Only selected papers are listed here. For other publications, please see the publication list. Recent publications are also available at the arXiv.
My very first work  "Thermodynamical description of incommensurate phase in prustite" (as an undergraduate student with V. L. Pokrovsky, then at the Landau Institute) described the Landau theory of the incommensurate phase in prustite (Ag3AsS3). In particular, effect of thermal fluctuations on the evolution of the incommensuration vector was analyzed. Unfortunately, we could not explain the "invar" effect: the lattice constant of the material mysteriously remains constant (the isobaric thermal expansion coefficient vanishes) throughout the incommensurate phase. The effect was also seen in other incommensurate materials, see, e.g., B. Sh. Bagautdinov and V. Sh. Shekhtman, "The invar effect and phase transitions in Cs2ZnI4 crystals", Phys. Sol. State, 41, 123 (1999). As far as I know, the puzzle of the invar effect remains unsolved.
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In the work  "Fluctuation-induced first order transition between the quantum Hall liquid and insulator" (with S.-C. Zhang, then my Ph.D. advisor at Stanford) we studied a field theory of relativistic charged boson minimally coupled with abelian Chern-Simons gauge field. This can be viewed as an effective model corresponding to quantum Hall to an Insulator transition in the presence of commensurated potential. The transition where the expectation value of the field φ vanishes is associated with the transition quantum Hall liquid to an Insulator. To avoid dimensional regularization, the calculation was done in a vicinity of the tricritical point (thus the need to study φ6 theory). A renormalization group analysis involving the second-loop β-function demonstrated that fluctuations drive the transition to first order. In regular QED this mechanism was invented by S. Coleman and E. Weinberg, see "Radiative Corrections as the Origin of Spontaneous Symmetry Breaking", Phys. Rev. D 7, 1888 (1973).
The study continued in the work  "Duality and Universality for the Chern-Simons bosons", where a closely related model was studied on a lattice. Namely, the model was the (Villain) lattice X−Y model coupled with the Chern-Simons field. In the assumption that the transitions are continuous, the analysis of two symmetries, "duality" (symmetry between particles and vortices, which results θ → −(2π)2/θ) and "flux attachment" (periodicity with respect to the mutual statistics of the particles, θ→θ+4π) demonstrated that the transitions in the models with any rational value of the Chern-Simons coefficient can be mapped onto only two universality classes.
The next paper in the series is  "Global symmetries of quantum Hall systems: lattice description". Here I studied the same model but with symmetry closer to that of the quantum Hall effect: the Lorentz invariance (more precisely, its Euclidean version) was assumed to be broken by non-zero charge and current densities, as well as external magnetic field. It turned out that the two transformations not only affect the Chern-Simons coupling θ (the statistics of the quasiparticles) but also the global currents and fluxes, which results in mappings for the conductivities σxx and σxy which are enticingly similar to the correspondence rules. However, because the statistics of the particles changes at the same time, the symmetry arguments alone were insufficient to prove the universality of the transitions.
This story continues in the section on edge states, where totally different methods were used. Also, the assumption that the transition is second order turned out to be not so innocuous: many closely related models feature first order transitions. Please see discussion on high-Tc stripes and lattice bosons below.
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If it was the statistics of quasiparticles which prevented the universality between different transitions, perhaps the universality in the experiment was due to the presence of disorder which would localize the quasiparticles and thus conveniently suppress their exchange statistics. To account for disorder seriously, in the case of integer quantum Hall effect an entirely different model is used, where incompressible quantum Hall liquid in two dimensions is separated into puddles by strong self-consistent potential. Then, at low temperatures the excitations within each puddle are gapped, and dissipative charge transport can only occur along the edges of the puddles. As a result, the quantum Hall to insulator transition driven by gating (electron density reduction) can be viewed as the (quantum) percolation transition.
In the paper  "Network of edge states: random Josephson junction array description" (with Karén Chaltikian), we constructed a version of the Chalker-Coddington model applicable for the fractional quantum Hall effect. The model consists of chiral (particles move one-way) edges connected by occasional saddle points where tunneling between the edges occur. We demonstrated that the duality of a sort works for each saddle point, no matter how complicated the system of edges is: if the tunneling of electrons through an insulating region between two nearby edges becomes too strong, one can reconnect the edges and consider the tunneling of quasiparticles in the transverse direction through quantum Hall liquid.
In fact, the quantum Hall symmetries, including the quantization of the Hall resistance well into the insulator phase, come out of a very similar classical model: a collection of chiral edges connected at saddle points [E. Shimshoni and A. Auerbach, "Quantized Hall insulator: Transverse and longitudinal transport" Phys. Rev. B 55, 9817 (1997)]. The only difference is that the quantum interference is suppressed. In the paper  "Hall Resistivity and Dephasing in the Quantum Hall Insulator" (with A. Auerbach), we demonstrated that the absence of the quantum interference is required: a simple numerical simulation (within a simpler Chalker-Coddington model applicable for integer quantum Hall effect) showed that the Hall resistance quantization disappeared as long as the effects of quantum interference were included. This indicates that some kind of phase-breaking mechanism is in play.
In the paper  "Coulomb interaction and delocalization in quantum Hall constrictions" (with A. Auerbach and E. Shimshoni), we tried to figure out whether this mysterious phase breaking could originate from the long-range Coulomb interaction. In short, the answer was no. However, we discovered that the duality between weak and strong tunneling remains in the presence of the Coulomb interactions. Namely, in an X-shaped geometry, duality relates junctions with opening angles α and (π − α). The Coulomb interaction effect is precisely cancelled in the self-dual geometry of a junction with α = π/2, where the tunneling problem becomes exactly solvable using the Wiener-Hopf method.
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Series of papers on correlated bilayers and multilayers with J. D. Naud and S. L. Sondhi. Please see our letter  "Fractional quantum Hall effect in infinite layer systems" for a short summary.
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