Sensitivity of non-Hermitian systems

Understanding the extreme sensitivity of the eigenvalues of non-Hermitian Hamiltonians to the boundary conditions is of great importance when analyzing non-Hermitian systems, as it appears generically and is intimately connected to the skin effect and the breakdown of the conventional bulk boundary correspondence. Here we describe a method to find the eigenvalues of one-dimensional one-band models with arbitrary boundary conditions. We use this method on several systems to find analytical expressions for the eigenvalues, which give us conditions on the parameter values in the system for when we can expect the spectrum to be insensitive to a change in boundary conditions. By stacking one-dimensional chains, we use the derived results to find corresponding conditions for insensitivity for some two-dimensional systems with periodic boundary conditions in one direction. This would be hard by using other methods to detect skin effect, such as the winding of the determinant of the Bloch Hamiltonian. Finally, we use these results to make predictions about the (dis)appearance of the skin effect in purely two-dimensional systems with open boundary conditions in both directions.Comment: 25 pages, 14 figures; v2: title change (previous title was `Stability of non-Hermitian systems') and some minor change

Biorthogonal Renormalization

The biorthogonal formalism extends conventional quantum mechanics to the non-Hermitian realm. It has, however, been pointed out that the biorthogonal inner product changes with the scaling of the eigenvectors, an ambiguity whose physical significance is still being debated. Here, we revisit this issue and argue when this choice of normalization is of physical importance. We illustrate in which settings quantities such as expectation values and transition probabilities depend on the scaling of eigenvectors, and in which settings the biorthogonal formalism remains unambiguous. To resolve the apparent scaling ambiguity, we introduce an inner product independent of the gauge choice of basis and show that its corresponding mathematical structure is consistent with quantum mechanics. Using this formalism, we identify a deeper problem relating to the physicality of Hilbert space representations, which we illustrate using the position basis. Apart from increasing the understanding of the mathematical foundations upon which many physical results rely, our findings also pave the way towards consistent comparisons between systems described by non-Hermitian Hamiltonians.Comment: 19 pages, 3 figure

Phase Transitions and Generalized Biorthogonal Polarization in Non-Hermitian Systems

Non-Hermitian (NH) Hamiltonians can be used to describe dissipative systems, and are currently intensively studied in the context of topology. A salient difference between Hermitian and NH models is the breakdown of the conventional bulk-boundary correspondence invalidating the use of topological invariants computed from the Bloch bands to characterize boundary modes in generic NH systems. One way to overcome this difficulty is to use the framework of biorthogonal quantum mechanics to define a biorthogonal polarization, which functions as a real-space invariant signaling the presence of boundary states. Here, we generalize the concept of the biorthogonal polarization beyond the previous results to systems with any number of boundary modes, and show that it is invariant under basis transformations as well as local unitary transformations. Additionally, we propose a generalization of a perviously-developed method with which to find all the bulk states of system with open boundaries to NH models. Using the exact solutions in combination with variational states, we elucidate genuinely NH aspects of the interplay between bulk and boundary at the phase transitions.Comment: 11 pages, 5 figure

Salmon Calcitonin Attenuates Some Behavioural Responses to Nicotine in Male Mice

The behavioural responses to nicotine involve appetite-regulatory hormones; however, the effects of the anorexigenic hormone amylin on reward-related behaviours induced by nicotine remain to be established. Previous studies have shown that the amylinergic pathway regulates behavioural responses to alcohol, amphetamine and cocaine. Here, we evaluated the effects of salmon calcitonin (sCT), an amylin and calcitonin receptor (CTR) agonist, on nicotine-induced locomotor stimulation and sensitisation as well as dopamine release in the nucleus accumbens (NAc) shell. Moreover, we investigated the effects of sCT on the acquisition and expression of nicotine-induced reward in the conditioned place preference (CPP) paradigm. Finally, we performed Western Blot experiments in an attempt to identify the levels of the amylin receptor components CTRa, CTRb, and RAMP1 in reward-related areas of mice responding differently to repeated injections of sCT and nicotine in the locomotor sensitisation test. We found that sCT blocked nicotine’s stimulatory and dopamine-releasing effects and prevented its ability to cause locomotor sensitisation. On the other hand, sCT did not alter nicotine-induced acquisition and expression of CPP. Lastly, sCT-nicotine treated mice from the locomotor sensitisation experiment displayed higher levels of total CTR, i.e. CTRa and CTRb together, in the reward-processing laterodorsal tegmental area (LDTg) of the brain compared to mice treated with vehicle-nicotine. Overall, the present data reveal that activation of CTR or/and amylin receptors attenuates certain nicotine-induced behaviours in male mice, further contributing to the understanding of appetite-regulatory peptides in reward regulation

Bulk-boundary correspondence and biorthogonality in non-Hermitian systems

In topological insulators, the bulk-boundary correspondence describes the relationship between the bulk invariant -- computed for a system with periodic boundary conditions -- and the number of topological boundary states in the corresponding system with open boundary conditions. This is a well-known property of these systems and is important for predicting how they will behave. In recent years, however, the modeling of dissipative and non-equilibrium systems using non-Hermitian Hamiltonians has become increasingly popular. These systems feature many novel phenomena; in particular the bulk-boundary correspondence breaks down since the spectrum of the system with periodic boundary conditions typically differs fundamentally from the spectrum of the system with open boundary conditions.         In this thesis, the behavior of the boundary states in non-Hermitian lattice models is studied. The framework of biorthogonal quantum mechanics is used to develop the biorthogonal bulk-boundary correspondence, which predicts the (dis)appearance of the boundary states in these systems. Closely related to the drastic change in spectra between boundary conditions is the non-Hermitian skin effect. This refers to the exponential localization of almost all eigenstates to the boundaries and is typically seen in non-Hermitian lattice models. How to predict this, and how to quantify the sensitivity of the spectrum to the boundary conditions are therefore questions that are also studied in this thesis.

Bulk-boundary correspondence and biorthogonality in non-Hermitian systems

In topological insulators, the bulk-boundary correspondence describes the relationship between the bulk invariant -- computed for a system with periodic boundary conditions -- and the number of topological boundary states in the corresponding system with open boundary conditions. This is a well-known property of these systems and is important for predicting how they will behave. In recent years, however, the modeling of dissipative and non-equilibrium systems using non-Hermitian Hamiltonians has become increasingly popular. These systems feature many novel phenomena; in particular the bulk-boundary correspondence breaks down since the spectrum of the system with periodic boundary conditions typically differs fundamentally from the spectrum of the system with open boundary conditions.         In this thesis, the behavior of the boundary states in non-Hermitian lattice models is studied. The framework of biorthogonal quantum mechanics is used to develop the biorthogonal bulk-boundary correspondence, which predicts the (dis)appearance of the boundary states in these systems. Closely related to the drastic change in spectra between boundary conditions is the non-Hermitian skin effect. This refers to the exponential localization of almost all eigenstates to the boundaries and is typically seen in non-Hermitian lattice models. How to predict this, and how to quantify the sensitivity of the spectrum to the boundary conditions are therefore questions that are also studied in this thesis.

Kvasikristaller : Klassificering, diffraktion och ytstudier

Quasicrystal is the term used for a solid that possesses an essentially discrete diffraction pattern without having translational symmetry. Compared to periodic crystals, this difference in structure gives quasicrystals new properties that make them interesting to study -- both from a mathematical and from a physical point of view. In this thesis we review a mathematical description of quasicrystals that aims at generalizing the well-established theory of periodic crystals. We see how this theory can be connected to the cohomology of groups and how we can use this connection to classify quasicrystals. We also review an experimental method, NIXSW (Normal Incidence X-ray Standing Waves), that is ordinarily used for surface structure determination of periodic crystals, and show how it can be used in the study of quasicrystal surfaces. Finally, we define the reduced lattice and show a way to plot lattices in MATLAB. We see that there is a connection between the diffraction pattern and the reduced lattice and we suggest a way to describe this connection

Bandstrukturer för topologiska kristallina isolatorer

Topological insulators and topological crystalline insulators are materials that have a bulk band structure that is gapped, but that also have toplogically protected non-gapped surface states. This implies that the bulk is insulating, but that the material can conduct electricity on some of its surfaces. The robustness of these surface states is a consequence of time-reversal symmetry, possibly in combination with invariance under other symmetries, like that of the crystal itself. In this thesis we review some of the basic theory for such materials. In particular we discuss how topological invariants can be derived for some specific systems. We then move on to do band structure calculations using the tight-binding method, with the aim to see the topologically protected surface states in a topological crystalline insulator. These calculations require the diagonalization of block tridiagonal matrices. We finish the thesis by studying the properties of such matrices in more detail and derive some results regarding the distribution and convergence of their eigenvalues