Browsing by Subject "Condensed matter theory"

Understanding the critical behavior near quantum critical points for strongly correlated quantum many-body systems remains intractable for the vast majority of scenarios. Challenges involve determining if a quantum phase transition is first- or second-order, and finding the critical exponents for second-order phase transitions. Learning about where second-order phase transitions occur and determining their critical exponents is particularly interesting, because each new second-order phase transition defines a new quantum field theory.

Quantum Monte Carlo (QMC) methods are one class of techniques that, when applicable, offer reliable ways to extract the nonperturbative physics near strongly coupled quantum critical points. However, there are two formidable bottlenecks to the applicability of QMC: (1) the sign problem and (2) algorithmic update inefficiencies. In this thesis, I overcome both these difficulties for a class of problems by extending the fermion bag approach recently developed by Shailesh Chandrasekharan to the Hamiltonian formalism and by demonstrating progress using the example of a specific quantum system known as the $t$-$V$ model, which exhibits a transition from a semimetal to an insulator phase for a single flavor of four-component Dirac fermions.

I adapt the fermion bag approach, which was originally developed in the context of Lagrangian lattice field theories, to be applicable within the Hamiltonian formalism, and demonstrate its success in two ways: first, through solutions to new sign problems, and second, through the development of new efficient QMC algorithms. In addressing the first point, I present a solution to the sign problem for the $t$-$V$ model. While the $t$-$V$ model is the simplest Gross-Neveu model of the chiral Ising universality class, the specter of the sign problem previously prevented its simulation with QMC for 30 years, and my solution initiated the first QMC studies for this model. The solution is then extended to many other Hamiltonian models within a class that involves fermions interacting with quantum spins. Some of these models contain an interesting quantum phase transition between a massless/semimetal phase to a massive/insulator phase in the so called Gross-Neveu universality class. Thus, the new solutions to the sign problem allow for the use of the QMC method to study these universality classes.

The second point is addressed through the construction of a Hamiltonian fermion bag algorithm. The algorithm is then used to compute the critical exponents for the second-order phase transition in the $t$-$V$ model. By pushing the calculations to significantly larger lattice sizes than previous recent computations ($64^2$ sites versus $24^2$ sites), I am able to compute the critical exponents more reliably here compared to earlier work. I show that the inclusion of these larger lattices causes a significant shift in the values of the critical exponents that was not evident for the smaller lattices. This shift puts the critical exponent values in closer agreement with continuum $4-\epsilon$ expansion calculations. The largest lattice sizes of $64^2$ at a comparably low temperature are reachable due to efficiency gains from this Hamiltonian fermion bag algorithm. The two independent critical exponents I find, which completely characterize the phase transition, are $\eta=.51(3)$ and $\nu=.89(1)$, compared to previous work that had lower values for these exponents. The finite size scaling fit is excellent with a $\chi^2/DOF=.90$, showing strong evidence for a second-order critical phase transition, and hence a non-perturbative QFT can be defined at the critical point.

Boundary quantum phase transitions are abrupt ground state transitions triggered by the change of the boundary conditions at single or multiple (but finite) points.

When boundary effects dominate, understanding boundary quantum phase transitions requires a deeper knowledge of strongly correlated electron systems that is beyond the widely applied mean field treatment.

Meanwhile, with strong boundary effect, most systems with boundary quantum phase transition can generally be considered as effectively zero-dimensional, with reservoir details ignored. Consequently, the critical features of boundary quantum phase transitions only involve long-time correlations instead of long-range ones.

On the other hand, different from the geometrical confinement of boundaries, dissipation or quantum noise widely exists along the entire system.

In bulk quantum phase transitions, dissipation decreases system coherence by reducing the long-range correlations.

This fact makes it plausible that dissipation destroys the critical behavior of the quantum critical points.

The effect of dissipation, however, remains unclear in boundary quantum phase transition systems due to their lack of long-range correlations.

In this thesis I thus focus on the effect of dissipation in boundary quantum phase transitions.

These studies are motivated and encouraged by recent experimental triumphs where dissipation is realized and precisely measured in mesoscopic systems, which provide experimental evidences to check theoretical researches.

This thesis involves multiple dissipative mesoscopic systems, including the dissipative two impurity Kondo, two channel Kondo, resonant level, and Anderson models.

To begin with, the effect of dissipation in two impurity Kondo model has been explored and we find that the presence of dissipation restores the quantum phase transition by reducing the unwanted charge tunneling process. We further provide the phase diagram for the system that has an exotic double-quantum-critical-point feature.

After that, the non-equilibrium $I$-$V$ feature of a dissipative resonant level model is studied.

This model has been experimentally proven to host a boundary quantum phase transition.

With different tuning parameters, we calculate the $I$-$V$ feature at both the quantum critical point and in the crossover regime analytically. The theoretical calculation agrees remarkably with the experimental data.

As the spinful version of the resonant level model, the dissipative Anderson model has multiple unique features, including the experimentally observed peak position shifting and dissipation dependent saturated peak conductance. Through renormalization group studies and mapping the model to the quantum Brownian motion model, we understand these features qualitatively.

As an example of the application of above research achievements, we study the stabilization of a Majorana zero mode with the quantum frustration in a dissipative resonant level model. The Majorana zero mode is known to be unstable against the coupling to its partner at the other end of the Majorana hosted nanowire.

We prove that the Majorana zero mode can be stabilized by coupling its partner to the quantum dot of a frustrated dissipative resonant level model, where an isolated impurity Majorana fermion is produced.

Finally, we study the relation between boundary quantum phase transitions and geometric phases. The calculation is carried out at the Toulouse point of a dissipative resonant level model.

Although it satisfies the criteria of bulk quantum phase transitions to host a non-trivial geometric phase, the dissipative resonant level model has zero geometric phase due to the identical zero geometric curvature. This phenomenon is generally explained by studying the geometric tensor of boundary quantum phase transition-hosted systems.