Fractal electronic spectra arising from the competition between lattice periodicity and magnetic flux are a fundamental hallmark of two-dimensional quantum systems. While the spectral properties of Hofstadter butterflies are well-documented, their thermodynamic response has remained remarkably unexplored. We present an original characterization of the electronic entropy Se and specific heat Ce, at half-filling, for square, honeycomb, and triangular lattices under a magnetic field. We demonstrate that these observables exhibit fast and slow magneto-thermo oscillations and pronounced magnetocaloric effects. We identify striking self-similarity in Se and Ce, tracing heart-shaped specific heat and tunnel-like entropy contours that repeat at specific lattice-dependent magnetic fluxes. Entropy minima at low temperatures act as fingerprints for the butterfly spines, resolving the underlying fractal spectra. These findings may establish thermal measurements as high-resolution spectroscopic probes, providing a robust framework for recognizing fractal signatures through thermodynamics in diverse nanostructures.

Quantum dot platforms have enabled the direct observation of Nagaoka ferromagnetism (NFM) in small arrays and noninfinite interaction strength. However, optimizing the cluster connectivity characteristics that yield a ground state with maximal spin and their robustness against magnetic fields remains unexplored. Employing exact diagonalization of the Hubbard Hamiltonian, we find a connection between the existence of kinetic ferromagnetism and graph theory descriptions. Algebraic connectivity (lambda 2) and Katz centrality (KC) are shown to be related to the spin correlation over the system. In square arrays, the onset of NFM is found to be tc/U lambda 22. In optimal cluster geometries, large lambda 2 and low KC fluctuation per site are found to enhance tc/U, extending the NFM phase while diminishing the strength of spin-correlation clouds. A perpendicular magnetic field introduces Aharonov-Bohm phases, and a critical flux for which NFM is destroyed. We further find that tuning the flux phase to pi results in a ground state that exhibits antiferromagnetic correlations (counter-Nagaoka state). Our results illustrate how NFM and polaron formation can be predicted from the array's connectivity (lambda 2 and KC), and how the introduction of flux results in the counterintuitive destruction of kinetic ferromagnetism in the system.

We study superconductivity and superfluid weight of the two-dimensional alpha-T3 lattice with on-site asymmetries, hosting an isolated quasiflat band with tunable bandwidth via a parameter alpha. Within a mean-field approximation of the attractive Hubbard model, we obtain the superconducting order parameters on the three inequivalent sublattices and show their strong dependence on alpha, interaction strength, and electron filling. At quasiflat band filling, a superconducting gap opens and grows power-law fast with interaction strength, instead of the usual slow exponential growth, due to diverging density of states. We calculate the superfluid weight from linear response theory and study its band dispersion and geometric contributions. While the conventional part proportional to band derivatives is suppressed in the quasiflat band regime, the contribution dominated by the quantum metric grows linearly for small interaction strength. We further demonstrate how tuning alpha enhances the quantum metric and thus the geometric superfluid weight especially near half filling, while increasing on-site asymmetries increases the conventional contribution by broadening the quasiflat band. We obtain the Berezinskii-Kosterlitz-Thouless transition temperature and demonstrate its strong dependence and enhancement with the parameter alpha. Our results establish a tunable flat band system, the alpha-T3 lattice model, as a candidate for tunable quantum geometry and superfluid weight and as a prototype of related behavior in tunable quantum materials.

In this work, we investigate a system of three entangled qubits within the XXX model, subjected to an external magnetic field in the z direction and incorporating an anisotropy term along the y axis. We explore the system's thermodynamics by calculating its magnetic susceptibility and analyzing how this quantity encodes information about entanglement. By deriving rigorous bounds for the susceptibility, we demonstrate that violations of these bounds serve as an entanglement witness. Our results show that anisotropy enhances entanglement, extending the temperature range over which it persists. Furthermore, when the system is traced over the degrees of freedom of two qubits, the reduced entropy of the remaining single qubit corresponds to that of a system in a thermal bath with temperature T>0K.

We report atomic-scale gating and visualization of local charge distribution within individual rare-earth-based molecular complexes on a metallic surface. The complexes are formed by a positively charged lanthanum ion coordinated to a (pcam)3 molecule and a negatively charged counterion trapped underneath via electrostatic interactions on a Au(111) surface. Local gating is performed by adding an additional negatively charged counterion to one side of the complex, which results in the redistribution of charges within the complex and a positive shift of the frontier orbitals. This is caused by the internal Stark effect induced by the added counterion. This effect is directly captured using tunneling spectroscopy and spectroscopic mapping at 5 K substrate temperature. The polarizability of the complex is corroborated by density functional theory and analytical calculations based on experimental findings. Furthermore, the influence of charge polarization on nearby complexes is investigated in a cluster purposely assembled using three complexes, which reveals maintaining the charge states as in single complexes. These findings will enable the design of robust charged rare-earth complexes to be tailored for potential solid-state applications.

We study kagome lattices with on-site and extended spin-singlet s-wave superconducting pairing and show that the inclusion of Rashba spin-orbit (RSO) interaction allows time-reversal-invariant topological superconducting states which support helical Majorana pairs at the edge. We calculate the Z2 topological invariant as a function of the pairing parameters for different chemical potentials. The rich phase diagrams reveal topological, nodal, and trivial superconducting states depending on the system parameters. We also consider a 2X2 time-reversal symmetry-breaking chiral flux phase, which has been demonstrated to be energetically favorable in the AV3Sb5 family of superconductors. Incorporating such symmetry-breaking order in our model leads to chiral Majorana edge states defined by a Chern number. We show how the RSO interaction allows for topological phases with even and odd Chern numbers for different system parameters. This work demonstrates how a simple s-wave kagome superconductor with RSO interaction can support helical and chiral Majorana edge states, and motivates the search for Majorana fermions in kagome superconductors.

A composite impurity in a metal explores different configurations, where its net magnetic moment may be screened by the electrons in the host. An interesting example is the two-stage Kondo (TSK) system where screening sets in with successively smaller energy scales. In contrast, the impurities may prefer a local singlet disconnected from the metal. This competition is decided by fine-tuning the couplings in the system, as has been studied before. A double quantum dot T-shape geometry, where a hanging dot is connected to current leads only via another dot, represents a flexible system in which these different regimes can be explored experimentally. It has been difficult, however, to clearly differentiate the two regimes. Here we provide a prescription to better identify the regime where the TSK occurs in such double dot geometry. The TSK regime requires a balance of the ratio t01/P0 between the interdot coupling (t01) and the coupling of the QD connected to the Fermi sea (P0). Above a certain value of this ratio, the system crosses over to a molecular regime, where the quantum dots form a local singlet and no Kondo screening occurs. Here we establish that there is a region in the t01 - P0 parameter space where a pure TSK regime occurs, i.e., where the properties of the second Kondo stage can be accurately described by a single impurity Anderson model with effective or renormalized parameters. By examining the magnetic susceptibility of the hanging QD, we show that a single parameter, Peff, can accurately simulate this susceptibility. This effective model also provides the hanging QD spectral function with great accuracy in a limited range of the t01 - P0 parameter space, thus defining the region where a true TSK regime occurs. We also show that in this parameter range, the spin correlations between both quantum dots show a universal behavior. Our results may guide experimental groups to choose parameter values that will place the system either in the TSK regime or in the crossover to the molecular regime.

In this work, we systematically study the electronic structure and stability of spin states of the [Fe-(tpy-ph)2]2+ molecule in both the gas phase and on a Au(111) substrate using density functional theory + U (DFT+U) calculations. We find that the stability of the Fe2+ ion's spin states predicted by the computations is significantly influenced by the Hubbard U parameter. In the gas phase, the low-spin (LS, S = 0) state is found to be energetically favorable for U(Fe) ≤ 3 eV, whereas the high-spin (HS, S = 2) state is stabilized for U(Fe) > 3 eV. Interaction with the Au(111) substrate is found to elevate the critical U for the spin-state transition to 3.5 eV. Additionally, we perform L-edge X-ray absorption spectroscopy (XAS) calculations for both HS and LS states. The calculated XAS suggests that the HS state more closely aligns with the experimental observations, indicating the potential coexistence of the HS state as the initial state during the X-ray excitation process. These findings enrich our understanding of spin-state dynamics in [Fe(tpy-Ph)2]2+.

Using the cumulant Green's functions method (CGFM), we study the single impurity Anderson model (SIAM). The CGFM starting point is the diagonalization of the SIAM Hamiltonian expressed in a semi-chain form containing N sites, viz., a correlated site (simulating an impurity) connected to the remaining N-1 uncorrelated conduction-electron sites. An exact solution can be obtained since the complete system has few sites. That solution is employed to calculate the atomic Green's functions and the approximate cumulants used to obtain the impurity and conduction Green's functions for the SIAM, and no self-consistency loop is required. We calculated the density of states, the Friedel sum rule, and the impurity occupation number, all benchmarked against results from the numerical renormalization group (NRG). One of the main insights obtained is that, at very low temperatures, only four atomic transitions contribute to generate the entire SIAM density of states, regardless of the number of sites in the chain and the model's parameters and different regimes: Empty orbital, mixed-valence, and Kondo. We also pointed out the possibilities of the CGFM as a valid alternative to describe strongly correlated electron systems like the Hubbard and t-J models, the periodic Anderson model, the Kondo and Coqblin-Schrieffer models, and their variants.

Understanding and quantifying entanglement entropy is crucial to characterize the quantum behaviors that drive phenomena in a variety of systems. Rare-earth spin complexes, with their unique magnetic properties, provide fertile ground for exploring these nonlocal correlations. In this work, we study Eu$^{2+}$ ions deposited on a Au(111) substrate, modeling finite clusters of large spin-moments using a Heisenberg Hamiltonian parameterized by first-principles calculations. Our analysis reveals a one-to-one correspondence between structures in the differential conductance profiles and changes in the von Neumann entanglement entropy of bipartite subsystems, influenced by probe-ion separation and applied magnetic fields. Distinct braiding patterns in the conductance profiles are shown to correspond to stepwise changes in the entanglement entropy, providing a new avenue for investigating quantum correlations. These results establish a foundation for experimentally probing and controlling entanglement in lanthanide-based systems, with potential applications in quantum technologies.

This study investigates the anisotropic effects on a system of three qubits with chain and ring topology, described by the antiferromagnetic Heisenberg XXX model subjected to a homogeneous magnetic field. We explore the Stirling and Otto cycles and find that easy-axis anisotropy significantly enhances engine efficiency across all cases. At low temperatures, the ring configuration outperforms the chain on both work and efficiency during the Stirling cycle. Additionally, in both topologies, the Stirling cycle achieves Carnot efficiency with finite work at quantum critical points. In contrast, the quasistatic Otto engine also reaches Carnot efficiency at these points but yields no useful work. Notably, the Stirling cycle exhibits all thermal operational regimes-engine, refrigerator, heater, and accelerator-unlike the quasistatic Otto cycle, which functions only as an engine or refrigerator.

We study effects of strain on the electronic properties of the kagome lattice in a tight-binding formalism with spin-orbit coupling (SOC). The degeneracy at the $\Gamma$ point evolves into a pair of emergent tilted Dirac cones under uniaxial strain, where the anisotropy and tilting of the bands depend on the magnitude and direction of the strain field. SOC opens gaps at the emergent Dirac points, making the flatband topological, characterized by a nontrivial $\mathbb{Z}_2$ index. Strains of a few percent drive the system into trivial or topological phases. This confirms that moderate strain can be used to engineer anisotropic Dirac bands with tunable properties to study new phases in kagome lattices.

We study a Kondo state that is strongly influenced by its proximity to an w-1/2 singularity in the metallic host density of states. This singularity occurs at the bottom of the band of a one-dimensional chain, for example. We first analyze the noninteracting system: A resonant state ed, located close to the band singularity, suffers a strong renormalization, such that a bound state (Dirac 8 function) is created below the bottom of the band in addition to a resonance in the continuum. When ed is positioned right at the singularity, the spectral weight of the bound state is 23, irrespective of its coupling to the conduction electrons. The interacting system is modeled using the single -impurity Anderson model, which is then solved using the numerical renormalization group method. We observe that the Hubbard interaction causes the bound state to suffer a series of transformations, including level splitting, transfer of spectral weight, appearance of a spectral discontinuity, changes in binding energy (the lowest state moves farther away from the bottom of the band), and development of a finite width. When ed is away from the singularity and in the intermediate valence regime, the impurity occupancy is lower. As ed moves closer to the singularity, the system partially recovers Kondo regime properties, i.e., higher occupancy and lower Kondo temperature TK. The impurity thermodynamic properties show that the local -moment (LM) fixed point is also strongly affected by the existence of the bound state. When ed is close to the singularity, the LM fixed point becomes impervious to charge fluctuations (caused by bringing ed close to the Fermi energy), in contrast to the LM suppression that occurs when ed is away from the singularity. We also discuss an experimental implementation that shows similar results to the quantum wire if the metallic host of the impurity is an armchair graphene nanoribbon.

We study the interplay of attractive electron interactions and topological states in strained kagome lattices with spin-orbit coupling via a Hubbard Hamiltonian in the mean-field approximation. In the unstrained lattice, there is a topological phase transition from a quantum spin Hall state to a charge density wave (CDW) with increasing interaction strength. Upon applying a uniform uniaxial strain to the lattice, we find a new phase with coexisting CDWs and topological states. For increasing interaction strength or strain, the system is driven into a pure CDW, signaling topological phase transitions. The directionality (nematicity) of the CDW is controlled by the direction of the applied strain. When $s$ wave electronic pairing is allowed, the system develops a superconducting order beyond a threshold attraction, which is totally suppressed by the onset of a CDW with increasing interaction. Most interestingly, moderate strain allows the coexistence of superconductivity and CDWs for a range of interaction values. This illustrates how electronic interactions and single-particle topological structures compete to create unusual correlated phases in kagome systems.

We study the physics of the strong-coupling Hubbard model in a kagome lattice ribbon under mechanical tension and half filling. It is known that in the absence of strain, the lattice symmetry of the system and strong electronic interactions induce magnetic frustration. As uniaxial strain is applied, the ribbon exhibits various configurations with energy oscillations that depend on the direction of the strain axis. The ground states are obtained by density-matrix renormalization-group calculations. We find that the system is characterized by strong antiferromagnetic bonds distributed throughout the lattice in directions and patterns that depend on the strain directions and may coexist with easily polarizable sites that are only weakly correlated to their neighbors. We identify frustration and correlation measures that follow the strain and interaction dependence of the system well. These results illustrate that strain-dependent magnetic susceptibility could be explored experimentally to help probe the role of symmetry and interactions in these systems.

Rare-earth complexes are vital for separation chemistry and useful in many advanced applications including emission and energy upconversion. Here, 2D rare-earth clusters having net charges are formed on a metal surface, enabling investigations of their structural and electronic properties on a one-cluster-at-a-time basis using scanning tunneling microscopy. While these ionic complexes are highly mobile on the surface at ≈100 K, their mobility is greatly reduced at 5 K and reveals stable and self-limiting clusters. In each cluster, a pair of charged rare-earth complexes formed by electrostatic and dispersive interactions act as a basic unit, and the clusters are chiral. Unlike other non-ionic molecular clusters formed on the surfaces, these rare-earth clusters show mechanical stability. Moreover, their high mobility on the surface suggests that they are in a 2D liquid-like state.

We explore the electronic entropy per particle $s$ and Seebeck coefficient $\mathcal{S}$ in zigzag graphene ribbons. Pristine and edge-doped ribbons are considered using tight-binding models to inspect the role of edge states in the observed thermal transport properties. As a bandgap opens when the ribbons are doped at one or both edges, due to asymmetric edge potentials, we find that $s$ and $\mathcal{S}$ signals are closely related to each other: both develop sharp dip-peak lineshapes as the chemical potential lies in the gap, while the ratio $s/\mathcal{S}$ exhibits a near constant value equal to the elementary charge $e$ at low temperatures. This constant ratio suggests that $\mathcal{S}$ can be seen as the transport differential entropy per charge, as suggested by some authors. Our calculations also indicate that measurement of $s$ and $\mathcal{S}$ may be useful as a spectroscopic probe of different electronic energy scales involved in such quantities in gapped materials.

We study a Kondo state that is strongly influenced by its proximity to an w^-1/2 singularity in the metallic host density of states. This singularity occurs at the bottom of the band of a 1D chain, for example. We first analyze the non-interacting system: A resonant state e_d, located close to the band singularity, suffers a strong `renormalization', such that a bound state is created below the bottom of the band in addition to a resonance in the continuum. When e_d is positioned right at the singularity, the spectral weight of the bound state is 2/3, irrespective of its coupling to the conduction electrons. The interacting system is modeled using the Single Impurity Anderson Model, which is then solved using the Numerical Renormalization Group method. We observe that the Hubbard interaction causes the bound state to suffer a series of transformations, including level splitting, transfer of spectral weight, appearance of a spectral discontinuity, changes in binding energy (the lowest state moves farther away from the bottom of the band), and development of a finite width. When e_d is away from the singularity and in the intermediate valence regime, the impurity occupancy is lower. As e_d moves closer to the singularity, the system partially recovers Kondo regime properties, i.e., higher occupancy and lower Kondo temperature T_K. The impurity thermodynamic properties show that the local moment fixed point is also strongly affected by the existence of the bound state. When e_d is close to the singularity, the local moment fixed point becomes impervious to charge fluctuations (caused by bringing e_d close to the Fermi energy), in contrast to the local moment suppression that occurs when e_d is away from the singularity. We also discuss an experimental implementation that shows similar results to the quantum wire, if the impurity's metallic host is an armchair graphene nanoribbon.

High spin systems, like those that incorporate rare-earth $4f$ elements (REEs), are increasingly relevant in many fields. Although research in such systems is sparse, the large Hilbert spaces they occupy are promising for many applications. In this work, we examine a one-dimensional linear array of europium (Eu) atoms on a Au(111) surface and study their electronic and magnetic excitations. Ab initio calculations using VASP with PBE+U are employed to study the structure. We find Eu atoms to have a net charge when on gold, consistent with a net magnetic momemt of $\simeq 3.5 \mu_B$. Examining various spin-projection configurations, we can evaluate first and second neighbor exchange energies in an isotropic Heisenberg model between spin-$\frac{7}{2}$ moments to obtain $J_1 \approx -1.2 \, \mathrm{K}$ and $J_2 \approx 0.2 \, \mathrm{K}$ for the relaxed-chain atomic separation of $a \approx 5$ $\mathrm{\dot{A}}$. These parameters are used to obtain the full spin excitation spectrum of a physically realizable four-atom chain. The large $|J_1|/J_2$ ratio results in a highly degenerate ferromagnetic ground state that is split by a significant easy plane single ion anisotropy of $0.6$ K. Spin-flip excitations are calculated to extract differential conductance profiles as those obtained by scanning tunneling microscopy techniques. We uncover interesting behavior of local spin excitations, especially as we track their dispersion with applied magnetic fields.

Superconductivity in graphene-based systems has recently attracted much attention, as either intrinsic behavior or induced by proximity to a superconductor may lead to interesting topological phases and symmetries of the pairing function. A prominent system considers the pairing to have chiral symmetry. The question arises as to the effect of possible spin-orbit coupling on the resulting superconducting quasiparticle (QP) spectrum. Utilizing a Bogolyubov-de Gennes (BdG) Hamiltonian, we explore the interplay of different interaction terms in the system, and their role in generating complex Berry curvatures in the QP spectrum, as well as non-trivial topological behavior. We demonstrate that the topology of the BdG Hamiltonian in these systems may result in the appearance of edge states along the zigzag edges of nanoribbons in the appropriate regime. For suitable chemical potential and superconducting pairing strength, we find the appearance of robust midgap states at zigzag edges, well protected by large excitation gaps and momentum transfer.