Two step I to II type transitions in layered Weyl semi-metals and their impact on superconductivity

Novel “quasi two dimensional” typically layered (semi) metals offer a unique opportunity to control the density and even the topology of the electronic matter. Along with doping and gate voltage, a robust tuning is achieved by application of the hydrostatic pressure. In Weyl semi-metals the tilt of the dispersion relation cones, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa ,$$\end{document}κ, increases with pressure, so that one is able to reach type II (\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa >1$$\end{document}κ>1starting from the more conventional type I Weyl semi-metals \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\kappa <1$$\end{document}κ<1. The microscopic theory of such a transition is constructed. It is found that upon increasing pressure the I to II transition occurs in two continuous steps. In the first step the cones of opposite chirality coalesce so that the chiral symmetry is restored, while the second transition to the Fermi surface extending throughout the Brillouin zone occurs at higher pressures. Flattening of the band leads to profound changes in Coulomb screening. Superconductivity observed recently in wide range of pressure and chemical composition in Weyl semi-metals of both types. The phonon theory of pairing including the Coulomb repulsion for a layered material is constructed and applied to recent extensive experiments on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$HfTe_{5}$$\end{document}HfTe5.

ically.In Section IV the phonon theory of pairing including the Coulomb repulsion for a layered material is applied to recent extensive experiments on Hf T e 5 under the hydrostatic pressure.The last Section contains conclusions and discussion.

A "UNIVERSAL" LATTICE MODEL OF LAYERED (TYPE I AND TYPE II) WEYL SEMI-METALS
Inter -layer hopping on honeycomb lattice A great variety of tight binding models were used to describe Weyl (Dirac) semimetals in 2D.Historically the first was graphene (type I, κ = 0) , in which electrons hope between the neighboring cites of the honeycomb lattice.Two Dirac cones appear at K and K crystallographic points in Brillouin one (BZ).Upon modification (gate voltage, pressure,intercalation) the hexagonal symmetry is lost, however a discrete chiral symmetry between two sublattices, denoted by I = A, B, ensures the 2D WSM.The tilted type I and even type II (κ > 1) WSM can be described by the same Hamiltonian with the tilt term added.We restrict the discussion to systems with the minimal two cones of opposite chirality and negligible spin orbit coupling.This model describes the compounds listed in Introduction and can be generalizable to more complicated WSM.This 2D model is extended to a layered system with interlayer distance d.The 2D WSM layers are separated by dielectric streaks with interlayer hopping neglected, so that they are coupled electromagnetically only [28].
The lateral atomic coordinates on the honeycomb lattice are r n = n 1 a 1 + n 2 a 2 , where lattice vectors are: ; The length of the lattice vectors a will be taken as the length unit and we also set = 1.The hopping Hamiltonian including the tilt term is: Here an integer l labels the layers.Operator ψ sA † nl is the creation operators with spin s =↑, ↓, while the density operator is defined as n nl = ψ sI † nl ψ sI nl .The chemical potential is µ, while t is the hopping energy.Each site has three neighbors separated by vectors δ 1 = 1 3 (a 1 − a 2 ) , δ 2 = − 1 3 (2a 1 + a 2 ) and δ 3 = 1 3 (a 1 + 2a 2 ).Dimensionless parameter κ determines the tilt of the Dirac cones along the a 1 direction [6].In the 2D Fourier space, , one obtains for Hamiltonian (for finite discrete reciprocal lattice N s ×N s ): Here k = k1 Ns b 1 + k2 Ns b 2 are the reciprocal lattice vectors and the matrix where From now on the hopping energy t will be our energy unit.
The free electrons part of the Matsubara action for Grassmanian fields ψ * sI kln therefore is: Here ω n = πT (2n + 1) is the Matsubara frequency.The Greens' function, g ss kn = δ ss g kn , of free electrons has the (sublattice) following matrix form: Now we turn to the interactions part of the Hamiltonian.

Coulomb repulsion
The electron-electron repulsion in the layered WSM on the lattice can be presented in the form, where v C n−n ,l−l is the "bare" Coulomb interaction between electrons.Making the 2D Fourier transform, one obtains, where with the in plane Coulomb repulsion being v 2D q = 2πe 2 q .Here is the inter -layer dielectric constant [34], while d is the interlayer distance.On the hexagonal lattice the exponential formula approximates the Coulomb repulsion well only away from the BZ boundaries.Near the boundaries the (periodic) potential is calculated numerically in SI3.The long range screening effect of the Coulomb interaction is effectively taken into account using the RPA approximation.Effect of pressure on the various parameters is discussed in the next section.

Pressure induced parameter modifications
While pressure turned out to be more experimentally accessible control parameter than the gate voltage, in the early works mentioned in Introduction typically the phase diagram was studied as a function of the chemical potential.Moreover in most recent experiments the hydrostatic pressure serves as a control parameter to induce topological transformations of the electronic matter in WSM.The parameter dependence of a microscopic model on pressure, is in principle derivable by the DFT and a corresponding adaptation of the elasticity theory [20].Although there exist a qualitative theoretical description of the pressure dependence of the Coulomb repulsion [30], electron-phonon coupling and the topology of the Fermi surface of these novel materials [31], it is difficult to determine quantitatively the tilt κ, inter layer spacing d, electron density and other parameters.Therefore we use an experimentally parametrized (see for example a comprehensive study [32]) dependence of these parameters on the pressure.In the present paper to describe a specific material Hf T e 5 as an example we utilize experimental results of ref. [19].Note that in many materials the robust electron gas exists only at certain pressure.
For not very large pressures (P < 15 GP a) several parameters dependencies can be accounted for as linear.In particular, the layer spacing are the tilt parameter are modified under pressure P as: The tilt parameter was estimated in ref. [20] for a wide range of κ.For layered Hf T e 5 the stress parameter is σ = 0.225A/GP a.The "ambient" value is d a = 7.7A.As noted above the electron gas exists [19] in this case only for P > 3GP a.For the tilt modulus κ a = −0.3 and γ = 0.15/GP a.
Measurements demonstrate that 3D electron density in the type I phase of layered WSM is exponential in pressure (for not very high pressures): It saturates upon approach to type II WSM.The ambient value is n a = 1.4×10 19 cm −3 , while β = 0.77/GP a.The two dimensional electron density in the layers is related to the measured density by n (P ) = n 3D (P ) d (P ).The influence on the interactions will be discussed in the next Section.Having described the model let us turn to the spectrum and topology of the Fermi surface for different pressures.

Topological phases of layered WSM
Upon increasing pressure the I to II transition occurs in two continuous steps.In the first step the cones of opposite chirality coalesce so that the chiral symmetry is restored, while the second transition to the Fermi surface extending throughout the Brillouin zone occurs at higher pressures.Fig. 1 describes the Fermi surface (blue areas depict the Fermi sea in upper contour plots) and dispersion relations (lower 3D plots) of three representative pressures value from the three phases.There are two branches (brown higher than green) crossing the Fermi level (blue plane).
The graphene -like dispersion relation for smallest value of pressure when the electron pockets exist, P = 3 GP a, κ = 0.15 (left panel in Fig. 1) represents the type I WSM below the chiral transition.A rhombic BZ (with coordinates k 1 and k 2 defined in Eq.( 3), yellow area covers the BZ) is chosen.Location of the cones (see a lower 3D plot) are close to crystallographic K ± points.There are two slightly tilted Dirac cones of opposite chirality.Increasing the pressure towards the chiral transition (see more plots in SI2) at P χ = 6.5 GP a, the two pockets of the Fermi surface become elongated and larger and eventually merge into a single pocket shown in the central figure for P = 8 GP a.The tilt parameter is already significant κ = 0.9.At yet larger pressure P = 8 GP a (right panel) the material becomes a type II WSM with large κ > 1.2.In this case FS envelops the BZ that topologically is torus.See the segment on the boundary k 2 = 0 = 2π/a.Obviously the upper band becomes flatter as the tilt (pressure) increases.Both the electron density and the density of states were calculated numerically for the Fermi distribution function at temperature T = 1K (the density at zero temperature corresponds to an area inside the FS) at various values of the chemical potential.Then the density is matched with those determined phenomenologically in the previous subsection.
The first topological transition: spontaneous chiral symmetry breaking At small pressures, 3 GP a < P < 4 GP a, the Fermi surface consists of two well separated Dirac cones of opposite chirality.The tilt does not affect the basic chiral symmetry of the honeycomb lattice: two sublattices are related by a reflection.The sixfold symmetry in undistorted graphene is of course typically broken down to the reflection symmetry only.When the tilted cones FS pockets merge at the transition P = P χ = 6.5 GP a (see the brown line in Fig. 2) the chiral symmetry of the ground state is restored.The overall chirality of the FS above P χ (a topological number) therefore is zero.Although we are not aware of a mathematical proof, this transition always precedes the I → II topological transition, see cyan line in Fig. 2. The chiral transition is also topological, but a more local sense: fracture of the Fermi surface like in graphene oxide [29] or Lifshitz transition in high T c cuprates like La 2−x Sr x CuO 4 .The I → II is more "exotic" [33]: it involves the global topology of the Fermi surface (it is a torus).The DOS at transition (the green curve in Fig. 2) has a finite maximum at which the derivative changes sign.

The second topological transition: I → II
The electron density in type I phase above the chiral transition grows quite fast, see red line in Fig. 2, so that at large pressures a significant part of BZ for one of the branches of spectrum is occupied.Eventually at P I→II = 9.9 GP a the growing single pocket envelops the BZ torus and thus FS splits again into two curves, see the right panel in Fig. 1.Density of electron saturates, while the DOS has another finite peak.The two transition lead to singularities in various physical quantities.In the next Section the screening of Coulomb interactions is discussed.

SCREENING IN LAYERED WEYL SEMI -METAL.
The screening in the layered system can be conveniently partitioned into the screening within each layer described by the polarization function Π qn and electrostatic coupling to carriers in other layers.We start with the former.

Polarization function of the electron gas in Layered WSM
In a simple Fermi theory of the electron gas in normal state with Coulomb interaction between the electrons in RPA approximation the Matsubara polarization is calculated as a simple minus "fish" diagram [28] in the form: Using the GF (see Eq.( 7)), one obtain: where Performing summation over m, one obtains: The polarization function however is strongly differ from the usual Lindhard expression for a parabolic band.
Screening due to electron gas in layered system Coulomb repulsion between electrons in different layers l and l within the RPA approximation is determined by the following integral equation: The polarization function Π qn in 2D was calculated in the previous subsection.This set of equations is decoupled by the Fourier transform in the z direction: where The screened interaction in a single layer therefore is is given by the inverse Fourier transform [28]: Considering screened Coulomb potential at the same layer l = l , the integration gives, where b qn = cosh (dq) − v 2D q Π qn sinh (dq).This formula is reliable only away from plasmons b qn > 1.It turns out that to properly describe superconductivity, one can simplify the calculation at low temperature by considering the static limit Π qn Π q0 .Consequently the potential becomes static: SUPERCONDUCTIVITY Superconductivity in WSM is caused by a conventional phonon pairing.The leading mode is an optical phonon mode assumed to be dispersionless with energy Ω.The effective electron-electron interaction due to the electronphonon attraction opposed by Coulomb repulsion (pseudo -potential) creates pairing below T c .Further we assume the singlet s-pairing channel and neglect the interlayer electrons pairing.It important to note that unlike in conventional 3D metal superconductors where a simplified pseudo -potential approach due to McMillan and other [27], in 2D and layered WSM, one have to resort to a more microscopic approach.

Effective attraction due to phonon exchange opposed by the effective Coulomb repulsion
The free and the interaction parts of the effective electron action ("integrating phonons"+RPA Coulomb interaction) [35] in the quasi-momentum -Matzubara frequency representation, S = S e + S int , Here n qln = p ψ * sI pln ψ sI q−p,l,n the Fourier transform of the electron density.The effective electron -electron coupling due to phonons is: where the bosonic frequencies are ω b m = 2πmT .The pressure dependence on the frequency is approximated as: For Hf T e 5 we take Ω a = 15meV and ζ = 0.005/GP a.

Nambu Green's functions and Gorkov equations
Normal and anomalous (Matsubara) intra layer Nambu Green's functions are defined by expectation value of the fields, ψ Is knl ψ * s J knl = δ ss G IJ kn and ψ Is knl ψ Js −k,−n,l = ε ss F IJ kn , while the gap function is where Gorkov equations [35]: This equation was solved numerically by iterations method.The momenta are discretized as q 1.2 = 2πj 1,2 /N s (where j 1,2 = −N s /2 ...(N s /2 − 1)) N s = 256 while the frequency cutoff was N T = 128 the interatomic in-plane distance a = 3.5A, electron-phonon coupling g = 140meV and the dielectric constant ε = 20.
The critical temperature as a function on the pressure is presented in Fig. 3.The blue points represent the T c when Coulomb repulsion is neglected.It clearly shows the spikes of the T c near the points of the both topological transformation of the electronic system caused by the hydrostatic pressure.It amplifies the dependence of the density of states (green line) in these points that can be understood from the approximate exponential BCS dependence, T c = Ωe −D(µ)g 2 .A more realistic model includes the Coulomb repulsion, see red points in Fig. 3.The critical temperatures are much smaller demonstrating that in the present case the repulsion plays the essential role.It turns out that it not possible to approximate this behavior using a simplistic pseudo -potential approach by McMillan [27] theory successfully applied to 3D good metals.

CONCLUSION
To summarize we have developed a theory of superconductivity in layered Weyl semi-metals under the hydrostatic pressure that properly takes into account the Coulomb repulsion.It is shown that in Weyl semi -metals the tilt of the dispersion relation cones, κ, increases with pressure, so that one is able to reach type II (κ > 1starting from the more conventional type I Weyl semi -metals , κ < 1).It is found that upon increasing pressure the I to II transition occurs in two continuous steps.In the first step the cones of opposite chirality coalesce so that the chiral symmetry is restored, while the second transition to the Fermi surface extending throughout the Brillouin zone occurs at higher pressures.We show that the critical temperature is a very robust tool to study these transformations of the electronic system.The critical temperature shows spike in the points of topological transformation repeating the density of the electron states.The generalization goes beyond the simplistic pseudo -potential approach by McMillan [27] theory.Superconductivity demonstrated significant effect of the Coulomb repulsion on the critical temperature.

FIG. 1 .
FIG. 1. Evolution of the Fermi surface topology as the pressure of Weyl semimetal increase.Parameters like the tilt κ (P ), electron density etc are given in Eqs.(11) of the Weyl semimetal.The upper raw depicts the Fermi surfaces of all three topological phases, while the lower row are the corresponding dispersion relation of both branches (brown and green surfaces) with respect to Fermi level (the blue plane).At relatively low pressure the FS consists of two small Dirac pockets.At intermediate pressures the two pockets merge into a single ellipsoidal large pocket (still type I).At very high pressures the electron liquid undergoes the type I to type II topological transition.

Fig. 2 FIG. 2 .
Fig.2gives the 2D electron density and the density of states as function of the chemical potential for Hamiltonian of the previous Section.

FIG. 3 .
FIG.3.The critical temperature Tc as function of the hydrostatic pressure P with (red points) and without (blue points) the Coulomb electron-electron interaction.The dependence has spikes near the points of topological transformations of the electronic system.Position of spikes coinsides with that of the density of states (the green curve).