Josephson junctions based on quantum dots offer a convenient tunability by means of local gates. Here we analyze a Josephson junction based on a serial double quantum dot in which the two dots are individually gated by phase-shifted microwave tones of equal frequency. We calculate the time-averaged current across the junction and determine how the phase shift between the drives modifies the current-phase relation of the junction. Breaking particle-hole symmetry on the dots is found to give rise to a finite average anomalous Josephson current with phase bias between the superconductors fixed to zero. This microwave gated weak link thus realizes a tunable “Floquet φ0 junction” with maximum critical current achieved for driving frequencies slightly off resonance with the subgap excitation energy. We provide numerical results supported by an analytical analysis for infinite superconducting gap and weak interdot coupling. We identify an interaction-driven 0−π transition of anomalous Josephson current as a function of driving phase difference. Finally, we show that this junction can be tuned so as to provide for complete rectification of the time-averaged Josephson current-phase relation.

Recent years have seen a growing interest in topological phases beyond the standard paradigm of gapped isolated systems. One recent direction is to explore topological features in non-Hermitian systems that are commonly used as effective descriptions of open systems. Another direction explores the fate of topology at critical points, where the bulk gap collapses. One interesting observation is that both systems, though very different, share certain topological features. For instance, both systems can host half-integer quantized winding numbers and have very similar entanglement spectra. Here we make this similarity explicit by showing the equivalence of topological invariants in critical systems with non-Hermitian point-gap phases, in the presence of sublattice symmetry. Also, the corresponding entanglement spectra show the same topological features. This correspondence may carry over to other features and even be helpful to deepen our understanding of non-Hermitian systems using our knowledge of critical systems and vice versa.

A central topic in condensed matter research during the last decades has been the study and classification of topological phases of matter. Topological insulators in particular, a subset of symmetry protected topological phases, have been investigated for over a decade. In recent years, several extensions to this formalism have been proposed to study more unconventional systems.In this thesis we explore two of these extensions, where key assumptions in the original formalism are removed. The first case is critical systems, which have no energy gap. Conventional topological invariants are discontinuous at topological transitions, and therefore not well-defined for critical systems. We propose a method for generalizing conventional topological invariants to critical systems and show robustness to disorder that preserves criticality. The second case involves non-Hermitian systems, which appear in effective descriptions of dissipation, where we study the entanglement spectrum and its connection to topological invariants. Furthermore, by introducing non-Hermiticity to critical systems we show how the winding numbers that characterize some topological phases of the non-Hermitian system, as well as topological signatures in the entanglement spectrum, can be obtained from the related critical model.

The entanglement spectrum is a useful tool to study topological phases of matter, and contains valuable information about the ground state of the system. Here, we study its properties for free non-Hermitian systems for both point-gapped and line-gapped phases. While the entanglement spectrum only retains part of the topological information in the former case, it is very similar to Hermitian systems in the latter. In particular, it not only mimics the topological edge modes, but also contains all the information about the polarization, even in systems that are not topological. Furthermore, we show that the Wilson loop is equivalent to the many-body polarization and that it reproduces the phase diagram for the system with open boundaries, despite being computed for a periodic system.

Various types of topological phenomena at criticality are currently under active research. In this paper we suggest to generalize the known topological quantities to finite temperatures, allowing us to consider gapped and critical (gapless) systems on the same footing. It is then discussed that the quantization of the topological indices, also at critically, is retrieved by taking the low-temperature limit. This idea is explicitly illustrated on a simple case study of chiral critical chains where the quantization is shown analytically and verified numerically. The formalism is also applied for studying robustness of the topological indices to various types of disordering perturbations.

The bulk polarization is a Z(2) topological invariant characterizing noninteracting systems in one dimension with chiral or particle-hole symmetries. We show that the bulk polarization can always be determined from the single-particle entanglement spectrum, even in the absence of symmetries that quantize it. In the symmetric case, the known relation between the bulk polarization and the number of virtual topological edge modes is recovered. We use the bulk polarization to compute Chern numbers in one and two dimensions, which illuminates their known relation to the entanglement spectrum. Furthermore, we discuss an alternative bulk polarization that can carry more information about the surface spectrum than the conventional one and can simplify the calculation of Chern numbers.

In this Licentiate thesis we give a brief review on the topics of topological

insulator and superconductor phases, the modern theory of polarization and

the entanglement spectrum, with a focus on one- and two-dimensional systems.

In the context of symmetry protected topological systems the bulk polarization

can be a topological invariant which characterizes the topological phase. By

the bulk-boundary correspondence the bulk polarization is known to be related

to the number of topological edge states, which is encoded in the entanglement

spectrum.

We study the general relation between the bulk polarization and the entanglement

spectrum and show how the bulk polarization can always be decoded

from the entanglement spectrum, even in the absence of symmetries that quantize

it. Applied to the topological case the known relation between the bulk

polarization and the number of topological edge states is recovered. Since the

bulk polarization is a geometric phase, we use it to compute Chern numbers

in one- and two-dimensional systems. The computation of these Chern numbers

is simplied by using an alternative bulk polarization constructed using

the entanglement spectrum. This alternative bulk polarization can also provide

more information about the topological features of the boundary than the

conventional bulk polarization.

I den har Licentiatavhandlingen ges en introduktion till faser i topologiska

isolatorer och supraledare, den moderna polarisationsteorin samt samman

atningsspektrumet, med fokus pa en- och tvadimensionella system. I

symmetriskyddade topologiska system kan bulkpolarisationen vara en topologisk

invariant som karaktariserar den topologiska fasen. Enligt bulkrandkorrespondensen

ar bulkpolarisationen relaterad till antalet topologiska

kanttillstand, som ar invavt i sammanatningsspektrumet.

Vi undersoker det allmanna forhallandet mellan bulkpolarisationen och samman

atningsspektrumet och visar hur man alltid kan hitta bulkpolarisationen

fran sammanatningsspektrumet, aven nar det inte nns nagra symmetrier

som kvantiserar det. I det topologiska fallet visar vi pa ett nytt satt

hur bulkpolarisationen beror pa antalet virtuella topologiska kanttillstand i

sammanatningsspektrumet. Eftersom bulkpolarisationen ar en geometrisk fas

anvander vi den for att berakna Cherntal i en- och tvadimensionella system.

Berakningen av Cherntalen forenklas genom att anvanda en alternativ bulkpolarisation

som konstrueras med hjalp av sammanatningsspektrumet och som

ocksa kan ge mer information om randens topologiska egenskaper an den vanliga

bulkpolarisationen.