Antiferromagnetism and chiral d-wave superconductivity from an effective t−J−Dt-J-D model for twisted bilayer graphene

Starting from the strong-coupling limit of an extended Hubbard model, we develop a spin-fermion theory to study the insulating phase and pairing symmetry of the superconducting phase in twisted bilayer graphene. Assuming that the insulating phase is an anti-ferromagnetic insulator, we show that fluctuations of the anti-ferromagnetic order in the conducting phase can mediate superconducting pairing. Using a self-consistent mean-field analysis, we find that the pairing wave function has a chiral d-wave symmetry. Consistent with this observation, we show explicitly the existence of chiral Majorana edge modes by diagonalizing our proposed Hamiltonian on a finite-sized system. These results establish twisted bilayer graphene as a promising platform to realize topological superconductivity

ELECTRON-ELECTRON INTERACTIONS IN DIRAC FERMIONS

Ph.DDOCTOR OF PHILOSOPH

Quantum tic tac toe.

We quantize the game of tic-tac-toe, by allowing superpositions of classical moves. To play the game, we require quantum moves so defined to be orthogonal to all previous moves, and to compute the weight a player has at a given site, we square the sum of the amplitudes at this site over all his moves. A player wins when the sum of weights along any of the eight straight lines we can draw in the 3 x 3 grid is greater than 3. We play the quantum tic-tac-toe first randomly, and then deterministically, to explore the impacts different opening moves, end games, and blocking strategies have on the outcome of the game. In contrast to the classical game of tic-tac-toe, the deterministic quantum game do not always end up in a draw, and do not always favour the starting player.Bachelor of Science in Physic

Re-Examining of Moffitt’s Theory of Delinquency through Agent Based Modeling

Moffitt’s theory of delinquency suggests that at-risk youths can be divided into two groups, the adolescence- limited group and the life-course-persistent group, predetermined at a young age, and social interactions between these two groups become important during the adolescent years. We built an agent-based model based on the microscopic interactions Moffitt described: (i) a maturity gap that dictates (ii) the cost and reward of antisocial behavior, and (iii) agents imitating the antisocial behaviors of others more successful than themselves, to find indeed the two groups emerging in our simulations. Moreover, through an intervention simulation where we moved selected agents from one social network to another, we also found that the social network plays an important role in shaping the life course outcome.Published versio

The distributions of the end-simulation adolescence-limited proportion when (a) <i>k</i> is increased or decreased by 50%, (b) <i>c</i> is increased or decreased by 50%, (c) <i>γ</i> is increased or decreased by 50%, and (d) Δ<i>e</i> is increased or decreased by 50%, compared against the benchmark distribution <i>F</i>(<b>p</b>₀) for the parameters shown in Table 1.

<p>The bin size for binning the adolescence-limited fraction is 1/<i>N</i> where <i>N</i> = 30 is the number of agents in one simulation. As we can see, the simulation outcome is most sensitive to the parameter <i>γ</i>, which is the rate of increase of the reward for antisocial behavior. Our model is next most sensitive to Δ<i>e</i>, the range of antisocial levels over which social mimicry can occur. Our model is very insensitive to the parameters <i>k</i>, which is how steeply the cost change with antisocial level and <i>c</i>, which is the proportionality constant that limits how fast agents can imitate each other.</p

The peer network structure at (a) <i>t</i> = 1 (age 7), (b) <i>t</i> = 100 (age 15), (c) <i>t</i> = 200 (age 23), and (d) <i>t</i> = 300 (age 32).

<p>The boxes represent the agents, and the labels of the boxes are the indices of the agents. The agents are ordered clockwise from the least antisocial to the most antisocial, and the color of the boxes indicate the antisocial levels of the agents, according to the colorbar at the bottom. All agents are connected to each other, but here we show only the directed connections whose weights are larger than 0.3. In this figure, we see that (a) the initial connections at age 7 are strong only between pro-social agents, and agents mostly imitate agents less antisocial than themselves. During the maturity gap at (b) age 15, social mimicry amongst agents strengthens most connections, and agents mostly imitate agents more antisocial than themselves. After entering adulthood at (c) age 23, the agents split into two groups, a pro-social group where agents imitate those less antisocial than themselves, and an antisocial group where agents imitate those more antisocial than themselves. These two groups become more distinct as time goes on, as we can see in (d) at age 32.</p

Intervention analysis comparing the final antisocial level of an agent in its native, rough network to his final antisocial level after he has been moved to a mild network.

<p>For each of the following four scenarios, we ran 50 simulations. In (a), a guaranteed adolescence-limited agent is moved from the rough network to the mild network, and we see that there is little change in his final antisocial level with or without intervention. In (b), a marginally adolescence-limited agent is moved from the rough network to the mild network, and we see a statistically significant reduction in his final antisocial level. In (c), a marginally life-course-persistent agent is moved from the rough network to the mild network. The reduction in final antisocial level is the largest in this scenario, even though there are cases where the intervention fails, and the agent remains life-course-persistent. Finally, in (d) we move a guaranteed life-course-persistent agent from the rough network to the mild network. Although in some cases we see reduction in the final antisocial level, the agent remains life-course-persistent.</p