Chalker-Coddington Network Model Research Articles

Engineering Plateau Phase Transition in Quantum Anomalous Hall Multilayers.

The plateau phase transition in quantum anomalous Hall (QAH) insulators corresponds to a quantum state wherein a single magnetic domain gives way to multiple domains and then reconverges back to a single magnetic domain. The layer structure of the sample provides an external knob for adjusting the Chern number C of the QAH insulators. Here, we employ molecular beam epitaxy to grow magnetic topological insulator multilayers and realize the magnetic field-driven plateau phase transition between two QAH states with odd Chern number change ΔC. We find that critical exponents extracted for the plateau phase transitions with ΔC = 1 and ΔC = 3 in QAH insulators are nearly identical. We construct a four-layer Chalker-Coddington network model to understand the consistent critical exponents for the plateau phase transitions with ΔC = 1 and ΔC = 3. This work will motivate further investigations into the critical behaviors of plateau phase transitions with different ΔC in QAH insulators.

Supermetal-insulator transition in a non-Hermitian network model

We study a non-Hermitian and non-unitary version of the two-dimensional Chalker-Coddington network model with balanced gain and loss. This model belongs to the class D^dagger with particle-hole symmetry^dagger and hosts both the non-Hermitian skin effect as well as exceptional points. By calculating its two-terminal transmission, we find a novel contact effect induced by the skin effect, which results in a non-quantized transmission for chiral edge states. In addition, the model exhibits an insulator to 'supermetal' transition, across which the transmission changes from exponentially decaying with system size to exponentially growing with system size. In the clean system, the critical point separating insulator from supermetal is characterized by a non-Hermitian Dirac point that produces a quantized critical transmission of 4, instead of the value of 1 expected in Hermitian systems. This change in critical transmission is a consequence of the balanced gain and loss. When adding disorder to the system, we find a critical exponent for the divergence of the localization length \nu \approx 1, which is the same as that characterizing the universality class of two-dimensional Hermitian systems in class D. Our work provides a novel way of exploring the localization behavior of non-Hermitian systems, by using network models, which in the past proved versatile tools to describe Hermitian physics.

Three-dimensional network model for strong topological insulator transitions

We construct a three-dimensional (3D), time-reversal symmetric generalization of the Chalker-Coddington network model for the integer quantum Hall transition. The novel feature of our network model is that in addition to a weak topological insulator phase already accommodated by the network model framework in the pre-existing literature, it hosts strong topological insulator phases as well. We unambiguously demonstrate that strong topological insulator phases emerge as intermediate phases between a trivial insulator phase and a weak topological phase. Additionally, we found a non-local transformation that relates a trivial insulator phase and a weak topological phase in our network model. Remarkably, strong topological phases are mapped to themselves under this transformation. We show that upon adding sufficiently strong disorder the strong topological insulator phases undergo phase transitions into a metallic phase. We numerically determine the critical exponent of the insulator-metal transition. Our network model explicitly shows how a semi-classical percolation picture of topological phase transitions in 2D can be generalized to 3D and opens up a new venue for studying 3D topological phase transitions.

Critical Behavior and Universal Signature of an Axion Insulator State.

Recently, the search for an axion insulator state in the ferromagnetic-3D topological insulator (TI) heterostructure and MnBi_{2}Te_{4} has attracted intense interest. However, its detection remains difficult in experiments. We systematically investigate the disorder-induced phase transition of the axion insulator state in a 3D TI with antiparallel magnetization alignment surfaces. It is found that there exists a 2D disorder-induced phase transition on the surfaces of the 3D TI which shares the same universality class with the quantum Hall plateau to plateau transition. Then, we provide a phenomenological theory which maps the random mass Dirac Hamiltonian of the axion insulator state into the Chalker-Coddington network model. Therefore, we propose probing the axion insulator state by investigating the universal signature of such a phase transition in the ferromagnetic-3D TI heterostructure and MnBi_{2}Te_{4}. Our findings not only show a global phase diagram of the axion insulator state, but also stimulate further experiments to probe it.

Numerical Study of a Dual Representation of the Integer Quantum Hall Transition.

We study the critical properties of the noninteracting integer quantum Hall to insulator transition (IQHIT) in a "dual" composite-fermion (CF) representation. A key advantage of the CF representation over electron coordinates is that at criticality CF states are delocalized at all energies. The CF approach thus enables us to study the transition from a new vantage point. Using a lattice representation of CF mean-field theory, we compute the critical and multifractal exponents of the IQHIT. We obtain ν=2.56±0.02 and η=0.51±0.01, both of which are consistent with the predictions of the Chalker-Coddington network model formulated in the electron representation.

Critical behavior at the integer quantum Hall transition in a network model on the kagome lattice

We study a network model on the Kagome lattice (NMKL). This model generalizes the Chalker-Coddington (CC) 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 CC model. We numerically compute the localization length index $\nu$ in the NKML. Our result $\nu= 2.658 \pm 0.046$ is close to CC 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.

The integer quantum Hall plateau transition is a current algebra after all

The scaling behavior near the transition between plateaus of the Integer Quantum Hall Effect (IQHE) has traditionally been interpreted on the basis of a two-parameter renormalization group (RG) flow conjectured from Pruisken's non-linear sigma model (NLσM). Yet, the conformal field theory (CFT) describing the critical point remained elusive, and only fragments of a quantitative analytical understanding existed up to now. In the present paper we carry out a detailed analysis of the current-current correlation function for the conductivity tensor, initially in the Chalker-Coddington network model for the IQHE plateau transition and then in its exact reformulation as a supersymmetric vertex model. We develop a heuristic argument for the continuum limit of the non-local conductivity response function at criticality and thus identify a non-Abelian current algebra at level n=4. Based on precise lattice expressions for the CFT primary fields we predict the multifractal scaling exponents of critical wavefunctions to be Δq=q(1−q)/4. The Lagrangian of the RG-fixed point theory for r retarded and r advanced replicas is proposed to be the GL(r|r)n=4 Wess-Zumino-Witten model deformed by a truly marginal perturbation. The latter emerges from the NLσM by a natural scenario of spontaneous symmetry breaking.

Localization-length exponent in two models of quantum Hall plateau transitions

Motivated by the recent numerical studies on the Chalker-Coddington network model that found a larger-than-expected critical exponent of the localization length characterizing the integer quantum Hall plateau transitions, we revisited the exponent calculation in the continuum model and in the lattice model, both projected to the lowest Landau level or subband. Combining scaling results with or without the corrections of an irrelevant length scale, we obtain $\nu = 2.48 \pm 0.02$, which is larger but still consistent with the earlier results in the two models, unlike what was found recently in the network model. The scaling of the total number of conducting states, as determined by the Chern number calculation, is accompanied by an effective irrelevant length scale exponent $y = 4.3$ in the lattice model, indicating that the irrelevant perturbations are insignificant in the topology number calculation.

Geometrically disordered network models, quenched quantum gravity, and critical behavior at quantum Hall plateau transitions

Recent high-precision results for the critical exponent of the localization length at the integer quantum Hall (IQH) transition differ considerably between experimental ($\nu_\text{exp} \approx 2.38$) and numerical ($\nu_\text{CC} \approx 2.6$) values obtained in simulations of the Chalker-Coddington (CC) network model. We revisit the arguments leading to the CC model and consider a more general network with geometric (structural) disorder. Numerical simulations of this new model lead to the value $\nu \approx 2.37$ in very close agreement with experiments. We argue that in a continuum limit the geometrically disordered model maps to the free Dirac fermion coupled to various random potentials (similar to the CC model) but also to quenched two-dimensional quantum gravity. This explains the possible reason for the considerable difference between critical exponents for the CC model and the geometrically disordered model and may shed more light on the analytical theory of the IQH transition. We extend our results to network models in other symmetry classes.

Gaussian free fields at the integer quantum Hall plateau transition

In this work we put forward an effective Gaussian free field description of critical wavefunctions at the transition between plateaus of the integer quantum Hall effect. To this end, we expound our earlier proposal that powers of critical wave intensities prepared via point contacts behave as pure scaling fields obeying an Abelian operator product expansion. Our arguments employ the framework of conformal field theory and, in particular, lead to a multifractality spectrum which is parabolic. We also derive a number of old and new identities that hold exactly at the lattice level and hinge on the correspondence between the Chalker–Coddington network model and a supersymmetric vertex model.

Pure scaling operators at the integer quantum Hall plateau transition.

Stationary wave functions at the transition between plateaus of the integer quantum Hall effect are known to exhibit multifractal statistics. Here we explore this critical behavior for the case of scattering states of the Chalker-Coddington network model with point contacts. We argue that moments formed from the wave amplitudes of critical scattering states decay as pure powers of the distance between the points of contact and observation. These moments in the continuum limit are proposed to be correlation functions of primary fields of an underlying conformal field theory. We check this proposal numerically by finite-size scaling. We also verify the conformal field theory prediction for a three-point function involving two primary fields.

Statistics of conductances and subleading corrections to scaling near the integer quantum Hall plateau transition

We study the critical behavior near the integer quantum Hall plateau transition by focusing on the multifractal (MF) exponents Xq describing the scaling of the disorder-average moments of the point contact conductance T between two points of the sample, within the Chalker-Coddington network model. Past analytical work has related the exponents Xq to the MF exponents of the local density of states (LDOS). To verify this relation, we numerically determine the exponents Xq with high accuracy. We thereby provide, at the same time, independent numerical results for the MF exponents for the LDOS. The presence of subleading corrections to scaling makes such determination directly from scaling of the moments of T virtually impossible. We overcome this difficulty by using two recent advances. First, we construct pure scaling operators for the moments of T which have precisely the same leading scaling behavior, but no subleading contributions. Secondly, we take into account corrections to scaling from irrelevant (in the renormalization group sense) scaling fields by employing a numerical technique (“stability map”) recently developed by us. We thereby numerically confirm the relation between the two sets of exponents, Xq (point contact conductances) and (LDOS), and also determine the leading irrelevant (corrections to scaling) exponent y as well as other subleading exponents. Our results suggest a way to access multifractality in an experimental setting.

Finite-Size Effects and Irrelevant Corrections to Scaling Near the Integer Quantum Hall Transition

We present a numerical finite-size scaling study of the localization length in long cylinders near the integer quantum Hall transition employing the Chalker-Coddington network model. Corrections to scaling that decay slowly with increasing system size make this analysis a very challenging numerical problem. In this work we develop a novel method of stability analysis that allows for a better estimate of error bars. Applying the new method we find consistent results when keeping second (or higher) order terms of the leading irrelevant scaling field. The knowledge of the associated (negative) irrelevant exponent y is crucial for a precise determination of other critical exponents, including multifractal spectra of wave functions. We estimate |y|>/~0.4, which is considerably larger than most recently reported values. Within this approach we obtain the localization length exponent 2.62±0.06 confirming recent results. Our stability analysis has broad applicability to other observables at integer quantum Hall transition, as well as other critical points where corrections to scaling are present.

Quantum Hall transitions: An exact theory based on conformal restriction

We revisit the problem of the plateau transition in the integer quantum Hall effect. Here we develop an analytical approach for this transition, and for other two-dimensional disordered systems, based on the theory of ``conformal restriction.'' This is a mathematical theory that was recently developed within the context of the Schramm-Loewner evolution which describes the ``stochastic geometry'' of fractal curves and other stochastic geometrical fractal objects in two-dimensional space. Observables elucidating the connection with the plateau transition include the so-called point-contact conductances (PCCs) between points on the boundary of the sample, described within the language of the Chalker-Coddington network model for the transition. We show that the disorder-averaged PCCs are characterized by a classical probability distribution for certain geometric objects in the plane (which we call pictures), occurring with positive statistical weights, that satisfy the crucial so-called restriction property with respect to changes in the shape of the sample with absorbing boundaries; physically, these are boundaries connected to ideal leads. At the transition point, these geometrical objects (pictures) become fractals. Upon combining this restriction property with the expected conformal invariance at the transition point, we employ the mathematical theory of ``conformal restriction measures'' to relate the disorder-averaged PCCs to correlation functions of (Virasoro) primary operators in a conformal field theory (of central charge $c=0$). We show how this can be used to calculate these functions in a number of geometries with various boundary conditions. Since our results employ only the conformal restriction property, they are equally applicable to a number of other critical disordered electronic systems in two spatial dimensions, including for example the spin quantum Hall effect, the thermal metal phase in symmetry class D, and classical diffusion in two dimensions in a perpendicular magnetic field. For most of these systems, we also predict exact values of critical exponents related to the spatial behavior of various disorder-averaged PCCs.

Integrable modification of the critical Chalker-Coddington network model

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.

Wave function multifractality and dephasing at metal–insulator and quantum Hall transitions

We analyze the critical behavior of the dephasing rate induced by short-range electron–electron interaction near an Anderson transition of metal–insulator or quantum Hall type. The corresponding exponent characterizes the scaling of the transition width with temperature. Assuming no spin degeneracy, the critical behavior can be studied by performing the scaling analysis in the vicinity of the non-interacting fixed point, since the latter is stable with respect to the interaction. We combine an analytical treatment (that includes the identification of operators responsible for dephasing in the formalism of the non-linear sigma-model and the corresponding renormalization-group analysis in 2 + ϵ dimensions) with numerical simulations on the Chalker–Coddington network model of the quantum Hall transition. Finally, we discuss the current understanding of the Coulomb interaction case and the available experimental data.

Spin-Flip Scattering at Quantum Hall Transition

We formulate a generalized Chalker-Coddington network model that describes the effect of nuclear spins on the two-dimensional electron gas in the quantum Hall regime. We find exact analytical expression for spin-dependent transmission coefficients of a charged particle through a saddle point potential in a perpendicular magnetic field. Spin-flip scattering creates a metallic state in a finite range around the critical energy of quantum Hall transition. As a result we find that the usual insulating phases with Hall conductance σxy=0,1,2 are separated by novel metallic phases.

Point-Contact Conductance in Asymmetric Chalker–Coddington Network Model

We study the transport properties of disordered two-dimensional electron systems with a perfectly conducting channel. We introduce an asymmetric Chalker–Coddington network model and numerically investigate the point-contact conductance. We find that the behavior of the conductance in this model is completely different from that in the symmetric model. Even in the limit of a large distance between the contacts, we find a broad distribution of conductance and a non-trivial power law dependence of the averaged conductance on the system width. Our results are applicable to systems such as zigzag graphene nano-ribbons where the numbers of left- and right-going channels are different.

Boundary Multifractality at the Integer Quantum Hall Plateau Transition: Implications for the Critical Theory

We study multifractal spectra of critical wave functions at the integer quantum Hall plateau transition using the Chalker-Coddington network model. Our numerical results provide important new constraints which any critical theory for the transition will have to satisfy. We find a nonparabolic multifractal spectrum and determine the ratio of boundary to bulk multifractal exponents. Our results rule out an exactly parabolic spectrum that has been the centerpiece in a number of proposals for critical field theories of the transition. In addition, we demonstrate analytically exact parabolicity of the related boundary spectra in the two-dimensional chiral orthogonal "Gade-Wegner" symmetry class.

Calculation of the conductance of a graphene sheet using the Chalker-Coddington network model

The Chalker-Coddington network model (introduced originally as a model for percolation in the quantum Hall effect) is known to map onto the two-dimensional Dirac equation. Here we show how the network model can be used to solve a scattering problem in a weakly doped graphene sheet connected to heavily doped electron reservoirs. We develop a numerical procedure to calculate the scattering matrix with the aide of the network model. For numerical purposes, the advantage of the network model over the honeycomb lattice is that it eliminates intervalley scattering from the outset. We avoid the need to include the heavily doped regions in the network model (which would be computationally expensive) by means of an analytical relation between the transfer matrix through the weakly doped region and the scattering matrix between the electron reservoirs. We test the network algorithm by calculating the conductance of an electrostatically defined quantum point contact and comparing with the tight-binding model of graphene. We further calculate the conductance of a graphene sheet in the presence of disorder in the regime where intervalley scattering is suppressed. We find an increase in conductance that is consistent with previous studies. Unlike the tight-binding model, the network model does not require smooth potentials in order to avoid intervalley scattering.