## Phase transitions in low dimensional systems via spontaneous symmetry breaking and topology

##### Doctoral thesis

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http://hdl.handle.net/11250/2364840##### Issue date

2015##### Metadata

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- Institutt for fysikk [1149]

##### Abstract

In nature there exists a fundamental distinction among particles regarding their statistics. Let us consider an isolated systems of two particles: two bosons are allowed to stay in the same physical state while two fermions are not. Electrons are fermions, they respond to the electromagnetic forces because they are charged.
Let us consider this question: is it possible to design a system of neutral bosons in such a way their Hamiltonian appears to be the same as that of a system of charged fermions under the action of an external magnetic field? This correspondence has been realized starting from 2008 in a set of brilliant experiments. Despite the equality of their Hamiltonians systems of Bosons and Fermions have still different statistics so it is open the exciting way of studying systems with fermionic Hamiltonians in the context of Bose-statistics. A striking phenomenon relative to bosonic systems is that ofcondensation. Bose-Einstein condensation (BEC) appears when the majority of particles of the system resides in the same state, the ground state. Condensation does not always occur there are limiting conditions related to the system’s dimensionality, the type of inter-particles interaction and the presence of external confinement potentials. In 2009 the BEC phase of bosons with the same Hamiltonian as charged fermions in an external magnetic field was experimentally detected for the fist time. The first part of my thesis presents the general features of the BEC phenomenon starting from its general occurrence in the free case till the illustration of the laser-atom interactions that allow to experimentally realize a system of bosons subjected to the same Hamiltonian of a system of fermions under a constant magnetic field. This part ends with the detailed discussion of a research article of mine and my supervisor, Phys. Rev. A 89, 061605(R), that studies the features of a two dimensional realization of a BEC of particles that have their motion coupled to their spin. The main results of this paper regard: the discussion of two different regimes of superfluidity one with finite condensation momentum the other with vanishing momentum, the discussion of the quantum phase transition among these two phases, the determination of the phase diagram of the superfluid phase, the determination of the sound velocities in the two regimes, the estimate of the BEC depletion at the transition among the two superfluid regimes.
The general topic of the second part of my thesis is that of topological states of matter in fermionic condensed matter systems. A state of matter is said to be topological when it has measurable quantities that are proportional to integer numbers called topological invariants. These quantities are robust within a certain extent against disorder, defects, deformations of the sample and weak interactions. To get an idea of what a topological features of a sample is let us consider the number of holes of a two dimensional surface. Let us take for example your favorite tennis headband, this is a two dimensional surface because you need only two angles to fix a point on it. You can stretch your headband or bend it, it still have one hole, but not cut it or knot it because the number of holes will change to zero or two. It was discovered at the beginning of the 80’ that the quantization of the transverse conductance of a two dimensional sample of electrons subjected to a perpendicular magnetic field has a topological nature, moreover in a finite sample the current flowing at the edge is insensitive to impurities, defects and weak inter particle interactions. This phenomenon is called quantum Hall (QH) effect. A related phenomenon proved almost ten years ago is the quantum spin Hall (QSH) effect. The QSH effect does not require an external magnetic field but it is an intrinsic state of matter consisting in the existence of a couple of counterpropagating currents of oppositely spin polarized electrons flowing along the edge of a two dimensional sample. In this case the topological invariant characterizing the system assumes only two values: zero or one. This effect has been detected in exotic junctions like a HgTe quantum wells. The second part of my thesis starts with the description of graphene (a two dimensional arrangement of carbon atoms on a honeycomb lattice) from the point of view of symmetries. It is relevant to consider if physical models related to graphene can support topological state. This brings to the study of silicone a close relative of graphene, it is an honeycomb arrangement of silicon atoms with staggered sublattices, that under the action of an external perpendicular electric fieldhas edges displaying the spin quantum Hall effect. The second part ends with a discussion of the research results of a preprint of mine and my supervisor regarding the topological phases of silicene under the action of an external static perpendicular electric field and in plane magnetic field and moreover with an applied circularly polarized electromagnetic field. The main achievements of this work regard: the determination of the topological phases of silicene under the applied external field, the determination of the features of the edge modes of a sample of finite size that are in agreement with the values of the invariants of each topological phase using the bulk-boundary correspondence.