We investigate the Lindblad dynamics of the reduced Loschmidt echo (RLE) in dissipative quadratic fermion systems. Focusing on the case of gain and loss dissipation, we derive general conditions for the persistence of nonanalyticities (so-called dynamical quantum phase transitions) in the time evolution of the RLE. We show that nonanalyticities that are present in the corresponding unitary dynamics can survive under purely gain or purely loss processes, but are completely smeared out as soon as both channels are active, even if one is infinitesimally small. These results hold for generic dissipative Gaussian evolutions, and are illustrated explicitly for the quench from the Néel state in the tight-binding chain, as well as for the quantum Ising chain. We also show that the subtle interplay between dissipative and unitary dynamics gives rise to a nested light-cone structure in the dynamics of the RLE, even in cases where this structure is not present in the corresponding unitary evolution, due to coherent cancellations in the phase structure of the wave function.

The discovery of chiral helical magnetism (CHM) in ${\text{Cr}}_{1/3}{\text{NbS}}_2$ and the stabilization of a chiral soliton lattice has attracted considerable interest in view of their potential technological applications. However, there is an ongoing debate regarding whether the sister compound, ${\text{Mn}}_{1/3}{\text{NbS}}_2$, which shares the same crystal structure, exhibits similar nontrivial properties which likewise rely on the lack of inversion symmetry at the magnetic ion. In this study, we conduct a comprehensive investigation of the magnetically ordered states of both compounds, using ${}^{53}\mathrm{Cr}, {}^{55}\mathrm{Mn}$, and ${}^{93}\mathrm{Nb}$ NMR. Our results, supported by density-functional calculations, detect in a high-quality single crystal of ${\text{Cr}}_{1/3}{\text{NbS}}_2$ all the signatures of the monoaxial CHM in a magnetic field, identifying it as a reference case for NMR. The detailed understanding of this prototypic behavior provides a reference for ${\text{Mn}}_{1/3}{\text{NbS}}_2$. Despite the much larger density of specific defects in this second material, we confirm the presence of a CHM phase in the Mn compound, characterized by a very large critical field for the forced ferromagnetic phase ($\approx 5\mathrm{T}$ for the applied field along $c$).

In this study, we examine how orbital degrees of freedom and Majorana surface states influence the magnetic penetration depth in the superconductor ${\mathrm{UTe}}_2$. Using a two-orbital model, we analyze pairing states belonging to the irreducible representations of the $D_{2h}$ crystal symmetry: $A_u, B_{1u}, B_{2u}$, and $B_{3u}$. For bulk nodal states such as $B_{2u}$, we find that the penetration depth for screening currents along the antinodal direction and the cylindrical axis scales as $T^2$, in strong contrast to the conventional $T^4$ law. This behavior originates from quasiparticles near the point nodes contributing to the interorbital paramagnetic current. We further show that Majorana surface states can dominate the low-temperature response. The fully gapped $A_u$ state hosts Majorana cones, which produce a $T^3$ dependence of the penetration depth when the ratio of penetration depth to coherence length ($\kappa$) is small. In contrast, the other pairing states exhibit Majorana Fermi arcs: The exponent is $n=2$ along the dispersive direction, while along the dispersionless direction it depends on whether the arcs terminate at end points. The exponent $n=2$ in the dispersive direction is robust, while it in the dispersionless direction relies on the presence or absence of the end points of the arcs and deviates from $n=2$ when end points are absent. Our results demonstrate that penetration-depth measurements provide a direct probe of Majorana surface states in low-$\kappa$ superconductors. For larger $\kappa$, the surface contribution becomes negligible and the temperature dependence is governed by bulk quasiparticles.

The pursuit of room-temperature superconductivity has generated growing interest in hydrogen-rich materials. However, the limited structural and compositional diversity of traditional binary superhydrides hinders further progress in this field. Ternary hydrides, with their enhanced structural flexibility and synergistic intermetallic interactions, provide a promising route to overcome these limitations. In this study, using $\mathrm{Y}{\mathrm{H}}_4$ as the parent material, we successfully synthesized two Y-Sc-H ternary hydrides by controlling laser-heating temperature and scandium doping concentration: the substitutionally doped stoichiometric $P4/mmm-\mathrm{YSc}{\mathrm{H}}_8$ and nonstoichiometric $I4/mmm-\left(\mathrm{Y},\mathrm{Sc}\right){\mathrm{H}}_4$. Electrical transport measurements revealed that $P4/mmm-\mathrm{YSc}{\mathrm{H}}_8$ exhibits a superconducting transition temperature ($T_{\mathrm{c}}$) of 113 K at 191.7 GPa, representing a 41% enhancement over $\mathrm{Y}{\mathrm{H}}_4$ (79 K at 179 GPa), with structural stability retained down to at least 160 GPa. In contrast, $I4/mmm-\left(\mathrm{Y},\mathrm{Sc}\right){\mathrm{H}}_4$ induces only a slight modification of the $T_{\mathrm{c}}$ of the parent $\mathrm{Y}{\mathrm{H}}_4$, which is consistent with the Anderson theorem. Electronic structure calculations reveal that the band structure of $P4/mmm-\mathrm{YSc}{\mathrm{H}}_8$ is substantially reconstructed due to the introduction of Sc $3d$ orbitals. This shifts the van Hove-like features closer to the Fermi level, increases the density of states at the Fermi level, and consequently enhances the electron-phonon coupling, ultimately leading to an elevated superconducting transition temperature $T_{\mathrm{c}}$. These findings demonstrate the effectiveness of elemental synergy strategies in designing high-$T_{\mathrm{c}}$ superhydrides.

Understanding the interplay between electronic correlations and band topology remains a central challenge in condensed matter physics, primarily hindered by a language-mismatch problem. While band topology is naturally formulated within a single-particle band theory, strong correlations typically elude such an effective one-body description. In this work, we bridge this gap by leveraging the ghost Gutzwiller (gGut) variational embedding framework, which introduces auxiliary quasiparticle degrees of freedom to recover an effective band-structure description of strongly correlated systems. This approach enables an interpretable and computationally efficient treatment of correlated topological phases, resulting in energy- and momentum-resolved topological features that are directly comparable to experimental spectra. We exemplify the advantages of this framework through a detailed study of the interacting Bernevig-Hughes-Zhang model. Not only does the gGut description reproduce established results, but it also reveals previously inaccessible aspects: most notably, the emergence of topologically nontrivial Hubbard bands hosting their own edge states, as well as possible ways to manipulate these through a finite magnetization. These results position the gGut framework as a promising tool for the predictive modeling of correlated topological materials.

The electromagnetic susceptibilities of an electron gas in crystals with broken space parity are considered. Because of the spin-orbit coupling, which is inherent in such compounds, their electromagnetic responses, in addition to the conductivity and the magnetic susceptibility also include some cross terms of magnetoelectric nature. It is found that almost all of these electromagnetic susceptibilities, as functions of frequency $\omega$ and wave vector $\mathbf{q}$, should contain nonreciprocal terms that are odd in the wave vector $\mathbf{q}$. Crystals of two types are considered: (i) polar symmetry crystals, in which the coupling between the electron momentum $\mathbf{p}$ and spin $\mathit{\sigma }$ has the form $H_{\mathrm{so}}=\alpha (\mathbf{p}\times \mathbf{c})·\mathit{\sigma }$ (where $\mathbf{c}$ is the polar axis), and (ii) crystals of a cubic space-odd symmetry, in which the Hamiltonian of the spin-orbit coupling in the simplest case has the form $H_{\mathrm{so}}^{\prime }={\alpha }^{\prime }\mathbf{p}·\mathit{\sigma }$. It is shown that effects of $H_{\mathrm{so}}$ and $H_{\mathrm{so}}^{\prime }$ on electromagnetic responses are noticeably different in spite of their rather similar form. All results are derived for the case of impure crystals in the limit of low frequencies $\omega \tau \ll 1$ and long wavelengths $q{v}_F\tau \ll 1$ (where $\tau$ is the mean collision time) when the main kinetic process responsible for the nonreciprocal terms in the susceptibilities is the spin nonconserving impurity scattering, that is, the same process that controls the spin relaxation in such crystals. The effect of the spin-orbit coupling on magnetic neutron scattering is briefly discussed.

Discoveries of both altermagnets and topological states have profoundly reshaped our understanding of condensed-matter physics and materials. However, tunable topological states in altermagnets remain largely unexplored. Here, we put forward the realization of second-order topological insulator (SOTI) and topological crystalline insulator (TCI) in two-dimensional altermagnets, and, in particular, demonstrate the feasibility of achieving a tunable topological phase transition with different bulk-boundary correspondence, i.e., from SOTI to TCI. As a concrete example, we consider the ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer to test the proposed scheme. The ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer is a prototypical material for altermagnets, and under equilibrium condition, it is a SOTI distinguished by well-localized nontrivial corner states. Remarkably, armed with preserved mirror symmetry ${\mathcal{M}}_{\mathit{z}}$, biaxial strain provides an effective means to engineer a topological phase transition in the ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer from SOTI to TCI, which is unambiguously confirmed by the calculated mirror Chern number ${\mathcal{C}}_{\mathcal{M}}$ = 1 and emergence of gapless edge states. Moreover, after the topological phase transition, an almost quantized plateau of the spin Hall conductivity ${\sigma }_{xy}^S$ is observed in the altermagnetic ${\mathrm{Fe}}_2{\mathrm{Se}}_2\mathrm{O}$ monolayer. Our work provides a concrete material platform for exploring the interplay between nontrivial band topology and altermagnetism, with promising implications for low-dissipation spintronics.

We present a theoretical framework for investigating superconducting proximity effect in altermagnet (AM)–superconductor (SC) heterostructures. In general, AMs, characterized by vanishing net magnetization but spin-split electronic spectra, provide a promising platform for realizing unconventional quantum phases. We consider a two-dimensional $d$-wave AM proximity coupled to a three dimensional ordinary $s$-wave SC. By integrating out the superconducting degrees of freedom, we derive an effective Hamiltonian that describes the proximity-induced modifications in the AM layer in the form of a self-energy. We then derive an effective Green's function to obtain the proximity-induced pairing amplitudes in the AM layer and classify the induced pairing amplitudes according to their parity, frequency, and spin. We find the presence of even-parity singlet and triplet pairing amplitudes in the AM layer. To achieve the odd-parity triplet components, important to realize topological superconductivity, we introduce a layer of Rashba spin-orbit coupling (RSOC) in the heterostructure. We analyze the band topology of this proximity-induced AM-RSOC layer and demonstrate the emergence of both weak and strong topological superconducting phases with edge-localized modes, characterized by winding number and Chern number. These findings highlight the role of AM–SC hybrid setup as a versatile platform for realizing odd-parity triplet pairings and engineering topological superconductivity in two dimensions.

We demonstrate a deterministic switching mechanism in collinear altermagnets driven by asymmetric sublattice spin currents. Unlike conventional antiferromagnets, where combined parity-time reversal symmetry enforces purely staggered sublattice spin torques, altermagnets host symmetry-protected nonrelativistic spin splitting that produces unequal torques on the two sublattices. Using doped ${\mathrm{FeSb}}_2$ as a representative $d$-wave altermagnet, our Landau-Lifshitz-Gilbert simulations show that these torques enable magnetic field-free and deterministic ${180}^{\circ }$ Néel vector reversal over picosecond timescale. The mechanism is generic to even-parity altermagnets and remains effective even in centrosymmetric, weak spin-orbit coupled altermagnets, where the Néel spin-orbit torque mechanism fails. Our results establish an experimentally accessible mechanism for switching of altermagnetic order, opening pathways for realizing ultrafast, low-power altermagnet spintronic devices.

We derive the self-consistent random phase approximations (sc-RPA) from the projective truncation approximation (PTA) for the equation of motion of two-time Green's function. The obtained sc-RPA applies to arbitrary temperature and recovers the Rowe's formalism at zero temperature. The PTA formalism not only rationalizes Rowe's formula, but also provides a general framework to extend sc-RPA. We implement the sc-RPA calculation for the one-dimensional spinless fermion model in the parameter regime of disordered ground state, with the $N$-representability constraints enforced. The obtained ground state energy, correlation function, and density spectral function agree well with existing results. The features of the Luttinger liquid ground state and the continuum/bound state in the spectral function are well captured. We discuss several issues concerning the approximations made in RPAs, difficulties of RPA for symmetric state, and the static component problem of PTA.

The layered $3d$ transition metal dichalcogenide ${\mathrm{CoTe}}_2$ is a topological Dirac type-II metal. However, the Co $3d$-bands in ${\mathrm{CoTe}}_2$ do not exhibit the expected correlation-induced band narrowing seen in CoO. We address this conundrum by studying the electronic structure of ${\mathrm{CoTe}}_2$ using hard x-ray photoemission spectroscopy (HAXPES), x-ray absorption spectroscopy (XAS) and resonant-PES. We quantify the on-site Coulomb energy $U_{dd}$ via single-particle partial density of states and the two-hole correlation satellite using valence band resonant-PES, and obtain $U_{dd}$ = 3.0 eV for ${\mathrm{CoTe}}_2$. Charge-transfer (CT) cluster model simulations of the experimental core-level Co $2p$ PES and $L$-edge XAS spectra of ${\mathrm{CoTe}}_2$ and CoO validate their contrasting electronic parameters: $U_{dd}$ and CT energy $\mathrm{\Delta }$ are (3.0 eV, −2.0 eV) for ${\mathrm{CoTe}}_2$, and (5.0 eV, 4.0 eV) for CoO, respectively. The $d\text{−}p$ hybridization strength $T_{eg}$ for ${\mathrm{CoTe}}_2<\mathrm{CoO}$, and indicates that the reduced $U_{dd}$ in ${\mathrm{CoTe}}_2$ is not due to $T_{eg}$. The increase in $d^n$ count $\sim 1$ by CT from ligand to Co site in ${\mathrm{CoTe}}_2$ is due to a negative $\mathrm{\Delta }$ and reduced $U_{dd}$. Yet, only because $U_{dd}>\left|\mathrm{\Delta }\right|, {\mathrm{CoTe}}_2$ becomes a topological metal with $p\to p$ type lowest energy excitations. The study reveals the connection between negative $\mathrm{\Delta }$ and reduced $U_{dd}$ required for setting up the electronic structure framework for achieving topological behavior via band inversion in the correlated metal ${\mathrm{CoTe}}_2$.

Angle-resolved photoemission spectroscopy combined with band structure calculations have shown that the layered transition metal dichalcogenide ${\mathrm{NiTe}}_2$ is a type-II Dirac semimetal. However, conflicting conclusions were reported regarding the role of electron correlations in ${\mathrm{NiTe}}_2$. We study core levels and valence band electronic structure of single-crystal ${\mathrm{NiTe}}_2$ using soft and hard x-ray photoemission spectroscopy, x-ray absorption spectroscopy (XAS), and Ni $2p-3d$ resonant photoemission spectroscopy to quantify electronic parameters in ${\mathrm{NiTe}}_2$. The Ni $3d$ on-site Coulomb energy ($U_{dd}$) is quantified from measurements of the Ni $3d$ single-particle density of states and the two-hole correlation satellite. The Ni $2p$ core level and $L$-edge XAS spectra are analyzed by charge-transfer (CT) cluster model calculations using the experimental $U_{dd}$, and it shows that ${\mathrm{NiTe}}_2$ exhibits a negative CT energy $\mathrm{\Delta }$. A comparative analysis of NiO $L$-edge XAS confirms its well-known strongly correlated CT insulator character, with a larger $U_{dd}$ and positive $\mathrm{\Delta }$. The $d\text{−}p$ hybridization strength $T_{eg}$ for ${\mathrm{NiTe}}_2<\mathrm{NiO}$ shows that $T_{eg}$ is not responsible for reducing $U_{dd}$ in ${\mathrm{NiTe}}_2$ compared to NiO. The negative $\mathrm{\Delta }$ and a reduced $U_{dd}$ lead to the increase in the $d^n$ count on the Ni site in ${\mathrm{NiTe}}_2$ by nearly one electron. However, importantly, since $U_{dd}>\left|\mathrm{\Delta }\right|$, a finite repulsive $U_{dd}$ results in pushing $d$ states away from Fermi level and this is required to make ${\mathrm{NiTe}}_2$ a moderately correlated Dirac semimetal with band inversion in the $p\text{−}p$-type lowest-energy excitations.

Electro-optic crystals—capable of efficient refractive index modulation via external electric fields—are essential components in modern optoelectronics. However, in wide band gap insulators, the electronic contribution to the electro-optic effect is typically modest. We propose that terahertz field–driven ionic displacements $\stackrel{\rightharpoonup }{u}\left(t\right)$ can effectively modulate the refractive index through the quadratic electro-optic effect $\beta {|\stackrel{\rightharpoonup }{u}\left(t\right)|}^2$, which circumvents the phase-matching condition required for the linear electro-optic effect. Taking the nonlinear optical crystal guanidinium tetrafluoroborate [$\mathrm{C}{\left(\mathrm{N}{\mathrm{H}}_2\right)}_3\mathrm{B}{\mathrm{F}}_4$]—composed of alternating layers of ${\left[\mathrm{C}{\left(\mathrm{N}{\mathrm{H}}_2\right)}_3\right]}^{+}$ and ${\left[\mathrm{B}{\mathrm{F}}_4\right]}^{-}$—as a prototype, our density functional perturbation theory calculations demonstrate that hydrogen (H) atomic displacements can be significantly enhanced via resonant absorption of terahertz photons. Specifically, terahertz-driven H atomic displacement efficiently tunes the refractive index through the quadratic electro-optic mechanism. Our work illustrates that in certain wide-band-gap insulators where the electronically contributed electro-optic effect is negligible, the quadratic electro-optic effect induced by terahertz-driven ionic motion offers an efficient approach for optical phase manipulation and frequency comb generation.

Bound states in the continuum (BICs) and their quasibound counterparts (QBICs) have emerged as powerful tools to engineer sharp resonances and extreme field confinement in wave systems. Here, we investigate the formation and tunability of BICs and QBICs for flexural waves in thin elastic plates decorated with a one-dimensional array of pointlike resonators. By designing a diatomic unit cell and varying the relative angle of the two resonators, we achieve precise control over the coupling between the guided modes of the plate and the radiation continuum. This simple geometric tuning enables the transition from a symmetry-protected BIC at the center of the Brillouin zone to a family of high-Q QBICs at finite Bloch wave vectors. We develop an analytical multiple-scattering model to compute the transmission and reflection coefficients of a plane flexural wave interacting with the array, and we demonstrate that the resulting resonances manifest as angle-dependent Fano line shapes in the scattering spectra. Furthermore, by employing a lumped-element description of the resonators, we propose a feasible physical realization of the system and validate our theoretical predictions against full finite element simulations, obtaining excellent quantitative agreement. Our results provide both a fundamental understanding and a practical route to engineer ultranarrow resonances and elastic metasurfaces with tailored dispersion, paving the way for advanced control of flexural waves in thin plates.

First-principles based crystal structure prediction (CSP) methods have proved to be an essential tool for the discovery of new materials. However, in solids close to displacive phase transitions, which are common in ferroelectrics, thermoelectrics, charge-density wave systems, or superconducting hydrides, the ionic contribution to the free energy and lattice anharmonicity become significant, limiting the capacity of CSP techniques to determine the thermodynamic stability of competing phases. While variational methods like the stochastic self-consistent harmonic approximation (SSCHA) accurately account for anharmonic lattice dynamics ab initio, their high computational cost makes them impractical for CSP. Machine-learning interatomic potentials offer accelerated sampling of the energy landscape compared to purely first-principles approaches, but their reliance on extensive training data and limited generalization restricts practical applications. Here, we propose an iterative learning framework combining evolutionary algorithms, atomic foundation models, and SSCHA to enable CSP with anharmonic lattice dynamics. Foundation models enable robust relaxations of random structures, drastically reducing the required training data. Applied to the highly anharmonic ${\mathrm{H}}_3\mathrm{S}$ system, our framework achieves good agreement with benchmarks based on density functional theory, accurately predicting phase stability and vibrational properties from 50 to 200 GPa. In PdH, our method is capable of revealing that the true ground state of the system is the rock-salt structure even if classical crystal structure predictions favor the zincblende structure. Importantly, we find that the statistical averaging in the SSCHA reduces the error in the free energy evaluation, avoiding the need for extremely high accuracy of machine-learning potentials. This approach bridges the gap between data efficiency and predictive power, establishing a practical pathway for CSP with anharmonic lattice dynamics.

Conventional perturbation theory, developed for Hermitian matrices, is inadequate for determining the eigenvalues and eigenvectors of bosonic Bogoliubov–de Gennes (BdG) Hamiltonians, whose dynamical structure leads to a non-Hermitian generalized eigenvalue problem. Here we extend the Hellmann-Feynman perturbation theory to bosonic BdG systems and formulate a unified framework applicable to both nondegenerate and nearly degenerate cases. The method provides systematic expansions for eigenvalues, eigenstates, and effective Hamiltonians, revealing that the nearly degenerate theory emerges naturally from the nondegenerate one. A key result of this work is that the perturbative framework gives access not only to the spectrum but also to geometric quantities. In particular, we obtain fully analytical expressions for the Berry curvature, including a previously overlooked contribution arising from the transformed basis. To demonstrate the robustness of the formalism, we apply it to both a ferromagnetic honeycomb lattice and a more complex triangular-lattice antiferromagnet described by a $6\times 6$ BdG Hamiltonian with multiple magnon bands and Dirac-point degeneracies. In both systems, the resulting effective $2\times 2$ Hamiltonians accurately capture the low-energy band structure and Berry curvature, in excellent agreement with full numerical diagonalization. These results establish a versatile and broadly applicable perturbation framework for bosonic BdG systems and provide new analytical insight into the geometric and topological properties of magnonic excitations.

The interplay of collective electronic orders represents a central theme in condensed matter physics, and kagome lattices offer fertile ground for studying their competition and coexistence. Although metal-decorated two-dimensional boron kagome lattices can be stabilized, their quantum phases remain largely unexplored. Here, we report a family of inverse-sandwich monolayers, $M_2{\mathrm{B}}_3$ ($M$ = Mg, Ca, Sr, Ba), in which a boron kagome sheet is sandwiched between two alkaline earth metal layers, and identify them as charge density wave (CDW) materials. We found that the CDW in ${\mathrm{Mg}}_2{\mathrm{B}}_3$ originates from the combined effects of Fermi surface nesting and electron-phonon coupling, whereas in ${\mathrm{Ca}}_2{\mathrm{B}}_3, {\mathrm{Sr}}_2{\mathrm{B}}_3$, and ${\mathrm{Ba}}_2{\mathrm{B}}_3$, it is driven by strong electron-phonon coupling between B $\sigma$ orbitals and in-plane vibrational modes. These instabilities lead to a 2 × 2 CDW ground state in ${\mathrm{Mg}}_2{\mathrm{B}}_3$ and 3 × 3 reconstructions in the heavier analogs, all of which are suppressed at room temperature by anharmonic lattice effects. Remarkably, ${\mathrm{Mg}}_2{\mathrm{B}}_3$ hosts a coexisting 2 × 2 CDW and superconductivity with a $T{}_{\text{C}}$ of 10.6 K. Under biaxial compressive strain, superconductivity coexists with nontrivial topology in the heavier compounds. Moreover, the nearly unchanged $T{}_{\text{C}}$ between the CDW ground state and the strained kagome phase implies weak competition between CDW and superconductivity. Our results establish kagome-based boron monolayers as a promising class of materials for exploring intertwined CDW, superconducting, and topological states in two dimensions.

We study superconductivity and superfluid weight of the two-dimensional $\alpha \text{−}{\mathcal{T}}_3$ 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 $\alpha \text{−}{\mathcal{T}}_3$ lattice model, as a candidate for tunable quantum geometry and superfluid weight and as a prototype of related behavior in tunable quantum materials.