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B. Andrei Bernevig

Publications and source records attributed to B. Andrei Bernevig.

At least 91 records · Page 5Linked to original sources
arXiv2024

Interacting Topological Quantum Chemistry in 2D: Many-body Real Space Invariants

The topological phases of non-interacting fermions have been classified by their symmetries, culminating in a modern electronic band theory where wavefunction topology can be obtained (in part) from momentum space. Recently, Real Space Invariants (RSIs) have provided a spatially local description of the global momentum space indices. The present work generalizes this real space classification to interacting 2D states. We construct many-body local RSIs as the quantum numbers of a set of symmetry operators on open boundaries, but which are independent of the choice of boundary. Using the $U(1)$ particle number, they yield many-body fragile topological indices, which we use to identify which single-particle fragile states are many-body topological or trivial at weak coupling. To this end, we construct an exactly solvable Hamiltonian with single-particle fragile topology that is adiabatically connected to a trivial state through strong coupling. We then define global many-body RSIs on periodic boundary conditions. They reduce to Chern numbers in the band theory limit, but also identify strongly correlated stable topological phases with no single-particle counterpart. Finally, we show that the many-body local RSIs appear as quantized coefficients of Wen-Zee terms in the topological quantum field theory describing the phase.

cond-mat.str-el↗
arXiv2024

The Thermoelectric Effect and Its Natural Heavy Fermion Explanation in Twisted Bilayer and Trilayer Graphene

We study the interacting transport properties of twisted bilayer graphene (TBG) using the topological heavy-fermion (THF) model. In the THF model, TBG comprises localized, correlated $f$-electrons and itinerant, dispersive $c$-electrons. We focus on the Seebeck coefficient, which quantifies the voltage difference arising from a temperature gradient. We find that the TBG's Seebeck coefficient shows unconventional (strongly-interacting) traits: negative values with sawtooth oscillations at positive fillings, contrasting typical band-theory expectations. This behavior is naturally attributed to the presence of heavy (correlated, short-lived $f$-electrons) and light (dispersive, long-lived $c$-electrons) electronic bands. Their longer lifetime and stronger dispersion lead to a dominant transport contribution from the $c$-electrons. At positive integer fillings, the correlated TBG insulators feature $c$- ($f$-)electron bands on the electron (hole) doping side, leading to an overall negative Seebeck coefficient. Additionally, sawtooth oscillations occur around each integer filling due to gap openings. Our results highlight the essential importance of electron correlations in understanding the transport properties of TBG and, in particular, of the lifetime asymmetry between the two fermionic species (naturally captured by the THF model). Our findings are corroborated by new experiments in both twisted bilayer and trilayer graphene, and show the natural presence of strongly-correlated heavy and light carriers in the system.

cond-mat.str-el↗
arXiv2024

Anderson Critical Metal Phase in Trivial States Protected by Average Magnetic Crystalline Symmetry

Transitions between distinct obstructed atomic insulators (OAIs) protected by crystalline symmetries, where electrons form molecular orbitals centering away from the atom positions, must go through an intermediate metallic phase. In this work, we find that the intermediate metals will become a scale-invariant critical metal phase (CMP) under certain types of quenched disorder that respect the magnetic crystalline symmetries on average. We explicitly construct models respecting average $C_{2z}T$, $m$, and $C_{4z}T$ and show their scale-invariance under chemical potential disorder by the finite-size scaling method. Conventional theories, such as weak anti-localization and topological phase transition, cannot explain the underlying mechanism. A quantitative mapping between lattice and network models shows that the CMP can be understood through a semi-classical percolation problem. Ultimately, we systematically classify all the OAI transitions protected by (magnetic) groups $Pm$, $P2'$, $P4'$, and $P6'$ with and without spin-orbit coupling, most of which can support CMP.

cond-mat.dis-nn↗
arXiv2024

Cryo-Near-Field Photovoltage Microscopy of Heavy-Fermion Twisted Symmetric Trilayer Graphene

Ever since the initial experimental observation of correlated insulators and superconductivity in the flat Dirac bands of magic angle twisted bilayer graphene, a search for the microscopic description that explains its strong electronic interactions has begun. While the seemingly disagreeing electronic transport and scanning tunneling microscopy experiments suggest a dichotomy between local and extended electronic orbitals, definitive experimental evidence merging the two patterns together has been much sought after. Here, we report on the local photothermoelectric measurements in the flat electronic bands of twisted symmetric trilayer graphene (TSTG). We use a cryogenic scanning near-field optical microscope with an oscillating atomic force microscopy (AFM) tip irradiated by the infrared photons to create a nanoscopic hot spot in the planar samples, which generates a photocurrent that we probe globally. We observe a breakdown of the non-interacting Mott formalism at low temperatures (10K), signaling the importance of the electronic interactions. Our measurements reveal an overall negative offset of the Seebeck coefficient and significant peaks of the local photovoltage values at all positive integer fillings of the TSTG's moiré superlattice, further indicating a substantial deviation from the classical two-band semiconductor Seebeck response. We explain these observations using the interacting topological heavy-fermion model. In addition, our data reveal a spatial variation of the relative interaction strength dependent on the measured local twist angle (1.2° - 1.6°). Our findings provide experimental evidence of heavy fermion behaviour in the topological flat bands of moiré graphene and epitomize an avenue to apply local thermoelectric measurements to other strongly correlated materials in the disorder-free limit.

cond-mat.mes-hall↗
arXiv2024

Evidence of heavy fermion physics in the thermoelectric transport of magic angle twisted bilayer graphene

It has been recently postulated, that the strongly correlated flat bands of magicangle twisted bilayer graphene (MATBG) can host coexisting heavy and light carriers. While transport and spectroscopic measurements have shown hints of this behavior, a more direct experimental proof is still lacking. Here, we explore the thermoelectric response of MATBG through the photo-thermoelectric (PTE) effect in gate-defined MATBG pn-junctions. At low temperatures, we observe sign-preserving, fillingdependent oscillations of the Seebeck coefficient at non-zero integer fillings of the moiré lattice, which suggest the preponderance of one carrier type despite tuning the Fermi level from hole to electron doping of the correlated insulator. Furthermore, at higher temperatures, the thermoelectric response provides distinct evidence of the strong electron correlations in the unordered, normal state. We show that our observations are naturally accounted for by the interplay of light and long-lived and heavy and short-lived electron bands near the Fermi level at non-zero integer fillings. Our observations firmly establish the electron and hole asymmetry of the correlated gaps in MATBG, and shows excellent qualitative agreement with the recently developed topological heavy fermion model (THF).

cond-mat.mes-hall↗
arXiv2023

Flat bands, strange metals, and the Kondo effect

Flat band materials such as the kagome metals or moiré superlattice systems are of intense current interest. Flat bands can result from the electron motion on numerous (special) lattices and usually exhibit topological properties. Their reduced bandwidth proportionally enhances the effect of Coulomb interaction, even when the absolute magnitude of the latter is relatively small. Seemingly unrelated to these cases is the large family of strongly correlated electron systems, which includes the heavy fermion compounds, cuprate and pnictide superconductors. In addition to itinerant electrons from large, strongly overlapping orbitals, they frequently contain electrons from more localized orbitals, which are subject to a large Coulomb interaction. The question then arises as to what commonality in the physical properties and microscopic physics, if any, exists between the two broad categories of materials? A rapidly increasing body of strikingly similar phenomena across the different platforms -- from electronic localization-delocalization transitions to strange metal behavior and unconventional superconductivity -- suggests that similar underlying principles could be at play. Indeed, it has recently been suggested that flat band physics can be understood in terms of Kondo physics. Inversely, the concept of electronic topology from lattice symmetry, which is fundamental in flat band systems, is enriching the field of strongly correlated electron systems where correlation-driven topological phases are increasingly being investigated. Here we elucidate this connection, survey the new opportunities for cross-fertilization in understanding across the platforms, and assess the prospect for new insights that may be gained into both the correlation physics and its intersection with electronic topology.

cond-mat.str-el↗
arXiv2023

Moiré Fractional Chern Insulators III: Hartree-Fock Phase Diagram, Magic Angle Regime for Chern Insulator States, the Role of the Moiré Potential and Goldstone Gaps in Rhombohedral Graphene Superlattices

We investigate in detail the $ν=+1$ displacement-field-tuned interacting phase diagram of $L=3,4,5,6,7$ layer rhombohedral graphene aligned to hBN (R$L$G/hBN). Our calculations account for the 3D nature of the Coulomb interaction, the inequivalent stacking orientations $ξ=0,1$, the effects of the filled valence bands, and the choice of `interaction scheme' for specifying the many-body Hamiltonian. We show that the latter has a dramatic impact on the Hartree-Fock phase boundaries and the properties of the phases, including for pentalayers (R5G/hBN) with large displacement field $D$ where recent experiments observed a Chern insulator at $ν=+1$ and fractional Chern insulators for $ν<1$. In this large $D$ regime, the low-energy conduction bands are polarized away from the aligned hBN layer, and are hence well-described by the folded bands of moiréless rhombohedral graphene at the non-interacting level. Despite this, the filled valence bands develop moiré-periodic charge density variations which can generate an effective moiré potential, thereby explicitly breaking the approximate continuous translation symmetry in the conduction bands, and leading to contrasting electronic topology in the ground state for the two stacking arrangements. Within time-dependent Hartree-Fock theory, we further characterize the strength of the moiré pinning potential in the Chern insulator phase by computing the low-energy $\mathbf{q}=0$ collective mode spectrum, where we identify competing gapped pseudophonon and valley magnon excitations. Our results emphasize the importance of careful examination of both the single-particle and interaction model for a proper understanding of the correlated phases in R$L$G/hBN.

cond-mat.str-el↗
arXiv2023

Moiré Fractional Chern Insulators II: First-principles Calculations and Continuum Models of Rhombohedral Graphene Superlattices

The experimental discovery of fractional Chern insulators (FCIs) in rhombohedral pentalayer graphene twisted on hexagonal boron nitride (hBN) has preceded theoretical prediction. Supported by large-scale first principles relaxation calculations at the experimental twist angle of $0.77^\circ$, we obtain an accurate continuum model of $n=3,4,5,6,7$ layer rhombohedral graphene-hBN moiré systems. Focusing on the pentalayer case, we analytically explain the robust $|C|=0,5$ Chern numbers seen in the low-energy single-particle bands and their flattening with displacement field, making use of a minimal two-flavor continuum Hamiltonian derived from the full model. We then predict nonzero valley Chern numbers at the $ν= -4,0$ insulators observed in experiment. Our analysis makes clear the importance of displacement field and the moiré potential in producing localized "heavy fermion" charge density in the top valence band, in addition to the nearly free conduction band. Lastly, we study doubly aligned devices as additional platforms for moiré FCIs with higher Chern number bands.

cond-mat.mes-hall↗
arXiv2023

Moiré Fractional Chern Insulators I: First-principles calculations and Continuum Models of Twisted Bilayer MoTe$_2$

Recent experiments observed fractional Chern insulators (FCI) in twisted bilayer MoTe$_2$ at zero magnetic field, yet even the single-particle model of this material is controversial, leading to unreliable predictions of the experimental phase diagram as discussed in [Yu et al., 2023]. In this light, we revisit the single-particle model of twisted bilayer MoTe$_2$. Utilizing large-scale density functional theory, we calculate the band structure of twisted AA-stacked bilayer MoTe$_2$ at various twist angles relevant to experiment. We find that a band inversion occurs near $4.41^\circ$ between the second and third bands. Our ab initio band structure is in qualitative agreement with [Wang et al., 2023], but shows important differences in the remote bands and in the $Γ$ valley. We incorporate two higher harmonic terms into the continuum model to capture the highest 3 valence bands per valley. We confirm that the two highest valence bands per valley have opposite Chern numbers with $|C|=1$ for small angles, and also use our model to predict a variety of Chern states in the remote bands accessible by displacement field. Finally, we perform DFT calculations and build models for the AB stacking configuration. Our work serves as a foundation for accurate determination of the correlated phases in twisted bilayer MoTe$_2$.

cond-mat.mes-hall↗
arXiv2023

Strong and Fragile Topological Dirac Semimetals with Higher-Order Fermi Arcs

Dirac and Weyl semimetals both exhibit arc-like surface states. However, whereas the surface Fermi arcs in Weyl semimetals are topological consequences of the Weyl points themselves, the surface Fermi arcs in Dirac semimetals are not directly related to the bulk Dirac points, raising the question of whether there exists a topological bulk-boundary correspondence for Dirac semimetals. In this work, we discover that strong and fragile topological Dirac semimetals exhibit 1D higher-order hinge Fermi arcs (HOFAs) as universal, direct consequences of their bulk 3D Dirac points. To predict HOFAs coexisting with topological surface states in solid-state Dirac semimetals, we introduce and layer a spinful model of an $s-d$-hybridized quadrupole insulator (QI). We develop a rigorous nested Jackiw-Rebbi formulation of QIs and HOFA states. Employing $ab\ initio$ calculations, we demonstrate HOFAs in both the room- ($α$) and intermediate-temperature ($α''$) phases of Cd$_{3}$As$_2$, KMgBi, and rutile-structure ($β'$-) PtO$_2$.

cond-mat.mes-hall↗
arXiv2023

Fractional Chern Insulators vs. Non-Magnetic States in Twisted Bilayer MoTe$_2$

Fractionally filled Chern bands with strong interactions may give rise to fractional Chern insulator (FCI) states, the zero-field analogue of the fractional quantum Hall effect. Recent experiments have demonstrated the existence of FCIs in twisted bilayer MoTe$_2$ without external magnetic fields -- most robust at $ν=-2/3$ -- as well as Chern insulators (CIs) at $ν=-1$. Although the appearance of both of these states is theoretically natural in an interacting topological system, experiments repeatedly observe nonmagnetic states (lacking FCIs) at $ν=-1/3$ and $-4/3$, a puzzling result which has not been fully theoretically explained. In this work, we perform Hartree-Fock and exact diagonalization calculations to test whether the standard MoTe$_2$ moiré model with the (greatly varying) parameter values available in the literature can reproduce the non-magnetic states at $ν=-1/3$ and $-4/3$ in unison with the FCI at $ν=-2/3$ and CI state at $ν= -1$. We focus on the experimentally relevant twist angles and, crucially, include remote bands. We find that the parameters proposed in [Wang et al. (2023)] can nearly capture the experimental phenomena at $ν=-1/3,-2/3,-1,-4/3$ simultaneously, though the predicted ground states at $ν=-1/3$ are still mostly fully-spin-polarized and a larger dielectric constant $ε>10$ than is typical of hexagonal boron nitride (h-BN) substrate $ε\sim 6$ is required. Our results show the importance of remote bands in identifying the competing magnetic orders and lay the groundwork for further study of the realistic phase diagram.

cond-mat.mes-hall↗
arXiv2023

Magic-Angle Twisted Symmetric Trilayer Graphene as Topological Heavy Fermion Problem

Recently, Ref. [1] reformulated magic-angle twisted bilayer graphene (MATBG) as a topological heavy fermion problem, and used this reformulation to provide a deeper understanding for the correlated phases at integer fillings. In this work, we generalize this heavy-fermion paradigm to magic-angle twisted symmetric trilayer graphene (MATSTG), and propose a low-energy $f-c-d$ model that reformulates MATSTG as heavy localized $f$ modes coupled to itinerant topological semimetalic $c$ modes and itinerant Dirac $d$ modes. Our $f-c-d$ model well reproduces the single-particle band structure of MATSTG at low energies for displacement field $\mathcal{E}\in[0,300]$meV. By performing Hartree-Fock calculations with the $f-c-d$ model for $ν=0,-1,-2$ electrons per Moiré unit cell, we reproduce all the correlated ground states obtain from the previous numerical Hartree-Fock calculations with the Bistritzer-MacDonald-type (BM-type) model, and we find additional new correlated ground states at high displacement field. Based on the numerical results, we propose a simple rule for the ground states at high displacement fields by using the $f-c-d$ model, and provide analytical derivation for the rule at charge neutrality. We also provide analytical symmetry arguments for the (nearly-)degenerate energies of the high-$\mathcal{E}$ ground states at all the integer fillings of interest, and make experimental predictions of which charge-neutral states are stabilized in magnetic fields. Our $f-c-d$ model provides a new perspective for understanding the correlated phenomena in MATSTG, suggesting that the heavy fermion paradigm of Ref. [1] should be the generic underpinning of correlated physics in multilayer moire graphene structures.

cond-mat.mes-hall↗
arXiv2023

Pb$_9$Cu(PO4)$_6$(OH)$_2$: Phonon bands, Localized Flat Band Magnetism, Models, and Chemical Analysis

In a series of recent reports, doped lead apatite (LK-99) has been proposed as a candidate ambient temperature and pressure superconductor. However, from both an experimental and theoretical perspective, these claims are largely unsubstantiated. To this end, our synthesis and subsequent analysis of an LK-99 sample reveals a multiphase material that does not exhibit high-temperature superconductivity. We study the structure of this phase with single-crystal X-ray diffraction (SXRD) and find a structure consistent with doped $\text{Pb}_{10}(\text{PO}_4)_6(\text{OH})_2$. However, the material is transparent which rules out a superconducting nature. From ab initio defect formation energy calculations, we find that the material likely hosts $\text{OH}^-$ anions, rather than divalent $\text{O}^{2-}$ anions, within the hexagonal channels and that Cu substitution is highly thermodynamically disfavored. Phonon spectra on the equilibrium structures reveal numerous unstable phonon modes. Together, these calculations suggest it is doubtful that Cu enters the structure in meaningful concentrations, despite initial attempts to model LK-99 in this way. However for the sake of completeness, we perform ab initio calculations of the topology, quantum geometry, and Wannier function localization in the Cu-dominated flat bands of four separate doped structures. In all cases, we find they are atomically localized by irreps, Wilson loops, and the Fubini-Study metric. It is unlikely that such bands can support strong superfluidity, and instead are susceptible to ferromagnetism (or out-of-plane antiferromagnetism) at low temperatures, which we find in ab initio studies. In sum, $\text{Pb}_{9}\text{Cu}(\text{PO}_4)_6(\text{OH})_2$ could more likely be a magnet, rather than an ambient temperature and pressure superconductor.

cond-mat.supr-con↗
arXiv2023

TBG as Topological Heavy Fermion: II. Analytical approximations of the model parameters

The recently-introduced Topological Heavy Fermion (THF) model [Phys. Rev. Lett. 129, 047601] of twisted bilayer graphene (TBG) aims to reconcile the quantum-dot-like electronic structure of the latter observed by scanning tunneling microscopy, with its electron delocalization seen in transport measurements. The THF model achieves this by coupling localized (heavy) fermions with anomalous conduction electrons. Originally, the parameters of the THF model were obtained numerically from the Bistritzer-Macdonald (BM) model of TBG [Phys. Rev. Lett. 129, 047601]. In this work, we derive analytical expressions for the THF model parameters as a function of the twist angle, the ratio between the tunneling amplitudes at the $AA$ and $AB$ regions ($w_0 / w_1$), and the screening length of the interaction potential. By numerically computing the THF model parameters across an extensive experimentally-relevant parameter space, we show that the resulting approximations are remarkably good, i.e. within the 30% relative error for almost the entire parameter space. At the single-particle level, the THF model accurately captures the energy spectrum of the BM model over a large phase space of angles and tunneling amplitude ratios. When interactions are included, we also show that the THF description of TBG is good around the magic angle for realistic values of the tunneling amplitude ratios ($0.6 \leq w_0/w_1 \leq 1.0$), for which the hybridization between the localized and conduction fermions $γ$ is smaller than the onsite repulsion of the heavy fermions $U_1$ (i.e. $|γ| < U_1$).

cond-mat.str-el↗
arXiv2023

Reply to "Comment on 'Nontrivial Quantum Geometry and the Strength of Electron-Phonon Coupling', arXiv:2305.02340, J. Yu, C. J. Ciccarino, R. Bianco, I. Errea, P. Narang, B. A. Bernevig"

This is our reply to "Comment on 'Nontrivial Quantum Geometry and the Strength of Electron-Phonon Coupling', arXiv:2305.02340, J. Yu, C. J. Ciccarino, R. Bianco, I. Errea, P. Narang, B. A. Bernevig" by Prof. Pickett, which focuses on the MgB$_2$ part of our work. We show that the entirety of the criticism in Prof. Pickett's comment pertaining to our work (arXiv:2305.02340) is invalid.

cond-mat.supr-con↗
arXiv2023

Kagome Materials I: SG 191, ScV$_6$Sn$_6$. Flat Phonon Soft Modes and Unconventional CDW Formation: Microscopic and Effective Theory

Kagome Materials with flat bands exhibit wildly different physical properties depending on symmetry group, and electron number. For the case of ScV$_6$Sn$_6$ in space group 191, we investigate the existence of a charge density wave (CDW) at vector $\bar{K}=(\frac{1}{3},\frac{1}{3},\frac{1}{3})$ and its relationship with the phonon behavior. The experimental findings reveal a $\sim$95K CDW without nesting/peaks in the electron susceptibility at $\bar{K}$. Notably, ScV$_6$Sn$_6$ exhibits a collapsed phonon mode at $H=(\frac{1}{3},\frac{1}{3},\frac{1}{2})$ and an imaginary flat phonon band in the vicinity of $H$. The soft phonon is attributed to triangular Sn ($Sn^T$) mirror-even vibrations along the $z$-direction. We develop a simple force constant model to describe the entire soft phonon dispersion. By employing a new (Gaussian) approximation of the hopping parameter, we demonstrate the renormalization of the phonon frequency and the consequent collapse of the $H$ phonon. Additionally, we propose an effective model with two order parameters (OPs) to explain the appearance of the CDW at $\bar{K}$, which competes with the collapsed phonon at $H$. Through comparisons with experimental data, we show that the $H$ OP undergoes a second-order phase transition while exhibiting substantial fluctuations, ultimately inducing the first-order transition of the $\bar{K}$ OP. Furthermore, we extend our analysis to the similar compound YV$_6$Sn$_6$, which lacks a CDW phase, attributing this difference to the participation of the heavier Y atom in the out-of-plane phonon behavior. Our comprehensive study not only elucidates the CDW in ScV$_6$Sn$_6$ but also presents a significant advancement in modeling complex electronic systems, fostering collaborations between ab-initio simulations and analytical approaches.

cond-mat.str-el↗
arXiv2023

$\mathbb Z_2$-Nontrivial Moiré Minibands and Interaction-Driven Quantum Anomalous Hall Insulators in Topological Insulator Based Moiré Heterostructures

We studied electronic band structure and topological property of a topological insulator thin film under a moiré superlattice potential to search for two-dimensional (2D) $\mathbb Z_2$ non-trivial isolated mini-bands. To model this system, we assume the Fermi energy inside the bulk band gap and thus consider an effective model Hamiltonian with only two surface states that are located at the top and bottom surfaces and strongly hybridized with each other. The moiré potential is generated by another layer of 2D insulating materials on top of topological insulator films. In this model, the lowest conduction (highest valence) mini-bands can be $\mathbb Z_2$ non-trivial when the minima (maxima) of the moiré potential approximately forms a hexagonal lattice with six-fold rotation symmetry. For the nontrivial conduction mini-band cases, the two lowest Kramers' pairs of conduction mini-bands both have nontrivial $\mathbb Z_2$ invariant in presence of inversion, while applying external gate voltages to break inversion leads to only the lowest Kramers' pair of mini-bands to be topologically non-trivial. The Coulomb interaction can drive the lowest conduction Kramers' mini-bands into the quantum anomalous Hall state when they are half-filled, which is further stabilized by breaking inversion symmetry. We propose the monolayer Sb$_{2}$ on top of Sb$_2$Te$_3$ thin films to realize our model based on results from the first principles calculations.

cond-mat.mes-hall↗
arXiv2023

Softening of a flat phonon mode in the kagome ScV$_6$Sn$_6$

The long range electronic modulations recently discovered in the geometrically frustrated kagome lattice have opened new avenues to explore the effect of correlations in materials with topological electron flat bands. The observation of the lattice response to the emergent new phases of matter, a soft phonon mode, has remained elusive and the microscopic origin of charge density waves (CDWs) is still unknown. Here, we show, for the first time, a complete melting of the ScV$_ 6$Sn$_ 6$ (166) kagome lattice. The low energy phonon with propagation vector $\frac{1}{3} \frac{1}{3} \frac{1}{2}$ collapses at 98 K, without the emergence of long-range charge order, which sets in with a propagation vector $\frac{1}{3} \frac{1}{3} \frac{1}{3}$. The CDW is driven (but locks at a different vector) by the softening of an overdamped phonon flat plane at k$_z$=$π$. We observe broad phonon anomalies in momentum space, pointing to (1) the existence of approximately flat phonon bands which gain some dispersion due to electron renormalization, and (2) the effects of the momentum dependent electron-phonon interaction in the CDW formation. Ab initio and analytical calculations corroborate the experimental findings to indicate that the weak leading order phonon instability is located at the wave vector $\frac{1}{3} \frac{1}{3} \frac{1}{2}$ of a rather flat collapsed mode. We analytically compute the phonon frequency renormalization from high temperatures to the soft mode, and relate it to a peak in the orbital-resolved susceptibility, obtaining an excellent match with both ab initio and experimental results, and explaining the origin of the approximately flat phonon dispersion. Our data report the first example of the collapse of a softening of a flat phonon plane and promote the 166 compounds of the kagome family as primary candidates to explore correlated flat phonon-topological flat electron physics.

cond-mat.str-el↗