On the theory of plateau-plateau transitions in Quantum Hall Effect

The Lagrangian (action) formulation of the Chalker-Coddington network model for plateauplateau transitions in quantum Hall effect is presented based on a model of fermions hopping on Manhattan Lattice (ML). The dimensionless Landauer resistance is considered and its average is calculated over the random U(1) phases with constant distribution on the circle. The Lagrangian of the resultant model on ML is found and the corresponding R-matrix is written down. It appeared, that this model is integrable, rising hope to investigate physics of plateauplateau transitions by the exact method of powerful Algebraic Bethe Ansatz (ABA). January 2003 e-mail:[email protected], [email protected] 1. The nature and the mechanism of plateau-plateau transitions in the Integer Hall Effect (IHE) is still one of open problems in the way of its complete understanding. In the article [1] J.Chalker and P.Coddington (CC) have introduced a network model to deal with this problem and have studied it numerically. The model is phenomenological and uses the Transfer Matrix formalism. It based on the quasi-classical picture of noninteracting electrons in two space dimensions moving along the boundaries of the droplets(edge currents), which are formed by domains of the constant perpendicular magnetic fields. This domains appear due to disorder character of the magnetic field, presence of which is crucial for the Quantum Hall Effect (QHE). Close to critical point the droplets approaching each other and the quantum tunnelling of edge electrons from one to other droplet becomes essential. This phenomena causes the appearance of de-localized state after disorder over random U(1) phases are taken into account. The careful numerical investigations of the critical conductivity and the correlation length index ν within the CC network model in a numerous articles [1, 2, 3, 4] have shown excellent coincidence with the well established experimental value ν ≃ 2.3 ± 0.1 ( may be precisely 7/3) [5, 6], which essentially has motivated the considerable interest to the model and stimulated it’s further investigation up to our days [7, 8, 9, 10, 11, 13, 14, 15, 16, 19, 20]. The main problem in this direction off course is the problem of formulation of the field theory, which corresponds to critical limit of CC model after quenched disorder is taken into account. But already before the disorder one needs to find the equivalence of CC Transfer Matrix formulation on a some kind of lattice. In the early stage of investigations of the CC model D.H.Lee [7] mapped original model to an antiferromagnetic spin chain. Later Lee, together with Wang [8], have extended the model to the replica limit of the associated Hubbard model. In the articles [10] authors have developed further this ideas by mapping the problem of localization to the problem of diagonalization of some one-dimensional non-Hermitian Hamiltonian of interacting bosons and fermions. But it is absolutely clear, that one should consider this mappings as an alternative description of he CC model, rather than exact correspondence. At the same time the technique of supersymmetry was used by M.Zirnbauer to average over the disorder in CC model in the article [11] , where he has reduced the problem to the supersymmetric σ-model and an antiferromagnetic supersymmetric spin-chain. In the articles [12] the U(1) CC model was extended to spin case to describe the spin QH transitions in a system of noninteracting quasi-particles in 2D. Authors of the articles [13] have used a supersymmetry representation of such models to obtain a mapping onto the 2D classical bond percolation transition and get some critical exponents and universal ratios analytically. It is also necessary to mention recent investigation of the chiral universality class of the localization problem and corresponding modification of the original network model [20]. In the interesting article [14] the authors, instead of mapping CC model into some kind of Hamiltonian, are analyzing the hierarchy of network models by real space renormalization technique in order to understand the nature of localization-delocalization transition. The multifractal properties of the generalized CC model was investigated in the articles [15]. Some variant of CC network model was used in the articles [16, 17, 18, 19] in order to study numerically (and analytically) the random bond 2D Ising model by mapping one onto the other. Analyzing carefully the mentioned upper articles, where authors have reduced the original Transfer Matrix formulation of the CC network model (just for which the correlation length 2 index was proved is coinciding with the experimental value of QH plateau transitions) to some kind of supersymmetric spin chain problem one can find out that their exact equivalence remains obscure. In this article, developing the results obtained in our previous works [21, 22, 23], we give the exact action (Lagrangian) formulation of the original CC Transfer Matrix model as a field theory on 2D lattice. First we show, that before the disorder over U(1) phases is taken into account the CC model is equivalent to some inhomogeneous modification of the XX model in the background of U(1) field. Basic element of this construction is the fermionized version of the standard R-operator of the XX model, but the Transfer Matrix is the staggered product of π/2-rotated R-operators. We prove, that in one particle sector the matrix elements of the Transfer Matrix of the defined model precisely reproduces the Transfer Matrix of the CC model. Then, by introducing the fermionic coherent states, we write down the action of the CC model on the 2D Manhattan Lattice (ML). Further,in order to investigate the presence of delocalized states, we consider the Landauer resistance in the model and take into account the disorder over U(1) phases. As it was argued in the articles [24], the averaged Landauer resistance in quasi-one dimensional systems defines the double of localization length of the theory. We have considered homogenous distribution of random U(1) phases over the circle, calculated the average 〈T ⊗ T †〉(T is the Transfer Matrix of the CC model) and found the R-matrix and the action of corresponding model. The result is written in terms of two type (spin up and spin down) Fermi fields. It appeared, that the R-matrix we have found is fulfilling the Yang-Baxter Equations (Y BE), which were defined in the article [22] for the models with staggered disposition of Rmatrices. This means that the resultant model is integrable. It rises hopes, that by use of powerful technique of Algebraic Bethe Ansatz (ABA) [25, 26] one can investigate and calculate the critical properties of the QHE exactly. 2.The basic element of our construction is the Ri,j-matrix(operator) of the Algebraic Bethe Ansatz technique, which acts on a direct product of the linear spaces Vi and Vj of the quantum states at the chain sites i and j respectively. Ři,j = Vi ⊗ Vj → V ′ i ⊗ V ′ j . (1) Let us attach the Fock spaces Vj of scalar fermions c + j , cj to each site of the chain and consider the operator forms of two types of R-matrices of the XX-model in braid formalism ر 2j,2j±1 = = a±n2jn2j±1 + a±(1− n2j)(1− n2j±1) + n2j(1− n2j±1) (2) + (a±a± + b±b±)n2j±1(1− n2j) + b±c + 2jc2j±1 + b±c + 2j±1c2j = : e[± + 2j±1c2j+b±c + 2jc2j±1+(a±−1)c + 2jc2j+(1−a±)c + 2j±1c2j±1] :, corresponding to two type of acts of scattering in the CC model, as it is drown in Figure1. In the expression (2) the symbol : : means normal ordering of fermionic operators in the even sites and anti-normal (hole) ordering in the odd sites. The convenience of this choose will be clear later. The dot lines in the picture represents standard view of the R-matrices, while solid lines are convenient in a language of fermions with hopping parameters a± and b±. Each