arXiv subjects

Balázs Hetényi

Publications and source records attributed to Balázs Hetényi.

At least 19 recordsLinked to original sources

arXiv2026

Fibonacci number systems and the localization criterion in the many-body Aubry-André model

We introduce an extension of the Fibonacci number system, we call the fractional Fibonacci number system, which interpolates between the Fibonacci number system (used for natural numbers) and the irrational base-$ϕ$ number system, which can be used to represent real numbers. The number system finds its use in interpreting the localization phase diagram of the many-body non-interacting Aubry-André model. For finite system sizes a localization criterion can be obtained if the particle density is written in the fractional Fibonacci number system. In the thermodynamic limit, the criterion remains, but in this case the particle density is expressed in base-$ϕ$. The nature of the thermodynamic limit is also discussed.

cond-mat.stat-mech↗

arXiv2026

Localization Transitions in a Half-Filled Helical Aubry-André Model

We study localization in a one-dimensional quasiperiodic lattice obtained by extending the Aubry-André model with an additional $N$th-neighbor hopping term of strength $J_{N}$. This long-range tunneling couples successive windings of an effective helical chain and introduces a second control parameter beyond the quasiperiodic potential strength $Δ$. Working with noninteracting fermions (typically at half filling), we diagnose the delocalization-localization transition using extensions of the modern theory of polarization. Specifically, we compute the polarization amplitudes of the many-body Slater-determinant ground state and construct a geometric Binder cumulant from polarization amplitudes. The critical potential where the localization transition happens is extracted from the sign change (zero crossing) of the geometric Binder cumulant. We map critical potential as a function of $J_N$ and the helical range $N$, finding that stronger helical hopping generally stabilizes the extended phase (shifting critical potential upward), while the $N$-dependence can display pronounced commensurability-induced spikes. We further compare the geometric Binder cumulant with the Fermi gap, which remains near zero at small values of potential and opens in the same parameter regime where the geometric Binder cumulant departs from extended phase. Finally, to take a controlled thermodynamic limit along Fibonacci system sizes, we employ a Zeckendorf-shift construction that fixes the many-body sector consistently as system size goes to infinity.

cond-mat.dis-nn↗

arXiv2026

From generating functions to the geometric Binder cumulant

We present an overview of the role of generating functions in quantum mechanical contexts, mainly in the modern theory of polarization and in the study of quantum phase transitions. Generating functions enable the derivation of moments and cumulants, quantities which characterize the fluctuations of an underlying probability distribution. In all of the cases we review, the fluctuations are those of a quantum system. We show that the original formalism for geometric phases, in which a quantum system is taken around an adiabatic cycle, can be extended to the case when degeneracy points are encountered along the cycle (quasiadiabatic cycles). The essential tool for this extension is a generalized Bargmann invariant which plays the role of a generating function. From the cumulants generated this way one can form ratios according to the Binder cumulant scheme in statistical mechanics. Such geometric Binder cumulants are sensitive to gap closure, as such, they are useful in identifying metal-insulator transitions, localization, and quantum phase transitions. We present example calculations on simple model systems, whose localization properties are well known, to validate to approach. We also complement our geometric Binder cumulant calculations with results for the fidelity susceptibility, a quantity directly related to the quantum geometry of the parameter space.

quant-ph↗

arXiv2026

Disorder averaging in random lattice models with periodic boundary conditions: Application to models with uncorrelated and correlated disorder

Periodic boundary conditions are not always used in the study of disordered systems, but it can be advantageous to apply them to mimick thermodynamically large systems. In this case, polarization and its cumulants can not be obtained directly, but through the tools of the modern theory of polarization. This theory casts the polarization in crystalline systems as a geometric phase, rather than an operator expectation value. We develop disorder averaging techniques within the context of this theory which can calculate the variance of the polarization, its higher order moments, and the excess kurtosis (or Binder cumulant). We also derive an indicator of delocalization based on the degeneracy as a function of boundary conditions. We apply the computational techniques to two model systems. To test localization, we use a one-dimensional disordered model which is fully Anderson localized. Our calculations verify this. We also apply our techniques to the one dimensional de Moura-Lyra model, developed to study power law correlated (controlled by a parameter, $α$) disorder. While this model is a pathological one, our method is validated. We also point out the significance of pairwise degeneracies found in the parameter range, $α>2$ and near the band center (or near half filling), where the model was conjectured to exhibit a mobility edge.

cond-mat.dis-nn↗

arXiv2025

Localized states and skin effect around non-Hermitian impurities in tight-binding models

We use the generalized Bloch theorem formalism of Alase {\it et al.} [{\it Phys. Rev. Lett.} {\bf 117} 076804 (2016)] to analyze simple one-dimensional tight-binding lattice systems connected by Hermitian bonds (all with the same hopping parameter $t$), but containing one bond impurity which can be either Hermitian or non-Hermitian. We calculate the band structure, the bulk-boundary correspondence indicator ($D_L(ε)$) and analyze the eigenvalues of the lattice translation operator ($z$), for each eigenstate. From the $z$ values the generalized Brillouin zone can be reconstructed. If the impurity is Hermitian (and $\mathcal{PT}$-symmetric), we find a parameter regime in which two localized edge states separate from the tight-binding band. We then simulate a non-Hermitian impurity by keeping hopping in one direction of the bond impurity the same as the rest of the tight-binding system, and varying only its reciprocal. Again, we find a region with localized edge states, but in this case the energy eigenvalues are purely imaginary. We also find that in this case the two zero energy eigenvectors coalesce, hence this system is an exceptional line. We then perform an interpolative scan between the above two scenarios and find that there is an intermediate region exhibiting a non-Hermitian skin effect. In this region a macroscopic fraction of states acquire complex energy eigenvalues and exhibit localization towards the impurity. Our numerical results are supported by a detailed analysis of the solutions of the boundary/impurity equation.

cond-mat.mes-hall↗

arXiv2025

A numerical study of the localization transition of Aubry-André type models

We use tools based on the modern theory of polarization for a numerical study of the localization transition of the Aubry-André model. In this model the spatial modulation of the potential, $α$, is an irrational number, which we approximate as the ratio of Fibonacci numbers, $F_{n+1}/F_n$, where $F_n=L$ is also the system size. We calculate the phase diagram as a function of particle density (filling) and potential strength $W$. We calculate the geometric Binder cumulant and also apply a renormalization approach. At any given finite system size we find that at many densities the transition occurs at or near $W=2t$ ($t$ denotes the hopping). This is where single particle states are known to localize. However, we also find "spikes", densites at which the transition occurs in the range $0<W<2t$. These spikes occur for densities at which there are no partially filled bands. As the system size (and both $F_n$ and $F_{n+1}$ in $α$) is increased the spikes tend towards zero, but the density at which they occur also changes slightly: they approach irrational numbers which can be written as Fibonacci ratios or sums thereof. For densities which are fixed ratios for all system sizes, the transition occurs at $W=2t$. We also study an extension of the original Aubry-André model with second nearest neighbor hoppings. This model also exhibits a distorted phase diagram compared to the original one, with spikes which do not necessarily tend to zero, but to finite values of $W$, determined by the modifed gap structure.

cond-mat.dis-nn↗

arXiv2024

Scaling of the bulk polarization in extended and localized phases of a quasiperiodic model

We study the finite size scaling of the bulk polarization in a quasiperiodic (Aubry-André) model using the geometric analog of the Binder cumulant. As a proof of concept we show that the geometric Binder cumulant method described here can reproduce the known literature values for the flat and raised cosine distributions, which are the two distributions that occur in the delocalized phase. For the Aubry-André model at half-filling the phase transition point is accurately reproduced. Not only is the correct size scaling exponent of the variance obtained in the extended and the localized phases, but the geometric Binder cumulant undergoes a sign change at the phase transition. We also calculate the state resolved Binder cumulant as a function of disorder strength to gain insight into the mechanism of the localization transition.

cond-mat.dis-nn↗

arXiv2023

Fluctuations, uncertainty relations, and the geometry of quantum state manifolds

The complete quantum metric of a parametrized quantum system has a real part (usually known as the Provost-Vallee metric) and a symplectic imaginary part (known as the Berry curvature). In this paper, we first investigate the relation between the Riemann curvature tensor of the space described by the metric, and the Berry curvature, by explicit parallel transport of a vector in Hilbert space. Subsequently, we write a generating function from which the complex metric, as well as higher order geometric tensors (affine connection, Riemann curvature tensor) can be obtained in terms of gauge invariant cumulants. The generating function explicitly relates the quantities which characterize the geometry of the parameter space to quantum fluctuations. We also show that for a mixed quantum-classical system both real and imaginary parts of the quantum metric contribute to the dynamics, if the mass tensor is Hermitian. A many operator generalization of the uncertainty principle results from taking the determinant of the complex quantum metric. We also calculate the quantum metric for a number of Lie group coherent states, including several representations of the $SU(1,1)$ group. In our examples non-trivial complex geometry results for generalized coherent states. A pair of oscillator states corresponding to the $SU(1,1)$ group gives a double series for its spectrum. The two minimal uncertainty coherent states show trivial geometry, but, again, for generalized coherent states non-trivial geometry results.

quant-ph↗

arXiv2022

Geometric cumulants associated with adiabatic cycles crossing degeneracy points: Application to finite size scaling of metal-insulator transitions in crystalline electronic systems

In this work we focus on two questions. One, we complement the machinary to calculate geometric phases along adiabatic cycles as follows. The geometric phase is a line integral along an adiabatic cycle, and if the cycle encircles a degeneracy point, the phase becomes non-trivial. If the cycle crosses the degeneracy point the phase diverges. We construct quantities which are well-defined when the path crosses the degeneracy point. We do this by constructing a generalized Bargmann invariant, and noting that it can be interpreted as a cumulant generating function, with the geometric phase being the first cumulant. We show that particular ratios of cumulants remain finite for cycles crossing a set of isolated degeneracy points. The cumulant ratios take the form of the Binder cumulants known from the theory of finite size scaling in statistical mechanics (we name them geometric Binder cumulants). Two, we show that the machinery developed can be applied to perform finite size scaling in the context of the modern theory of polarization. The geometric Binder cumulants are size independent at gap closure points or regions with closed gap (Luttinger liquid). We demonstrate this by model calculations for a one-dimensional topological model, several two-dimensional models, and a one-dimensional correlated model. In the case of two dimensions we analyze to different situations, one in which the Fermi surface is one-dimensional (a line), and two cases in which it is zero dimensional (Dirac points). For the geometric Binder cumulants the gap closure points can be found by one dimensional scaling even in two dimensions. As a technical point we stress that only certain finite difference approximations for the cumulants are applicable, since not all approximation schemes are capable of extracting the size scaling information in the case of a closed gap system.

cond-mat.str-el↗

arXiv2021

Scaling and renormalization in the modern theory of polarization: application to disordered systems

We develop a scaling theory and a renormalization technique in the context of the modern theory of polarization. The central idea is to use the characteristic function (also known as the polarization amplitude) in place of the free energy in the scaling theory and in place of the Boltzmann probability in a position-space renormalization scheme. We derive a scaling relation between critical exponents which we test in a variety of models in one and two dimensions. We then apply the renormalization to disordered systems. In one dimension the renormalized disorder strength tends to infinity indicating the entire absence of extended states. Zero(infinite) disorder is a repulsive(attractive) fixed point. In two and three dimensions, at small system sizes, two additional fixed points appear, both at finite disorder, $W_a$($W_r$) is attractive(repulsive) such that $W_a<W_r$. In three dimensions $W_a$ tends to zero, $W_r$ remains finite, indicating metal-insulator transition at finite disorder. In two dimensions we are limited by system size, but we find that both $W_a$ and $W_r$ decrease significantly as system size is increased.

cond-mat.dis-nn↗

arXiv2021

Calculating the polarization in bi-partite lattice models: application to an extended Su-Schrieffer-Heeger model

We address the question of different representation of Bloch states for lattices with a basis, with a focus on topological systems. The representations differ in the relative phase of the Wannier functions corresponding to the diffferent basis members. We show that the phase can be chosen in such a way that the Wannier functions for the different sites in the basis both become eigenstates of the position operator in a particular band. A key step in showing this is the extension of the Brillouin zone. When the distance between sites within a unit cell is a rational number, $p/q$, the Brillouin extends by a factor of $q$. For irrational numbers, the Brillouin zone extends to infinity. In the case of rational distance, $p/q$, the Berry phase "lives" on a cyclic curve in the parameter space of the Hamiltonian, on the Brillouin zone extended by a factor of $q$. For irrational distances the most stable way to calculate the polarization is to approximate the distance as a rational sequence, and use the formulas derived here for rational numbers. The use of different bases are related to unitary transformations of the Hamiltonian, as such, the phase diagrams of topological systems are not altered, but each phase can acquire different topological characteristics when the basis is changed. In the example we use, an extended Su-Schrieffer-Heeger model, the use of the diagonal basis leads to toroidal knots in the Hamiltonian space, whose winding numbers give the polarization.

cond-mat.str-el↗

arXiv2020

Interaction driven polarization shift in the $t-V-V'$ lattice fermion model at half filling: emergent Haldane phase

We study the $t-V-V'$ model in one dimension at half-filling. It is known that for large enough $V$ fixed, as $V'$ is varied, the system goes from a charge-density wave into a Luttinger liquid, then a bond-order, and then a second charge density wave phase. We find that the Luttinger liquid state is further split into two, separating parts with distinct values of the many-body polarization Berry phase. Inside this phase, the variance of the polarization is infinite in the thermodynamic limit, meaning that even if the polarization differs, it would not be measurable. However, in the gapped phases on each side of the Luttinger liquid, the polarization takes a different measurable value, implying topologically distinction. The key difference is that the large-$V'$ phases are link-inversion symmetric, while the small-$V'$ one is site-inversion symmetric. We show that large-$V'$ phase can be related to an $S=1$ spin chain, and exhibits many features of the Haldane phase. The lowest lying states of the entanglement spectrum display different degeneracies in the two cases, and we also find string order in the large-$V'$ phase. We also study the system under open boundary conditions, and suggest that the number of defects is related to the topology.

cond-mat.str-el↗

arXiv2019

Generalized Aubry-André-Harper model with modulated hopping and $p$-wave pairing

We study an extended Aubry-Andr{é}-Harper model with simultaneous modulation of hopping, on-site potential, and $p$-wave superconducting pairing. For the case of commensurate modulation of $β= 1/2$ it is shown that the model hosts four different types of topological states: adiabatic cycles can be defined which pump particles, two types of Majorana fermions, or Cooper pairs. In the incommensurate case we calculate the phase diagram of the model in several regions. We characterize the phases by calculating the mean inverse participation ratio and perform multi-fractal analysis. In addition, we characterize whether the phases found are topologically trivial or not. We find an interesting critical extended phase when incommensurate hopping modulation is present. The rise between the inverse participation ratio in regions separating localized and extended states is gradual, rather than sharp. When, in addition, the on-site potential modulation is incommensurate, we find several sharp rises and falls in the inverse participation ratio. In these two cases all different phases exhibit topological edge states. For the commensurate case we calculate the evolution of the Hofstadter butterfly and the band Chern numbers upon variation of the pairing parameter for zero and finite on-site potential. For zero on-site potential the butterflies are triangular-like near zero pairing, when gap-closure occurs, they are square-like, and hexagonal-like for larger pairing, but with the Chern numbers switched compared to the triangular case. For the finite case gaps at quarter and three-quarters filling close and lead to a switch in Chern numbers.

cond-mat.dis-nn↗

arXiv2019

Quantum phase transitions from analysis of the polarization amplitude

In the modern theory of polarization, polarization itself is given by a geometric phase. In calculations for interacting systems the polarization and its variance are obtained from the polarization amplitude. We interpret this quantity as a discretized characteristic function and derive formulas for its cumulants and moments. In the case of a non-interacting system, our scheme leads to the gauge-invariant cumulants known from polarization theory. We study the behavior of such cumulants for several interacting models. In a one-dimensional system of spinless fermions with nearest neighbor interaction the transition at which gap closure occurs can be clearly identified from the finite size scaling exponent of the variance. When next nearest neighbor interactions are turned on a model with a richer phase diagram emerges, but the finite size scaling exponent is still an effective way to identify the localization transition.

cond-mat.str-el↗

arXiv2018

Variational study of the interacting spinless Su-Schrieffer-Heeger model

We study the phase diagram and the total polarization distribution of the Su-Schrieffer-Heeger model with nearest neighbor interaction in one dimension at half-filling. To obtain the ground state wave-function, we extend the Baeriswyl variational wave function to account for alternating hopping parameters. The ground state energies of the variational wave functions compare well to exact diagonalization results. For the case of uniform hopping for all bonds, where it is known that an ideal conductor to insulator transition takes place at finite interaction, we also find a transition at an interaction strength somewhat lower than the known value. The ideal conductor phase is a Fermi sea. The phase diagram in the whole parameter range shows a resemblance to the phase diagram of the Kane-Mele-Hubbard model. We also calculate the gauge invariant cumulants corresponding to the polarization (Zak phase) and use these to reconstruct the distribution of the polarization. We calculate the reconstructed polarization distribution along a path in parameter space which connects two points with opposite polarization in two ways. In one case we cross the metallic phase line, in the other, we go through only insulating states. In the former case, the average polarization changes discontinuously after passing through the metallic phase line, while in the latter the distribution "walks across" smoothly from one polarization to its opposite. This state of affairs suggests that the correlation acts to break the chiral symmetry of the Su-Schrieffer-Heeger model, in the same way as it happens when a Rice-Mele onsite potential is turned on.

cond-mat.str-el↗

arXiv2018

Statistics and dynamics of the center of mass coordinate in a quantum liquid

Motivated by recent experiments in ultracold gases, we focus on the properties of the center of mass coordinate of an interacting one dimensional Fermi gas, displaying several distinct phases. While the variance of the center of mass vanishes in insulating phases such as phase separated and charge density wave phases, it remains finite in the metallic phase, which realizes a Luttinger liquid. By combining numerics with bosonization, we demonstrate that the autocorrelation function of the center of mass coordinate is universal throughout the metallic phase. It exhibits persistent oscillations and its short time dynamics reveal important features of the quantum liquid, such as the Luttinger liquid parameter and the renormalized velocity. The full counting statistics of the center of mass follows a normal distribution already for small systems. Our results apply to non-integrable systems as well and are within experimental reach for e.g. carbon nanotubes and cold atomic gases.

cond-mat.str-el↗

arXiv2018

Extended Creutz ladder with spin-orbit coupling: a one-dimensional analog of the Kane-Mele model

We construct a topological ladder model, one-dimensional, following the steps which lead to the Kane-Mele model in two dimensions. Starting with a Creutz ladder we modify it so that the gap closure points can occur at either $k = π/ 2$ or $-π/2$. We then couple two such models, one for each spin channel, in such a way that time-reversal invariance is restored. We also add a Rashba spin-orbit coupling term. The model falls in the CII symmetry class. We derive the relevant $2\mathbb{Z}$ topological index, calculate the phase diagram and demonstrate the existence of edge states. We also give the thermodynamic derivation of the quantum spin Hall conductance (Středa-Widom). Approximate implementation of this result indicates that this quantity is sensitive to the topological behavior of the model.

cond-mat.str-el↗

arXiv2018

Topological insulation in a ladder model with particle-hole and reflection symmetries

A two-legged ladder model, one dimensional, exhibiting the parity anomaly is constructed. The model belongs to the $C$ and $CI$ symmetry classes, depending on the parameters, but, due to reflection, it exhibits topological insulation. The model consists of two superimposed Creutz models with onsite potentials. The topological invariants for both cases are {\it mirror winding numbers,} which are nonzero individually in the topological phase, but sum to zero overall in both phases. We demonstrate the presence of edge states and quantized Hall response in the topological region. Our model exhibits two distinct topological regions, distinguished by the different types of reflection symmetries.

cond-mat.str-el↗