Abstract:
Extensive body of work has shown that for the model of a non-interacting electron in a random potential there is a quantum critical point for dimensions greater than two---a metal-insulator transition. This model also plays an important role in the plateau-to-plateu transition in the integer quantum Hall effect, which is also correctly captured by a scaling theory. Yet, in neither of these cases the ground state energy shows any non-analyticity as a function of a suitable tuning parameter, typically considered to be a hallmark of a quantum phase transition, similar to the non-analyticity of the free energy in a classical phase transition. Here we show that von Neumann entropy (entanglement entropy) is non-analytic at these phase transitions and can track the fundamental changes in the internal correlations of the ground state wave function. In particular, it summarizes the spatially wildly fluctuating intensities of the wave function close to the criticality of the Anderson transition. It is likely that all quantum phase transitions can be similarly described.

Abstract:
The dimensional crossover in a spin-$S$ nearest neighbor Heisenberg antiferromagnet is discussed as it is tuned from a two-dimensional square lattice, of lattice spacing $a$, towards a spin chain by varying the width $L_y$ of a semi-infinite strip $L_x\times L_y$. For integer spins and arbitrary $L_y$, and for half integer spins with $L_y/a$ an arbitrary even integer, explicit analytical expressions for the zero temperature correlation length and the spin gap are given. For half integer spins and $L_y/a$ an odd inetger, it is shown that the $c=1$ behavior of the $SU(2)_1$ WZW fixed point is squeezed out as the width $L_y\to \infty$; here $c$ is the conformal charge. The results specialized to $S=1/2$ are relevant to spin-ladder systems.

Abstract:
Josephson plasma frequency is computed within the interlayer tunneling theory and compared against the experimental results in Tl_2Ba_2CuO_y and La_{2-x}Sr_xCuO_4. It is shown that the theoretical estimates are fully consistent with the recent experiments.

Abstract:
The thermal disordering of the $d$-density wave, proposed to be the origin of the pseudogap state of high temperature superconductors, is suggested to be the same as that of the statistical mechanical model known as the 6-vertex model. The low temperature phase consists of a staggered order parameter of circulating currents, while the disordered high temperature phase is a power-law phase with no order. A special feature of this transition is the complete lack of an observable specific heat anomaly at the transition. There is also a transition at a even higher temperature at which the magnitude of the order parameter collapses. These results are due to classical thermal fluctuations and are entirely unrelated to a quantum critical point in the ground state. The quantum mechanical ground state can be explored by incorporating processes that causes transitions between the vertices, allowing us to discuss quantum phase transition in the ground state as well as the effect of quantum criticality at a finite temperature as distinct from the power-law fluctuations in the classical regime. A generalization of the model on a triangular lattice that leads to a 20-vertex model may shed light on the Wigner glass picture of the metal-insulator transition in two-dimensional electron gas. The power-law ordered high temperature phase may be generic to a class of constrained systems and its relation to recent advances in the quantum dimer models is noted.

Abstract:
High temperature superconductivity in cuprate superconductors remains an unsolved problem in theoretical physics. The same statement can also be made about a number of other superconductors that have been dubbed unconventional. What makes these superconductors so elusive is an interesting question in itself. The present manuscript focuses on the recent magnetic oscillation experiments and how they fit into the broader picture. Many aspects of these experiments can be explained by Fermi liquid theory; the key issue is the extent to which this is true. If true, the entire paradigm developed over the past three decades must be reexamined. A critical analysis of this issue has necessitated a broader analysis of questions about distinct ground states of matter, which may be useful in understanding other unconventional superconductors.

Abstract:
A selected set of topics in quantum phase transition is discussed. It includes dissipative quantum phase transitions, the role of disorder, and the relevance of quantum phase transition to measurement theory in quantum mechanics.

Abstract:
I comment on two recent papers on Kerr effect as evidence of gyrotropic order in cuprates, and I suggest that the arguments may not be sound. The difficulty is that in practically all cases the wave vector $k_{z}$ perpendicular to the copper-oxygen plane is not a good quantum number. This appears to be problematic for arXiv:1212.2698, whereas in arXiv:1212.2274 the symmetry arguments may turn out to be robust, but the microscopic picture is wanting. Thus, the Kerr effect in cuprates remains a puzzle, as there is little doubt that the arguments presented against time reversal symmetry breaking appear to be rather strong in both of these papers on experimental grounds.

Abstract:
The interlayer tunneling mechanism of the cuprate high temperature superconductors involves a conversion of the confinement kinetic energy of the electrons perpendicular to the CuO-planes ($c$-axis) in the normal state to the pair binding energy in the superconducting state. This mechanism is discussed and the arguments are presented from the point of view of general principles. It is shown that recent measurements of the $c$-axis properties support the idea that the electrons substantially lower their $c$-axis kinetic energy upon entering the superconducting state, a change that is nearly impossible in any conventional mechanism. The proper use of a $c$-axis conductivity sum rule is shown to resolve puzzles involving the penetration depth and the optical measurements.

Abstract:
We revisit the interlayer tunneling theory of high temperature superconductors and formulate it as a mechanism by which the striking systematics of the transition temperature within a given homologous series can be understood. We pay attention not only to the enhancement of pairing, as was originally suggested, but also to the role of competing order parameters that tend to suppress superconductivity, and to the charge imbalance between inequivalent outer and inner CuO2 planes in a unit cell. Calculations based on a generalized Ginzburg-Landau theory yield results that bear robust and remarkable resemblance to experimental observations.