arXiv subjects

Frank Marsiglio

Publications and source records attributed to Frank Marsiglio.

At least 19 recordsLinked to original sources

arXiv2026

A Superconducting Peierls Instability

The Peierls instability is a foundational mechanism in condensed matter physics, showing how the electron--phonon interaction can transform a simple metal into an insulating state with a new lattice periodicity. In a one-dimensional (1D) metal the Peierls instability follows from the enhanced electronic response at wavevector $Q=2k_F$ (connecting the Fermi points) producing a Kohn anomaly: a softening of the phonon mode at the same wavevector. Condensation of this mode then generates the tell-tale Peierls periodic lattice distortion and charge-density wave (CDW) that gaps the electronic spectrum. Here we show an analogous instability at the edge of a two-dimensional (2D) superconductor, where a dispersing Andreev bound state (ABS) hosts Bogoliubov Fermi points at $\pm k_c$. The enhanced quasiparticle response at the connecting wavevector $Q=2k_c$ couples directly to pairing-fluctuations and produces a superconducting Kohn anomaly: a softening of a pairing mode at the same wavevector. Condensation of this mode then generates an edge pair-density wave (PDW) that gaps the ABS. We refer to this as a superconducting Peierls instability and identify the boundary quasiparticle structure that enables it. As a concrete realization, we consider a square-lattice extended Hubbard model whose mixed-symmetry $s+d+ip$ state hosts a dispersing ABS with zero-energy crossings at finite edge momenta. Using self-consistent Bogoliubov--de Gennes (BdG) calculations we show that the order parameter develops an edge PDW, with the wavevector set by the superconducting Kohn anomaly, that gaps the ABS crossings. Our results identify a novel mechanism for spontaneous translation-symmetry breaking in a superconductor. As this superconducting Peierls mechanism does not require an extensive zero-energy flat band or topologically protected edge states, it may apply broadly to unconventional superconductors.

cond-mat.supr-con↗

arXiv2026

Analytical Solution to the Kronig-Penney Model with Harmonic Oscillator Wells: Insights to Tight-Binding

The celebrated Kronig-Penney model traditionally has been formulated with square well potentials representing atomic centres. Here, we use a slightly more realistic potential, the truncated harmonic oscillator, in lieu of square well potentials, and solve the model analytically. We derive the energy dispersion and wave functions for this model. This configuration has some important similarities and differences compared to the usual model. In particular, we write the governing equation in a form suggestive of the tight-binding approximation, as can be done for the usual model. In this way, it is straightforward to derive an expression for the tunneling amplitude used in tight-binding in terms of the harmonic oscillator potential parameters.

quant-ph↗

arXiv2026

Coherent-state ansatz for the Holstein polaron in one and two dimensions

The Holstein model often serves as an archetype for electron-phonon interactions and polaron formation in solids. However, precise descriptions of the Holstein polaron are difficult when the phonon frequency is small and the electron-phonon coupling is strong, due to the presence of many phonons in the ground state. We present a semi-analytical approximation that consists of a variational ansatz with clouds of phonons surrounding the electron in the form of coherent states. This becomes particularly simple and exact in the Lang-Firsov limit. We determine the domain of validity away from this limit, and further explore the improvement achieved with a removal of the requirement that the phonon clouds form coherent states. Both approximations work extremely well at strong coupling, and both work surprisingly well also at weak coupling. The coherent-state ansatz provides a simple and intuitive picture of the polaron ground-state wavefunction, and in addition predicts accurate values for the ground-state energy and effective mass.

cond-mat.str-el↗

arXiv2025

A simple quantum dot: numerical and variational solutions

We describe a simple quantum dot that consists of two crossed two-dimensional troughs. As such there is no potential well; nonetheless, this geometry gives rise to a bound state, centred on the point at which these troughs cross one another. This problem is interesting both because the existence of a bound state may surprise students and because it can be solved using a variety of computational techniques, including matrix mechanics, finite differences, and mode matching. We present these methods and show how the mode-matching method in this case provides the most accurate solution to the problem. Additionally, the mode-matching method can be used to generate a simple wave function that yields the lowest energy known to date to arise out of an analytical variational solution for this problem.

cond-mat.mes-hall↗

arXiv2025

Mixed-symmetry superconductivity and the energy gap

The symmetry of the superconducting order parameter, or simply the ``gap'', provides certain constraints on the actual mechanism that gives rise to pairing and ultimately to superconductivity. In this work we show how superconducting phases with mixed singlet-triplet symmetries can arise below $T_c$ for a generic tight-binding model. We first examine the 1D case to better illustrate the prevalence of symmetry-breaking transitions below $T_c$, and then the more realistic 2D case. In both cases we illustrate the implication for spectroscopic investigations of the energy gap by calculating the density of states for different temperatures below $T_c$. We find that the structure of the density of states near $T_c$ can vary dramatically from its form near $T=0$. A complete picture of the superconducting symmetry can only be attained if measurements are made over the entire temperature range.

cond-mat.supr-con↗

arXiv2024

Investigation of Floquet engineered non-Abelian geometric phase for holonomic quantum computing

Holonomic quantum computing (HQC) functions by transporting an adiabatically degenerate manifold of computational states around a closed loop in a control-parameter space; this cyclic evolution results in a non-Abelian geometric phase which may couple states within the manifold. Realizing the required degeneracy is challenging, and typically requires auxiliary levels or intermediate-level couplings. One potential way to circumvent this is through Floquet engineering, where the periodic driving of a nondegenerate Hamiltonian leads to degenerate Floquet bands, and subsequently non-Abelian gauge structures may emerge. Here we present an experiment in ultracold $^{87}$Rb atoms where atomic spin states are dressed by modulated RF fields to induce periodic driving of a family of Hamiltonians linked through a fully tuneable parameter space. The adiabatic motion through this parameter space leads to the holonomic evolution of the degenerate spin states in $SU(2)$, characterized by a non-Abelian connection. We study the holonomic transformations of spin eigenstates in the presence of a background magnetic field, characterizing the fidelity of these single-qubit gate operations. Results indicate that while the Floquet engineering technique removes the need for explicit degeneracies, it inherits many of the same limitations present in degenerate systems.

quant-ph↗

arXiv2021

The bound-state solutions of the one-dimensional pseudoharmonic oscillator

We study the bound states of a quantum mechanical system consisting of a simple harmonic oscillator with an inverse square interaction, whose interaction strength is governed by a constant $α$. The singular form of this potential has doubly-degenerate bound states for $-1/4\leqα<0$ and $α>0$; since the potential is symmetric, these consist of even and odd-parity states. In addition we consider a regularized form of this potential with a constant cutoff near the origin. For this regularized potential, there are also even and odd-parity eigenfunctions for $α\geq-1/4$. For attractive potentials within the range $-1/4\leqα<0$, there is an even-parity ground state with increasingly negative energy and a probability density that approaches a Dirac delta function as the cutoff parameter becomes zero. These properties are analogous to a similar ground state present in the regularized one-dimensional hydrogen atom. We solve this problem both analytically and numerically, and show how the regularized excited states approach their unregularized counterparts.

quant-ph↗

arXiv2021

Vortex-line topology in iron-based superconductors with and without second-order topology

The band topology of a superconductor is known to have profound impact on the existence of Majorana zero modes in vortices. As iron-based superconductors with band inversion and $s_{\pm}$-wave pairing can give rise to time-reversal invariant second-order topological superconductivity, manifested by the presence of helical Majorana hinge states in three dimensions, we are motivated to investigate the interplay between the second-order topology and the vortex lines in both weak- and strong-Zeeman-field regimes. In the weak-Zeeman-field regime, we find that vortex lines far away from the hinges are topologically nontrivial in the weakly doped regime, regardless of whether the second-order topology is present or not. However, when the superconductor falls into the second-order topological phase and a topological vortex line is moved close to the helical Majorana hinge states, we find that their hybridization will trivialize the vortex line and transfer robust Majorana zero modes to the hinges. Furthermore, when the Zeeman field is large enough, we find that the helical Majorana hinge states are changed into chiral Majorana hinge modes and thus a chiral second-order topological superconducting phase is realized. In this regime, the vortex lines are always topologically trivial, no matter how far away they are from the chiral Majorana hinge modes. By incorporating a realistic assumption of inhomogeneous superconductivity, our findings can explain the recent experimental observation of the peculiar coexistence and evolution of topologically nontrivial and trivial vortex lines in iron-based superconductors.

cond-mat.supr-con↗

arXiv2020

The functional-integral approach to Gaussian fluctuations in Eliashberg theory

The Eliashberg theory of superconductivity is based on a dynamical electron-phonon interaction as opposed to a static interaction present in BCS theory. The standard derivation of Eliashberg theory is based on an equation of motion approach, which incorporates certain approximations such as Migdal's approximation for the pairing vertex. In this paper we provide a functional-integral-based derivation of Eliashberg theory and we also consider its Gaussian-fluctuation extension. The functional approach enables a self-consistent method of computing the mean-field equations, which arise as saddle-point conditions, and here we observe that the conventional Eliashberg self energy and pairing function both appear as Hubbard-Stratonovich transformations. An important consequence of this fact is that it provides a systematic derivation of the Cooper and density-channel interactions in the Gaussian fluctuation response. We also investigate the strong-coupling fluctuation diamagnetic susceptibility near the critical temperature.

cond-mat.supr-con↗

arXiv2020

Mixed temperature-dependent order parameters in the extended Hubbard model

The extended Hubbard model can host s-wave, d-wave and p-wave superconducting phases depending on the values of the on-site and nearest-neighbour interactions. Upon detailed examination of the free energy functional of the gap in this model, we show that these symmetries are often dependent on temperature. The critical points of this functional are constrained by symmetry and allow us to formulate stringent conditions on the temperature profile of the gap function, applicable to other models as well. We discuss the finite temperature phase diagram of the extended Hubbard model, and point out the existence of symmetry transitions below $T_c$. Understanding the nature of these transitions is crucial to assessing the symmetry of unconventional superconductors.

cond-mat.supr-con↗

arXiv2020

The bound-state solutions of the one-dimensional hydrogen atom

The one-dimensional hydrogen atom is an intriguing quantum mechanics problem that exhibits several properties which have been continually debated. In particular, there has been variance as to whether or not even-parity solutions exist, and specifically whether or not the ground state is an even-parity state with infinite negative energy. We study a "regularized" version of this system, where the potential is a constant in the vicinity of the origin, and we discuss the even- and odd-parity solutions for this regularized one-dimensional hydrogen atom. We show how the even-parity states, with the exception of the ground state, converge to the same functional form and become degenerate for $x > 0$ with the odd-parity solutions as the cutoff approaches zero. This differs with conclusions derived from analysis of the singular (i.e., without regularization) one-dimensional Coulomb potential, where even-parity solutions are absent from the spectrum.

quant-ph↗

arXiv2020

Edge Localized Schrödinger Cat States in Finite Lattices via Periodic Driving

Floquet states have been used to describe the impact of periodic driving on lattice systems, either using a tight-binding model, or by using a continuum model where a Kronig-Penney-like description has been used to model spatially periodic systems in one dimension. A number of these studies have focused on finite systems, and results from these studies are distinct from those of infinite lattice systems as a consequence of boundary effects. In the case of a finite system, there remains a discrepancy in the results between tight-binding descriptions and continuous lattice models. Periodic driving by a time-dependent field in tight-binding models results in a collapse of all quasienergies within a band at special driving amplitudes. In the continuum model, on the other hand, a pair of nearly-degenerate edge bands emerge and remain gapped from the bulk bands as the field amplitude increases. We resolve these discrepancies and explain how these edge bands represent Schrödinger cat-like states with effective tunneling across the entire lattice. Moreover, we show that these extended cat-like states become perfectly localized at the edge sites when the external driving amplitude induces a collapse of the bulk bands.

cond-mat.mes-hall↗

arXiv2020

First- and Second-Order Topological Superconductivity and Temperature-Driven Topological Phase Transitions in the Extended Hubbard Model with Spin-Orbit Coupling

The combination of spin-orbit coupling with interactions results in many exotic phases of matter. In this Letter, we investigate the superconducting pairing instability of the two-dimensional extended Hubbard model with both Rashba and Dresselhaus spin-orbit coupling within the mean-field level at both zero and finite temperature. We find that both first- and second-order time-reversal symmetry breaking topological gapped phases can be achieved under appropriate parameters and temperature regimes due to the presence of a favored even-parity $s+id$-wave pairing even in the absence of an external magnetic field or intrinsic magnetism. This results in two branches of chiral Majorana edge states on each edge or a single zero-energy Majorana corner state at each corner of the sample. Interestingly, we also find that not only does tuning the doping level lead to a direct topological phase transition between these two distinct topological gapped phases, but also using the temperature as a highly controllable and reversible tuning knob leads to different direct temperature-driven topological phase transitions between gapped and gapless topological superconducting phases. Our findings suggest new possibilities in interacting spin-orbit coupled systems by unifying both first- and higher-order topological superconductors in a simple but realistic microscopic model.

cond-mat.supr-con↗

arXiv2020

Landau Levels and the Issue of Gauge Invariance in Confined Spaces

We examine the behaviour of a charged particle in a two-dimensional confining potential, in the presence of a magnetic field. The confinement serves to remove the otherwise infinite degeneracy, but additional ingredients are required to produce sensible results. We treat both circular and square geometries, and in the latter we explicitly demonstrate the gauge invariance of the energy levels and wave function amplitudes. Both bulk states and edge states are examined, and in the latter case, with sufficiently high quantum numbers we achieve significant differences in the square and circular geometries. Results are achieved using straightforward matrix mechanics, in a manner that is accessible to novices in the field.

cond-mat.mes-hall↗

arXiv2020

Majorana corner flat bands in two-dimensional second-order topological superconductors

In this paper we find that confining a second-order topological superconductor with a harmonic potential leads to a proliferation of Majorana corner modes. As a consequence, this results in the formation of Majorana corner flat bands which have a fundamentally different origin from that of the conventional mechanism. This is due to the fact that they arise solely from the one-dimensional gapped boundary states of the hybrid system that become gapless without the bulk gap closing under the increase of the trapping potential magnitude. The Majorana corner states are found to be robust against the strength of the harmonic trap and the transition from Majorana corner states to Majorana flat bands is merely a smooth crossover. As a harmonic trap can potentially be realized in heterostructures, this proposal paves a way to observe these Majorana corner flat bands in an experimental context.

cond-mat.supr-con↗

arXiv2018

The effect of strong electron-rattling phonon coupling on some superconducting properties

Using the Eliashberg theory of superconductivity we have examined several properties of a model in which electrons are coupled only to rattling phonon modes represented by a sharp peak in the electron-phonon coupling function. Our choice of parameters was guided by experiments on $β$-pyrochlore oxide superconductor KOs$_{2}$Os$_{6}$. We have calculated the temperature dependence of the superconducting gap edge, the quasiparticle decay rate, the NMR relaxation rate assuming that the coupling between the nuclear spins and the conduction electrons is via a contact hyperfine interaction which would be appropriate for the O-site in KOs$_{2}$Os$_{6}$, and the microwave conductivity. We examined the limit of very strong coupling by considering three values of the electron-phonon coupling parameter $λ=$ 2.38, 3, and 5 and {\em did not} assume that the rattler frequency $Ω_{0}$ is temperature dependent in the superconducting state. We obtained a very unusual temperature dependence of the superconducting gap edge $Δ(T)$, very much like the one extracted from photoemission experiments on KOs$_2$O$_6$.

cond-mat.supr-con↗

arXiv2018

Enhancement of Superconducting $T_c$ due to the Spin-orbit Interaction

We calculate the superconducting $T_c$ for a system which experiences Rashba spin-orbit interactions. Contrary to the usual case where the electron-electron interaction is assumed to be wave vector-independent, where superconductivity is suppressed by the spin-orbit interaction (except for a small region at low electron or hole densities), we find an enhancement of the superconducting transition temperature when we include a correlated hopping interaction between electrons. This interaction originates in the expansion of atomic orbitals due to electron-electron repulsion and gives rise to superconductivity only at high electron (low hole) densities. When superconductivity results from this interaction it is enhanced by spin-orbit coupling, in spite of a suppression of the density of states. The degree of electron-hole asymmetry about the Fermi surface is also enhanced.

cond-mat.supr-con↗

arXiv2017

Microscopic origin of the Drude-Smith model

The Drude-Smith model has been used extensively in fitting the THz conductivities of nanomaterials with carrier confinement on the mesoscopic scale. Here, we show that the conventional 'backscattering' explanation for the suppression of low-frequency conductivities in the Drude-Smith model is not consistent with a confined Drude gas of classical non-interacting electrons and we derive a modified Drude-Smith conductivity formula based on a diffusive restoring current. We perform Monte Carlo simulations of a model system and show that the modified Drude-Smith model reproduces the extracted conductivities without free parameters. This alternate route to the Drude-Smith model provides the popular formula with a more solid physical foundation and well-defined fit parameters.

cond-mat.mes-hall↗