This Letter presents variational ground-state and excited-state wave functions which describe the condensation of a two-dimensional electron gas into a new state of matter.

It is shown that the quantization of the Hall conductivity of two-dimensional metals which has been observed recently by Klitzing, Dorda, and Pepper and by Tsui and Gossard is a consequence of gauge invariance and the existence of a mobility gap. Edge effects are shown to have no influence on the accuracy of quantization. An estimate of the error based on thermal activation of carriers to the mobility edge is suggested.There has been considerable interest in the remarkable observation made recently by von Klitzing, Dorda, and Pepper' and by Tsui and Gossard that, under suitable conditions, the Hall conductivity of an inversion layer is quantized to better than one part in 105 to integral muitiples of e2/h. The singularity of the result lies in the apparent total absence of the usual dependence of this quantity on the density of mobile electrons, a sample-dependent parameter, As it has been proposed' to use this effect to define a new resistance standard or to refine the known value of the fine-structure constant, an important issue at present is to what accuracy the quantization is exact, particularly in the regime of high impurity density, Some light has been shed on this question by the renormalized weak-scattering calculations of Ando, ' who has sho~n that the presence of an isolated impurity does not affect the Hall current. A similar result has been obtained recently by Prange, who has shown that an isolated 5-function impurity does not affect the Hall conductivity to lowest order in the drift velocity u =cE/H, even though it binds a localized state, because the remaining delocalized states carry exactly enough extra current to compensate for its loss. The exactness of these results and their apparent insensitivity to the type or location of the impurity suggest that the effect is due, ultimately, to a fundamental principle. In this communication, we point out that it is, in fact, due to the long-range phase rigidity characteristic of a supercurrent, and that quantization can be derived from gauge invariance and the existence of a mobility gap.We consider the situation illustrated in Fig. 1, of a ribbon of two-dimensional metal bent into a loop of circumference L, and pierced every~here by a magnetic field Ho normal to its surface. The density of states of this system, also illustrated in Fig. 1, consists, in the absence of disorder, of a sequence of 8 functions, one for each Landau level. These broaden, in the presence of disorder, into bands of extended states separated by tails of localized ones.We consider the disordered case with the Fermi level in a mobility gap, as shown.We wish to relate the total current I carried around the loop to the potential drop V from one edge to another. This current is equal to the adiabatic derivative of the total electronic energy U of the system with respect to the magnetic flux $ through the loop. This may be obtained by differentiating with respect to a uniform vector potential A pointing around the loop, in the manner BU~9U 04 LOA This derivative is nonzero only by vi...

We propose that the enigmatic pseudogap phase of cuprate superconductors is characterized by a hidden broken symmetry of d x 2 Ϫy 2-type. The transition to this state is rounded by disorder, but in the limit that the disorder is made sufficiently small, the pseudogap crossover should reveal itself to be such a transition. The ordered state breaks time-reversal, translational, and rotational symmetries, but it is invariant under the combination of any two. We discuss these ideas in the context of ten specific experimental properties of the cuprates, and make several predictions, including the existence of an as-yet undetected metal-metal transition under the superconducting dome.

We present evidence that the ground state of the frustrated Heisenberg antiferromagnet in two dimensions is well described by a fractional quantum Hall wave function for bosons. This is compatible with the resonating-valence-bond concept of Anderson in being a liquid with neutral spin--, ' excitations. Our results suggest strongly that the resonating-valence-bond and fractional quantum Hall states are the same thing. We also argue that the excitation spectrum has an energy gap. This makes the boson energy bands disperse down as one moves away from the center of the Brillouin zone. To

Bernevig et al. Reply:We provide a Reply to the preceding Comment [1] by Greiter and Schuricht (GS). Let us first stress that there is no doubt about the mathematical correctness of our derivation. We now show that our interpretation of the physics of spinons is correct. Against the existence of spinon attraction, GS recall that spinons constitute an ''ideal gas of half fermions'' [2]. In reality, spinons feel a statistical interaction associated with a rule for progressively filling single-particle states. The fully dressed S matrix for spinons [3] in the Haldane-Shastry model (HSM) [4] takes the trivial form S i times the identity I. (This is well known and we derived it on p. 9 of the second paper of Ref. [5], which is equivalent to computing the S matrix associated with exchange statistics of semions-which can also be done by asymptotic Bethe ansatz -there is an extra i in our formula: it comes from a misprint in the final version of the Letter.) Triviality of the S matrix means that spinons are alleged asymptotic states of the HSM. Their long-distance behavior is not affected by their dynamical short-range attraction. In our Letter, we explicitly show the short-distance effects of the interaction on two-spinon wave functions [5]. We now consider the points raised by GS separately.GS claim our interpretation of the p mn e i as twospinon relative wave function is ambiguous, because the p mn 's are defined by expanding the (overcomplete) set of states in the (basis of) energy eigenfunctions mn . This statement is false. In real space, spinons are nonlocal excitations, with typical size of the order of the lattice step [5]. The 's should be thought of as the lattice version of two-particle coherent states. The overcompleteness of the 's is, therefore, the lattice version of the usual overcompleteness of coherent states. Two spinons at the same site correspond to a localized spin-1 excitation, unambiguously described by z 1 ; . . . ; z N=2ÿ1 [5]. The physical meaning of p mn 1 as the probability enhancement when the two spinons are at the same site (and form the spin-1 excitation) is hence absolutely unambiguous, at odds to GS's statement that the jp mn j 2 's ''cannot be interpreted as probability distribution.'' We also find a similar short-range attraction between a spinon and a holon (in the supersymmetric t ÿ J model with 1=r 2 interaction) although the states of a localized spinon and a localized holon do not form an overcomplete set [5]. We wish to remark that using overcomplete sets of states to treat nonlocal excitations as quantum-mechanical particles is in fact widely used for Laughlin quasiholes, a fact which Greiter correctly and explicitly points out in a prior publication [6]. The spinon situation is no different.GS assert that ''the second argument of [Bernevig, Giuliano, and Laughlin] is that the last term in their expression for the energy of the two-spinon states represents 'a negative interaction contribution that becomes negligibly small in the thermodynamic limit'.'' We clearly state that th...

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We discuss recent developments in our understanding of matter, broadly construed, and their implications for contemporary research in fundamental physics.

The electron transport properties of well-contacted individual single-walled carbon nanotubes are investigated in the ballistic regime. Phase coherent transport and electron interference manifest as conductance fluctuations as a function of Fermi energy. Resonance with standing waves in finite-length tubes and localized states due to imperfections are observed for various Fermi energies. Two units of quantum conductance 2G(0) = 4e(2)/h are measured for the first time, corresponding to the maximum conductance limit for ballistic transport in two channels of a nanotube.