The E ects of Disorder and Interaction in Metallic Systems.

I also like to thank the faculty here: Jason Alicea, Olexei Motrnich, Xie Chen, and Alexei Kitaev. I am grateful for the cooperation of Prashant Kumar and conversations with Srinivas Raghu, Hart Goldman and Alex Thomson. I would also like to thank Shu-Yu Ho for many conversations about computational technical issues and Jyong-Hao Chen for discussion about 1D physics technique, pointing out insightful bosonization references and sharing work application information.

I would like to thank Yu-An Chen for discussing my first year coursework as well as his kindness in picking me up from LAX the first time I came to the US. I would also like to thank Yen-Yung Chang and Yu-Chen Hsu for their hospitality and accommodations the first night I arrived in Pasadena. I also thank my families in Taiwan, including my grandparents, uncles, aunts, and cousins, who support me in their own ways. Third, we study a single 2D Dirac fermion at nit density, subjected to a quenched random magnetic field.

INTRODUCTION

The effect of the renormalization of the parameter (σxx, σxy) in this case of the integer quantum Hall number is more involved. If one takes the midpoint between the two plateaus as the critical point of the quantum Hall transition, then the prediction from Pruisken's RG analysis seems to be consistent with the truth of the metallic nature of the critical point. Indeed, numerous numerical and experimental evidences suggest that while the integer quantum Hall plateau can be explained by non-interacting theory, the presence of interacting elements becomes essential in the vicinity of the critical point.

For the bosoinc system, the superconductor-insulator transition (SIT) is one of the typical examples. Regarding the experimental signature of NFL, the linear temperature dependent resistance is one of the key features. Therefore, the deviation from the Wiedemann-Franz law is the evidence of the breakdown of quasi-particles.

Figure 1.1: The conductivity evolution via scaling theory, gure is taken from the original paper[1]

BIBLIOGRAPHY

SCALING AND DIFFUSION OF DIRAC COMPOSITE FERMIONS

In contrast to fermionic models, only random mass perturbations M(x) resulted in accessible diffusive fixed points of the XY model. Moreover, the observed IQHT and SIT appear to be sensitive to the precise nature of the Coulomb interaction ([43, 77] and references therein). Critical exponents are found for the fermionic dual" of the XY model with random mass perturbation.

In our approach, we are unable to approach the infinite point z" found in the study of the dissipative XY model in [3]. In the remainder of the main text, we drop the superscripts B and R for notional clarity. the term of leading order in the large Nf expansion provides a poor approximation of the critical exponents of pure isolation points [13].

Table 2.1 summarizes the transformations of the operators that appear in (2.7) under charge-conjugation C and time-reversal T symmetries

APPENDIX

As discussed in the main text, we choose the dynamical critical exponent z in such a way that the fermion velocity v does not run, i.e. the velocity beta function is zero. We use φ1 to parameterize the screening of the perturbation described in Appendix2.C. φ1 = 0 means the screening is ignored; φ1 = 1 means the screening is included. In the absence of the Coulomb interaction, the equality of the bearing corrections in δ1 and δ2 is a coincidence, making βg0, βgj independent of the bearing corrections.

When the Coulomb interaction is included, βg0 receives corrections from the gauge part, while βgj does not. Quantization of the Chern-Simons level and the gauge self-energy level in 3D implies. 2.124) Consequently, the renormalizations of κ, wx, σe and De are controlled by their engineering dimensions. From the action (2.100), we can read Feynman's rules for the different types of perturbations.

The 2π factor always cancels with the 1/2π accompanying any frequency integral R dω. 2.140) The minus sign comes from the fermion loop. Diagrams of type B3, B4 can be obtained directly from the 3-point vertex corrections in Annexes (2.E) and (2.F) with symmetry factor 2 (count upper or lower vertices), so we do not need to recalculate them here. Model for a Quantum Hall Eect without Landau Levels: Condensed Matter Realization of the "Parity Anomaly".

Relationship between the correlation dimensions of multifractal wavefunctions and spectral measurements in integer quantum Hall systems. Crossover from the non-universal scaling regime to the universal scaling regime in quantum Hall plateau transitions. Scaling in the plateau-to-plateau transition: A direct connection of Quantum Hall systems to the Anderson localization model.

Transition from quantum-Hall to insulator in topological insulator films with ultra-low carrier density and a hidden phase of the zero Landau level.

Figure 2.B.1: Diagrams contributing to δ 1 , δ 2 .

COMPOSITE FERMION NONLINEAR SIGMA MODELS

We begin with a description of the excitation function for disorder-averaged products of delayed and advanced composite fermion Green's functions. Within the replica approach (see, e.g., disorder-averaged delayed and advanced HLR composite fermion Green's functions obtained from the disorder average of the path integral,. The corresponding action appearing in the generating functional of Dirac composite fermion delayed and advanced Green s functions have exactly the same form as (3.11) with the replacement of the mass matrices by

The composite Dirac fermion mass mD changes sign under a particle-hole transformation of the (2+1)d theory [40]. Σ is a logarithmic divergence that we absorb in a review of the Fermi energy and the composite fermion mass. This implies the lack of importance of the mD mass in the composite Dirac fermion average rate theory near the hall diusive integer quantum transition; it represents an emergent particle-hole symmetry in HLR theory.

To this end, we include additional scalar potential randomness V0(x) coupling to ψ†ψ in (3.4), independent of the anti-correlated vector and scalar potential variances in (3.5) which have zero mean and variance W0. Such a chiral coupling clearly violates particle-hole symmetry in the Dirac composite fermion theory.) The contribution of V0(x) to the disorder-averaged action. Our result shows how particle-hole symmetry can emerge in the HLR composite fermion theory and gives further evidence for the possible IR equivalence of the Dirac and HLR theories. Alternatively, if particle–hole symmetry does not emerge, we expect either a gapped insulator or a diusive metal (at least in the vicinity of the particle–hole symmetric limit at θ=π).

In this appendix, we detail the NLSM calculation for Q, which is sketched in 3.3. Self-duality of an integer quantum Hall transition to an insulator: Description of composite fermions. On localization in the theory of the quantized Hall effect: a two-dimensional realization of the θ-vacuum.

Criticality of two-dimensional disordered Dirac fermions in the unitary class and the universality of the integer quantum Hall transition.

RANDOM MAGNETIC FIELD AND THE DIRAC FERMI SURFACE

An alternative study of the Dirac theory using (non-)Abelian bosonization by Ludwig et al. We first take the limit of the theory focusing on the low-energy uctuations around the Dirac Fermi surface before incorporating disorder. We then move on to a longitudinal conductivity calculation, finding that the conductivity varies continuously with the strength of the disturbance.

The chemical potential is finite and non-zero, and we set the speed of the Dirac fermion to unity. Therefore, for excitations with p ∼ kF, depending on the sign of µ, the low-energy effective theory holds only the particles or antiparticles. This is an order of magnitude smaller than the contribution of the random magnetic field that we will focus on in the rest of this article.

The random vector potential is eliminated from the eective action using the non-dimensional symmetry of the Fermi surface. This is the discrete form of the action (4.51) that we will use in the next section. In general, in the presence of quenched disorder V with unit-normalized distribution P[V], the disorder mean of the correlation function of a physical observable O is demonstrated to be.

In this paper, we instead use the exact solution of the low-energy efficiency theory presented in 4.2 to perform the clutter averaging directly. The disorder average of the product of the U and C matrices is calculated as in the previous section. A study of the effects of various leading corrections to the effective theory (4.23) could potentially clarify the limit (4.78).

This question is relevant to the expected universality of conduction in the quantum phase transition [63].

CURRENT ALGEBRA APPROACH TO 2D INTERACTING CHIRAL METALS

We identify U(N) with the improved IR symmetry of the chiral metal, with N → ∞ equal to the number of points on the Fermi surface. The additional spatial dimension of the 2d chiral metal arises from the U(N) degrees of freedom of the WZW model. The solvability of the perturbed WZW model at k = 1 (when the theory is equivalent to a free Fermi gas) extends to level k > 1.

This means that g should be considered as a right-moving element of the composition LU(N)/U(N), i.e. the loop group of U(N) modulo arbitrary x-independent matricesW(t)[67]. This algebra (at k = 1) is different from the commutation algebra of density operators of the free chiral metal (see Appendix 5.A). In this section we determine the two-point correlation functions of the single-particle fermion operator, the U(1) number density and the U(1) current in the interacting chiral metals.

Because the jump matrix M(x)M†(0) is an oscillating function of x, it does not contribute to the scaling of the two-point function for large|x|. We now calculate the two-point function k = 1 density directly using the free fermion representation of the theory: In this section we therefore concentrate on the two-point correlation function of the U(1) current along the y-direction.

The extra spatial dimension arises from the U(N) degrees of freedom of the WZW model. The k > 1 theories are interaction generalizations, in which the symmetries of the k = 1 theory are maintained. Here it is the U(N) symmetry (which includes U(1) number conservation and discrete translational invariance along the y direction) and continuous translational invariance along the x direction of the free 2d chiral metal that is preserved stick to non-zero interaction k > 1.

In this appendix we derive the commutation algebra of the Fourier modes of the density operator of the chiral metal.

Figure 5.2.1: (a) Stack of 2d integer quantum Hall states with layer separation δ = 1

SUMMARY OF THIS THESIS