Resonant multi-gap superconductivity at room temperature near a Lifshitz topological transition in sulfur hydrides

Maria Vittoria Mazziotti RICMASS Rome International Center for Materials Science, Superstripes Via dei Sabelli 119A, 00185 Roma, Italy Department of Mathematics and Physics, University Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy Roberto Raimondi Department of Mathematics and Physics, University Roma Tre, via della Vasca Navale 84, 00146 Roma, Italy Antonio Valletta Italian National Research Council CNR, Institute for Microelectronics and Microsystems IMM, via del Fosso del Cavaliere, 100, 00133 Roma, Italy Gaetano Campi Institute of Crystallography, CNR, via Salaria Km 29.300, I-00015 Monterotondo, Roma, Italy Antonio Bianconi RICMASS Rome International Center for Materials Science, Superstripes Via dei Sabelli 119A, 00185 Roma, Italy Institute of Crystallography, CNR, via Salaria Km 29.300, I-00015 Monterotondo, Roma, Italy National Research Nuclear University MEPhI (Moscow Engineering Physics Institute), 115409 Moscow, Russia

Abstract

The maximum critical temperature for superconductivity in pressurized hydrides appears at the top of superconducting domes in $T_{c}$ versus pressure curves at a particular pressure, which is not predicted by standard superconductivity theories. The a high-order anisotropic van Hove singularity near the Fermi level observed in band structure calculations of pressurized sulfur hydride, typical of a supermetal, has been associated with the array of metallic hydrogen wires modules forming a nanoscale heterostructure at atomic limit called superstripes phase. Here we propose that pressurized sulfur hydrides behave as a heterostructure made of a nanoscale superlattice of interacting quantum wires with a multicomponent electronic structure. We present first-principles quantum calculation of a universal superconducting dome where $T_{c}$ amplification in multi-gap superconductivity is driven by the Fano-Feshbach resonance due to configuration interaction between open and closed pairing channels, i.e., between multiple gaps in the BCS regime, resonating with a single gap in the BCS-BEC crossover regime. In the proposed three dimensional (3D) phase diagram the critical temperature shows a superconducting dome where $T_{c}$ is a function of two variables (i) the Lifshitz parameter ( $\eta$ ) measuring the separation of the chemical potential from the Lifshitz transition normalized by the inter-wires coupling and (ii) the effective electron phonon coupling (g) in the appearing new Fermi surface including phonon softening. The results will be of help for material design of room temperature superconductors at ambient pressure.

^†^†preprint: AIP/123-QED

I Introduction

I.1 Phenomenological overview

Pressurized sulfur hydride $H_{3}S$ , with $T_{c}$ = $203\ K$ at $162\ GPa$ [Drozdov et al., 2015], has reached in 2015 the record for the highest critical temperature, held before by cuprate perovskites since 1986 [Bednorz and Müller, 1988,Gao et al., 1994,Yamamoto et al., 2015]. This discovery has been followed by superconductivity in lanthanum hydrides with $T_{c}$ above $260\ K$ [Somayazulu et al., 2019,Drozdov et al., 2019], in yttrium hydrides with $T_{c}=243\ K$ [Troyan et al., 2021,Kong et al., 2021], and in a ternary carbonaceous sulfur hydride $CSH_{x}$ [Snider et al., 2020] reaching room temperature. X-ray diffraction, using focused synchrotron radiation, has shown the crystalline $Im\bar{3}m$ lattice symmetry of $H_{3}S$ above $103\ GPa$ [Einaga et al., 2016; Goncharov et al., 2017; Duan et al., 2017; Kruglov et al., 2017; Duan et al., 2017] and X-ray absorption spectroscopy has provided information on the local structure of yttrium hydrides [Purans et al., 2021].

Recent experimental results show an anomalous superconductivity phase [Troyan et al., 2021,Kong et al., 2021], while conventional superconductivity [Eliashberg, 1960,Dynes, 1972], considering only the pairing of the superconducting electrons via electron-phonon coupling (Cooper pairs) and a single-gap superconductivity paradigm, has been used to predict and to explain the high critical temperature in pressurized hydrides since the early days [Duan et al., 2014, 2015; Durajski and Szczęśniak, 2017; Gorkov and Kresin, 2018; Kostrzewa et al., 2020].

The old single-gap paradigm was found to be incompatible with band structure calculations of $H_{3}S$ in the pressure range where the critical temperature shows its maximum value, $T_{c\ max}$ . In fact band-structure calculations [Bianconi and Jarlborg, 2015a, b; Jarlborg and Bianconi, 2016], show that:

(i)

the applied pressure induces an increasing compressive lattice strain which pushes an incipient density of states (DOS) peak, due to a van Hove singularity (vHS), to higher energy until it crosses the Fermi level, [Bianconi and Jarlborg, 2015a] as confirmed by several authors [Quan and Pickett, 2016,Souza and Marsiglio, 2017];
(ii)

multiple Fermi surfaces coexist in different spots of the $k$ -space, [Bianconi and Jarlborg, 2015b];
(iii)

the Migdal approximation ${E_{Fn}}$ >> ${\hbar\omega_{0}}$ in the appearing nth Fermi-surface spot breaks down near the Lifshitz transition [Jarlborg and Bianconi, 2016];
(iv)

the anomalous pressure-dependent isotope coefficient [Jarlborg and Bianconi, 2016] strongly deviates from the single-band constant value predicted by the standard BCS theory.

The Fermi energy ${E_{Fn}}$ in the appearing new nth Fermi-surface spot at the Lifshitz transition and the energy width of the vHS singularity are of the order of the energy of the optical phonon ${\hbar\omega_{0}}$ = $160\ meV$ , observed by Capitani et al.[Capitani et al., 2017] in pressurized $H_{3}S$ infrared spectra. The latter show a Fano lineshape with the characteristic strong asymmetry indicating its interference with electronic degrees of freedom at the Fermi level [Fano, 1935,Fano, 1961].

The vHS at the Fermi energy, by using band-structure calculations, has been attributed to an electronic band of s orbitals originating from the network of hydrogen chains with short $H-H$ hydrogen bonds [Jarlborg and Bianconi, 2016]. The lattice compressive strain, due to increasing pressure, induces the energy shift of the vHS. The latter crosses the chemical potential yielding a Lifshitz transition for the appearing of a new small Fermi-surface spot [Bianconi and Jarlborg, 2015a], while the other Fermi surfaces contribute to the featureless weak broad background of the density of states. The Lifshitz transition belongs to the class of electronic topological transitions [Lifshitz et al., 1960; Volovik, 2017; Volovik and Zhang, 2017; Volovik, 2018] for the appearing of a new Fermi surfaceof the 2.5 order for standard Fermi gases. These transitions become first order showing arrested phase separation for strongly interacting fermions [Kugel et al., 2008,Bianconi et al., 2015].

The critical temperature as a function of pressure in $H_{3}S$ and $CSH_{x}$ is shown in Fig.1. The external pressure induces a compressive strain, shown in panel (a) of Fig.1. The strain is given by $\epsilon=100(a-a_{0})/a_{0}$ , where $a_{0}$ is the lattice constant at $P=103\ GPa$ , where the $Im\bar{3}m$ lattice symmetry appears because of a structural phase transition. The superconducting $dome$ over the strain range $0.5<\epsilon<4$ shows the maximum $T_{c}$ at $\epsilon=2.5$ . Panel (b) of Fig.1 shows the superconducting $dome$ observed in $CSH_{x}$ [Snider et al., 2020]. Comparing Panels (a) and (b) we notice that the maximum $T_{c}$ is higher in the superconducting dome of $CSH_{x}$ while its width is smaller.

I.2 The Fano-Feshbach resonance in multi-gap superconductivity

The paradigm shift to multi-gap superconductivity including the key role of Majorana exchange interaction between different condensates [Bianconi, 2003,Vittorini-Orgeas and Bianconi, 2009,Palumbo, Marcelli, and Bianconi, 2016] has been proposed since 2015 [Bianconi and Jarlborg, 2015a]. The Bogoljubov formulation of superconductivity, beyond the attractive BCS force between two electrons via the exchange of a phonon, includes also the attractive Majorana or repulsive Heisenberg exchange interactions [Bogoljubov, Tolmachov, and Širkov, 1958] as in nuclear matter. In the latter the forces which are commonly assumed in the phenomenological proton-neutron Hamiltonian include

(i)

the Heisenberg exchange operator for particles which exhibit antisymmetric states, which interchanges both position and spin coordinates [Heisenberg, 1933];
(ii)

the Majorana exchange operator for particles which exhibit symmetric states, which interchanges the positions of the particles, leaving their spin directions unaffected [Majorana, 1933];
(iii)

the nuclear force, resulting from the exchange of mesons between neighboring nucleons (Yukawa type) [Yukawa, 1935].

The nuclear force has been called also the short range Wigner force, applied exclusively to non-exchange forces to account for the explanation of the large binding energy of He⁴ in comparison with the deuteron, as well as of the main features of neutron-proton scattering. The interplay of these forces in the many-body quantum physical description of nuclear matter has been the object of extended studies [Iachello, 2013; Feshbach and Iachello, 1974; Blatt and Weisskopf, 1979].

The proposed scenario of multi-gap superconductivity including exchange interactions near a Lifshitz transition in pressurized $H_{3}S$ [Bianconi and Jarlborg, 2015b] was previously proposed by Bianconi, Perali and Valletta (BPV) for other non BCS superconductors like hole-doped cuprates [Perali et al., 1996; Valletta et al., 1997; Bianconi et al., 1998; Bianconi, 2006; Perali et al., 2012], diborides [Bianconi et al., 2001a; Bianconi, 2003; Perali et al., 2004; Perali, Pieri, and Strinati, 2004; Bianconi, 2005], iron-doped superconductors [Innocenti et al., 2010a, b; Bianconi, 2012; Innocenti and Bianconi, 2013; Bianconi, 2013a], organics [Mazziotti et al., 2017; Pinto et al., 2020], metallic nanoscale multilayers with nodal lines where the spin-orbit interaction plays a key role in the $T_{c}$ amplification [Mazziotti et al., 2021a], superconductivity [Bianconi et al., 2014] at the interface of oxide perovskites which can host also Majorana fermions [Mazziotti et al., 2018]. The BPV theory focuses on the quantum Fano-Feshbach resonance due to the configuration interaction between the open and the closed scattering channels [Bianconi, 2003,Vittorini-Orgeas and Bianconi, 2009,Palumbo, Marcelli, and Bianconi, 2016]. The Fano-Feshbach resonance was first proposed theoretically in atomic physics by Fano in 1935 [Fano, 1935,Fano, 1961] and extended by Feshbach in 1962 in the many-body physics of nuclear matter [Feshbach, 1962], where it is called shape resonance and it is described by the multi-channel optical model.

Refer to caption — Figure 1: Panel a: The superconducting $dome$ of $H_{3}S$ for $P>120\ GPa$ with the critical temperature $T_{c\ max}=203K$ at its top at $P_{opt}=162\ GPa$ with half width of about $43\ GPa$ . Panel b: The superconducting $dome$ of $CSH_{x}$ for $P>220\ GPa$ with $T_{c\ max}=287K$ at $P_{opt}=265\ GPa$

In the quantum theory of the many-body systems, made of different electronic components, the Fano-Feshbach resonance appears when the Fermi wavelength of one of the components is of the order of the system size as in nuclear matter [Feshbach, 1983,Feshbach, 2014] and in condensed matter at the nanoscale [Miroshnichenko, Flach, and Kivshar, 2010]. Shape resonances have been found in the final states of X-ray absorption near edge structure where the photoelectron wavelength is of the order of interatomic distance and the electronic multiple scattering resonance is degenerate with the continuum [Bianconi et al., 1978; Bianconi, 1980; Bianconi et al., 1982].

The exchange interaction between condensates was included in the theories for overlapping bands by Suhl, Matthias, and Walker (SMW) [Suhl, Matthias, and Walker, 1959], Moskalenko [Moskalenko, 1959] and Kondo [Kondo, 1963], even though they assumed, in a first approximation, that all intraband pairing channels in each of the $n$ bands were in the BCS regime with ( $E_{Fn}\gg\omega_{0}$ ). Furthermore the exchange term for interband pair transfer was assumed to be a constant parameter with no energy or momentum dependence. Therefore the above theories of overlapping bands could not include Fano-Feshbach resonances. Indeed, the Fano-Feshbach resonance in the Bogoliubov superconductivity theory of multi-gap superconductors is due to the configuration interaction between different pairing channels in different Fermi surfaces [Mazziotti et al., 2021b] with exchange of pairs between the first condensate in the BCS regime and second condensate in the BCS-BEC crossover regime. In the BCS-BEC regime the momentum and energy dependence of the exchange interaction between different coexisting gaps plays a key role, while it is neglected in the anisotropic Eliashberg theory of multi-gap superconductors. On the contrary, in the BPV theory [Perali et al., 1996], the Fano-Feshbach resonance between a $first$ pairing channel (called closed channel) in the BCS-BEC crossover regime, and the open pairing channels (called open channels) in other large Fermi surfaces in the BCS regime has been calculated from the overlap of the wave-functions of the electron pairs in different bands. The latter are determined by the subtle overlap of the wave-functions of pairs in superlattices of interacting 1D or 2D units. In ultracold fermion gases the Fano-Feshbach resonance has been applied in 2004 to generate unconventional fermion superfluids with a very large ratio of $T_{c}/T_{F}$ [Zwierlein et al., 2004,Zhang et al., 2004]. The quantum amplification mechanism at Fano-Feshbach resonance near a Lifshitz transition is generated by the quantum interference of pairing between:

(i)

electrons in the new appearing small Fermi surface with low Fermi energy and Fermi wavelength $\lambda_{F}$ of the order of the system size;
(ii)

electrons in other Fermi surfaces with very high Fermi energies and very short Fermi wavelength $\lambda_{F}$ .

At optimum $T_{c}$ in the closed pairing channel, $\lambda_{F}$ is larger but close to the superconducting coherence length $k_{F}\xi_{0}\sim 10$ [Uemura et al., 1989,Pistolesi and Strinati Calvanese, 1994].

I.3 Choice of the model

Here we propose that the experimental superconducting $dome$ , given by the curves of the critical temperature $T_{c}$ as a function of pressure P in Fig.1 in sulfur hydrides, is the smoking gun of the Fano-Feshbach resonance between pairing channels driven by the variable lattice strain [Perali et al., 1996,Mazziotti et al., 2017,Agrestini et al., 2003] which tunes the chemical potential at a topological Lifshitz transition.

The Fano-Feshbach resonance, in multi-gap superconductivity in $H_{3}S$ , is supported by the unusual behavior of the isotope coefficient. Indeed, the isotope coefficient decreases from $2.37$ to $0.31$ in the range going from the threshold to the top of the superconducting $dome$ [Jarlborg and Bianconi, 2016; Szczesniak and Durajski, 2017, 2018; Drozdov et al., 2019], deviating markedly from the value $0.5$ , predicted by the single-band BCS theory. A similar anomalous behavior of the isotope coefficient has been found in the superconducting $dome$ of cuprate perovskites [Perali et al., 1997],[Bianconi et al., 1998],[Perali et al., 2012,Innocenti and Bianconi, 2013].

A nanoscale heterostructure is expected to appear in compounds made by combining several chemical elements which leads to competing orders of electronic degrees of freedom. In pressurized hydrides the nanoscale heterogeneity is determined by the local lattice structure controlled by the effects of the lattice strain resulting from the interplay between lattice misfit-strain (or chemical pre-compression) and the external pressure. These heterostructures are a particular case of a supersolid stripes crystal [Bianconi, Innocenti, and Campi, 2013] called superstripes [Bianconi, 2013a, b], which can be realized with optical lattices in ultracold gases [Masella et al., 2019].

The superconductivity in the superstripes phase with coexisting different localised and delocalised electronic components moving in complex nanoscale heterostructures of low dimensional (quasi one-dimensional 1D) structural units (called chains or stripes or ladders) has been found in:

(i)

A15 intermetallics which have held the record for the highest superconducting $T_{c}=23.2\ K$ from 1973 to 1986. A15 intermetallics like $Nb_{3}Ge$ and $Nb_{3}Sn$ have the same average crystal symmetry $Im\bar{3}m$ as $H_{3}S$ [Mazziotti et al., 2021b] and show complex textures made of a metallic 3D network of interacting 1D metallic Nb chains [Testardi, 1975].
(ii)

hole doped cuprate perovskites where 2D networks of extrinsic stripes with different local lattice distortions [Bianconi et al., 1991, 1996a, 1996b] appear at nanoscale in the $CuO_{2}$ atomic layers facilitated by the polymorphism of perovskite structures. These form metamorphic lattice stripes in mismatched material systems [Gavrichkov et al., 2019] whose mismatch is tuned by the lattice misfit strain [Agrestini et al., 2003,Bianconi et al., 2000,Di Castro et al., 2000].
(iii)

superconducting organics like doped p-terphenyl [Pinto et al., 2020], where 1D-wires of short hydrogen bonds have been observed by X-ray diffraction [Barba et al., 2018] and it was proposed that the high- $T_{c}$ is driven by the Fano-Feshbach resonance in the nanoscale superlattice of quantum wires [Mazziotti et al., 2017].

Superlattices of two-dimensional (2D) metallic quantum wells at nanoscale have been found in:

(i)

diborides made of stacks of boron layers intercalated by magnesium [Bauer et al., 2001; Agrestini et al., 2001; Bianconi et al., 2001b; Agrestini et al., 2004; Di Castro et al., 2002];
(ii)

iron-based perovskite superconductors, iso-structural with electron doped cuprates, [Ricci et al., 2009,Ricci et al., 2010], which are made of stacks of iron atomic layers and the tuning of the chemical potential near the Lifshitz transition has been clearly seen in ARPES experiments [Bianconi, 2013a,Kordyuk, 2018,Pustovit and Kordyuk, 2016].

In this work we present the theoretical prediction of the superconducting dome for room-temperature superconductivity in pressurized hydrides due to a Fano-Feshbach resonance near a Lifshitz transition in the frame of the multi-gap superconductivity scenario discussed recently by several authors [Guidini et al., 2016; Cariglia et al., 2016; Doria, Cariglia, and Perali, 2016; Bussmann-Holder et al., 2016; Valentinis, Van Der Marel, and Berthod, 2016; Bussmann-Holder et al., 2017, 2019; Chubukov and Mozyrsky, 2018; Salasnich et al., 2019; Kagan and Bianconi, 2019; Tajima, Perali, and Pieri, 2020; Vargas-Paredes et al., 2020].

A key feature of our approach is the inclusion of the exchange integrals, obtained by the overlap of the wave-functions of electrons in different Fermi surfaces. We evaluate them by solving the Schrödinger equation for a lattice heterostructure including the renormalisation of the chemical potential at the Lifshitz transition with the opening of a new superconducting gap, controlled by the constraint of the number density equation. The results provide a significant step in understanding room-temperature superconductivity and the physical origin of the superconducting dome. Moreover, the results indicate a road map for the material design of artificial mesoscopic heterostructures made of nanoscale quantum wires which can be used by material scientists to synthesize new room-temperature superconductors at ambient pressure.

II The van Hove Singularity in the superstripes phase

We propose that in agreement with [Bianconi, 1996,Bianconi, 2001,Bianconi, 1994] a heterostructure at atomic limit made of superconducting quantum wires running in the $x$ -direction intercalated by spacers of thickness $W$ with periodicity $d=W+L$ (in the $z$ -direction). The metallic wires are separated by spacers which generate a potential barrier of amplitude V [Bianconi, 1996, 2001, 1994], where the parameters values are chosen in order to capture the main features of the DOS at the van Hove singularity near the Fermi level of $H_{3}S$ . The lineshape of the DOS peak shows the features of a high-order anisotropic van Hove singularity typical of a supermetal [Isobe and Fu, 2019], In this superstripes phase the superconductivity is calculated using the BPV approach first proposed for striped cuprate perovskites [Perali et al., 1996, 1997; Bianconi et al., 1997; Valletta et al., 1997; Bianconi et al., 1998; Bianconi, 2006; Perali et al., 2012].

We present numerical calculations of the multi-gap superconductivity $dome$ of the critical temperature as a function of pressure where we change both $i)$ the proximity of the chemical potential to the Lifshitz transition, $ii)$ the electron-phonon coupling for electrons in the upper subband and the $iii)$ the phonon softening for increasing electron-phonon coupling g in the disappearing Fermi surface.

The proximity to the Lifshitz transition is measured by the Lifshitz parameter $\eta$ given by the energy difference between the chemical potential and the energy $E_{L}$ of the topological Lifshitz transition, which is the band-edge energy of the highest energy subband, normalized to the transversal energy dispersion, $\Delta E$ , between the 1D metallic chains

\eta=\frac{\mu-E_{L}}{\Delta E}.

The applied external pressure induces the variation of either $i)$ of the Lifshitz parameter $\eta$ and $ii)$ of the electron-phonon coupling joint with phonon energy softening of the particular phonon mode coupled with the electrons in the small Fermi surface spot in the new appearing subband. In the heterostructure of quantum wires the electrons along the $x$ -direction are free, while along the $z$ -direction they are subjected to a periodic potential. Hence, the eigenfunctions $\psi_{nk_{z}}(z)$ and the eigenvalues $E_{n}(k_{z})$ , along the confinement direction, can be computed only numerically by solving a corresponding Kronig-Penney model. The solution of the eigenvalues equation gives the electronic dispersion for the $n$ subbands. Indeed, in the heterostructure of quantum wires, the quantum-size effects give a multiband electronic structure where the subband with higher energy shows a two-dimensional behavior due to hopping between 1D-chains. For the numerical calculation of the superconducting dome of $H_{3}S$ we start with the evaluation of the DOS peak in [Jarlborg and Bianconi, 2016] at $E_{F}$ by using a model of 1D-chains corresponding to the chains with short H-H bonds, as shown in Fig.2. We have designed an artificial nanoscale heterostructure at atomic limit made of quantum wires of width $L=0.85$ nm, spacers of width $W=0.55$ nm separated by a potential barrier V=4.16 eV which reproduces the van Hove singularity in a range of 500 meV around the Fermi energy $H_{3}S$ i.e., within the energy cut-off of the pairing interaction relevant for the emergence of superconductivity. We consider the models A and B for the heterostructure characterized by different coupling between the quantum wires i.e., with different transversal dispersion $\Delta E$ obtained by changing the effective mass in the spacers and in the wires. The model A of the superlattice of wires is characterized by $\Delta E$ = $274\ meV$ transversal dispersion as it was found in $H_{3}S$ . The model B is characterized by a smaller transversal dispersion $\Delta E=145\ meV$ , obtained by increasing the effective mass in the barrier. A smaller dispersion is expected to give a sharper superconducting $dome$ and a higher maximum critical temperature close to room temperature. The DOS peak and the partial DOS for the model A and model B as a function of the Lifshitz parameter are plotted in panel (a) and panel (b) of Fig.3 respectively. Panel (c) of Fig.3 shows the Lifshitz transition for the appearing of a new Fermi surface, called of type (I), tuning the chemical potential near the band-edge ( $\eta_{1}$ ) of the subband with a critical point where a new 2D Fermi surface spot appears. The second type of Lifshitz transition (type II) occurs at the opening of a neck in the small Fermi surface with the appearing of a singular nodal point which gives the sharp DOS maximum at $\eta_{2}$ (Fig.3 panel (c)) at the crossover between the 2D and 1D topology. The nearly flat portion of the DOS between (type I) and (type II) Lifshitz transitions in Fig.3 correspond with the regime where a small 2D Fermi surface with a low Fermi energy appears. While in previous theoretical descriptions of the Fano-Feshbach resonance near the Lifshitz transition the electron-phonon coupling $g_{nn}$ was assumed to be constant, in this work we take into account that the external pressure changes either the Lifshitz parameter ( $\eta$ ) and the electron-phonon coupling $g_{nn}$ in the appearing Fermi surface in the upper subband and the renormalized phonon energy $\tilde{\omega}_{0}$ shows the softening according to the Migdal relation:

\tilde{\omega}_{0}=\omega_{0}\sqrt{1-{\color[rgb]{0,0,0}2\times\max_{n}g_{nn}}}.

(1)

The relation contains the coupling constant for the metal forming the superconducting layers for $g<0.5$ , and it is used here to qualitatively estimate the effect of the coupling constant on the phonon frequency in the appearing Fermi surface as it is shown in Fig.4. In our theory, the variable $\tilde{\omega}_{0}$ is also the cut-off energy of the pairing interaction in the Bogoliubov gap equation which changes with $\eta$ . We have fixed, for the case A, $\omega_{0}=330\ meV$ in order to reproduce with moderate intraband electron phonon coupling, $0.3<g<0.33$ , the experimental phonon frequency, $\tilde{\omega}_{0}=160\ meV$ [Capitani et al., 2017], measured in pressurized $H_{3}S$ at $150\ GPa$ For the case B we have fixed $\omega_{0}=225\ meV$ , in order to get $\tilde{\omega}_{0}=\Delta E$ = $145\ meV$ with moderate intraband electron phonon coupling $g=0.25$ .

In the case of organic superconductors [Mazziotti et al., 2017] it has been shown that the amplification of the critical temperature in heterostructures of quantum wires and a narrow superconducting dome occurs where the coupling in the appearing new nth Fermi surface is larger than the in other Fermi surfaces and the interband coupling is small. In our model $g_{nn^{\prime}}$ is the superconducting dimensionless coupling constant for the three-band system which has a matrix structure that depends on the band indices $n$ and $n^{\prime}$ .

In this work we confirm previous results [Mazziotti et al., 2017]: in fact the superconducting dome with a sharp drop of $T_{c}$ at both sides of the maximum with a stronger Fano-Feshbach anti-resonance is generated by a weak intra-band coupling for the Cooper pairing channel $g_{nn}$ and weak inter-band exchange channels $g_{nn^{\prime}}$ . The Fano-Feshbach resonance increases the maximum value of $T_{c}$ at the top of the $dome$ increasing the intra-band coupling for the Cooper pairing channel $g_{nn}$ in the new appearing or disappearing Fermi surface. Moreover the maximum of the critical temperature is expected to increase where the phonon energy giving the energy cut off for the pairing processes is of the same order as the hopping energy between the wires $\Delta E=\tilde{\omega}$ .

When the Lifshitz parameter is tuned between the band-edge and the van Hove singularity, a new Fermi surface appears with a very small number density of electrons in the strong coupling limit. The condensates in the other Fermi surfaces (first and second subband) have a very high Fermi energy and therefore are in the adiabatic regime and coexist with a third condensate in the small Fermi surface where the classical BCS approximations are violated. In the models (A) and (B) for 0< $\eta$ <1 a new closed 2D Fermi surface appears as shown in band structure calculations [Bianconi and Jarlborg, 2015a] for $H_{3}S$ around 160 GPa where the maximum critical temperature is observed at a top of a $dome$ . For this heterostructure we assume that quantum size effects are not negligible and the electron hopping in the transverse direction is finite so that the quantum wires can be considered to be interacting wires in the metallic phase while very weakly interacting wires are in the localization limit. This is reflected in the spectrum that appears to split into $n$ subbands characterized by quantized values of the transverse moment that depends on the band index and the dimension of the wires.

In the heterostructure of quantum wires the electrons along the $x$ -direction are free, while along the $z$ direction they are subjected to a periodic potential $V(z)$ :

V(z)=V\sum^{\infty}_{-\infty}\theta(W/2-|m\lambda_{p}-z|).

(2)

In the periodic potential we assume that the full single-particle wave-function can be written as

\psi_{n,\mathbf{k},\alpha}(\mathbf{r})=\frac{1}{\sqrt{L_{x}L_{z}}}e^{ik_{x}x}\psi_{nk_{z}}(z)\bm{\chi}_{\alpha},

(3)

where $L_{x}$ and $L_{z}$ are the spatial dimensions of the system, $n$ is the band index, $\mathbf{k}=(k_{x},k_{z})$ is the wavevector, and $\bm{\chi}_{\alpha}$ is the spinor part with spin $\alpha=\uparrow$ or $\downarrow$ . The corresponding energy eigenvalues, independents from the spin, are given by

\varepsilon_{n}(\mathbf{k})=\frac{\hbar^{2}}{2m_{x}}k^{2}_{x}+E_{n}(k_{z}).

(4)

The eigenfunctions $\psi_{nk_{z}}(z)$ and the eigenvalues $E_{n}(k_{z})$ are computed numerically by solving a corresponding Kronig-Penney model. The solution of the eigenvalues equation gives the electronic dispersion for the $n$ subbands.

III multi-gap superconductivity beyond BCS

In the multi-gap superconducting scenario the exchange integral for pairs of electrons in different bands plays a key role for the $T_{c}$ amplification while it is neglected in the single-band BCS theory.

The pairing interaction is assumed to be originated from an electron-electron contact interaction with a cut-off equal to the renormalized phonon energy $\tilde{\omega}_{0}$ . The pairing interaction takes then a generalized BCS form

U_{{\bf k}{\bf k^{\prime}}}^{nn^{\prime}}=\tilde{U}_{k_{z}k^{\prime}_{z}}^{nn^{\prime}}\theta(\tilde{\omega}_{0}-|\xi_{n,{\bf k}}|)\theta(\tilde{\omega}_{0}-|\xi_{n^{\prime},{\bf k^{\prime}}}|)

(5)

where $\xi_{n,{\bf k}}=\varepsilon_{n}(\mathbf{k})-\mu$ . The $\tilde{U}_{k_{z}k^{\prime}_{z}}^{nn^{\prime}}$ coupling constants, which in the original BCS model are structureless, originate from the matrix elements between exact eigenstates of the superlattice, and depend on the wave vectors $k_{z}$ and $k_{z}^{\prime}$ in the superlattice direction as well as on the band indices $n$ and $n^{\prime}$ . This induces a structure in the k-dependent interband coupling interaction for the electrons that determines the quantum interference between electron pairs wave functions in different subbands or minibands of the superlattice. The generalized couplings can be expressed as

\tilde{U}_{k_{z}k^{\prime}_{z}}^{nn^{\prime}}=-{\color[rgb]{0,0,0}U_{0}^{nn^{\prime}}}\ I_{k_{z}k^{\prime}_{z}}^{nn^{\prime}},

where $-{\color[rgb]{0,0,0}U_{0}^{nn^{\prime}}}$ is the original attractive contact interaction, that we allow to have a dependence from the band index of the electron pairs, and $I_{k_{z}k^{\prime}_{z}}^{nn^{\prime}}$ is the pair superposition integral, calculated considering the interference between electronic wave functions in different subbands [Innocenti et al., 2010a].

I_{k_{z}k^{\prime}_{z}}^{nn^{\prime}}=\frac{1}{L_{x}L^{2}_{z}}\int\psi_{nk_{z}}^{*}(z)\psi_{n-k_{z}}^{*}(z)\psi_{n^{\prime}k^{\prime}_{z}}(z)\psi_{n^{\prime}-k^{\prime}_{z}}(z)dz.

Notice that for vanishing $V$ , the amplitude of the periodic potential, the overlap integrals would reduce to the standard BCS form $I_{k_{z}k^{\prime}_{z}}^{nn^{\prime}}=(L_{x}L_{z})^{-1}$ independent of the $k_{z}$ and $k_{z}^{\prime}$ wave vectors as well. We emphasize that the exchange interaction is not constant but depends not only on the wave vector along z but also on the band index, therefore it has a matrix structure. For later reference and to compare with the homogeneous case, it is useful to introduce the standard dimensionless coupling constant $g_{nn^{\prime}}=U_{0}^{nn^{\prime}}N_{0}$ , where $N_{0}$ is the two-dimensional density of states. The non diagonal terms ( $nn^{\prime}$ ) of the superposition integral are calculated for model A and are shown in Fig.5.

The self-consistent equation for the superconducting gap at zero temperature can then be written as

\Delta_{nk_{z}}=-\frac{1}{2}\sum_{n^{\prime},\mathbf{k}^{\prime}}\frac{U^{nn^{\prime}}_{{\bf k}{\bf k^{\prime}}}\Delta_{n^{\prime}k^{\prime}_{z}}}{\sqrt{(\varepsilon_{n^{\prime}}(\mathbf{k^{\prime}})-\mu)^{2}+|\Delta_{n^{\prime}k^{\prime}_{z}}|^{2}}},

(6)

In order to take into account the renormalisation of the chemical potential and charge densities in each subband when a new superconducting gap appears in a single subband, the joint Bogoliubov gap equation and the charge density equation have been solved where the charge density $\rho$ and the chemical potential in the superconducting phase are related by

\rho=\frac{1}{L_{x}L_{z}}\sum_{n\mathbf{k}}\bigg{(}1-\frac{\varepsilon_{n}(\mathbf{k})-\mu}{\sqrt{(\varepsilon_{n}(\mathbf{k})-\mu)^{2}+\Delta^{2}_{n,k_{z}}}}\bigg{)}\theta(\mu-\varepsilon_{n}(\mathbf{k})),

(7)

The joint solution of the gap equation (3) and the density equation (4) is essential in order to correctly describe the multi-gap superconductivity near the Lifshitz transition where the gap in the upper subband approaches to the Bose-Einstein condensation.

The superconducting critical temperature is calculated by iteratively solving the linearized equation

\Delta_{n\mathbf{k}}=-\frac{1}{2}\sum_{n^{\prime}\mathbf{k}^{\prime}}U_{{k}_{z}{k^{\prime}}_{z}}^{nn^{\prime}}\Delta_{n^{\prime}k^{\prime}_{z}}\frac{\tanh\left(\frac{(\varepsilon_{n^{\prime}}(\mathbf{k}^{\prime})-\mu)}{2T_{c}}\right)}{(\varepsilon_{n^{\prime}}(\mathbf{k}^{\prime})-\mu)}

(8)

until the vanishing solution is reached with increasing temperature.

Here we present a case of Fano-Feshbach resonant superconductivity giving a superconducting $dome$ where the top of the $dome$ reaches the high temperature range $200<T_{c}\ max<300K$ of pressurized hydrides much larger than $T_{c}\ max=123K$ calculated in a previous work for cuprates and organics [Mazziotti et al., 2017]. This result is obtained by the resonance regime by increasing gaps anisotropy where the two gaps differ by a sizable factor in the range 2.9-3.9, at the top of the dome where the coupling strength in a small Fermi surface spot is in the range 0.3<g<0.42 and the phonon energy scale determines not only a large prefactor for the critical temperature, but it also induces a large width of the resonance.

Here, following Ref.[Mazziotti et al., 2017], the superconducting dome is generated by considering the case where the first and the second subband are in a weak coupling regime because the Fermi level is very far from the band edge. Therefore in our model we fixed the values of $U_{0}^{nn^{\prime}}$ in order to have the following values for the dimensionless coupling constants: $g_{11}=g_{22}=0.10$ . On the contrary the coupling in the third subband $g_{33}=g$ is considered to be variable because the Fermi level is tuned around the band edge. In fact we expect that the electron-phonon coupling for the upper subband should be enhanced because of a Kohn anomaly or because the interplay with the formation of a charge density wave (CDW) in a narrow momentum region around the CDW wave vector. In parallel, we vary the cut-off energy $\tilde{\omega}_{0}$ according to the Migdal relation [Migdal, 1958].

In Fig.6 the panel A1 (B1) shows the values of the gap ratio, $2\Delta/T_{c}$ , for the second and third subband for the A (B) case as a function of the Lifshitz parameter $\eta$ . The panel A2 (B2) in Fig.6 shows the trend of the isotope coefficient for the A (B) case as a function of the Lifshitz parameter $\eta$ . All these graphs were obtained at a fixed coupling value equal to $g=1/4$ for the case (A) and at $g=1/3$ for the case (B). At the Lifshitz transition for the appearing of a new Fermi surface, $\eta=0$ , the value of the gap ratio $2\Delta/T_{c}$ is close to the predicted value the BCS theory ( $2\Delta/T_{c}=3.5$ ) but for $0.5<\eta<1=0$ we see strong deviations from the BCS single gap prediction. In fact, $2\Delta_{2}/T_{c}$ reaches a very small value, between 0 and 1, while $2\Delta_{3}/T_{c}$ remains approximately constant at the BCS value. A similar scenario was observed in magnesium diboride [Innocenti et al., 2010a] due to the exchange integral for pairs transfer between the second and third subbands.
While the BCS theory predicts that the isotope coefficient should be constant close to the value of 0.5 we see that the isotope coefficient shows a strong maximum of the Lifshitz transition $\eta=0$ , and a minimum at $0.5<\eta<1=0$ near the topological Lifshitz transition for opening a neck in the Fermi surface of the third subband, These theoretical predictions are in agreement with the experiments showing that the isotope coefficient shows an anomalous trend as a function of pressure in pressurized sulfur hydrides [Jarlborg and Bianconi, 2016]. I

In Fig.7 we show the trend of the critical temperature as a function of the ratio between the gap in the third subband and the gap in the second subband for the case A in panel A and for the case B in panel B. The results clearly show that the maximum critical temperature is reached with the highest anisotropy between the gaps. In fact the graph shows that the maximum of $T_{c}$ is reached when the ratio $\Delta_{3}/\Delta_{2}$ is maximum. This figure shows clearly that room-temperature superconductivity is reached by increasing electron-phonon coupling in the a small Fermi surface spot pushing up the gap in the appearing Fermi surface due to the third subband $\Delta_{3}$ while the gap $\Delta_{2}$ in the second Fermi surface with large Fermi energy remains small because the electron-phonon coupling remains small. These results show the predicted effect of Fano-Feshbach resonance driven by the exchange interaction between closed (strong) pairing channels in the third subband and open (weak) pairing channels in the second subband

IV Superconducting dome

In Fig.8 we plot the critical temperature as a function of the Lifshitz parameter in both semi-logarithmic and linear scales for variable values of the electron-phonon (e-ph) coupling in the upper subband. In the linear scale we see a variable superconducting dome where $T_{c\ max}$ increases with $g$ increasing up to $g=0.4$ and it decreases in the range $0.4<g<0.5$ because the phonon softening goes to zero at $g=0.5$ . In the case (A) the maximum value of the critical temperature is $250\ K$ , therefore it explain the superconductivity in $H_{3}S$ . While the maximum $T_{c}$ in case (B) reaches $330\ K$ showing the possibility of room-temperature superconductivity. The plots in semi-logarithmic scale show the typical form of the Fano-Feshbach anti-resonance which becomes more relevant as $g$ increases. Fig.9 shows the variation of the critical temperature $T_{c}$ at constant $\eta$ and the variable electron-phonon coupling $g$ for the A case. In the anti-resonant regime $-1<\eta<0$ we observe a clear feature of the Fano-Feshbach resonance. In fact, at the low energy side of the Fano-Feshbach resonance between closed and open channels, the negative interference gives the observed $T_{c}$ minimum appearing at $\eta=-0.34$ where the critical temperature decreases with increasing e-ph coupling, on the contrary for $\eta>0$ $T_{c}$ increases with increasing e-ph coupling up to $g=0.4$ .

Fig.10 shows the critical temperature $T_{c}$ (g, $\eta$ ) as a function of two variables: $(i)$ the e-ph coupling in the third subband (g), where $g$ is the reduced Allen-Dynes electron-phonon coupling ( $\lambda$ /(1+ $\lambda$ )) [Dynes, 1972] and $(ii)$ the Lifshitz energy parameter ( $\eta$ ). The critical temperature $T_{c}$ is calculated by the BPV approach including the superconducting shape resonance between multiple gaps. The maximum $T_{c}$ of the dome occurs in the $(\eta,g)$ plane at the point $(1,0.4)$ $i.e.,$ at the Lifshitz transition for neck disrupting, at $\eta$ =1, which is associated with a transition of the topology of the small appearing Fermi surface from 1D at higher energy to 2D topology at lower energy. The universal superconductive dome obtained in this figure is needed to understand the experimental dome observed in the experimental curves of the critical pressure versus pressure $T_{c}$ (P) of sulfur hydrides. In fact the external pressure induces a joint variation of both the energy position of the chemical potential with respect to the band-edge (the Lifshitz parameter $\eta$ ) as well as the electron-phonon coupling $g$ in the upper subband along a line in the ( $\eta$ , $g$ ) plane. The variable electron-phonon coupling is associated with the softening of the phonon mode energy coupled with electrons in the upper subband according to the Migdal relation. Therefore the experimental curve of $T_{c}$ vs pressure shown in Fig.1 for a particular pressurized hydride is determined by different cuts of the universal superconductive dome determined by the particular pathway in the ( $\eta$ , $g$ ) plane driven by variable pressure.

In order to reproduce room-temperature superconductivity in $CSH_{x}$ we have numerically evaluated the gaps and critical temperature for the case (B) where we have decreased the hopping between the wires to simulate the modified spacer material in $CSH_{x}$ in comparison with $H_{3}S$ . Therefore we have used the transversal dispersion $\Delta E=145$ meV to simulate the superconducting dome of $CSH_{x}$ . The results are shown in Fig.10 and Fig.11. In the case B the critical temperature $T_{c\ max}$ at the top of the $superconducting$ $dome$ reaches room-temperature superconductivity.

Tuning the chemical potential, $\mu$ , in the proximity of the band-edge, the superconducting system reaches different regimes which are distinguished by the Lifshitz parameter. At the Lifshitz transition for appearing of the new Fermi surface spot the Fermi level in the hot spot is very low and therefore the few electrons there are strongly coupled with lattice phonons showing the Khon anomaly and softening with superconductivity competing with charge density wave (CDW) and phase separation as it has been observed in doped diborides [Bauer et al., 2001,Agrestini et al., 2004] which show phonon softening at the maximum $T_{c}$ [Simonelli et al., 2009].

In the Lifshitz transition for the topological transition of the type opening a neck the Fano-Feshbach resonance gives the maximum $T_{c}$ . In fact the BCS condensate, made of the majority of electrons in the first subband, coexists with a minority of electrons in the second subband forming a condensate in BCS-BEC crossover [Guidini et al., 2016; Ochi et al., 2021]. These results show that the maximum critical temperature in the multi-gap superconducting scenario can reaches room-temperature superconductivity driven by the exchange interaction between different condensates, neglected in the BCS approximation.

V Conclusions

In conclusion, we have shown that the theory of multi-gap superconductivity in a superlattice of nanoscale stripes, which was first proposed for high temperature superconductivity in hole doped cuprate perovskites [Perali et al., 1996; Valletta et al., 1997; Bianconi et al., 1998; Bianconi, 2006; Perali et al., 2012], could provide a quantitative description of room temperature superconductivity in pressurized hydrides. We have calculated the superconducting domes for two different cases of the heterostructure of quantum stripes with larger and smaller hopping between stripes where the critical temperature is determined by both the Lifshitz parameter and the variable electron-phonon coupling in the appearing Fermi surface. The key point of our work is the solution of the Bogoljubov gap equations in a multi-gap system including the Fano-Feshbach resonance driven by the variable exchange interaction between condensates, which is usually neglected in the standard Migdal-Eliashberg theory. We have shown that multiple gaps in large Fermi surfaces with high Fermi energy in the weak coupling regime can be amplified by exchange interaction with a large gap in the strong coupling regime in a small Fermi surface spot. We have presented cases where the Fano-Feshbach resonance appears by tuning the chemical potential near an electronic topological Lifshitz transition in heterostructures of quantum wires. We have presented two different heterostructures of quantum wires where the critical temperature reaches the $200<T_{c}<300K$ range.

Acknowledgements.

We thank the staff of Department of Mathematics and Physics of Roma Tre University, the Computing Center of Institute of Microelectronics and Microsystems IMM of Italian National Research Council CNR and Supertripes-onlus for financial support of this research project.

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