Type II Nodal line Semimetal

Recently, topological semimetals become hot topic in condensed matter physics, including Dirac semimetal, Weyl semimetal, and nodal line semimetal (NLSM). In this paper, a new type of node- line semimetal - type-II NLSM is proposed based on a two-band cubic lattice model. For type-II NLSM, the zero energy bulk states have a closed loop in momentum space but the (local) Weyl cones on nodal line become tilted. The effect of magnetic field and that of correlation on type-II NLSM are studied. In particular, after considering repulsive interaction and additional spin degrees of freedom, different types of long range magnetic orders appear in bulk states. In addition, the interaction-induced ferromagnetic order of surface states may exist. At critical point between type-I NLSM and type-II NLSM, arbitrary tiny interaction induces ferromagnetic order due to a flat band at Fermi surface.


I. INTRODUCTION
Recently, topological semimetals have attracted considerable eyes of researchers. Compared to topological insulator, topological semimetals have gapless bulk states and topologically protected surface Fermi arc states. There exist different types of topological semimetals, such as Dirac semimetal (DSM) [1,2], Weyl semimetal (WSM) [3][4][5][6], and nodal line semimetal (NLSM) [7][8][9][10]. WSM was proposed to separate a single Dirac node into two Weyl nodes by breaking either time reversal symmetry or inversion symmetry. The surface states of Weyl semimetal become Fermi arc between a pair of Weyl points with opposite chiralities. Morever, Weyl semimetals have been found in experiments such as TaAs family [11][12][13][14]. Nodal line semimetal is a threedimensional graphene-like system with low-energy relativistic excitations, but the band touches are closed loop in momentum space instead of points. The surface states of node-line semimetal have drumheadlike surface flat bands. The node-line semimetal is also realized in experiments (For example Ca 3 P 2 [15] and Cu 3 PdN [16]).
In addition, new types of WSMs are proposed which are called type-II Weyl semimetal [17] and Hybrid (type-1.5) Weyl semimetal [18,19]. In these types of WSMs, Lorentz invariance of low-energy excitations is broken. As a result, the nodes are tilted along given directions (see FIG.1(a) and (b)) and the transport properties become anisotropic. There are many remarkable phenomena appearing in type-II WSMs, such as the anisotropic negative magnetoresistance effect caused by Landau level collapsion [20,21] and the existence of tilted surface states [19]. In Hybrid (type-1.5) WSM, because the remaining symmetry (inversion symmetry, time reversal symmetry or mirror symmetry) of two nodes is broken, one Weyl node belongs to type-I and the other Weyl node belongs to type-II. These new types of WSMs attracted plenty of studies in past two years.
In this paper, based on a tight-binding model, we point out that there exists a new type of NLSM named type-II NLSM. For type-II NLSM, the zero energy bulk states have a closed loop in momentum space but the (local) Weyl cones on nodal line become tilted (see FIG.1(c) and (d)). In sec.II and sec.III, we introduce a two-band tightbinding model that describes type-II NLSM. In sec.IV, we study the effect of magnetic field on type-II NLSM and show the Landau level collapsion in this system. Next, we study the correlation effect on type-II NLSM and the interaction-induced magnetic order of NLSM is found in sec.V. An interesting result is at critical point between type-I NLSM and type-II NLSM, arbitrary tiny interaction induces ferromagnetic order (FM) due to a flat band at Fermi surface. Finally, we give the conclusion and propose an experimental realization in sec.VI. The electron and hole pockets touch, and the dispersions become anisotropic. (c) An illustration of low energy dispersion of type-I NLSM that has a closed loop in momentum space. The low energy effective excitation of every node on the nodalline is also linear. (d) An illustration of low energy dispersion of type-II NLSM that also has a closed loop in momentum space. Due to the tilted linear dispersion, the valence and conduction bands are asymmetry.

II. THE NODAL LINE HAMILTONIAN IN REAL SPACE ON CUBIC LATTICE
Firstly, we start with a nodal line semimetal from a three dimensional (3D) tight-binding Hamiltonian on cubic lattice that is given by where a = 1, 2 is the orbital degree of freedom.ĉ i,a is the annihilation operator of the electron at the site i with orbital degree of freedom. t x/y/z are the nearest neighbor hoppings in x/y/z direction, t ′ xz /t ′ yz are the orbital-flip hoppings in xoz/yoz plane. t xy /t z0 are the effective Zeeman field. k 0 determines the radius of the nodal line. k x0 = 0.4π, k y0 = 0.4π are to eliminate the Weyl points.δ 1/2/3 are the nearest vectors which are (a 0 , 0, 0) , (0, a 0 , 0) , (0, 0, a 0 ) ,b 1/2/3/4 are the next nearest vectors which are (a 0 , 0, a 0 ) , (a 0 , 0, −a 0 ) , (0, a 0 , a 0 ) , (0, a 0 , −a 0 ) . The lattice constant a 0 is set to be unit. It is obvious that not only the inversion symmetry but also the time-reversal symmetry are broken.
Using Fourier transformation, we obtain the Hamiltonian in momentum space with where Then, the spectrum for free fermions is derived as In the following parts of the paper, the hopping parameters are set to t x = t y = t z = t xy = t z0 = t. There is a drumhead inside the nodal line. The parameters are k0 = π/2, t ′ xz = t ′ yz = 0.5t.
Next, we study the nodal line of the nodal line semimetal. In k x -k y plane, the nodal line satisfy the equation of cos k x + cos k y = 1 + cos k 0 FIG.2(a) shows the spectrum at k z = 0. For this case, the nodal line locates at k x -k y plane with the the radius of k 0 = π/2, t ′ xz = t ′ yz = 0.5t. Additionally, we study the surface state of the nodal line semimetal. We  [16,22], and the fermi surface like a disk in the core of the BZ.

III. TYPE II NODAL LINE SEMIMETAL
In this part, a new type of NLSM named type-II NLSM is proposed. To get a typical type-II NLSM, we add a new term into the original model as C is a coefficient that determines the type of a NLSM. |C| = 1 is a critical point: when |C| < 1, the NLSM belongs to type-I nodal line SM; when |C| > 1, the NLSM belongs to type-II nodal line SM. At the critical point |C| = 1, NLSM has a flat band at Fermi surface as FIG.3 (h) (the red region). For the case of |C| > 1, one of the energy bands reverses. The sign of coefficient C denotes the tilting orientation: When C > 0, the tilting of the spectra towards to the center of the node-line, while away from the center when We can see clearly that the tilting of the nodal line towards to the center of the nodal line when C > 0; while away from the center when C < 0, (g)-(i) are Fermion surface of the bulk system for |C| = 0.6, 1.0 and 1.5. We then study the topological properties of Type-II nodal line SM. The topological protected surface state is a hallmark of topological system. In type-II nodal line semimetal, the surface states show similar behavior of the nodal states on bulk system -the surface states can also tilted and becomes 'type-II'. In tilted NLSM, the surface states are shown in FIG.4 which are top views from z axis for lowest two bands near Fermi surface. In FIG.4, the coefficient C > 0 for (a)-(c), while C < 0 for (d)-(f ). Due to the tilting effect for the case of C = 0, the drumhead-like surface flat band like FIG.2 (b) disappears and instead by a dispersive one. Thus, the surface states in NLSM can also be tilted like nodal line in bulk, which is similar as type-II Weyl semimetal [17].
We discuss the evolution of Fermi surface of lowest energy band of bulk states. In type-I NLSM with C = 0, the Fermi surface of bulk states is a circle at µ = 0 (here µ is the chemical potential). At the critical point |C| = 1, one band of NLSM becomes flat, which leads to a tilted surface state. While the Fermi surface of surface states is a disk at µ = 0 when C = 0. At the critical point |C| = 1, it becomes a flat band with a hole in the center like FIG.4 (h). In addition, we also calculate the density of states (DOS). The expression for calculating DOS is Here η is an infinite small quantity and real, ω is the energy level. After considering the tilting effect on the spectra, the DOS changes correspondingly. In FIG.5(a) there always exists a sharp peak at E = 0 due to the flat band states for type-II NLSM. In FIG.5(b), for k z = 0, owing to the existence of bulk flat band, there exists a sharp peak at E = 0 for the case of |C| = 1.

IV. EFFECT OF MAGNETIC FIELD ON TYPE-II NODAL LINE SEMIMETAL
In type-II Weyl semimetal, the negative magnetic effect (NME) becomes anisotropic. The failure of NME in the prescribed direction is caused by the collapsion of Landau level [21]. We now show that the collapsion of Landau level also appears in nodal line semimetal.
We add the magnetic field along x-direction, i.e., B = Bx and A = (0, Bz/2, −By/2) , then use the usual Peierls substitutions k x →k z − eBy/2, k y →k y + eBz/2. We introduce the ladder operators wherek x = −i ∂ x ,k y = −i ∂ y . These operators rise and fall the Landau levels of free electrons as and where |n, k x is the free electrons Landau level wavefunction. When an electron occupies the state |n, k x , it rounds in circles in y-z plane. The translation invariance along x-direction is preserved so that k x is still a good quantum number. We expanse Hamiltonian near nodal line and only keep first-order terms, considering the perturbation along radial (∆k R ) and tangential (∆k T ) directions of the nodal line. After a unitary transformation between two coordinates, we have ∆k x σ x + ∆k y σ y = ∆k T σ T + ∆k R σ R where σ T = σ x sin θ − σ y cos θ, σ R = σ x cos θ + σ y sin θ and k T = k x sin θ − k y cos θ, k R = k x cos θ + k y sin θ, and θ is the intersection angle with x-axis in x-y plane. Then, the Hamiltonian variation induced by the perturbation is which is independent of k T because of there is no dispersion along nodal line. As magnetic field is applied along x-direction, and A T (tangential directions of A) is irrelevant, we focus tangential component of magnetic field B sin θ. The corresponding Landau levels near the nodal line becomes where v 0 = 2k 0 , α = 1 − β 2 , β = C, e is elementary charge, is Planck constant. In type-I region, the zeroth level E = 0 is maintained; in type-II region |C| > 1, 1 − β 2 < 0, so that α is imaginary and the expression is invalid. This corresponds to collapsing of Landau levels mentioned in Ref. [21]. The zeroth Landau level also disappears. In FIG.6, we also give the numerical results with different tilting strengthes C. There are two flat bands near nodal line when |C| < 1, which correspond to zeroth Landau level. When |C| > 1, the flat bands disappears, and the system becomes metal which is similar to Weyl semimetal [17][18][19].

V. CORRELATION EFFECT ON TYPE-II NODAL LINE SEMIMETAL
In this part, we study the correlation effect on type-II NLSM by considering an on-site repulsive interaction.
Then the Hamiltonian is rewritten as where H 0,↑ and H 0,↓ are the Hamiltonians of Eq.(1) after considering the spin degree of freedom.n i,τ,a = c † i,τ,aĉ i,τ,a is the operator of particle number with two spin degrees of freedom τ and two orbital degrees of freedom a, U is the on-site Coulomb repulsive interaction strength and µ is the chemical potential. In the global phase diagram, there are two kinds of quantum phase transitions: one is the quantum phase transition between a long range ordered state and a phase without the long range order, the other is metalinsulator transition that is characterized by the condition of zero fermion's energy gaps.
Because the orbital SU(2) rotation symmetry is broken, when considering the repulsive interaction, magnetic order of spin degree of freedom may appears and the corresponding spin SU(2) rotation symmetry is spontaneously broken. By the mean field theory, the ferromagnetic (FM) order of spin degree of freedom for bulk states is denoted by where n is the number of particles, and we only consider the half-filling case for n = 1. τ = 1 represents spin up and τ = −1 represents spin down. M F is the FM order parameter of spin degree of freedom. We can write the self-consistent equations as After Fourier transformation, the self-consistent equations in momentum space can be rewritten as where θ (x) is a step-up function and θ (x) = 1 for x > 0 and θ (x) = 0 for x < 0, N is the number of the unit cells and with µ ef f = µ − U 2 . At the mean field level, we can also define other long range orders: the antiferromagnetic (AF) order of spin degree of freedom for bulk states where M AF is the AF order parameter of spin degree of freedom; the ferromagnetic (FM) order of orbital degree of freedom for bulk states where M ′ F is FM order parameter of orbital degree of freedom; the antiferromagnet (AF) order of orbital degree of freedom for bulk states where M ′ AF is AF order parameter of orbital degree of freedom. These numerical calculations are the same as the FM case of spin degree of freedom.
Then by using mean field approach, we obtain the global phase diagram for different NLSMs with different tilting strengthes C in FIG.7. In FIG.7,  quantum phase transition between a long range ordered state and a phase without the long range order, the other is metal-insulator transition that is characterized by the condition of zero fermion's energy gaps.
In the global phase diagram, a remarkable result is about the magnetic phase transition at C = 1. For the case of C = 1, there exists a flat band Fermi surface (See FIG.3(h)). As a result, a very tiny repulsive interaction will induce an FM order of spin degree of freedom (See the result in FIG.7). In FIG.8, we also plot the magnetization, the energy gap and the ground state energy via the repulsive interaction for the cases C = 0.6, C = 1.0 and C = 1.7, respectively. The first, second and third rows represent magnetization, the energy gap and the ground state energy respectively. Different columns represent different tilting strengthes. In these figures, we use different colored lines represent different phases, like blue line represents nodal line SM-FM, red line represents Spin AFM-M, cyan line represents Spin AFM-I, green line represents Orbital Ferrimagnetic-I and magenta line represents nodal line FM-I. We use black dotted lines to distinguish different magnetic order phases.
Next, we consider the correlated effect on surface states and show the interaction-induced surface orders in the NLSMs. Because the orbital SU(2) rotation symmetry is broken and the antiferromagnetic order of spin degree of freedom for surface states is not well defined, we focus on ferromagnetic order of spin degree of freedom for surface states.
Because the nodal line locates at k x -k y plane, we con-   sider a system with periodic boundary conditions (PBC) along x and y-direction, but open boundary conditions (OBC) along z-direction. Now, due to SU(2) spin rotation symmetry, the ansatz of FM order of spin degree of freedom is the same as Eq. (16). Along z-direction, the system have 10 lattice site like Fig.9. Because there is no translation symmetry along z-direction, we must calculate the mean field ansatz of FM order site-by-site. After considering inverse symmetry, there are five different cases to calculate. In Fig.10  Beyond the critical tilting point C = 1, one of the energy bands of surface states reverses. See Fig.12. For different tilting strengthes, with the increase of interaction, the shape of Fermi surface for surface states changes, and finally the system becomes an insulator.

VI. CONCLUSION
In this paper, we pointed out that there exists a new type of node-line semimetal -type-II NLSM based on a two-band cubic lattice model. We studied the effect of magnetic field on type-II NLSM and found the Landau level collapsion in this system. After considering repulsive interaction and additional spin degree of freedom, different magnetic orders appear in the bulk states and ferromagnetic order exist in surface states. At critical point between type-I NLSM and type-II NLSM, arbitrary tiny interaction induces ferromagnetic order due to a flat band at Fermi surface.
Finally, we propose an experimental setup to realize the NLSM on optical lattice. The model discussed in this paper includes complex-valued nearest and next nearest neighbor hopping in cubic lattice. Hopefully this can be realized in a three-dimensional optical lattice with two components of Fermi atoms such as 6 Li and 40 K. The real-valued hopping can be induced by kinetic which could be tuned by change the potential depth and the imaginary-valued hopping could be induced by a twophoton Raman process or shaking lattice. Similar system in one dimension and two dimensions had been realized recently [23,24].