We study the localization properties in the transition from a two-dimensional electron gas at zero magnetic field into an integer quantum Hall (QH) liquid. By carrying out a direct calculation of the localization length for a finite size sample using a transfer matrix technique, we systematically investigate the field and disorder dependences of the metal-insulator transition in the weak field QH regime. We obtain a different phase diagram from the one conjectured in previous theoretical studies. In particular, we find that: (1) the extended state energy Ec for each Landau level (LL) is always linear in magnetic field; (2) for a given Landau level and disorder configuration there exists a critical magnetic field Bc below which the extended state disappears; (3) the lower LLs are more robust to the metal-insulator transition with smaller Bc. We attribute the above results to strong LL coupling effect. Experimental implications of our work are discussed.PACS numbers: 71.30.+h, 73.20.Jc, 73.40.Hm It is very important to understand the localization properties in the transition from two-dimensional electron gas (at zero magnetic field) into an integer quantum Hall liquid [1]. According to the scaling theory of localization [2] all electrons in a two-dimensional system are localized in the absence of magnetic field. When the twodimensional electron system is subject to a strong perpendicular magnetic field, the energy spectrum becomes a series of impurity broadened Landau levels. Extended state appears in the center of each Landau band, while states at other energies are localized. This gives rise to the integer quantum Hall effect. The interesting issue is to understand the evolution of the extended states in the weak field regime as the magnetic field goes to zero where all extended states disappear.There could be two scenarios for the fate of the extended states as B → 0. The first one was proposed by Kivelson, Lee, and Zhang [3] in their global phase diagram of the quantum Hall effect. According to this phase diagram, in a strongly disordered quantum Hall system, the extended states stay with the center of the Landau bands at strong magnetic field, but float up in energy at small magnetic field and go to infinity as B → 0. This phase diagram is consistent with the semiclassical argument put forth by Khmelnitskii [4] and Laughlin [5].In this letter, we are proposing an alternative scenario for the behavior of the extended states at weak magnetic field limit. In our picture, each extended state is simply destroyed by strong disorder at a critical magnetic field instead of floating up in energy. By carrying out a direct calculation of the localization length for a finite size sample using a transfer matrix technique, we systematically investigate the field and disorder dependence of the metal-insulator transition in the weak field quantum Hall regime. We find that: (1) the extended state energy E c for each Landau level (LL) is always linear in magnetic field; (2) for a given Landau level and disorder configurat...

We study disorder induced transition between quantum Hall states and insulator state in a lattice model. We nd the positions of the extended states do not change as disorder strength is varied. As a consequence, there are direct transitions from all Landau level quantum Hall states to insulator states, in contrast to the global phase diagram from early studies based on continuous models. We also provide the microscopic understanding of the transition in terms of the topological properties of the system. PACS numbers: 71.30.+h, 73.20.Jc, 73.40.Hm It is well known that all electrons in a two-dimensional system are localized in the absence of magnetic eld according to the scaling theory of localization 1]. When the two-dimensional electron system is subject to a strong perpendicular magnetic eld, the energy spectrum becomes a series of impurity broadened Landau levels. The external magnetic eld breaks the time-reversal symmetry and as a consequence, extended state appears in the center of each Landau band while states away from the Landau band centers are still localized. This gives rise to the integer quantum Hall e ect 2]. Recently there have been much interest to understand the evolution of the extended states as the magnetic eld goes to zero or equivalently as the disorder increases such that eventually all the extended states should disappear 3{7].Earlier work by Laughlin 8] and Khmelnitskii 9] concluded that the extended state energy for the nth Landau level (LL) behaves in the following form:where ! c is the cyclotron frequency and is impurity scattering time. According to this formula, E c n deviates from the linear magnetic eld dependence when ! c 1 and starts to oat up as B decreases or as disorder increases. Based on the oating up picture from continuous models by Laughlin and Khmelnitskii, Kivelson, Lee and Zhang 10] proposed the global phase diagram for transitions between the quantum Hall states (QHS) and insulator (or localized) states. There are two important consequences from the global phase diagram: (i) as disorder increases for a xed eld, no direct transition between higher LL (n 6 = 0) QHS and insulator is allowed; (ii) as B decreases for xed disorder, reentry behavior should be observed in quantized Hall e ect, e.g. = 1 ! = 2 ! = 1.In an earlier work 11] by us and Niu, we found that the oating-up picture is not valid in the lattice model. We concluded that: (1) the extended state energy E c for each Landau level is always linear in magnetic eld; (2) for a given Landau level and disorder con guration there exists a critical magnetic eld B c below which the extended state disappears; (3) the lower LLs are more robust to the metal-insulator transition with smaller B c . We attributed the above results to strong LL coupling e ect.In this paper, we expand our study on disorder induced transition between quantum Hall states and insulator state in a lattice model. By carrying out nite-size scaling calculations of localization length, we demonstrate that the positions of the extended states do not ...

We study the order parameter symmetry in bi-layer cuprates such as YBaCuO, where interesting π phase shifts have been observed in Josephson junctions. Taking models which represent the measured spin fluctuation spectra of this cuprate, as well as more general models of Coulomb correlation effects, we classify the allowed symmetries and determine their associated physical properties. π phase shifts are shown to be a general consequence of repulsive interactions, independent of whether a magnetic mechanism is operative. While it is known to occur in d-states, this behavior can also be associated with (orthorhombic) s-symmetry when the two sub-band gaps have opposite phase. Implications for the magnitude of Tc are discussed.PACS numbers: 74.20.Mn, 74.50.+r, The observation in YBCO of unusual Josephson junction behavior 1-4 is one of the most important experimental results to emerge from the cuprate literature in recent years. Here in a Josephson SQUID experiment the two junctions are configured so that their normals lie along the a and b axes of the CuO 2 plane. This measurement has been widely interpreted as support for a d-symmetry of the order parameter, as well as for a magnetic mechanism for superconductivity. In this paper we show that both of these inferences may be inappropriate. For notational precision, throughout this paper we use the terms s-( or d-) symmetry to correspond to states which have the same (or opposite) sign under a π/2 rotation. Thus the d-states under consideration are not necessarily of the specific d x 2 −y 2 form.The gap equation for bi-layer systems has been studied earlier in the context of a magnetic mechanism for superconductivity 5,6 . There it was observed that the d-symmetric state of the single layer problem, is transformed to a pair of in-phase d states on each of the two sub-bands, and that these compete with a pair of outof-phase s-states. Here we take the problem to a greater level of generality, establishing that this situation persists for a wide class of repulsive interactions, which are unrelated to the antiferromagnetic spin fluctuation picture. Alternate classes of the order parameter symmetry are also generated. These correspond to in-phase s-states and out-of-phase d-states. We establish how the relative stability of the two competing states is affected by changes in bandstructure, orthorhombicity, and hole filling.It should not be surprising that d-states have a more general origin beyond the antiferromagnetic spin exchange models. In a one layer cuprate, the lattice symmetry requires that all gap states are either even (s-) or odd (d-) under a π/2 rotation. In bi-layer systems, these one layer states generalize naturally to a pair of even or odd, in-phase or out-of-phase states, associated with each of the two sub-bands. Thus, as one of two allowed states, d-symmetry should be widespread, and independent of the microscopic details of the model.In the presence of both intra-and inter-layer interactions (V and V ⊥ ), the weak coupling BCS gap equation becomes a set...

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We extend previous work on pre-formed pair models of superconductivity to incorporate Coulomb correlation effects. For neutral systems, these models have provided a useful scheme which interpolates between BCS and Bose Einstein condensation with increasing coupling and thereby describes some aspects of pseudo-gap phenomena. However, charge fluctuations (via the plasmon, ωp) significantly modify the collective modes and therefore the interpolation behavior. We discuss the resulting behavior of the pseudo-gap and thermodynamic quantities such as Tc, χ and Cv as a function of ωp.The role of the pseudo-gap 1 in the high T c cuprates is emerging as an important indicator of the nature of the superconductivity as well as the normal state. There are two widely discussed but competing explanations for pseudo-gap effects but no clear and decisive experiments to support one scenario over the other. Early observations associated the pseudo-gap with magnetic pairing 2 above T c (often called the "spin gap"). It is now clear, however, that some form of normal state pairing is seen in photoemission as well as charge transport data. Moreover, at least in the photoemission data the pseudo-gap appears to have the d-wave symmetry 1 of the ordered state and this leads naturally to the association of this "gap" with precursor superconductivity. 3-5 This second scenario is further supported by the observation of low dimensionality and short coherence lengths in high T c superconductors, which suggests important deviations from ideal mean field or BCS transitions. Indeed, the approach of the present paper assumes the precursor superconductivity scenario, in large part because it is important to establish, at least as a base-line, the extent to which such superconducting "fluctuation" effects may be responsible for pseudo-gap behavior.Among those models which subscribe to a precursor superconductivity scenario there are additionally two rather distinct viewpoints. Emery and Kivelson 5 have argued that the pseudo-gap state of the cuprates is similar to that observed in granular films where phase coherence is not fully established, although large regions of the material have a well established superconducting amplitude. Because it is small, in some sense, in the cuprates their approach focuses on n/m * or alternatively on the plasma frequency ω p as the key "phase stiffness" parameter. Alternatively, others 3,4,6,7 have focused on the observed small size of the superconducting correlation length ξ to argue for important corrections to BCS theory associated with pre-formed or nearly-formed pairs 8 which exist well above T c and therefore give rise to significant pseudo-gap effects. The present paper is based on the viewpoint that in the cuprates the characteristic parameter of the charge degrees of freedom, n/m * or equivalently ω p , should be treated on a relatively equal footing with the correlation length, ξ.To study the role of Coulomb interactions on pseudogap phenomena, we adopt a natural microscopic framework which incoporates ch...

In the composite fermion model of the fractional quantum Hall e ect, composite fermions experience, in addition to the usual potential disorder, also a magnetic ux disorder. Motivated by this, we investigate the localization properties of a single fermion in two dimensions, moving in the presence of both potential and magnetic ux disorders, but with a non-zero average magnetic eld. It is found that the exponent characterizing the divergence of the localization length is not changed upon the addition of the ux disorder, provided it is not too large, suggesting that the transitions between fractionally quantized Hall plateaus belong to the same universality class as those between the integrally quantized ones. PACS numbers: 73.40.Hm In the absence of a magnetic eld, an arbitrary weak random potential localizes all states of an electron in two dimensions 1]. In contrast, existence of extended states, separated by bands of localized states, is crucial for the explanation of the quantum Hall e ect 2]. Numerical studies 3,4] have made a convincing case that the localization length of an electron, con ned to two dimensions and exposed to a strong magnetic eld, diverges at a single energy near the center of each Landau level, and the divergence is characterized by an exponent := jE E c j ;(1) where 7=3, independent of the Landau level (LL) index. No analytical derivation of the value of the exponent yet exists. Experimentally, the width of the transition region (e.g., the width of the conductance peak) in the integer quantum Hall regime (IQHE) regime has been found to vanish in a range of temperature as T , where 0:42, independent of the LL in question 5]. This is consistent with the theoretical value of in a variable range hopping theory 6], which obtains = 1= (in other words, writing = 1=z , this work shows that the dynamical exponent z is equal to unity for the Coulomb interaction). Direct measurements of the localization length are also in agreement with7=3 7]. At rst glance, the nature of localization would appear to be much more complex in the fractional quantum Hall e ect (FQHE) regime, since interactions are an intimate part of the physics here and cannot be neglected. However, in the composite fermion model 8], the essential part of the interelectron interaction is incorporated through the formation of composite fermions, which themselves can be treated as non-interacting to a good approximation. An e ective single-particle description of the strongly correlated electron state thus becomes possible in terms of composite fermions. This paper studies the scaling of the localization length in the FQHE regime in the framework of a non-interacting composite fermion model. While the dynamics of composite fermions is similar to that of electrons in a weaker magnetic eld, an essential di erence is that the composite fermions also experience a random ux disorder in addition to the usual random potential disorder 9{11]. The main conclusion of the paper is that the exponent is the same for non-interacting composite fer...