arXiv subjects

Domenico Monaco

Publications and source records attributed to Domenico Monaco.

At least 19 recordsLinked to original sources

arXiv2026

The Colored Hofstadter Butterfly as a Many-Body Quantum Hall Phase Diagram

We prove that the colored Hofstadter butterfly has a many-body interpretation for a broad class of weakly interacting lattice fermion systems. Starting from a spectral gap of a Hofstadter-like one-particle Hamiltonian at arbitrary magnetic flux $b$, we construct an open region in the three-dimensional parameter space $(b,\mu,\lambda)$ of magnetic field, chemical potential, and interaction strength on which the infinite-volume interacting system has locally unique gapped ground states. The construction combines quasi-adiabatic continuation in the interaction strength with denominator-independent magnetic perturbation estimates, and therefore covers both commensurate and incommensurate fluxes, where no finite magnetic unit cell exists. On connected uniformly gapped regions meeting the non-interacting plane $\lambda=0$, we prove a many-body gap-labeling theorem: the Hall conductivity appearing in the macroscopic Ohm's law is constant and quantized, satisfying $2\pi\sigma^{\mathrm{H}}\in\mathbb{Z}$. Thus the integer colors of the non-interacting Hofstadter butterfly persist as Hall-conductivity labels of interacting quantum Hall phases.

math-ph↗

arXiv2026

The Zak phase in topologically insulating chains: invariants and limitations

In this work we investigate the topological content of the Zak phase in one-dimensional translation-invariant topological insulators endowed with time-reversal, particle-hole and/or chiral symmetries, extending results from \cite{Monaco_2023}. We analyze the extent to which the Zak phase captures the topology of all Altland--Zirnbauer--Cartan (AZC) symmetry classes in $1$D. Building on the framework of fibered Hamiltonians and spectral projections, we construct symmetric Bloch bases adapted to the discrete symmetries of the system and define a $\mathbb{Z}_2$-valued topological invariant $\mathrm{I}^{(\mathrm{AZC-class})}(H)$ obtained from the abelian Zak phase. Moreover, we demonstrate that in symmetry classes admitting a quaternionic structure, i.e. anti-unitary symmetries squaring to minus the identity, the Zak phase is further constrained, leading to the vanishing of the $\mathbb{Z}_2$ invariant mentioned above. This highlights the sensitivity of the Zak phase to additional geometric structures of the manifold of occupied energy states, as well as its limitations in being an effective marker for topological phases of insulating chains. As an example, we discuss the case of generalized Kitaev chains with arbitrary finite-range hopping and single or multiple chiral channels, and show how the Zak phase only retains partial information about their different topological phases.

math-ph↗

arXiv2026

Universality of the Hall conductivity for a weakly interacting magnetic fermionic gas in the Hartree-Fock approximation

We consider a two-dimensional gas of interacting fermions in presence of an external constant magnetic field: the system is extended and homogeneous, and thus assumed to be invariant under magnetic translations. Working within the Hartree-Fock approximation, we analyze the system directly in the thermodynamic limit by solving a self-consistent fixed-point equation for the one-particle density matrix. We prove that, provided that the interactions among fermions are sufficiently weak, there exists a unique one-particle density matrix that solves the self-consistency condition. By choosing the Fermi-Dirac distribution as the function in the fixed-point equation, this approach can describe both positive and zero-temperature cases. At zero temperature and when the chemical potential of the non-interacting system lies in a spectral gap of the free Landau operator, our self-consistent solution is an orthogonal projection (an "interacting" effective Fermi projection). We prove that its integrated density of states varies linearly with the external magnetic field, provided the interaction is weak enough: the slope of this variation is quantized and independent of the interaction. According to the St\v{r}eda formula, this can be seen as yet another expression of the universality of the quantum Hall effect in weakly interacting systems, at least within the Hartree-Fock approximation.

math-ph↗

arXiv2026

Zak phase and bulk-boundary correspondence in a generalized Dirac-Kronig-Penney model

We investigate the topological properties of a generalized Dirac--Kronig--Penney model, a continuum one-dimensional model for a relativistic quantum chain. By tuning the coupling parameters this model can accommodate five Altland--Zirnbauer--Cartan symmetry classes, three of which (AIII, BDI and D) support non-trivial topological phases in dimension one. We characterize analytically the spectral properties of the Hamiltonian in terms of a spectral function, and numerically compute the Zak phase to probe the bulk topological content of the insulating phases. Our findings reveal that, while the Zak phase is quantized in classes AIII and BDI, it exhibits non-quantized values in class D, challenging its traditional role as a topological marker in continuum settings. We also discuss the bulk-boundary correspondence for a truncated version of the chain, analyzing how the emergence of edge states depends on both the truncation position and the boundary conditions. In classes AIII and BDI, we find that the Zak phase effectively detects edge states as a relative boundary topological index, although the correspondence is highly sensitive to the parameters characterizing the truncation.

quant-ph↗

arXiv2026

Vanishing of power-law corrections to Kubo's formula for the Hall current at incommensurate magnetic fields

We consider a non-interacting electron gas confined to a two-dimensional crystal by the action of a perpendicular magnetic field; in the one-particle approximation, the dynamics of the system is modelled by a spectrally gapped Bloch-Landau Hamiltonian. No commensurability condition is assumed between the magnetic flux per unit cell and the quantum of magnetic flux. We construct a non-equilibrium almost-stationary state (NEASS) which "dresses" the equilibrium Fermi projection on states below the spectral gap, and models the state of the system after the addition of a weak external electric field of strength $\varepsilon \ll 1$. Having in mind applications to the integer quantum Hall effect, we probe the response of a current operator in the direction transverse to that of the applied electric field, and show that the resulting current density in the NEASS is linear in $\varepsilon$, with no power-law corrections. The linear response coefficient, namely the Hall conductivity, is computed in terms of the equilibrium Fermi projection via the double-commutator formula, in accordance with the prediction from Kubo's linear response theory. Our results generalize the methods and findings of [Lett. Math. Phys. 112 (2022), 91] to the setting of uniform magnetic fields with incommensurate magnetic flux per unit cell, and to lattice-periodic perturbation of such magnetic fields.

math-ph↗

arXiv2025

Splitting obstructions and $\mathbb{Z}_2$ invariants in time-reversal symmetric topological insulators

The Fu-Kane-Mele $\mathbb{Z}_2$ index characterizes two-dimensional time-reversal symmetric topological phases of matter. We shed some light on some features of this index by investigating projection-valued maps endowed with a fermionic time-reversal symmetry. Our main contributions are threefold. First, we establish a decomposition theorem, proving that any such projection-valued map admits a splitting into two projection-valued maps that are related to each other via time-reversal symmetry. Second, we provide a complete homotopy classification theorem for these maps, thereby clarifying their topological structure. Third, by means of the previous analysis, we connect the Fu-Kane-Mele index to the Chern number of one of the factors in the previously-mentioned decomposition, which in turn allows to exhibit how the $\mathbb{Z}_2$-valued topological obstruction to constructing a periodic and smooth Bloch frame for the projection-valued map, measured by the Fu-Kane-Mele index, can be concentrated in a single pseudo-periodic Kramers pair.

math-ph↗

arXiv2025

Derivation of Hartree-Fock Dynamics and Semiclassical Commutator Estimates for Fermions in a Magnetic Field

We study the quantum dynamics of a large number of interacting fermionic particles in a constant magnetic field. In a coupled mean-field and semiclassical scaling limit, we show that solutions of the many-body Schr\"odinger equation converge to solutions of a non-linear Hartree-Fock equation. The central ingredient of the proof are certain semiclassical trace norm estimates of commutators of the position and momentum operators with the one-particle density matrix of the solution of the Hartree-Fock equation. In a first step, we prove their validity for non-interacting initial data in a magnetic field by generalizing a 2020 result of Fournais and Mikkelsen. We then propagate these bounds from the initial data along the Hartree-Fock flow to arbitrary times.

math-ph↗

arXiv2024

Near linearity of the macroscopic Hall current response in infinitely extended gapped fermion systems

We consider an infinitely extended system of fermions on a $d$-dimensional lattice with (magnetic) translation-invariant short-range interactions. We further assume that the system has a locally unique gapped ground state. Physically, this is a model for the bulk of a generic topological insulator at zero temperature, and we are interested in the current response of such a system to a constant external electric field. Using the non-equilibrium almost-stationary states approach, we prove that the longitudinal current density induced by a constant electric field of strength $\varepsilon$ is of order $\mathcal{O}(\varepsilon^\infty)$, i.e. the system is an insulator in the usual sense. For the Hall current density we show instead that it is linear in $\varepsilon$ up to terms of order $\mathcal{O}(\varepsilon^\infty)$. The proportionality factor $\sigma_\mathrm{H}$ is by definition the Hall conductivity, and we show that it is given by a generalization of the well known double commutator formula to interacting systems. As a by-product of our results, we find that the Hall conductivity is constant within gapped phases, and that for $d=2$ the relevant observable that "measures" the Hall conductivity in experiments, the Hall conductance, not only agrees with $\sigma_{\mathrm{H}}$ in expectation up to $\mathcal{O}(\varepsilon^\infty)$, but also has vanishing variance. A notable difference to several existing results on the current response in interacting fermion systems is that we consider a macroscopic system exposed to a small constant electric field, rather than to a small voltage drop.

math-ph↗

arXiv2024

Effective quantum dynamics for magnetic fermions

We show how to derive an effective nonlinear dynamics, described by the Hartree-Fock equations, for fermionic quantum particles confined to a two-dimensional box and in presence of an external, uniform magnetic field. The derivation invokes the Dirac-Frenkel principle. We discuss the validity of this effective description with respect to the many-body Schr\"odinger dynamics for small times and for weak interactions, and also in regards to the number of particles.

math-ph↗

arXiv2023

A $\mathbb{Z}_2$ invariant for chiral and particle-hole symmetric topological chains

We define a $\mathbb{Z}_2$-valued topological and gauge invariant associated to any 1-dimensional, translation-invariant topological insulator which satisfies either particle-hole symmetry or chiral symmetry. The invariant can be computed from the Berry phase associated to a suitable basis of Bloch functions which is compatible with the symmetries. We compute the invariant in the Su-Schrieffer-Heeger model for chiral symmetric insulators, and in the Kitaev model for particle-hole symmetric insulators. We show that in both cases the $\mathbb{Z}_2$ invariant predicts the existence of zero-energy boundary states for the corresponding truncated models.

math-ph↗

arXiv2023

Topology vs localization in synthetic dimensions

Motivated by recent developments in quantum simulation of synthetic dimensions, e.g. in optical lattices of ultracold atoms, we discuss here $d$-dimensional periodic, gapped quantum systems for $d \le 4$, with focus on the topology of the occupied energy states. We perform this analysis by asking whether the spectral subspace below the gap can be spanned by smooth and periodic Bloch functions, corresponding to localized Wannier functions in position space. By constructing these Bloch functions inductively in the dimension, we show that if they are required to be orthonormal then in general their existence is obstructed by the first two Chern classes of the underlying Bloch bundle, with the second Chern class characterizing in particular the 4-dimensional situation. If the orthonormality constraint is relaxed, we show how $m$ occupied energy bands can be spanned by a Parseval frame comprising at most $m+2$ Bloch functions.

math-ph↗

arXiv2022

From charge to spin: analogies and differences in quantum transport coefficients

We review some recent results from the mathematical theory of transport of charge and spin in gapped crystalline quantum systems. The emphasis will be in transport coefficients like conductivities and conductances. As for the former, those are computed as appropriate expectations of current operators in a non-equilibrium almost-stationary state (NEASS), which arises from the perturbation of an equilibrium state by an external electric field. While for charge transport the usual double-commutator Kubo formula is recovered (also beyond linear response), we obtain formulas for appropriately-defined spin conductivities which are still explicit but more involved. Certain "Kubo-like" terms in these formulas are also shown to agree with corresponding contributions to the spin conductance. In addition to that, we employ similar techniques to show a new result, namely that even in systems with non-conserved spin there is no generation of spin torque, that is the spin torque operator has an expectation in the NEASS which vanishes faster than any power of the intensity of the perturbing field.

math-ph↗

arXiv2022

Symmetric Fermi projections and Kitaev's table: topological phases of matter in low dimensions

We review Kitaev's celebrated "periodic table" for topological phases of condensed matter, which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections which depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for 0- and 1-dimensional systems, and recover the (weak and strong) topological invariants proposed by Kitaev and others.

math-ph↗

arXiv2021

Purely linear response of the quantum Hall current to space-adiabatic perturbations

Using recently developed tools from space-adiabatic perturbation theory, in particular the construction of a non-equilibrium almost stationary state, we give a new proof that the Kubo formula for the Hall conductivity remains valid beyond the linear response regime. In particular, we prove that, in quantum Hall systems and Chern insulators, the transverse response current is quantized up to any order in the strength of the inducing electric field. The latter is introduced as a perturbation to a periodic, spectrally gapped equilibrium Hamiltonian by means of a linear potential; existing proofs of the exactness of Kubo formula rely instead on a time-dependent magnetic potential. The result applies to both continuum and discrete crystalline systems modelling the quantum (anomalous) Hall effect.

math-ph↗

arXiv2020

Spin Hall conductivity in insulators with non-conserved spin

We study the linear response of a spin current to a small electric field in a two-dimensional crystalline insulator with non-conserved spin. We adopt the spin current operator proposed in [J. Shi et al., Phys. Rev. Lett. 96, 076604 (2006)], which satisfies a continuity equation and fits the Onsager relations. We use the time-independent perturbation theory to present a formula for the spin Hall conductivity, which consists of a "Chern-like" term, reminiscent of the Kubo formula obtained for the quantum Hall systems, and a correction term that accounts for the non-conservation of spin. We illustrate our findings on the Bernevig-Hughes-Zhang model and the Kane-Mele model for time-reversal symmetric topological insulators and show that the correction term scales quadratically with the amplitude of the spin-conservation-breaking terms. In both models, the spin Hall conductivity deviates from the quantized value when spin is not conserved.

cond-mat.mes-hall↗

arXiv2020

Středa formula for charge and spin currents

We consider a 2-dimensional Bloch--Landau--Pauli Hamiltonian for a spinful electron in a constant magnetic field subject to a periodic background potential. Assuming that the $z$-component of the spin operator is conserved, we compute the linear response of the associated spin density of states to a small change in the magnetic field, and identify it with the spin Hall conductivity. This response is in the form of a spin Chern marker, which is in general quantized to a half-integer, and to an integer under the further assumption of time-reversal symmetry. Our result is thus a generalization to the context of the quantum spin Hall effect to the well-known formula by Středa, which is formulated instead for charge transport.

math-ph↗

arXiv2020

Beyond Diophantine Wannier diagrams: Gap labelling for Bloch-Landau Hamiltonians

It is well known that, given a $2d$ purely magnetic Landau Hamiltonian with a constant magnetic field $b$ which generates a magnetic flux $φ$ per unit area, then any spectral island $σ_b$ consisting of $M$ infinitely degenerate Landau levels carries an integrated density of states $\mathcal{I}_b=M φ$. Wannier later discovered a similar Diophantine relation expressing the integrated density of states of a gapped group of bands of the Hofstadter Hamiltonian as a linear function of the magnetic field flux with integer slope. We extend this result to a gap labelling theorem for any $2d$ Bloch-Landau operator $H_b$ which also has a bounded $\mathbb{Z}^2$-periodic electric potential. Assume that $H_b$ has a spectral island $σ_b$ which remains isolated from the rest of the spectrum as long as $φ$ lies in a compact interval $[φ_1,φ_2]$. Then $\mathcal{I}_b=c_0+c_1φ$ on such intervals, where the constant $c_0\in \mathbb{Q}$ while $c_1\in \mathbb{Z}$. The integer $c_1$ is the Chern marker of the spectral projection onto the spectral island $σ_b$. This result also implies that the Fermi projection on $σ_b$, albeit continuous in $b$ in the strong topology, is nowhere continuous in the norm topology if either $c_1\ne0$ or $c_1=0$ and $φ$ is rational. Our proofs, otherwise elementary, do not use non-commutative geometry but are based on gauge covariant magnetic perturbation theory which we briefly review for the sake of the reader. Moreover, our method allows us to extend the analysis to certain non-covariant systems having slowly varying magnetic fields.

math-ph↗

arXiv2019

The Haldane model and its localization dichotomy

Gapped periodic quantum systems exhibit an interesting Localization Dichotomy, which emerges when one looks at the localization of the optimally localized Wannier functions associated to the Bloch bands below the gap. As recently proved, either these Wannier functions are exponentially localized, as it happens whenever the Hamiltonian operator is time-reversal symmetric, or they are delocalized in the sense that the expectation value of $|\mathbf{x}|^2$ diverges. Intermediate regimes are forbidden. Following the lesson of our Maestro, to whom this contribution is gratefully dedicated, we find useful to explain this subtle mathematical phenomenon in the simplest possible model, namely the discrete model proposed by Haldane (Phys. Rev. Lett. 61, 2017 (1988)). We include a pedagogical introduction to the model and we explain its Localization Dichotomy by explicit analytical arguments. We then introduce the reader to the more general, model-independent version of the dichotomy proved in (Commun. Math. Phys. 359, 61-100 (2018)), and finally we announce further generalizations to non-periodic models.

math-ph↗