arXiv subjects

Bruno Mera

Publications and source records attributed to Bruno Mera.

29 records · Page 2Linked to original sources

arXiv2021

Kähler geometry and Chern insulators: Relations between topology and the quantum metric

We study Chern insulators from the point of view of Kähler geometry, i.e. the geometry of smooth manifolds equipped with a compatible triple consisting of a symplectic form, an integrable almost complex structure and a Riemannian metric. The Fermi projector, i.e. the projector onto the occupied bands, provides a map to a Kähler manifold. The quantum metric and Berry curvature of the occupied bands are then related to the Riemannian metric and symplectic form, respectively, on the target space of quantum states. We find that the minimal volume of a parameter space with respect to the quantum metric is $π|\mathcal{C}|$, where $\mathcal{C}$ is the first Chern number. We determine the conditions under which the minimal volume is achieved both for the Brillouin zone and the twist-angle space. The minimal volume of the Brillouin zone, provided the quantum metric is everywhere non-degenerate, is achieved when the latter is endowed with the structure of a Kähler manifold inherited from the one of the space of quantum states. If the quantum volume of the twist-angle torus is minimal, then both parameter spaces have the structure of a Kähler manifold inherited from the space of quantum states. These conditions turn out to be related to the stability of fractional Chern insulators. For two-band systems, the volume of the Brillouin zone is naturally minimal provided the Berry curvature is everywhere non-negative or nonpositive, and we additionally show how the latter, which in this case is proportional to the quantum volume form, necessarily has zeros due to topological constraints.

cond-mat.mes-hall↗

arXiv2021

Interferometric geometry from symmetry-broken Uhlmann gauge group with applications to topological phase transitions

We provide a natural generalization of a Riemannian structure, i.e., a metric, recently introduced by Sjöqvist for the space of non degenerate density matrices, to the degenerate case, i.e., the case in which the eigenspaces have dimension greater than or equal to 1. We present a physical interpretation of the metric in terms of an interferometric measurement. We apply this metric, physically interpreted as an interferometric susceptibility, to the study of topological phase transitions at finite temperatures for band insulators. We compare the behaviors of this susceptibility and the one coming from the well-known Bures metric, showing them to be dramatically different. While both infer zero temperature phase transitions, only the former predicts finite temperature phase transitions as well. The difference in behaviors can be traced back to a symmetry breaking mechanism, akin to Landau-Ginzburg theory, by which the Uhlmann gauge group is broken down to a subgroup determined by the type of the system's density matrix (i.e., the ranks of its spectral projectors).

quant-ph↗

arXiv2020

The product of two independent Su-Schrieffer-Heeger chains yields a two-dimensional Chern insulator

We provide an extensive look at Bott periodicity in the context of complex gapped topological phases of free fermions. In doing so, we remark on the existence of a product structure in the set of inequivalent phases induced by the external tensor product of vector bundles -- a structure which has not yet been explored in condensed-matter literature. Bott periodicity appears in the form of a generalized Dirac monopole built out of a given phase, which is equivalent to the product of a Dirac monopole phase with that same given phase. The complex K-theory cohomology ring is presented as a natural way to store the information of these phases, with a grading corresponding to the number of Clifford symmetries modulo $2$. The Künneth formula allows us to derive the result that, for band insulators, the Su-Schrieffer-Heeger (SSH) chain in one dimension allows one to generate the K-cohomology of the $d$-dimensional Brillouin zone. In particular, we find that the product of two SSH chains in independent momentum directions yields a two-dimensional Chern insulator. The results obtained relate the associated topological phases of charge-conserving band insulators and their topological invariants in all spatial dimensions in a unified way.

math-ph↗

arXiv2020

On the minmax regret for statistical manifolds: the role of curvature

Model complexity plays an essential role in its selection, namely, by choosing a model that fits the data and is also succinct. Two-part codes and the minimum description length have been successful in delivering procedures to single out the best models, avoiding overfitting. In this work, we pursue this approach and complement it by performing further assumptions in the parameter space. Concretely, we assume that the parameter space is a smooth manifold, and by using tools of Riemannian geometry, we derive a sharper expression than the standard one given by the stochastic complexity, where the scalar curvature of the Fisher information metric plays a dominant role. Furthermore, we derive the minmax regret for general statistical manifolds and apply our results to derive optimal dimensional reduction in the context of principal component analysis.

math.ST↗

arXiv2020

Localization anisotropy and complex geometry in two-dimensional insulators

The localization tensor is a measure of distinguishability between insulators and metals. This tensor is related to the quantum metric tensor associated with the occupied bands in momentum space. In two dimensions and in the thermodynamic limit, it defines a flat Riemannian metric over the twist-angle space, topologically a torus, which endows this space with a complex structure, described by a complex parameter $τ$. It is shown that the latter is a physical observable related to the anisotropy of the system. The quantity $τ$ and the Riemannian volume of the twist-angle space provide an invariant way to parametrize the flat quantum metric obtained in the thermodynamic limit. Moreover, if by changing the couplings of the theory, the system undergoes quantum phase transitions in which the gap closes, the complex structure $τ$ is still well defined, although the metric diverges (metallic state), and it is fixed by the form of the Hamiltonian near the gap closing points. The Riemannian volume is responsible for the divergence of the metric at the phase transition.

cond-mat.mes-hall↗

arXiv2019

Information geometric analysis of long range topological superconductors

The Uhlmann connection is a mixed state generalisation of the Berry connection. The latter has a very important role in the study of topological phases at zero temperature. Closely related, the quantum fidelity is an information theoretical quantity which is a measure of distinguishability of quantum states. Moreover, it has been extensively used in the analysis of quantum phase transitions. In this work, we study topological phase transitions in 1D and 2D topological superconductors with long-range hopping and pairing amplitudes, using the fidelity and the quantity $Δ$ closely related to the Uhlmann connection. The drop in the fidelity and the departure of $Δ$ from zero signal the topological phase transitions in the models considered. The analysis of the ground state fidelity susceptibility and its associated critical exponents are also applied to the study of the aforementioned topological phase transitions.

cond-mat.supr-con↗

arXiv2019

Topologically protected quantization of work

The transport of a particle in the presence of a potential that changes periodically in space and in time can be characterized by the amount of work needed to shift a particle by a single spatial period of the potential. In general, this amount of work, when averaged over a single temporal period of the potential, can take any value in a continuous fashion. Here we present a topological effect inducing the quantization of the average work. We find that this work is equal to the first Chern number calculated in a unit cell of a space-time lattice. Hence, this quantization of the average work is topologically protected. We illustrate this phenomenon with the example of an atom whose center of mass motion is coupled to its internal degrees of freedom by electromagnetic waves.

quant-ph↗

arXiv2018

Dynamical phase transitions at finite temperature from fidelity and interferometric Loschmidt echo induced metrics

We study finite-temperature Dynamical Quantum Phase Transitions (DQPTs) by means of the fidelity and the interferometric Loschmidt Echo (LE) induced metrics. We analyse the associated dynamical susceptibilities (Riemannian metrics), and derive analytic expressions for the case of two-band Hamiltonians. At zero temperature the two quantities are identical, nevertheless, at finite temperatures they behave very differently. Using the fidelity LE, the zero temperature DQPTs are gradually washed away with temperature, while the interferometric counterpart exhibits finite-temperature Phase Transitions (PTs). We analyse the physical differences between the two finite-temperature LE generalisations, and argue that, while the interferometric one is more sensitive and can therefore provide more information when applied to genuine quantum (microscopic) systems, when analysing many-body macroscopic systems, the fidelity-based counterpart is a more suitable quantity to study. Finally, we apply the previous results to two representative models of topological insulators in 1D and 2D.

cond-mat.stat-mech↗

arXiv2017

Boltzmann-Gibbs states in topological quantum walks and associated many-body systems: Fidelity and Uhlmann parallel transport analysis of Phase Transitions

We perform the fidelity analysis for Boltzmann-Gibbs-like states in order to investigate whether the topological order of 1D fermionic systems at zero temperature is maintained at finite temperatures. We use quantum walk protocols that are known to simulate topological phases and the respective quantum phase transitions for chiral symmetric Hamiltonians. Using the standard approaches of the fidelity analysis and the study of edge states, we conclude that no thermal-like phase transitions occur as temperature increases, i.e., the topological behaviour is washed out gradually. We also show that the behaviour of the Uhlmann geometric factor associated to the considered fidelity exhibits the same behaviour as the latter, thus confirming the results obtained using the previously established approaches.

quant-ph↗

arXiv2017

The Uhlmann connection in fermionic systems undergoing phase transitions

We study the behaviour of the Uhlmann connection in systems of fermions undergoing phase transitions. In particular, we analyse some of the paradigmatic cases of topological insulators and superconductors in dimension one, as well as the BCS theory of superconductivity in three dimensions. We show that the Uhlmann connection signals phase transitions in which the eigenbasis of the state of the system changes. Moreover, using the established fidelity approach and the study of the edge states, we show the absence of thermally driven phase transitions in the case of topological insulators and superconductors. We clarify what is the relevant parameter space associated with the Uhlmann connection so that it signals the existence of order in mixed states.

quant-ph↗

arXiv2015

A theorem regarding families of topologically non-trivial fermionic systems

We introduce a Hamiltonian for fermions on a lattice and prove a theorem regarding its topological properties. We identify the topological criterion as a $\mathbb{Z}_2-$ topological invariant $p(\textbf{k})$ (the Pfaffian polynomial). The topological invariant is not only the first Chern number, but also the sign of the Pfaffian polynomial coming from a notion of duality. Such Hamiltonian can describe non-trivial Chern insulators, single band superconductors or multiorbital superconductors. The topological features of these families are completely determined as a consequence of our theorem. Some specific model examples are explicitly worked out, with the computation of different possible topological invariants.

cond-mat.mes-hall↗