Infinite matrix product states, boundary conformal field theory, and the open Haldane-Shastry model

We show that infinite Matrix Product States (MPS) constructed from conformal field theories can describe ground states of one-dimensional critical systems with open boundary conditions. To illustrate this, we consider a simple infinite MPS for a spin-1/2 chain and derive an inhomogeneous open Haldane-Shastry model. For the spin-1/2 open Haldane-Shastry model, we derive an exact expression for the two-point spin correlation function. We also provide an SU($n$) generalization of the open Haldane-Shastry model and determine its twisted Yangian generators responsible for the highly degenerate multiplets in the energy spectrum.Comment: 5+7 pages, 4 figures, published version, a typo in the twisted Yangian generators corrected (thanks to the authors of arXiv:1511.08613 for pointing out this typo

Majorana Edge States in Interacting Two-chain Ladders of Fermions

In this work we study interacting spinless fermions on a two-chain ladder with inter-chain pair tunneling while single-particle tunneling is suppressed at low energy. The model embodies a $\mathbb{Z}_2$ symmetry associated with the fermion parity on each chain. We find that when the system is driven to the strong-coupling phase by the pair tunneling, Majorana excitations appear on the boundary. Such Majorana edge states correspond to two-fold degeneracy of ground states distinguished by different fermion parity on each chain, thus representing a generalization of one-dimensional topological superconductors. We also characterize the stability of the ground state degeneracy against local perturbations. Lattice fermion models realizing such effective field theory are discussed.Comment: 6 pages, 1 figur

Parent Hamiltonians for lattice Halperin states from free-boson conformal field theories

We introduce a family of many-body quantum states that describe interacting spin one-half hard-core particles with bosonic or fermionic statistics on arbitrary one- and two-dimensional lattices. The wave functions at lattice filling fraction $\nu=2/(2m+1)$ are derived from deformations of the Wess-Zumino-Witten model $\mathfrak{su}(3)_1$ and are related to the $(m+1,m+1,m)$ Halperin fractional quantum Hall states. We derive long-range SU(2) invariant parent Hamiltonians for these states which in two dimensions are chiral $t$-$J$-$V$ models with additional three-body interaction terms. In one dimension we obtain a generalisation to open chains of a periodic inverse-square $t$-$J$-$V$ model proposed in [Z. N. C. Ha and F. D. M. Haldane, Phys. Rev. B $\mathbf{46}$, 9359 (1992)]. We observe that the gapless low-energy spectrum of this model and its open-boundary generalisation can be described by rapidity sets with the same generalised Pauli exclusion principle. A two-component compactified free boson conformal field theory is identified that has the same central charge and scaling dimensions as the periodic bosonic inverse-square $t$-$J$-$V$ model.Comment: 19 pages, 2 figures. v2: minor corrections and partial rewriting of section IV B

Symmetry-protected intermediate trivial phases in quantum spin chains

Symmetry-protected trivial (SPt) phases of matter are the product-state analogue of symmetry-protected topological (SPT) phases. This means, SPt phases can be adiabatically connected to a product state by some path that preserves the protecting symmetry. Moreover, SPt and SPT phases can be adiabatically connected to each other when interaction terms that break the symmetries protecting the SPT order are added in the Hamiltonian. It is also known that spin-1 SPT phases in quantum spin chains can emerge as effective intermediate phases of spin-2 Hamiltonians. In this paper we show that a similar scenario is also valid for SPt phases. More precisely, we show that for a given spin-2 quantum chain, effective intermediate spin-1 SPt phases emerge in some regions of the phase diagram, these also being adiabatically connected to non-trivial intermediate SPT phases. We characterize the phase diagram of our model by studying quantities such as the entanglement entropy, symmetry-related order parameters, and 1-site fidelities. Our numerical analysis uses Matrix Product States (MPS) and the infinite Time-Evolving Block Decimation (iTEBD) method to approximate ground states of the system in the thermodynamic limit. Moreover, we provide a field theory description of the possible quantum phase transitions between the SPt phases. Together with the numerical results, such a description shows that the transitions may be described by Conformal Field Theories (CFT) with central charge c=1. Our results are in agreement, and further generalize, those in [Y. Fuji, F. Pollmann, M. Oshikawa, Phys. Rev. Lett. 114, 177204 (2015)].Comment: 7 pages, 5 figures, 1 table, revised version. Accepted in PR

All spin-1 topological phases in a single spin-2 chain

Here we study the emergence of different Symmetry-Protected Topological (SPT) phases in a spin-2 quantum chain. We consider a Heisenberg-like model with bilinear, biquadratic, bicubic, and biquartic nearest-neighbor interactions, as well as uniaxial anisotropy. We show that this model contains four different effective spin-1 SPT phases, corresponding to different representations of the $(\mathbb{Z}_2 \times \mathbb{Z}_2) + T$ symmetry group, where $\mathbb{Z}_2$ is some $\pi$-rotation in the spin internal space and $T$ is time-reversal. One of these phases is equivalent to the usual spin-1 Haldane phase, while the other three are different but also typical of spin-1 systems. The model also exhibits an $SO(5)$-Haldane phase. Moreover, we also find that the transitions between the different effective spin-1 SPT phases are continuous, and can be described by a $c=2$ conformal field theory. At such transitions, indirect evidence suggests a possible effective field theory of four massless Majorana fermions. The results are obtained by approximating the ground state of the system in the thermodynamic limit using Matrix Product States via the infinite Time Evolving Block Decimation method, as well as by effective field theory considerations. Our findings show, for the first time, that different large effective spin-1 SPT phases separated by continuous quantum phase transitions can be stabilized in a simple quantum spin chain.Comment: 7 pages, 6 figures, revised version. To appear in PR

Momentum polarization: an entanglement measure of topological spin and chiral central charge

Topologically ordered states are quantum states of matter with topological ground state degeneracy and quasi-particles carrying fractional quantum numbers and fractional statistics. The topological spin $\theta_a=2\pi h_a$ is an important property of a topological quasi-particle, which is the Berry phase obtained in the adiabatic self-rotation of the quasi-particle by $2\pi$. For chiral topological states with robust chiral edge states, another fundamental topological property is the edge state chiral central charge $c$. In this paper we propose a new approach to compute the topological spin and chiral central charge in lattice models by defining a new quantity named as the momentum polarization. Momentum polarization is defined on the cylinder geometry as a universal subleading term in the average value of a "partial translation operator". We show that the momentum polarization is a quantum entanglement property which can be computed from the reduced density matrix, and our analytic derivation based on edge conformal field theory shows that the momentum polarization measures the combination $h_a-\frac{c}{24}$ of topological spin and central charge. Numerical results are obtained for two example systems, the non-Abelian phase of the honeycomb lattice Kitaev model, and the $\nu=1/2$ Laughlin state of a fractional Chern insulator described by a variational Monte Carlo wavefunction. The numerical results verifies the analytic formula with high accuracy, and further suggests that this result remains robust even when the edge states cannot be described by a conformal field theory. Our result provides a new efficient approach to characterize and identify topological states of matter from finite size numerics.Comment: 13 pages, 8 figure

Entanglement and SU(n) symmetry in one-dimensional valence bond solid states

Here we evaluate the many-body entanglement properties of a generalized SU(n) valence bond solid state on a chain. Our results follow from a derivation of the transfer matrix of the system which, in combination with symmetry properties, allows for a new, elegant and straightforward evaluation of different entanglement measures. In particular, the geometric entanglement per block, correlation length, von Neumann and R\'enyi entropies of a block, localizable entanglement and entanglement length are obtained in a very simple way. All our results are in agreement with previous derivations for the SU(2) case.Comment: 4 pages, 2 figure