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arXiv:1103.3368 · Physical Review B

Integrable modification of the critical Chalker-Coddington network model

Yacine Ikhlef, Paul Fendley, John Cardy

Published Oct 3, 2011• 21 citations
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Abstract

We consider the Chalker-Coddington network model for the integer quantum Hall effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models with two loop flavors. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra and parameterized by the loop fugacity $n$. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-gas theory, whereas the other two branches (3,4) couple the two loop flavors, and relate to an $\mathrm{SU}{(2)}_{r}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{r}/\mathrm{SU}{(2)}_{2r}$ Wess-Zumino-Witten (WZW) coset model for the particular values $n=\ensuremath{-}2\mathrm{cos}[\ensuremath{\pi}/(r+2)]$, where $r$ is a positive integer. The truncated Chalker-Coddington model is the $n=0$ point of branch 4. By numerical diagonalization, we find that its universality class is neither an analytic continuation of the WZW coset nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.

DOI
10.1103/physrevb.84.144201

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We consider the Chalker-Coddington network model for the integer quantum Hall effect, and examine the possibility of solving it exactly. In the supersymmetric path integral framework, we introduce a truncation procedure, leading to a series of well-defined two-dimensional loop models with two loop flavors. In the phase diagram of the first-order truncated model, we identify four integrable branches related to the dilute Birman-Wenzl-Murakami braid-monoid algebra and parameterized by the loop fugacity nnn. In the continuum limit, two of these branches (1,2) are described by a pair of decoupled copies of a Coulomb-gas theory, whereas the other two branches (3,4) couple the two loop flavors, and relate to an SU(2)r\ifmmode×\else\texttimes\fiSU(2)r/SU(2)2r\mathrm{SU}{(2)}_{r}\ifmmode\times\else\texttimes\fi{}\mathrm{SU}{(2)}_{r}/\mathrm{SU}{(2)}_{2r}SU(2)r​\ifmmode×\else\texttimes\fiSU(2)r​/SU(2)2r​ Wess-Zumino-Witten (WZW) coset model for the particular values n=\ensuremath−2cos[\ensuremathπ/(r+2)]n=\ensuremath{-}2\mathrm{cos}[\ensuremath{\pi}/(r+2)]n=\ensuremath−2cos[\ensuremathπ/(r+2)], where rrr is a positive integer. The truncated Chalker-Coddington model is the n=0n=0n=0 point of branch 4. By numerical diagonalization, we find that its universality class is neither an analytic continuation of the WZW coset nor the universality class of the original Chalker-Coddington model. It constitutes rather an integrable, critical approximation to the latter.
The supersymmetric reformulation of physical observables in the Chalker-Coddington model (CC) for the plateau transition in the integer quantum Hall effect leads to a reformulation of its critical properties in terms of a 2D non-compact loop model or a 1D non-compact gl(2|2) spin chain. Following a proposal by Ikhlef, Fendley and Cardy [11], we define and study a series of truncations of these loop models and spin chains, involving a finite and growing number of degrees of freedom per site. The case of the first truncation is solved analytically using the Bethe-ansatz. It is shown to exhibit many of the qualitative features expected for the untruncated theory, including a quadratic spectrum of exponents with a continuous component, and a normalizable ground state below that continuum. Quantitative properties are however at odds with the results of simulations on the CC model. Higher truncations are studied only numerically. While their properties are found to get closer to those of the CC model, it is not clear whether this is a genuine effect, or the result of strong finite-size corrections.
We show that the localization transition in the integer quantum Hall effect as described by the Chalker-Coddington network model is quantum critical. We first map the anisotropic network model to the problem of diagonalizing a one-dimensional non-Hermitian noncompact supersymmetric lattice Hamiltonian of interacting bosons and fermions. Its behavior is investigated numerically using the density matrix renormalization group method, and critical behavior is found at the plateau transition. This result is confirmed by a generalization of the Lieb-Schultz-Mattis theorem.
We study a network model on the kagome lattice (NMKL). This model generalizes the Chalker-Coddington network model for the integer quantum Hall transition. Unlike random network models we studied earlier, the geometry of the kagome lattice is regular. Therefore, we expect that the critical behavior of the NMKL should be the same as that of the Chalker-Coddington model. We numerically compute the localization length index $\ensuremath{\nu}$ in the NKML. Our result $\ensuremath{\nu}=2.658\ifmmode\pm\else\textpm\fi{}0.046$ is close to Chalker-Coddington model values obtained in a number of recent papers. We also map the NMKL to the Dirac fermions in random potentials and in a fixed periodic curvature background. The background turns out irrelevant at long scales. Our numerical and analytical results confirm our expectation of the universality of critical behavior on regular network models.
We continue our work on lattice models of webs, which generalise the well-known loop models to allow for various kinds of bifurcations [1] , [2] . Here we define new web models corresponding to each of the rank-two spiders considered by Kuperberg [3] . These models are based on the A 2 , G 2 and B 2 Lie algebras, and their local vertex configurations are intertwiners of the corresponding q -deformed quantum algebras. In all three cases we define a corresponding model on the hexagonal lattice, and in the case of B 2 also on the square lattice . For specific root-of-unity choices of q , we show the equivalence to a number of three- and four-state spin models on the dual lattice. The main result of this paper is to exhibit integrable manifolds in the parameter spaces of each web model. For q on the unit circle, these models are critical and we characterise the corresponding conformal field theories via numerical diagonalisation of the transfer matrix. In the A 2 case we find two integrable regimes. The first one contains a dense and a dilute phase, for which we have analytic control via a Coulomb gas construction, while the second one is more elusive and likely conceals non-compact physics . Three particular points correspond to a three-state spin model with plaquette interactions, of which the one in the second regime appears to present a new universality class. In the G 2 case we identify four regimes numerically. The B 2 case is too unwieldy to be studied numerically in the general case, but it found analytically to contain a simpler sub-model based on generators of the dilute Birman-Murakami-Wenzl algebra.
A careful study of the supersymmetric version of Pruisken’s nonlinear σ model for the integer quantum Hall effect is presented. The lattice regularized model is cast in Hamiltonian form by taking the anisotropic limit and interpreting the topological density as an alternating sum of Wess-Zumino terms. It is argued that the relevant large-scale physics of the model is preserved by projection of the quantum Hamiltonian on its sector of degenerate strong-coupling ground states. For values of the Hall conductivity close to e 2 /2h (mod e 2 /h), where a delocalization transition occurs, this yields the Hamiltonian of a quantum superspin chain which is closely related to an anisotropic version of the Chalker-Coddington model. The relation implies that the ratio of magnetic length over potential correlation length is an irrelevant parameter at the transition. The superspin chain resembles a 1d isotropic antiferromagnet with spin 1/2. It has an alternating structure which however permits an invariance under translation by one site. The conductance coefficients of a quantum Hall system with N small contacts translate into N-superspin correlation functions which are governed by conformal invariance. The superspin formalism provides a framework for studying the crossover from classical to quantum percolation. It does not however encompass the frequency-dependent correlations of wave amplitudes at criticality.
We construct and study a chiral Sachdev-Ye (SY) model consisting of NNN chiral SU(M)1\mathrm{SU}{(M)}_{1}SU(M)1​ Wess-Zumino-Witten (WZW) models with current-current interactions among each other, which generalizes the 0+1d0+1\mathrm{d}0+1d quantum chaotic SY spin model into 1+1d1+1\mathrm{d}1+1d chiral system with anyon excitations. Each WZW model hosts Abelian anyons as charge excitations, and may arise as the chiral edge theory of 2+1d2+1\mathrm{d}2+1d gapped topological phases. We solve the chiral SY model in two limits which show distinct quantum dynamics. The first limit is the case with uniform interactions at any integers NNN and MMM, which is integrable and decomposes into a chiral SU(M)N\mathrm{SU}{(M)}_{N}SU(M)N​ WZW model and its coset with a different speed of light. When N=M=2N=M=2N=M=2, the model maps to a free Majorana fermion model. The second limit is the large NNN and MMM limit with random interactions, which is solvable to the leading 1NM\frac{1}{NM}NM1​ order, and exhibits many-body quantum chaos in the out-of-time-ordered correlation of anyons. As the interaction strength approaches the upper limit preserving the chirality, the leading velocity-dependent Lyapunov exponent of the model saturates the maximal chaos bound $2\ensuremath{\pi}/\ensuremath{\beta}$ at temperature ${\ensuremath{\beta}}^{\ensuremath{-}1}$.