Theoretical Quantum Condensed Matter Physics

Some topics we are currently interested in:

Quantum Hall Transitions: Transitions between different quantum Hall plateaus are paradigmatic in the theory of quantum criticality in 2+1D. Experimental evidence indicates that both integer and fractional quantum Hall transitions exhibit the same critical exponents, yet traditional theoretical models handle them differently, leading to a disconnect between theory and experiment.

We have developed a unified framework based on composite-fermions, which incorporates both electron-electron interactions and disorder, offering a promising explanation for the observed "super-universality". While certain arguments support this notion of super-universality, further research is needed for a more careful exploration the effects of gauge-field fluctuations close to the mean-field quantum critical point.

As complementary directions to the above, we are currently exploring more microscopic models and network model based approaches.

Correlations in Topological Phases: Topological ground states cannot be continuously connected to product states, which have zero correlations for certain well-defined local observables, such as the spin operator in an Ising ferromagnet with all spins pointing up. This suggests the presence of a minimum/lower bound on correlations in topological phases determined by topological properties.

Recently, we investigated two such lower bounds in 2+1D, which emerge in the long-wavelength expansion of the static structure factor (equal-time density-density correlation function). Notably, the term quadratic in the wavenumber is bounded below by the Hall conductivity. We discovered that a large class of model wavefunctions and Hamiltonians saturate this bound, explaining why it was previously overlooked in the literature.

The second bound we studied is the Haldane bound, which provides a lower limit on the quartic term in the static structure factor, determined by the Hall viscosity of the phase. Our numerical simulations showed that this bound is not saturated, except by very specific conformal block wavefunctions, disproving certain conjectures in prior work.

We are interested in exploring more general forms of these bounds across a variety of topological phases and developing quantum field theories that accommodate these constraints.

Non-Fermi Liquids: Non-Fermi liquids (NFLs) are examples of exotic gapless matter in two-dimensions, arising from the effects of strong interactions. Key questions remain regarding the nature of quasi-particles in these systems, particularly whether strong interactions lead to their breakdown or if they retain some features of conventional Fermi-liquid behavior.

In our previous numerical work, we investigated the low-energy excitations of the half-filled lowest Landau level on an infinitely long cylinder with finite circumference. We found that the composite-fermion theory provides a remarkably accurate quantitative description of these excitations. Moving forward, we aim to explore the nature of quasi-particles by devising methods that approach the true 2D limit, allowing us to better distinguish the NFL behavior from that of a more conventional FL of composite-fermions.

Vortex Metal in 2+1D: In two spatial dimensions, the scaling theory of localization predicts that metallic phases should not exist in the presence of impurities. Yet, such phases are commonly observed in proximity to superconductors (SC) and are believed to consist of vortices in the SC order parameter. Addressing this fundamental problem requires a framework that incorporates both interactions and disorder, and no satisfactory mean-field approach has been discovered to date.

We are currently developing new theoretical approaches aimed at elucidating the nature of these metallic phases, with the hope of resolving this long-standing problem.