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arXiv:cond-mat/9704115 · Physical review. B, Condensed matter

Effective theory for midgap states in doped spin-ladder and spin-Peierls systems: Liouville quantum mechanics

David G. Shelton, Alexei M. Tsvelik

Published Jun 1, 1998• 49 citations
arXivHTMLPDF

Abstract

In gapped spin-ladder and spin-Peierls systems the introduction of disorder, for example by doping, leads to the appearance of low-energy midgap states. The fact that these strongly correlated systems can be mapped onto one-dimensional noninteracting fermions provides a rare opportunity to explore systems that have both strong interactions and disorder. In this paper we will show that the statistics of the zero energy midgap wave functions ${\ensuremath{\psi}}_{0}(x)$ in these models can be effectively described by Liouville quantum mechanics. This enables us to calculate averages over disorder of the products ${\ensuremath{\psi}}_{0}^{2}{(x}_{1}){\ensuremath{\psi}}_{0}^{2}{(x}_{2})\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\psi}}_{0}^{2}{(x}_{N})$ (the explicit calculation is performed for $N=2,3)$. We find that while these midgap states are typically weakly correlated, their disorder averaged correlations are power law. This discrepancy arises because the correlations are not self-averaging and averages of the wave functions are dominated by anomalously strongly correlated configurations; a fact that is not always appreciated in the literature.

DOI
10.1103/physrevb.57.14242

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In gapped spin-ladder and spin-Peierls systems the introduction of disorder, for example by doping, leads to the appearance of low-energy midgap states. The fact that these strongly correlated systems can be mapped onto one-dimensional noninteracting fermions provides a rare opportunity to explore systems that have both strong interactions and disorder. In this paper we will show that the statistics of the zero energy midgap wave functions \ensuremathψ0(x){\ensuremath{\psi}}_{0}(x)\ensuremathψ0​(x) in these models can be effectively described by Liouville quantum mechanics. This enables us to calculate averages over disorder of the products \ensuremathψ02(x1)\ensuremathψ02(x2)\ensuremath⋅\ensuremath⋅\ensuremath⋅\ensuremathψ02(xN){\ensuremath{\psi}}_{0}^{2}{(x}_{1}){\ensuremath{\psi}}_{0}^{2}{(x}_{2})\ensuremath{\cdot}\ensuremath{\cdot}\ensuremath{\cdot}{\ensuremath{\psi}}_{0}^{2}{(x}_{N})\ensuremathψ02​(x1​)\ensuremathψ02​(x2​)\ensuremath⋅\ensuremath⋅\ensuremath⋅\ensuremathψ02​(xN​) (the explicit calculation is performed for N=2,3)N=2,3)N=2,3). We find that while these midgap states are typically weakly correlated, their disorder averaged correlations are power law. This discrepancy arises because the correlations are not self-averaging and averages of the wave functions are dominated by anomalously strongly correlated configurations; a fact that is not always appreciated in the literature.
Quasi-one-dimensional spin-Peierls and spin-ladder systems are characterized by a gap in the spin-excitation spectrum, which can be modeled at low energies by that of Dirac fermions with a mass. In the presence of disorder, these systems can still be described by a Dirac fermion model, but with a random mass. Some peculiar properties, like the Dyson singularity in the density of states, are well known and attributed to creation of low-energy states due to the disorder. We take one step further and study single-particle correlations by means of Berezinskii's diagram technique. We find that, at low energy $\ensuremath{\epsilon}$, the single-particle Green function decays in real space like $G(x,\ensuremath{\epsilon})\ensuremath{\propto}{(1/x)}^{3/2}$. It follows that at these energies the correlations in the disordered system are strong --- even stronger than in the pure system without the gap. At distances $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}\ensuremath{\propto}{\mathrm{ln}}^{2}(1/\ensuremath{\epsilon})$, the Green function crosses over to a standard exponential decay. The scale ${L}_{\ensuremath{\epsilon}}$ is different from the Thouless localization length ${\ensuremath{\lambda}}_{\ensuremath{\epsilon}}\ensuremath{\propto}\mathrm{ln}(1/\ensuremath{\epsilon})$. Additionally, we study the effects of boundaries on the local density of states. We find that the latter is logarithmically (in the energy) enhanced close to the boundary. This enhancement decays into the bulk as 1/x1/\sqrt{x}1/x​ saturating at $x\ensuremath{\sim}{L}_{\ensuremath{\epsilon}}.$ We also discuss some implications of these results for the spin systems and their relation to the investigations based on the real-space renormalization group approach.
The influence of nonmagnetic doping on the thermodynamic properties of two-leg S = 1/2 spin ladders is studied in this paper. It is shown that, for a weak interchain coupling, the problem can be mapped onto a model of random mass Dirac (Majorana) fermions. We investigate in detail the structure of the fermionic states localized at an individual mass kink (zero modes) in the framework of a generalized Dirac model. The low-temperature thermodynamic properties are dominated by these zero modes. We use the single-fermion density of states, known to exhibit the Dyson singularity in the zero-energy limit, to construct the thermodynamics of the spin ladder. In particular, we find that the magnetic susceptibility χχχ diverges at T→0T\to 0T→0 as 1/Tln⁡2(1/T)1/T\ln^2(1/T)1/Tln2(1/T), and the specific heat behaves as C∝1/ln⁡3(1/T)C\propto 1/\ln^3(1/T)C∝1/ln3(1/T). The predictions on magnetic susceptibility are consistent with the most recent results of quantum Monte Carlo simulations on doped ladders with randomly distributed impurities. We also calculate the average staggered magnetic susceptibility induced in the system by such defects.
A proper theoretical description of doped antiferromagnets is until now an unresolved issue. In this work we study chain and ladder systems, and in particularly we focus on the physically most interesting two leg ladder. Our starting point is the Spin Fermion model in which an antiferromagnetic background of localized spins interacts with mobile holes via a rotation invariant Kondo-like term. The interesting region of the phase diagram is that close to the Mott-insulator transition where the doping delta is zero. At zero doping in fact the system is an insulator and the spins organize themselves in a spin liquid, a rotational invariant state characterized by a finite correlation length and an energy gap (a spin gap) above the ground state. Such a state is also the ground state of a field theory, the non linear sigma model (NLsigmaM) which is recognized as the low energy effective theory for antiferromagnetic spin ladder. The first question we answer is how such a spin state evolves as one moves off from the Mott phase by increasing the doping. Integrating out the fermions in our model we obtain an effective theory for the spins which we are able to evaluate in the continuum limit. The effective theory is again a NLsigmaM with coupling constants which depend on the concentration of dopant holes. In contrast to existing mean field calculation our theory predicts a lowering of the spin gap with doping and a consequent increase in the correlation length. Indeed a lowering of the spin gap due to doping is also observed in numerical simulation and on NMR experiments on $Sr14-xCaxCu24O41 with which we obtain very good agreement. Secondly we concentrated on the behavior of the fermions. The general paradigm of interacting fermions in one dimension is that of the Luttinger Liquid characterized by bosonic excitations and spin charge separation. By generalizing our approach we are able to access the one particle fermion propagator as a quantity averaged over the NLsigmaM which controls the spin background. We see that in the limit of zero doping the quasiparticle weight Z is non-zero in a neighborhood of the Fermi energy. This in turn implies that the Luttinger liquid parameter Krho goes to one as the doping delta goes to zero as was first argued by Schulz. Our stronger result allows us to assert that in the very low doping regime the fermions constitute a Fermi liquid.
We study the dimensional crossover from one to two dimensions in half-filled and lightly doped, weakly interacting N-leg Hubbard ladders. In this case, the Hubbard ladders are equivalent to an N-band model. Using renormalization-group techniques, we find that the half-filled ladders exhibit (in the spin sector) an odd-even effect only below a crossover energy ${E}_{c}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}\ensuremath{\alpha}\mathrm{exp}(\ensuremath{\gamma}N)]$ $(\ensuremath{\alpha}\ensuremath{\ll}1$ and $\ensuremath{\gamma}\ensuremath{\sim}1$ depend on the interaction strength and on the hopping matrix elements): Below Ec,{E}_{c},Ec​, the dominant interactions take place within band pairs $(j,N+1\ensuremath{-}j)$ [and within the band (N+1)/2(N+1)/2(N+1)/2 for N odd], such that even-leg ladders are an insulating spin liquid, while odd-leg ladders have one gapless spinon mode. In contrast, above the energy scale Ec,{E}_{c},Ec​, all bands are interacting with each other and the system is a two-dimensional- (2D)-like (insulating) antiferromagnet; we obtain an analytical expression for the Hamiltonian which is similar to the 2D Heisenberg antiferromagnet. Bosonization techniques show that in the charge sector the Mott insulator is as well below and above Ec{E}_{c}Ec​ of the same type as in N/2N/2N/2 half-filled two-leg ladders. Doping away from half-filling, we find that the effect of an increasing doping $\ensuremath{\delta}$ is very similar to decreasing the number of legs N: In both cases interactions between unpaired bands are suppressed and thereby the antiferromagnetic correlations reduced. The resulting band pairs form then insulating spin liquids and when doped, there is a spin gap, but phase coherence exists only within the band pairs. At higher doping levels ${\ensuremath{\delta}}_{c}={\ensuremath{\delta}}_{c}(N),$ phase coherence between all band pairs sets in and the system becomes a 2D-like d-wave superconductor $[{\ensuremath{\delta}}_{c}(N)\ensuremath{\rightarrow}0$ for $\stackrel{\ensuremath{\rightarrow}}{N}\ensuremath{\infty}].$
In the first part of the present Thesis, the ground state (GS) properties of the quasi-one-dimensional AB₂ Hubbard model are investigated taking the effects of charge and spin quantum fluctuations on equal footing. Using a functional integral approach, combined with a perturbative expansion in the strong-coupling regime, we obtain the Lagrangian density associated with the charge (Grassmann fields) and spin [SU(2) gauge fields] degrees of freedom. In the strong-coupling regime, we derive a perturbative low-energy theory suitable to describe the ferrimagnetic phase at half filling and the phases in the hole-doped regime. At half filling, a perturbative spin-wave analysis allows us to find the GS energy, sublattice magnetizations, and total spin per unit cell in the Lieb ferrimagnetic GS of the effective quantum Heisenberg model, in very good agreement with previous results. In the challenging hole doping regime away from half filling, we derive the corresponding t-J Hamiltonian. Under the assumption that charge and spin quantum correlations are decoupled, the evolution of the second-order spin-wave modes in the doped regime unveils the occurrence of spatially modulated spin structures and the emergence of phase separation in the presence of resonatingvalence-bond (RVB) states. We also calculate the doping-dependent GS energy and total spin per unit cell, including both Zeeman and orbital contributions, in which case it is shown that the spiral ferrimagnetic order collapses at a critical hole concentration. Notably, our analytical results in the doped regime are in very good agreement with density matrix renormalization group (DMRG) studies, where our assumption of spin-charge decoupling is numerically supported by the formation of charge-density waves in anti-phase with the modulation of the magnetic structure. In the second part, motivated by analogy with photonic lattices, we examine the edge states of a one-dimensional trimer lattice in the phases with and without inversion symmetry protection. In contrast to the Su-Schrieffer-Heeger model, we show that the edge states in the inversion-symmetry broken phase of the trimer model turn out to be chiral, i.e., instead of appearing in pairs localized at opposite edges they can appear at a single edge. Interestingly, these chiral edge states remain robust to large amounts of disorder. In addition, we use the Zak phase to characterize the emergence of degenerate edge states in the inversion-symmetric phase of the trimer model. Furthermore, we capture the essentials of the whole family of trimers through a mapping onto the commensurate off-diagonal Aubry-André-Harper model, which allows us to establish a direct connection between chiral edge modes in the two models, including the calculation of Chern numbers.We thus suggest that the chiral edge modes of the trimer lattice have a topological origin inherited from this effective mapping. Also, we find a nontrivial connection between the topological phase transition point in the trimer lattice and the one in its associated two-dimensional parent system, in agreement with results in the context of Thouless pumping in photonic lattices.
We develop a nonperturbative method to calculate the density of states (DOS) $\ensuremath{\rho}(\ensuremath{\omega})$ of the fluctuating gap model describing the low-energy physics of electrons on a disordered Peierls chain. For a real order parameter field we calculate $\ensuremath{\rho}(0)$ (i.e., the DOS at the Fermi energy) exactly as a functional of the disorder for a chain of finite length LLL. Averaging $\ensuremath{\rho}(0)$ with respect to a Gaussian probability distribution of the fluctuating Peierls order parameter, we show that for $L\ensuremath{\rightarrow}\ensuremath{\infty}$ the average $〈\ensuremath{\rho}(0)〉$ diverges for any finite value of the correlation length above the Peierls transition. Pseudogap behavior emerges only if the Peierls order parameter is finite and sufficiently large.