Weyl Mott Insulator

Relativistic Weyl fermion (WF) often appears in the band structure of three dimensional magnetic materials and acts as a source or sink of the Berry curvature, i.e., the (anti-)monopole. It has been believed that the WFs are stable due to their topological indices except when two Weyl fermions of opposite chiralities annihilate pairwise. Here, we theoretically show for a model including the electron-electron interaction that the Mott gap opens for each WF without violating the topological stability, leading to a topological Mott insulator dubbed Weyl Mott insulator (WMI). This WMI is characterized by several novel features such as (i) energy gaps in the angle-resolved photo-emission spectroscopy (ARPES) and the optical conductivity, (ii) the nonvanishing Hall conductance, and (iii) the Fermi arc on the surface with the penetration depth diverging as approaching to the momentum at which the Weyl point is projected. Experimental detection of the WMI by distinguishing from conventional Mott insulators is discussed with possible relevance to pyrochlore iridates.

where σ are Pauli matrices acting on spin degrees of freedom, n k is the density operator n k = ψ † k ψ k , and U is the magnitude of the repulsive interaction. The repulsive interaction is infiniteranged in the real space, which can be represented in a local way in the momentum space.
Thanks to the locality in the momentum space, the Green's function can be exactly computed for this Hamiltonian as follows.

First we perform a unitary transformation
that diagonalizes the single-particle part of the Hamiltonian as Then the Green's function is transformed as where T τ denotes the time ordering and c kα (τ ) = e τ H c kα e −τ H .
In this basis, the Hamiltonian is diagonalized as where |0 is the vacuum state and h(k) = |h(k)|. Thus the expectation value is written as In the right hand side of the equation for b ka (τ )b † ka , the forth term vanishes and other terms are nonzero when a = a . Thus we obtain Therefore, the imaginary-time Green's function is given by (S15) A. Green's function for T = 0 In the zero temperature (β → ∞), the above Green' function reduces to (S17) In the original basis, the Green's function is given by (S18)

SII. FERMI ARC
In this section, we study the Fermi arc in the WMIs. Nonvanishing topological indices for the WMIs indicate that the Fermi arc remains in the WMIs, which we verify in the following. Since our model [Eq. (S1)] is diagonalized at each k-point and hence we can consider an effective two-dimensional model for each k z -sector when the system is periodic in the z-direction. Because of the bulk-boundary correspondence, we expect that "edge channels" for each k z form a Fermi arc. More explicitly, one can obtain the surface bound state from the effective Hamiltonian by replacing the momenta k x , k y with the derivatives −i∂ x , −i∂ y . Away from the plane k z = ±k 0z , the surface state is almost unchanged from the noninteracting case. The nontrivial issue is how the surface state behaves as k z approaches ±k 0z . Specifically, the problem is whether the penetration depth of the surface states diverges or not with k z → ±k 0z .
Intuitively, the finite gap U indicates that the length scale ξ remains finite, i.e., ξ ∼ = v F /U .
However, it turns out not when one studies the effective Hamiltonian in Eq. (S19) and the asymptotic behavior of the surface bound state as |x| → ∞ (here we assume k y = 0) by tentatively taking the limit of |k x | |k z |. In this limit, H eff ∼ = (v F + U 2 |k z | −1 )[−i∂ x σ 1 +k z σ 3 ], which indicates that the penetration depth diverges with ξ = |k z | −1 . In any case, the length scale is determined by |k z | −1 even when we take into account of the higher orders in ∂ x .
Therefore, the surface bound states penetrate into the bulk as k z approaches to ±k 0z .

SIII. OPTICAL CONDUCTIVITY
We study the optical conductivity σ(ω) for a single WF described h(k) = v F k. In the following, we set the Fermi velocity v F = 1, which can be always restored by the dimension analysis.

A. Matrix elements
Here we calculate matrix elements that we will need in evaluation of conductivities, i.e., ±|σ i |± . We first parameterize the direction of the momentum as n = (sin θ cos φ, sin θ cos φ, cos θ). (S20) Then the wave functions that diagonalize the Hamiltonian are written as The matrix elements are given by In the evaluation of the optical conductivity, we need In the evaluation of the Hall conductivity as a function of k z , we need We first focus on the conductivity σ(ω) for the zero temperature. The Green's function is given by with n = k/|k|. The optical conductivity is given by for a > 0, b > 0, we perform the summation over iω m for the above equation and obtain After the analytic continuation iΩ → ω + i , only the pole at k = ω − U 2 contributes to the imaginary part of the k-integral. Thus, we obtain where we used the formula Im 1 k−a−i = πδ(a). Hence, the optical conductivity for the zero temperature is given by By restoring the unit of e 2 / and the Fermi velocity v F , we end up with

Poles of σ(q, ω)
Let us study the locus of the poles of the two-particle correlation function that contribute to the conductivity σ(q, ω) for nonzero q. From Eq. (S35) and setting a = |k| + U 2 and b = |k + q| + U 2 , the poles of iωm tr[G(k, iω m )σ x G(k + q, iω m + iΩ)σ x ] can be read off as ω = a + b = |k| + |k + q| + U . By using the formula |k| + |k + q| ≥ |q| and restoring the Fermi velocity v F , the lower bound of the poles is given by (S40)

C. Finite temperature
In this section, we calculate the optical conductivity σ(ω) in the finite temperature. In doing so, we consider contributions from interband and intraband transitions separately as

Interband transition
The interband contribution to Q(iΩ) is given by where This is reduced to After the analytic continuation iΩ → ω + i , poles that contribute to the imaginary part of the k-integral are Thus the interband contribution to the optical conductivity at the finite temperature is given by The minus sign for the term in the last line arises because the pole k = U −ω−i 2 locates in the lower half plane while other poles locate in the upper half plane.

Intraband transition
The intraband contribution to Q(iΩ) is given by where and After performing an analytic continuation iΩ m → ω + i and taking a limit q → 0, we obtain the intraband contribution to the optical conductivity We note that we used the equation n F ( ) = n F (− ) in the first term, and the forth term in Eq. (S49) can be discarded after analytic continuation because of a factor δ(ω + U ).
We show the temperature dependence of weights of peaks at ω = 0 and U in Fig. S1.

D. Temperature dependence of Drude weight
We study the behavior of the Drude weight in the limit T → 0. The Drude weight is given by the coefficient of δ(ω) in Eq. (S50) as In the noninteracting case (U = 0), the Drude weight behaves as W Drude ∝ T 2 . This is obtained from a crude estimation by replacing the factor e −βk +e βk (e −βk +e βk +2) 2 in the integrand with 1 for k < T and with 0 otherwise. On the other hand, in the case of strong interactions (U → ∞), the Drude weight behaves as W Drude ∝ e −β U 2 T 2 . Thus the Drude weight is suppressed exponentially as the interaction U increases.

E. Hall conductivity
We study the Hall conductivity for a fixed value of k z . The Hall conductivity has a nonzero contribution from a combination Here, we set the momentum transfer as q = 0 because we focus on the dc Hall conductivity.
We note that other combinations of current matrices vanish. After we integrate over the direction φ of (k x , k y ) = k (cos φ, sin φ) in current matrices [Eq. (S30)], the expectation value is given by With the Fourier transformation, we obtain By performing analytic continuation and taking the zero frequency limit, we obtain the Hall conductivity σ xy (k z ) = 1 (2π) 2 k dk Re( where k is the radial coordinate for (k x , k y ) and k = k + k 2 z . In the zero temperature limit, the Hall conductivity reduces to If we restore the unit of e 2 / , the Hall conductivity is given by which remains quantized into ±e 2 /2h in the WMI.

SIV. STABILITY OF THE MOTT GAP
We study the stability of the Mott gap against the interaction We consider the self-energy arising in the second order of this interaction, with the density-density correlation function This is explicitly written as If the instability for the Mott gap were present, the gap should close at k = 0 by the consideration from the rotation symmetry. Therefore, we focus on the self-energy for k = 0.
By summing over Matsubara frequencies and setting k = 0, we obtain After performing an integration over k , the terms |k − q| and k · (k − q) no longer have a dependence on the angle of q, because they only depend on the relative angle between k and q. Then the only term depending on the angle of q after the k integration is n q · σ, which vanishes upon the integration over the angle of q. Thus the self-energy Σ(k = 0, iω) is diagonal with respect to the spin degrees of freedom. Furthermore, the imaginary part of Σ(k, ω) (after the analytic continuation) appears only at ω = |k | + |k − q| + |q| + 3U 2 ≥ 3U and ω = |k | + |k − q| − |q| + U 2 ≥ U 2 ; The imaginary part of Σ(k = 0, ω) is zero for ω < U 2 . Therefore, the gap of U 2 in the Green's function is stable against the inclusion of the interaction H C .
We note that the the perturbation theory with respect to V (q) is valid because of the absence of the infrared divergence. In the case of the contact quartic interaction V (q) = V , we notice that the infrared divergence does not appear for iω = 0 because of the gap of U 2 in the energy denominator. In the case of the repulsive Coulomb interaction V (q) = 4πe 2 q 2 , the infrared divergence is also absent, because the density-density correlation function behaves Π(q, iΩ) ∝ q 2 for small q and Ω, and the integral is convergent around q = 0.