conformal towers of the critical lines, including that of singlets inside the spin gap on the Ising line.We consider the spin-1 chain Hamiltonian:On top of the standard Heisenberg coupling J 1 , it includes two of the three interactions that appear in nextto-leading order in the strong coupling expansion of the two-band Hubbard model: the NNN interaction J 2 and a three-site interaction with coupling strength J 3 . (The biquadratic interaction (S i · S i+1 ) 2 has been omitted for simplicity.) We set J 1 = 1 throughout the paper and concentrate on the case J 2 , J 3 ≥ 0.Let us first summarize the main results obtained using extensive DMRG simulations and exact diagonalizations (ED). The phase diagram as a function of J 2 and J 3 consists of three phases, each of which may be schematically illustrated by a diagram with lines indicating valence bond singlets formed between various site, (see Fig. 1): a Haldane phase with one valence bond per J 1 bond, a next-nearest neighbor (NNN)-Haldane phase with one valence-bond per J 2 bond, and a dimerized phase with two valence-bonds on every other J 1 bond. Th...

The phase diagram of the spin-1 chain with bilinear-biquadratic and next-nearest neighbor interactions, recently investigated by Pixley, Shashi and Nevidomskyy [Phys. Rev. B 90, 214426 (2014)], has been revisited in the light of results we have recently obtained on a similar model. Combining extensive Density Matrix Renormalization Group (DMRG) simulations with conformal-field theory arguments, we confirm the presence of the three phases identified by Pixley et al, a Haldane phase, a next-nearest neighbor (NNN) Haldane phase, and a dimerized phase, but we come to significantly different conclusions regarding the nature of the phase transitions to the dimerized phase: i) We provide numerical evidence of a continuous Ising transition between the NNN-Haldane phase and the dimerized phase; ii) We show that the tri-critical end point, where the continuous transition between the Haldane phase and the dimerized phase turns into a first order transition, is distinct from the triple point where the three phases meet; iii) Finally, we demonstrate that the tri-critical end point is in the same Wess-Zumino-Witten (WZW) SU(2)2 universality class as the continuous transition line that ends at this point. I. MOTIVATIONTwo years ago, the phase diagram of the bilinearbiquadratic spin-1 chain with next-nearest neighbor interaction has been mapped out by Pixley, Shashi and Nevidomskyy 1 . It consists of three phases, and the nature of the phase transitions has been determined using Density Matrix Renormalization Group (DMRG) and field-theory arguments. More recently, we have investigated a similar model in which the biquadratic interaction is replaced by a three-site interaction that provides the appropriate generalization of the spin-1/2 Majumdar-Ghosh chain 2 . Much to our surprise, while the competing phases are the same as for the model with biquadratic interaction -Haldane, NNN-Haldane (called NNN-AKLT in Ref.[1]) and dimerized -we came to significantly different conclusions regarding the transitions between them. The aim of this comment is to re-investigate the nature of the phase transitions in the model with biquadratic interactions along the lines of Ref.[2]. As we will see, this leads to a new phase diagram that turns out to be qualitatively similar to that of the model with three-site interactions. II. PHASE DIAGRAMThe J 1 −J 2 −J b model is described by the Hamiltonian: Below, with the help of extensive density matrix renormalization group (DMRG) 3-6 calculations, we will demonstrate that: i) The phase transition between the NNN-Haldane phase and the dimerized phase is continuous and in the Ising universality class, and not first order;ii) The continuous WZW SU(2) 2 transition starts at the Takhtajan-Babujian (TB) point and terminates at a tricritical point that is distinct from the triple point; iii) Beyond the tri-critical point, the phase transition between the Haldane phase and the dimerized phase is first order; iv) The tri-critical point is in the same WZW SU(2) 2 universality class as the critical line that ends...

We show that, in certain circumstances, exact excitation energies appear as locally siteindependent (or flat) modes if one records the excitation spectrum of the effective Hamiltonian while sweeping through the lattice in the variational Matrix Product State formulation of the Density Matrix Renormalization Group (DMRG), a remarkable property since the effective Hamiltonian is only constructed to target the ground state. Conversely, modes that are very flat over several consecutive iterations are systematically found to correspond to faithful excitations. We suggest to use this property to extract accurate information about excited states using the standard ground state algorithm. The results are spectacular for critical systems, for which the low-energy conformal tower of states can be obtained very accurately at essentially no additional cost, as demonstrated by confirming the predictions of boundary conformal field theory for two simple minimal modelsthe transverse-field Ising model and the critical three-state Potts model. This approach is also very efficient to detect the quasi-degenerate low-energy excitations in topological phases, and to identify localized excitations in systems with impurities. Finally, using the variance of the Hamiltonian as a criterion, we assess the accuracy of the resulting Matrix Product State representations of the excited states.