Topology and interaction effects in one-dimensional systems

Doctoral thesis (2018)

With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades ... [more ▼]

With the discovery of the integer quantum Hall effect by von Klitzing and collaborators in 1980, the mathematical field of topology entered the world of condensed matter physics. Almost three decades later, this eventually led to the theoretical prediction and the experimental realization of many intriguing topological materials and topology-based devices. In this Ph.D. thesis, we will study the interplay between topology and another key topic in condensed matter physics, namely the study of inter-particle interactions in many-body systems. This interplay is analyzed from two different perspectives. Firstly, we studied how the presence of electron-electron interactions affects single-electron injection into a couple of counter-propagating one-dimensional edge channels. The latter appear at the edges of topologically non-trivial systems in the quantum spin Hall regime and they can also be engineered by exploiting the integer quantum Hall effect. Because of inter-channel interactions, the injected electron splits up into a couple of counter-propagating fractional excitations. Here, we carefully study and discuss their properties by means of an analytical approach based on the Luttinger liquid theory and the bosonization method. Our results are quite relevant in the context of the so-called electron quantum optics, a fast developing field which deeply exploits the topological protection of one-dimensional edge states to study the coherent propagation of electrons in solid-state devices. As an aside, we also showed that similar analytical techniques can also be used to study the time-resolved dynamics of a Luttinger liquid subject to a sudden change of the interaction strength, a protocol known as quantum quench which is gaining more and more attention, especially within the cold-atoms community. Secondly, we study how inter-particle interactions can enhance the topological properties of strictly one-dimensional fermionic systems. More precisely, the starting point is the seminal Kitaev chain, a free-fermionic lattice model which hosts exotic Majorana zero-energy modes at its ends. The latter are extremely relevant in the context of topological quantum computation because of their non-Abelian anyonic exchange statistics. Here we show that, by properly adding electron-electron interactions to the Kitaev chain, it is possible to obtain lattice models which feature zero-energy parafermionic modes, an even more intriguing generalization of Majoranas. To this end, we develop at first an exact mapping between Z4 parafermions and ordinary fermions on a lattice. We subsequently exploit this mapping to analytically obtain an exactly solvable fermionic model hosting zero-energy parafermions. We study their properties and numerically investigate their signatures and robustness even when parameters are tuned away from the exactly solvable point. [less ▲]

Universal scaling of quench-induced correlations in a one-dimensional channel at finite temperature

in SciPost Physics (2018)

It has been shown that a quantum quench of interactions in a one-dimensional fermion system at zero temperature induces a universal power law ∝t−2 in its long-time dynamics. In this paper we demonstrate ... [more ▼]

It has been shown that a quantum quench of interactions in a one-dimensional fermion system at zero temperature induces a universal power law ∝t−2 in its long-time dynamics. In this paper we demonstrate that this behaviour is robust even in the presence of thermal effects. The system is initially prepared in a thermal state, then at a given time the bath is disconnected and the interaction strength is suddenly quenched. The corresponding effects on the long times dynamics of the non-equilibrium fermionic spectral function are considered. We show that the non-universal power laws, present at zero temperature, acquire an exponential decay due to thermal effects and are washed out at long times, while the universal behaviour ∝t−2 is always present. To verify our findings, we argue that these features are also visible in transport properties at finite temperature. The long-time dynamics of the current injected from a biased probe exhibits the same universal power law relaxation, in sharp contrast with the non-quenched case which features a fast exponential decay of the current towards its steady value, and thus represents a fingerprint of quench-induced dynamics. Finally, we show that a proper tuning of the probe temperature, compared to that of the one-dimensional channel, can enhance the visibility of the universal power-law behaviour. [less ▲]

Energy and momentum distribution of fractional excitations in helical systems

Poster (2017, September 05)

Quench-induced entanglement and relaxation dynamics in Luttinger liquids

in Physical Review. B (2017), 96

We investigate the time evolution towards the asymptotic steady state of a one-dimensional interacting system after a quantum quench. We show that at finite times the latter induces entanglement between ... [more ▼]

We investigate the time evolution towards the asymptotic steady state of a one-dimensional interacting system after a quantum quench. We show that at finite times the latter induces entanglement between right- and left-moving density excitations, encoded in their cross-correlators, which vanishes in the long-time limit. This behavior results in a universal time decay ∝t−2 of the system spectral properties, in addition to nonuniversal power-law contributions typical of Luttinger liquids. Importantly, we argue that the presence of quench-induced entanglement clearly emerges in transport properties, such as charge and energy currents injected in the system from a biased probe and determines their long-time dynamics. In particular, the energy fractionalization phenomenon turns out to be a promising platform to observe the universal power-law decay ∝t−2 induced by entanglement and represents a novel way to study the corresponding relaxation mechanism. [less ▲]

Transport Properties as a Tool to Study Universal Quench-induced Dynamics in 1D Systems

Scientific Conference (2017, August 07)

The study of the relaxation process that follows a quantum quench in 1D systems still represents an open research field. Here we consider a sudden change of the interparticle interaction and we identify a ... [more ▼]

The study of the relaxation process that follows a quantum quench in 1D systems still represents an open research field. Here we consider a sudden change of the interparticle interaction and we identify a peculiar correlator of the system whose behavior is directly and deeply affected by the quench-induced dynamics. Interestingly, it features a universal power-law decay in time. Unfortunately, such a universal decay, although present, turns out to be subleading in intrinsic properties of the system such as the non- equilibrium spectral function. We thus consider a tunnel coupling of the system with a biased tip in order to be able to study also transport properties, namely the charge and energy current flowing from the tip to the system after the quench. In these quantities the universal power-law emerges clearly, especially if one focuses on energy current and its fractionalization into a right- and left- moving components. In particular, we show that the presence of a transient in the energy fractionalization ratio is a direct hallmark of the quench-induced relaxation. Within the setup we have considered, time-dependent transport properties are thus promoted to useful and promising tools to access the mechanisms at the base of the out-of- equilibrium dynamics following quantum quench. [less ▲]

Fractionalization of charge and energy after electron injection in 1D helical systems

Scientific Conference (2017, March 20)

The possibility to inject a single electron into ballistic 1D conductors is at the basis of the new and fast developing field of electron quantum optics. In this respect, helical edge states of ... [more ▼]

The possibility to inject a single electron into ballistic 1D conductors is at the basis of the new and fast developing field of electron quantum optics. In this respect, helical edge states of topological insulators can be used as electronic waveguides and would be an ideal playground [1,2]. Here we thus study and characterize the tunneling of a single electron from a mesoscopic capacitor into a couple of interacting helical edge channels [3]. The injection process leads to the creation of a pair of fractional excitations travelling in opposite directions. Their charge and energy profiles are analyzed. We also show that the energy partitioning between the two fractional excitations depends both on the interaction strength and on the injection parameters. Interestingly, this allows for a situation in which energy and charge mainly flow in opposite directions. In addition, such peculiar behavior of energy partitioning suggests that it can be also used as a tool to probe features of out-of-equilibrium systems [4]. [1] G. Fève et al., Science 316, 1169 (2007) [2] D. Ferraro et al., PRB 89, 075407 (2014) [3] A. Calzona et al., PRB 94, 035404 (2016) [4] A. Calzona et al., arXiv:1610.04492 [less ▲]

Energy fractionalization after single electron injection into an interacting helical liquid

Poster (2016, September)