We study the boundary charge Q B of generic semi-infinite one-dimensional insulators with translational invariance and show that nonlocal symmetries (i.e., including translations) lead to rational quantizations p/q of Q B .In particular, we find that (up to an unknown integer) the quantization of Q B is given in integer units of 1 2 ρ and 1 2 ( ρ -1), where ρ is the average charge per site (which is a rational number for an insulator).This is a direct generalization of the known half-integer quantization of Q B for systems with local inversion or local chiral symmetries to any rational value.Quite remarkably, this rational quantization remains valid even in the presence of short-ranged electron-electron interactions as well as static random disorder (breaking translational invariance).This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions.We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models.Furthermore, for the special case of half-filling ρ = 1 2 , we present explicit results in single-channel and nearest-neighbor hopping models and identify Weyl semimetal physics at gap closing points.Our general framework also allows us to shed new light on the well-known rational quantization of soliton charges at domain walls.
We study the boundary charge Q B of generic semi-infinite one-dimensional insulators with translational invariance and show that nonlocal symmetries (i.e., including translations) lead to rational quantizations p/q of Q B .In particular, we find that (up to an unknown integer) the quantization of Q B is given in integer units of 1 2 ρ and 1 2 ( ρ -1), where ρ is the average charge per site (which is a rational number for an insulator).This is a direct generalization of the known half-integer quantization of Q B for systems with local inversion or local chiral symmetries to any rational value.Quite remarkably, this rational quantization remains valid even in the presence of short-ranged electron-electron interactions as well as static random disorder (breaking translational invariance).This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions.We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models.Furthermore, for the special case of half-filling ρ = 1 2 , we present explicit results in single-channel and nearest-neighbor hopping models and identify Weyl semimetal physics at gap closing points.Our general framework also allows us to shed new light on the well-known rational quantization of soliton charges at domain walls.
We study the boundary charge QB of generic semi-infinite one-dimensional insulators with translational invariance and show that nonlocal symmetries (i.e., including translations) lead to rational quantizations p/q of QB. In particular, we find that (up to an unknown integer) the quantization of QB is given in integer units of 1/2¯ρ and 1/2(¯ρ−1), where ρ is the average charge per site (which is a rational number for an insulator). This is a direct generalization of the known half-integer quantization of QB for systems with local inversion or local chiral symmetries to any rational value. Quite remarkably, this rational quantization remains valid even in the presence of short-ranged electron-electron interactions as well as static random disorder (breaking translational invariance). This striking stability can be traced back to the fact that local perturbations in insulators induce only local charge redistributions. We establish this result with complementary methods including density matrix renormalization group calculations, bosonization methods, and exact solutions for particular lattice models. Furthermore, for the special case of half-filling ¯ρ=1/2, we present explicit results in single-channel and nearest-neighbor hopping models and identify Weyl semimetal physics at gap closing points. Our general framework also allows us to shed new light on the well-known rational quantization of soliton charges at domain walls.
We use analytic (mean-field) and density-matrix renormalization group (DMRG) methods to study a simplified one-dimensional version of the SSH-Holstein model of electron-phonon interactions with one electron per site (half filling) where, as a function of the ratio of the two couplings, three distinct insulating phases arise - a site-centered charge-density-wave (CDW), a bond-centered bond-density-wave (BDW), and an intermediate ferroelectric (inversion-symmetry breaking) phase with coexisting CDW and BDW order. In the intermediate phase our DMRG results establish the existence of point-like soliton excitations with irrational charge (that depends on a ratio of coupling constants) and either spin 1/2 or spin 0, and a surprisingly light effective mass.
The boundary charge that accumulates at the edge of a one-dimensional single-channel insulator is known to possess the universal property that its change under a lattice shift towards the edge by one site is given by the sum of the average bulk electronic density and a topologically invariant contribution, restricted to the values 0 and $\ensuremath{-}1$ [Pletyukhov et al., Phys. Rev. B 101, 165304 (2020)]. This quantized contribution is associated with particle-hole duality, ensures charge conservation, and fixes the mod(1) ambiguity appearing in the modern theory of polarization. In the present paper we generalize the above-mentioned single-channel results to the multichannel case by employing the technique of boundary Green's functions. We show that the topological invariant associated with the change in boundary charge under a lattice shift in multichannel models can be expressed as a winding number of a certain combination of components of bulk Green's functions as a function of the complex frequency, as it encircles the section of the energy axis that corresponds to the occupied part of the spectrum. We observe that this winding number is restricted to values ranging from $\ensuremath{-}{N}_{c}$ to zero, where Nc is the number of channels (orbitals) per site. Furthermore, we consider translationally invariant one-dimensional multichannel models with an impurity and introduce topological indices which correspond to the quantized charge that accumulates around said impurity. These invariants are again given in terms of winding numbers of combinations of components of bulk Green's functions. Through this construction we provide a rigorous mathematical proof of the so-called nearsightedness principle formulated by Kohn [Kohn, Phys. Rev. Lett. 76, 3168 (1996)] for noninteracting multichannel lattice models.
It is well known that integrable models associated to rational R matrices give rise to certain non-Abelian symmetries known as Yangians. Analogously boundary symmetries arise when general but still integrable boundary conditions are implemented, as originally argued by Delius, Mackay, and Short from the field theory point of view, in the context of the principal chiral model on the half-line. In the present study we deal with a discrete quantum mechanical system with boundaries, that is the N site gl(n) open quantum spin chain. In particular, the open spin chain with two distinct types of boundary condition known as soliton preserving and soliton nonpreserving is considered. For both types of boundaries we present a unified framework for deriving the corresponding boundary nonlocal charges directly at the quantum level. The nonlocal charges are simply coproduct realizations of particular boundary quantum algebras called boundary or twisted Yangians, depending on the choice of boundary conditions. Finally, with the help of linear intertwining relations between the solutions of the reflection equation and the generators of the boundary or twisted Yangians we are able to exhibit the exact symmetry of the open spin chain, namely we show that a number of the boundary nonlocal charges are in fact conserved quantities.
The opening of a charge gap driven by interaction is a fingerprint of the transition to a Mott insulating phase. In strongly correlated low-dimensional quantum systems, it can be associated to the ordering of hidden non-local operators. For Fermionic 1D models, in the presence of spin–charge separation and short-ranged interaction, a bosonization analysis proves that such operators are the parity and/or string charge operators. In fact, a finite fractional non-local parity charge order is also capable of characterizing some two-dimensional Mott insulators, in both the Fermionic and the bosonic cases. When string charge order takes place in 1D, degenerate edge modes with fractional charge appear, peculiar of a topological insulator. In this article, we review the above framework, and we test it to investigate through density-matrix-renormalization-group (DMRG) numerical analysis the robustness of both hidden orders at half-filling in the 1D Fermionic Hubbard model extended with long range density-density interaction. The preliminary results obtained at finite size including several neighbors in the case of dipolar, screened and unscreened repulsive Coulomb interactions, confirm the phase diagram of the standard extended Hubbard model. Besides the trivial Mott phase, the bond ordered and charge density wave insulating phases are also not destroyed by longer ranged interaction, and still manifest hidden non-local orders.
Local moment formation is ubiquitous in disordered semiconductors such as Si:P, where it is observed both in the metallic and the insulating regimes. Here, we focus on local moment behavior in disordered insulators, which arises from short-ranged, repulsive electron-electron interactions. Using density matrix renormalization group and strong-disorder renormalization group methods, we study paradigmatic models of interacting insulators: one-dimensional Hubbard chains with quenched randomness. In chains with either random fermion hoppings or random chemical potentials, both at and away from half-filling, we find exponential decay of disorder-averaged charge and fermion two-point correlations, but power-law decay of disorder-averaged spin correlations that are indicative of the random singlet phase. The numerical results can be understood qualitatively by appealing to the large-interaction limit of the Hubbard chain, in which a remarkably simple picture emerges.