Phase diagram of three dimensional disordered nodal-line semimetals: weak localization to Anderson localization

Nodal-line semimetals are new members of the topological materials family whose experimental characterization has seen recent progress using both ARPES and quantum oscillation measurements. Here, we theoretically study the presence of a disorder-induced phase transition in a cubic lattice nodal-line semimetal using numerical diagonalization and spectral calculations. In contrast to the 3D nodal-point semimetals, we found that nodal-line semimetals do not display a stable disordered semimetal phase, as an infinitely weak disorder can lead to a diffusive metal phase. The absence of a semimetal phase is also reflected in the quadratic relationship of the electronic specific heat at low temperatures. Furthermore, we illustrate that a localization transition occurs under the influence of strong disorder, shifting the material from a weakly localized diffusive metal state to an Anderson insulator. This transition is substantiated by calculating the adjacent gap ratio and the typical density of states.


Introduction
During the last decades, research on topological phase has become a blossoming branch of solid state physics [1][2][3][4][5].Depending on whether the bulk states are gapped or gapless, topological materials can be generally divided into topological insulators [5,6]and topological semimetal [7,8].In contrast to normal metals, topological semimetals support a gapless phase in which the conduction and valence bands exhibit a linear dispersion relationship near their contact points within the Brillouin zone (BZ).As a result, low-energy excitations can be accurately described by a massless Dirac equation.Materials with this exotic phase exhibit unique properties, such as surface states protected by topology [7][8][9][10] and the 3D quantum hall effect [11][12][13][14].In particular, when the Fermi level is placed at the convergence of the two bands, the density of states (DOS) vanishes, but the spectrum is gapless.Consequently, they are called topological semimetals because their behavior is intermediate between metals and insulators.The paradigmatic examples of the topologically nontrivial gapless band structure are Dirac and Weyl semimetals [7][8][9][10][15][16][17][18], which have attracted considerable attention due to the unique surface states and the chiral transport.Because of the ubiquitous presence of disorder in real solid state materials, a precise understanding of the disorder effects is a substantial demand that has been in progress since the mid-1980 s .When impurities are added to a normal metal, the electrons are diffused under the scattering of the impurities, leading to the diffused metal phase (DM) [19,47].Anderson localization(AL) [48] arises as the impurity strength reaches a level where electrons can no longer propagate.In semimetals with linear band, the quadratically vanishing DOS at zero energy (ρ(E) ∼ E 2 ) and the linear three-dimensional energy band dispersion yields the electron self-energy Σ(E, k → 0) ∼ E 2 which is negligible compared to E around zero energy.Therefore, in 3D point-node semimetals, the vanishing DOS is maintained at finite disorder strength so the electrons can move without scattering and the system is in the semimetal phase (SM).When the disorder exceeds a certain value, a quantum phase transition occurs, beyond which the DOS is finite and the system is in the diffusive metal phase.The average DOS (ADOS) at zero energy acts as an order parameter for describing this transition [22,43,49].At a higher disorder strength, the 3D point-node semimetals display an Anderson localization which can be captured by the typical DOS (TDOS) [42,50,51] that probes its extended or localized nature.In contrast to the 3D case, mean-field and perturbative field theoretical calculations have shown that d = 2 is the lower critical dimension of the SM-DM transition and 2D point-node semimetals do not have the perturbatively irrelevant SM phase [40][41][42][43][44][45][46]52].In fact, an infinitesimal amount of disorder leads to AL in 2D semimetals [42] which is consistent with Anderson localization theory.Besides, the non-perturbative rare region effect would induce avoided quantum criticality in disordered 3D Dirac and Weyl semimetals [25,[53][54][55], thus the quantum critical point (QCP) does not exist.However, this effect is outside the scope of our discussion in this paper.
Recently, nodal-line semimetal (NLS) has reinvigorated research interests in topological semimetals [7,.The crossings of the conduction band and the valance band form lines or closed loops whose ends meet at the BZ boundary or wind into closed loops inside the BZ, with each loop carrying a π Berry flux.Moreover, it has unique surface states whose zero energy forms nearly a flat band called drumhead-like surface states (DSS) [59,85,86].Subsequently, efforts were made to detect the DSSs and various candidates for NLS have been reported [57][58][59][60][61][62][63][64][65][66][87][88][89][90][91][92].These experiments have now raised the exciting prospect of studying the physics of the disorder effect.Theoretically, the effects of the disorder have been studied in several works under certain impurity structures [93,94].Nonetheless, it is still an important open question to figure out how many distinct quantum phases indeed exist in a dirty 3D NLS system with chiral symmetry.
In this paper, we study the disorder effect in 3D NLS numerically.By using the kernel polynomial method (KPM) [95], we calculate the ADOS and the TDOS on a sufficiently large cubic lattice with different disorder distributions [42].We show that the DOS becomes finite for an infinitesimally weak disorder strength, which agrees with the theoretical predictions [7,94].In contrast to a previous study which claims that there exists a multifractal semimetal phase [93] , this result indicates that unlike point-node semimetals, the SM phase in NLS is unstable under the disorder which preserves chiral symmetry.The NLS will go into the DM phase and acts like the normal metal even with an infinitesimally weak disorder.Moreover, we calculated the electron specific heat scaling with temperatures by taking different disorder strength in the DM phase, which also proves that the QCP of SM and DM is W = 0. Furthermore, the adjacent gap ratio ⟨r⟩ has been computed, and the ratio's mean value is ⟨r⟩ ≃ 0.53, which indicates that the DM phase of NLS belongs to the orthogonal class and possesses the weak localization effect.At a higher disorder strength, the DM phase will turn into the AL phase.The results of the typical DOS ρ t and the adjacent gap ratio ⟨r⟩ consistently indicates that the phase transition critical disorder strength from DM to AL phase is W l /t ≃ 9.2.
The remainder of the paper is organized as follows.The model Hamiltonian and the proposed method are introduced in section 2 and the numerical results and discussion on the behavior of the DOS are presented in section 3. A summary and discussion are given in section 4.

Model and method
We consider the following Hamiltonian on a cubic lattice as where c r is a two-component spinor composed of an electron at site r.µ = e x , e y , e z are the unit vectors along x, y, z directions.σ x , σ z are the Pauli matrix representing the orbital degrees of freedom.t is the hopping strength, and m controls the major radius of the nodal line.In the absence of disorder and set t = 1, we obtain the full Hamiltonian of the NLS model by Fourier transform of the bulk lattice-space Hamiltonian to momentum-space : The dispersion relation can be easily obtained as: Setting k x = 0 and m = 1.5, the conduction and the valence band degenerate at a line forming a nodal loop as shown in figure 1(a).In all the text, we will take t = 1 and m = 1.5 without loss of generality.In the low energy limit, the Fermi energy forms a torus-shaped Fermi surface that encircles the nodal line.The area enclosed by the projection of the nodal loop on the surface BZ forms DSS which manifests its topological properties.
The nodal-line semimetal possesses an antiunitary symmetry Γ = σ z K with K as the complex conjugation, such that According to symmetry classification, Γ is a spinless time-reversal symmetry and belongs to the orthogonal class [96,97] which will be used in section 3.2.The DOS can be directly evaluated from the imaginary part of the retarded Green function as where G(k) = ⟨k| 1 E−H−iδ |k⟩ is the Green function without disorder.Substitute the dispersion relationship in equation ( 5) and finish the integral numerically we can get DOS ∼ |E|.To study the effect of the disorder,the Hanmiltonian becomes We consider two types of the onsite random scalar potential V(r) ,the Gaussian distribution with zero mean and variance W 2 and the uniform distribution randomly distributed between [−W, W] which are widely used in the calculation of impurities induced phase transitions.As shown below, our aim is to obtain the ADOS where i is the lattice index, and the TDOS [42,95] where i is the lattice index and N is the number of the randomly picked lattice sites.The ⟨. ..⟩ denotes a disorder average.By definition, ADOS is the DOS after averaging over different disorder realizations.TDOS is the product of a series of local densities of states.TDOS is equal to 0 when the DOS at some lattice sites is zero, indicating that electrons move only at some lattice sites and the system is localized.Thanks to the numerically exact KPM [95], we are capable to calculate the two DOS at sufficiently large sizes(L = 10 2 , which is difficult to diagonalize in 3D systems).We have used KPM expansion order N c = 512 for the results presented on the paper so that the DOS is smooth (see the appendix for the numerical details).

Absence of SM
An important feature of SM is the vanishing DOS at the band touching energy in weak disorder strength.The transport of low energy excitations is ballistic without scattering which has been discovered in Weyl and Dirac semimetals.Unfortunately, this phase is absent in NLS.As shown in figures 1(b) and (c), the scaling relation ρ ∼ |E| n changes with disorder strength.In the clean limit, the DOS of the NLS is characterized by a linear relation DOS ∼ |E| that vanishes at zero energy.Tuning the model away from the clean limit by varying the disorder strength, the DOS at zero energy ρ(0) becomes finite immediately in both two types of disorder distributions, which manifests the sensitivity of the NLS to chiral disorder.For sufficiently large system sizes, ρ(0) converges in L (see appendix B).In order to understand the difference between the point-node semimetal and the NLS explicitly, we compare our results with the case of the Weyl system in [42,43].A point-node system begins with a quadratic relation ρ ∼ E 2 in the clean limit.As W approaches the transition point W c , the ADOS sharpens up and has a scaling formula ρ(E) ∼ |E| d/z−1 with z = 1.5 the dynamical exponent [43].This leads ρ(E) ∼ |E| at W c , before which the ρ(0) vanishes and the system is in the semimetal phase [42,43].But in the NLS (figure 1(b)), the DOS begins its variation with a linear relation n = 1, displaying an apparent absence of the disorder irrelevant region from n = 2 to n = 1.This is consistent with the transfer-matrix calculation and self-consistent Born study which suggest that an infinitesimally small disorder drives the nodal line Dirac semimetal in the clean limit to the metal [7,94].It is worth noting that our results are quite different from [93], which is attributed to the different type of disorder which makes our system belong to different symmetry classes, and the specific reason is worth further exploration.Now that we have calculated the DOS, it is straightforward to obtain the electron specific heat C v (T) = ∂<E> ∂T (where < E > is the free energy of the system) by numerically integrating the DOS using the following formula [43] From the standard textbook [98], we know that electron specific heat C v scales as C v ∼ T and ρ(E) ∼ const in the DM phase.For a point-node semimetal, the linear dispersion relation gives ρ(E) ∼ |E| d−1 in the clean limit where d is the dimension of the system.Substituting it into equation ( 9), we can get ) ( e ) .(10) To use the special integral where n is positive integer,we set x = E/T.Finishing the integral we can get electron specific heat C v scales as C v ∼ T d in the SM phase.In contrast to them, the electron specific heat of the 3D NLS has a non-Fermi-liquid behavior C v ∼ T 2 in the clean limit as shown in figure 2(a).It is interesting to mention that at the transition point of the disordered point-node semimetal, the well-established one-parameter scaling theory also gives [43,49] C v ∼ T 2 in the vicinity of the QCP W c where ρ(E) ∼ |E|.Here in our NLS system, we have the quadratic relation in the clean limit.With the increase of W, the scaling approaches linear at low temperatures which characterizes the appearance of DM phase as shown in figure 2(b).We performed a linear fitting on the low-temperature part and found that, at low temperatures, the system has the same linear relation as that of the metal.Above a certain critical temperature, this linear relation no longer holds.The range of the linear relation grows larger with the increase of W. This is reasonable since the curve ρ(E) − E becomes flat with the increase of W, as shown in figure 1(b).Note that the picture we depicted above for NLS with disorder is similar to the point-node semimetal beyond the QCP W c where the system starts the DM phase.Given the theoretic co-dimension 2 = 3 − 1 within K theory analysis [7,99], it seems reasonable to compare the NLS with the 2D point-node which has the same co-dimension 2 = 2 − 0. As a 3D semimetal, we find that NLS behaves more like a 2D point-node semimetal which also possesses.ρ(E) ∼ |E| relation [43] and obeys the C v ∼ T 2 relation in the clean limit.However, they are completely different in the localization phase.An infinitesimal amount of disorder leads to an AL in the 2D point-node system [42].In contrast, NLS goes through an interesting process, as we will show below.

Weak localization to Anderson localization
To further understand the localization effects, let us first discuss the properties and the difference between the TDOS and the ADOS.From the definition, we can see that the ADOS captures all the extended and localized states.On the contrary, only the extended states contribute to the TDOS because of the product relation.Consequently, the ratio M edge = ρ t (E)/ρ a (E) manifests the location of the mobility edge separating the DM and AL phases [42].As shown in figure 3(a), ρ t (0) and ρ a (0) increase concomitantly at low W because it is the disorder that generates the DOS in the vicinity of E = 0.The mismatch of the ADOS and the TDOS indicates the appearance of the localized states.For larger values of disorder, the ρ t (0) goes to zero around (W l /t ≈ 9.2) and the DM phase undergoes an Anderson localization.
To estimate the localization transition and compare our KPM results with the well-established classification of the diffusive metal phase, we compute the adjacent gap ratio [100][101][102] by using where δ n = E n − E n−1 is the spacing of neighboring eigenvalues E n .We diagonalize the Hamiltonian for smaller system sizes L = 10, 11, 12, 13, 14 and Gaussian distribution disorder.For each L, we averaged over 1000 disorder realizations.Since we are interested in E ∼ 0, we only focused on 1/20 of the eigenstates centered around the middle of the band.The one-parameter finite-size scaling scenario for the proposed diffusive-to-insulating phase transition would have these traces of different L all cross at W l .From this data, we estimate that W l /t ≈ 9.2 is in agreement with the results in figure 3(a).For the uncorrelated Poisson spectrum, the probability distribution of this ratio r is P(r) = 2/(1 + r) 2 so its mean value is ⟨r⟩ = 2 ln 2 ≈ 0.386 [102].According to the classification of the ensembles of random matrix, there are three symmetry classes with different ⟨r⟩ which are unitary at ⟨r⟩ ≈ 0.602, orthogonal at ⟨r⟩ ≈ 0.536 and symplectic at ⟨r⟩ ≈ 0.676 [100].The symmetry of the system directly determines the weak localization or anti-localization of the DM.Our generalized line-node model has time-reversal symmetry and spin-rotational symmetry, so it belongs to the orthogonal class where weak localization is expected.The numerically determined ⟨r⟩ is shown in figure 3(b), its mean value is 0.53 for the weak disorder, which is consistent with the theoretical prediction of weak localization in short-ranged disordered line-node semimetal [96].

Conclusion
To conclude, we have shown that the SM phase in line-node semimetal is unstable against small disorder.We have also calculated the electron-specific heat and shown that it has a scaling behavior C v ∼ T 2 in clean limit.Away from the clean limit, the nodal-line system behaves as a diffusive metal that can undergo weak localization to Anderson localization, we have supplemented this with an explicit adjacent gap ratio calculation.Besides, in real nodal-line semimetals, 3D weak localization will dominate the transport, which is consistent with the theoretical prediction in our paper.More precise calculations appear to be made on the long-range disorder and the weak antilocalization it may cause.The relation between the phase transition of semimetal and the appearance of exceptional lines in non-hermitian physics [103] is also an open question to study.

Figure 1 .
Figure 1.(a) is the band structure of the line-node model (1) at kx = 0 with periodic boundary conditions.(b) and (c) show the ADOS of NLS calculated by KPM.In(b) we plot ρ(E) versus energy E of linear size L = 100 at KPM expansion order Nc = 512 with periodical boundary conditions and Gaussian distribution.The arrow indicating increasing disorder strength from 0 to 3.0t for E close to zero energy.100 disorder realizations were used for each value W of the disorder.(c) is the ADOS at zero energy ρ(0) as a function of disorder strength in two typies of disorder distributions, which clearly shows that there is no SM phase and ρ(0) is finite even for an infinitesimal amount of disorder in line-node compared to the point-node obtained in[43].

Figure 2 .
Figure 2. Numerically (KPM) calculated Cv as a function of T for a system size L = 60.(a) shows in the clean limit, Cv − T 2 relation.(b) is the Cv − T relation with the change of disorder strength from W = 0 to W = 2 in Gaussian disbution which shows Cv ∼ T 2 in the clean limit and Cv ∼ T in the DM over some range of T. The dashed lines are the fits for low temperatures which depict Cv ∼ T. The critical temperatures at which the fits no longer apply increase with W.

Figure 3 .
Figure 3. Phase diagram of three dimensional NLS.(a) shows ρa(0) and ρt(0) as functions of disorder strength for L = 60 and Gaussian distribution.The TDOS and the ADOS become finite when the disorder turned on.The division between them shows the existence of localized states.Finally, ρt(0) goes to zero at W l ≈ 9.2, signaling the Anderson localization transition.(b) is ⟨r⟩ as a function of W for different system sizes, showing that the transition from orthogonal to Poisson becomes sharper as L increases and all the curves cross at W l /t ≈ 9.2.(c) is the M edge as a function of energy and disorder strength for L = 60.The blue and the black regions stand for metallic and localized states, and the blank region is in the band gap, which clearly shows the existence of the energy dependent mobility edge.

Figure 4 .
Figure 4. Convergence of ADOS and TDOS.(a) and (c) show the low energy dependence of ρ(E, Nc) in Uniform and Gaussian distribution respectively for various Nc with L = 100 and W/t = 2.0, showing that there are bigger deviations in the ADOS for larger Nc.The low energy peak is a result of the line-node states.(b) shows Nc dependence of ρ(0, L, Nc) converges in Nc for Nc ⩾ 512.(d) is the Anderson localization QCP from the TDOS as a function of the Chebyshev expansion order Nc for L = 30 in Gaussian distribution.We find the localization transition in three dimensions is converging to W l /t ≈ 9.2.

Figure 5 .
Figure 5. DOS at zero energy ρ(0) as a function of the inverse system size 1/L up to L = 120 on a log-log scale.Results for the our NLS model(left) and a Weyl semimetal (right).As the impurity strength increases from 0 to 1.0.

Figure 6 .
Figure 6.DOS at zero energy ρ(0) as a function of the impurity strength W. Results for the our NLS model(left) and a Weyl semimetal (right).Insert shows the result at low W for comparison.