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Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program


Year Degree Awarded


Month Degree Awarded


First Advisor

Nikolay Prokof'ev

Second Advisor

Boris Svistunov

Third Advisor

Jonathan Machta

Fourth Advisor

Murugappan Muthukumar

Subject Categories

Atomic, Molecular and Optical Physics | Condensed Matter Physics | Other Physics | Quantum Physics | Statistical, Nonlinear, and Soft Matter Physics


A quantum critical point (QCP) is a point in the phase diagram of quantum matter where a continuous phase transition takes place at zero temperature. Low-dimensional quantum critical systems are strongly correlated, therefore hosting nontrivial emergent phenomena. In this thesis, we first address two decades-old problems on quantum critical dynamics. We then reveal two novel emergent phenomena of quantum critical impurity problems. In the first part of the thesis, we address the linear response dynamics of the $(2+1)$-dimensional $O(2)$ quantum critical universality class, which can be realized in the ultracold bosonic system near the superfluid (SF) to Mott insulator (MI) transition in two dimensions. The first problem we address is about the fate of the massive Goldstone (Higgs) mode in the two-dimensional relativistic theory. Using large-scale Monte Carlo simulations and numerical analytical continuation, we obtain universal spectral functions in SF, MI and normal quantum critical liquid phases and reveal that they all have a relatively sharp resonant peak before saturating to the critical plateau behavior at higher frequencies. The universal resonance peak in SF reveals a critically-defined massive Goldstone boson, while the peaks in the last two phases are beyond the predictions of previous theories. The second problem we address is to controllably calculate one of the most fundamental transport properties---optical conductivity---in the quantum critical region. We precisely determine the conductivity on the quantum critical plateau, $\sigma(\infty)=0.359(4)\sigma_Q$ with $\sigma_Q$ the conductivity quantum. For the first time, the shape of the $\sigma(i\omega_n)- \sigma(\infty)$ function in the Matsubara representation is accurate enough to compare a holographic gauge-gravity duality theory for transport properties [Myers, Sachdev, and Singh, Phys. Rev. D 83, 066017 (2011)] to the reality. We find that the theory---in the original form---can not account for our data, thereby inspiring the theorists to modify the corresponding holographic theory. The second part of this thesis discusses two exotic impurity states hosted by quantum critical environments. The first one is the halon, a novel critical state of an impurity in $O(N\ge2)$ quantum critical environment. We find that varying the impurity-environment interaction leads to a boundary quantum critical point (BQCP) between two competing ground states with charges differing by $\pm 1$. In the vicinity of the BQCP, the halon phenomenon emerges. The hallmark of the halon physics is that a well-defined integer charge carried by the impurity gets fractionalized into two parts: a microscopic core with half-integer charge and a critically large halo carrying a complementary charge of $\pm 1/2$. The halon can be generalized to other incompressible quantum-critical environments with particle-hole symmetry. The second novel phenomenon we reveal is termed "trapping collapse". We address a simple fundamental question of how many repulsively interacting bosons can be localized by a trapping potential. We find that under rather generic conditions, for both weakly and strongly repulsive particles, in two and three dimensions---but not in one-dimension!---this potential well can trap infinitely many bosons. Even hard-core repulsive interactions do not prevent this effect from taking place. Our results imply that an attractive impurity in a generic SF-MI quantum critical environment can carry divergent charges.