Artificial electric field and electron hydrodynamics

In the electron dynamics in quantum matter, the Berry curvature of the electronic wave function provides the artificial magnetic field (AMF) in momentum space, which leads to non-trivial contributions to transport coefficients. It is known that in the presence of electron-electron and/or electron-phonon interactions, there is an extra contribution to the electron dynamics due to the artificial electric field (AEF) in momentum space. In this work, we construct hydrodynamic equations for the electrons in time-reversal invariant but inversion-breaking systems and find the novel hydrodynamic coefficients related to the AEF. Furthermore, we investigate the novel linear and non-linear transport coefficients in presence of the AEF.


I. INTRODUCTION
Transport properties of electrons in quantum matter reflect the nature of the quasi-particle interactions and possible quantum interference effects. The Berry curvature of the electron wave function is the prominent example of the quantum correction to the semi-classical equation of motion of electrons. It stems from a topological property of the electronic wave function in momentum space. Various unusual linear and non-linear transport coefficients have been discussed as the Berry curvature effect in the quasi-particle dynamics, which enters as an artificial magnetic field (AMF) in momentum space [1,2]. Recently, the effect of the AMF on the electron hydrodynamic equations for time-reversal invariant but inversionsymmetry breaking systems is studied in great detail [3]. For example, it is pointed out that the Poiseuille flow [4] , is modified in a non-trivial way. Such effects would be of great interest to both high energy and condensed matter physics [5].
In the electron hydrodynamics [6], it is assumed that the electron-electron scattering rate 1/τ ee is much greater than other scattering rates such as the electron-phonon 1/τ ep and electron-impurity 1/τ imp scattering rates. The strong electron-electron scattering establishes local equilibrium so that local temperature and chemical potential are well-defined. It is generally hard to achieve this regime in real materials, where typically 1/τ ep (1/τ imp ) dominates the high (low) temperature regime. Strongly interacting electrons in ultra-pure systems, however, may offer such a hydrodynamic regime, where a window of temperature exists for 1/τ ee 1/τ ep , 1/τ imp . Much attention has been paid to graphene, PdCoO 2 , and WP 2 as possible candidate materials [7][8][9][10].
In the presence of interactions, it has been known that the semi-classical electron dynamics is affected by the artificial electric field (AEF), which may be regarded as a generalized Berry phase effect in frequency-momentum space [11]. In addition to the effect of AMF on transport coefficients, we then have to consider the influence of AEF on the electron transport. In the hydrodynamic regime, the momentum relaxation rate is small by definition and it may be considered in the Boltzmann equation via a phenomenological parameter 1/τ rm . For example, the small momentum relaxation of electrons may occur due to the weak electron-phonon interactions, which may also be a source of the AEF. [12] In this work, we investigate the electron hydrodynamics by taking into account both AMF and AEF on equal footing. For concreteness, we consider the systems, where time-reversal symmetry is preserved, but the inversion symmetry is broken. We demonstrate that the AEF provides unexpected novel transport and hydrodynamic coefficients. Some explicit examples of the AEF effects on transport and electron hydrodynamics are shown.
The rest of the paper is organized as follows. In section II, we derive the hydrodynamic equations from the equation of motion and the Boltzmann equation by taking into account both AMF and AEF. In section III, an explicit example of the AEF effect in the presence of a weak electron-phonon interaction is shown and the corresponding transport coefficients are computed. In section IV, we show that the Poiseuille flow becomes fully threedimensional in the presence of the AEF.

II. HYDRODYNAMIC EQUATION WITH AEF
In this section, we investigate the contribution of AEF in the Boltzmann equation, and its consequences in hydrodynamic coefficients. To do so, we start with the Boltzmann equation in relaxation time approximation. We construct the constitutive relations for stress tensor and momentum to find the hydrodynamic equation for hydrodynamic velocity variable u. Finally, we find the transport current expressed in terms of hydrodynamic variables and investigate the transport coefficients in a spacially uniform solution.

A. AEF and equation of motion
To derive the hydrodynamic equations, we start with semi-classical equations of motions and Boltzmann equation. Both AEF and AMF can be incorporated in the equation of motion as follows. [11] Here, E is the external electric field, n is the band index and v n = ∂ n (p) ∂ p , where n (p) is the energy dispersion. Ω and E are the AMF and AEF respectively. For Abeline gauge field A α µ = u α | ∂ ∂kµ |u α where |u α is the Bloch wave function, AMF and AEF are defined by Ω α The Boltzmann equation that describes the evolution of electron distribution function is given by where f (t, r, p) is the electron distribution function and C[f ] is the collision term. By using Eq(1) and Eq(2) in Eq(3) one can find the contribution of AMF and AEF in the Boltzmann equation.

B. Derivation of hydrodynamic equation
In the following, we consider the systems, where their band structure near the Fermi level is constructed of several equivalent valleys with an isotropic parabolic dispersion with mass m [13]. To obtain the hydrodynamic equation for the total momentum, we need to multiply the equation by momentum and integrate over the momentum space. We consider the collision term where the first term is related to the collisions that conserve momentum, and the second term is related to the collisions that relax the momentum which we parametrize it with f (t,r,p) τmr in relaxation time approximation. By integrating over the momentum, the conserved momentum term vanishes, and we can find the following as a hydrodynamic equation. (see the appendix for more details).
where we can define momentum, stress tensor and density respectively P = [dp] pf , n = [dp]f .
When there are well defined local temperature and chemical potential, the distribution function f can be written as which is the Fermi-Dirac distribution function, and [dp] = When we are in the hydrodynamic regime, we can express these quantities in terms of hydrodynamic variables velocity u, chemical potential µ and temperature T . As a result, in a noncentrosymmetric metals, we find that the presence of the AEF and AMF leads to the following expressions of the momentum and stress tensor where ρ is the mass density, P is the pressure and the coefficients C il and G ijk are two anomalous coefficients. The C il is reported in [3] and G ijk is a novel transport coefficient which is related to the AEF as follows The G ijk is anti-symmetric under exchanging first two indices, G ijk = −G ijk . Also it is even under time-reversal ( G ijk = G ijk ) and odd under Inversion symmetry (G ijk = −G ijk ). It means, in a system that is invariant under both of these symmetries, G ijk vanishes.
Knowing the constitutive relation of hydrodynamic quantities, momentum density and stress tensor, we can find the hydrodynamic equation for u using Eq(5) Here the transport coefficients F il are D il are from AMF as reported in [3] and I ijk , G ijk are the novel transport coefficients resulting from AEF that have the following forms These coefficients are related to G ijk as G ijk ∼ ∂G ijk ∂µ and I ijk ∼ ∂G ijk ∂T so they have same symmetries as G ijk ; both G ijk and I ijk are anti-symmetric tensors under exchanging first two indexes and they are even under timereversal and odd under Inversion symmetry.

C. Transport current
One way to investigate system's response to external sources such as electric field E = Re[Ẽe iωt ] and ∇T , is to study the transport current J, which is known as [14][15] where M is an orbital magnetization. By expanding the terms in hydrodynamic variables we find the following expression for the transport current.
As an example, we can look at the uniform solution of Eq(12) and find the linear and non-linear transport coefficients in presences of external electric field and temperature gradient. We can define the transport current as i is the non-linear current. By considering the uniform solution of Eq(12) we can find the on-shell current, we can then find the linear and non-linear transport coefficients. We can write the current as J = J D + J anom , where J D is the standard Drude current, and the second term is the anomalous current. We show that in the presence of AEF, there is an additional contribution to J anom , which we define as J E . Other contributions to J anom coming from AMF is investigated in [3].
where we can define novel transport coefficients as follows.
All these transport coefficients correspond to nonlinear response.

III. EFFECT OF AEF ON THE TRANSPORT IN A TWO-DIMENSIONAL SYSTEM WITH ELECTRON-PHONON INTERACTION
In the following, we explain the origin of these transport coefficients and discuss the consequences by investigating an example. We consider a 2D Hamiltonian model, and we find the AEF due to the electron-phonon interaction. Finally, we investigate the AEF in this model, and find analytic expressions for new transport coefficients in a specific limit.
In the systems with electron-phonon interaction, strictly speaking, we need to consider another Boltzmann equation for phonon distribution function for selfconsistency. As mentioned in the [16], however, phonons in the hydrodynamic regime are much slower than electrons, so most of the contributions to the transport coefficients come from electrons. As a result, we will ignore changes in phonon distribution function and consider them at equilibrium.

A. Free Hamiltonian
We consider a 2D system that has two valleys located at finite momentum positions, K and K . The low-energy effective Hamiltonian near these points is given by [17]: where α = ± is related to the valley index. The dispersion relation for this model can be written as where k = k 2 x + k 2 y , λ(k) = (vk) 2 + ∆ 2 and γ = ±, γ = + is for the conduction band and γ = − is for the valence band. Also the eigenvectors can be parametrized as where FIG. 1. The second order diagrams contributing to the selfenergy

B. electron-phonon interaction
Now we consider the electron-phonon interaction as the following and where b q is the bosonic field related to phonons, ω 0 is a constant frequency, ψ α (q) is the fermionic field related to electrons and [g(q)] αβ is the electron-phonon coupling. For simplicity we assume [g(q)] αβ ≈ [g(0)] αβ and because the valleys are located far from each other in k-space, the electron-phonon interaction cannot scatter one electron form a valley to another, then [g(0)] αβ ≈ gδ αβ . Due to the electron-phonon interaction, the renormalized effective Lagrangian can be written asL(k, ω) = H 0 (k) +σ(k, ω) whereσ(k, ω) is the real part of the selfenergy corresponding to the diagrams in Fig.[1]. which we can write as the following where G (0) (k, iω n ) is the free electron propagator and D (0) (k, iω n ) is the free phonon propagator Here ω n is the Matsubara frequency, η is the small positive number and |u j (k) are the eigenvectors of the H 0 Hamiltonian. By summing over Matsubara frequencies we can find the following expression for the self energy.
If we analytically continue the imaginary-time self energy, we can find the life-time corresponding to the imaginary part of self-energy and the real partσ(k, ω). In the limit |ω − µ| ω 0 and T → 0, the imaginary part of the self energy vanishes but the real part remains finite even at T = 0 [12] σ(k, ω) =g 2 where e γ (k) = γ (k) − µ. By using the results in section A we can write the projection operator as Using the equation above, we can rewriteσ(ω, k) aŝ where The coefficient of τ y is zero because the Hamiltonian is invariant under k x → −k x and then the integral is odd under this symmetry. These coefficients can be calculated numerically as shown in Fig.[2].

C. AEF and Transport coefficients
For any two band systems, we can expand the effective Lagrangian in terms of Pauli matriceŝ where and C(k, ω) = αsk y + S 0 (ω) , We use the definition of the AMF and AEF in [12] to compute these quantities.
where U is the unitary operator which diagonalizes the effective Lagrangian. and Using Eq(45) and Eq(44), we can find the following equation for AEF (48) Using the definition of N (k, ω) in Eq(41), we can find the AEF Also, AMF can be calculated as where all the expressions should be evaluated at ω = (0) γ,p . AMF is matched with the results in [3] for a model without electron-phonon interaction. One can see the frequency dependence of the N is crucial to have a non-zero AEF and that comes from electron-phonon interaction in our model. To simplify the calculations, we can choose µ = 0. Also in weak strain limit, we can approximate the dispersion in Eq(22) as α γ (k) ∼ (p+pα) 2 2m + ∆ + O( s v ) 2 , where m = ∆/v 2 and p α = (0, αs∆/v 2 ). Finally, we can now estimate the magnitude of the AEF on the Fermisurface ω = µ.
In this model, all the contributions in AEF comes from S α 2 (ω) because as one can see in Fig.[2], ∂S α 1 (ω) ∂ω vanishes at ω = 0. By using these approximations, we can find the analytic expressions for the transport coefficients in section II at T = 0.

IV. EFFECT OF AEF ON THREE-DIMENSIONAL POISEUILLE FLOW
In this part, we consider a 3D model with an external electric field in the y direction. The system is bounded in the x direction by the width w. To consider boundary effects, the viscosity term is introduced [18]: where Here ν is the shear viscosity and ξ is the bulk viscosity, which we are going to ignore. Now we use an ansatz as a solution, which is u x = u z = 0 and u y = u y (x). So the hydrodynamic equation Eq(12) becomes: The solution for the above equation with the boundary condition u( w 2 ) = u(− w 2 ) = 0 is given by: which is a standing wave solution in the x direction, where and the vorticity is: Now when we have the vorticity, we can compute the on-shell current by using Eq (16): As a result we can see that there are contributions in all directions, which come from AEF.
The G xyy and G zyy terms are non-linear contributions to the currents, G zxy and C il terms are linear contributions.

V. CONCLUSION AND OUTLOOK
We demonstrate that the AEF introduces a number of novel hydrodynamic and non-linear transport coefficients in the time-reversal invariant systems with broken inversion symmetry. In the example of a two-dimensional electron system, we show how these novel transport coefficients arise from the electron-phonon interaction. For instance, it is shown that there is a non-linear transport current along the x-direction when the electric field is applied along the y-direction, that is J x = G xyy E y E y with a finite G xyy . In similar spirit, the Poiseuille flow in the three-dimensional system in a constriction would allow non-uniform (non-linear) transport currents in all three directions. This is in contrast to the usual case, where the non-uniform current exists only along the direction of the applied electric field or even to the case when the AMF effect is included, where there exist the Poiseuille flow in two directions via the presence of the finite vorticity field [3].
Our work sheds light on novel non-linear transport and hydrodynamic phenomena in ultra-pure stronglyinteracting electron systems. Such systems are great platforms for the discovery of the intricate quantum effects associated with the rather elusive AEF. It will also be interesting to explore further consequences of the AEF in other non-linear electromagnetic responses, both theoretically and experimentally.  Research 2, 032021 (2020).
where Ω is the Berry curvature, E is an artificial electric field, E is electric field. we can see this as a two equation with two unknown variablesk n andṙ n which need to be solved. Usingk in Eq(A2) and Eq(A1) we can find: When we findk andṙ we can write the Boltzmann equation: We can find hydrodynamic equations by multiplying the above equation by momentum and integrate over the momentum space that we can find the following equation which is the extended hydrodynamic equation for quasi-conserved quantity, momentum.
where we can define momentum density and modified stress tensor as follows In the right hand side of the Eq(A4), we consider the collision term where the first term is related to the collisions that conserve momentum, and the second term is related to the collisions that relax the momentum. So after integration the first term vanishes and we can parametrize the second term with f (t,r,p) τmr in relaxation time approximation.
To find the constitutive relations for momentum and stress tensor we expand the distribution function in terms of hydrodynamic variables. In the following we assume that the underlying effective theory is invariant under Galilean To find a relation for the momentum we use Eq(A7) where n = [dp]f 0 (p). By the same approach we find the constitutive relation for stress tensor using Eq(A8), which we rewrite it as where n is the band index and the first term in the rhs of the above equation is the standard terms for stress tensor in hydrodynamic regime [dp]p i v j = ρu i u j + P δ ij . The second term is the anomalous part which we are going to investigate in the following. We denote the first anomalous part as Π Ω ij which means the AMF contributions to the stress tensor.
up to the second order in u and E we have The second term is zero due to time-reversal symmetry(Ω l → −Ω l , p → −p) and the third term is zero because the sum of berry charge over each valley is zero. Finally where In the following, without loss of generality, we can drop the band index and finally we sum over all bands. For the second part of the anomalous term Π E ij we have The first term is odd under Inversion symmetry and even under Time-reversal. The second,third and forth terms are odd under Time-reversal and even under Inversion. So if we consider Time-reversal invariant Noncentrosymmetric system then only the first term is non-zero. Finally we can write the new contribution to the stress tensor as the following where By finding all the contributions we can write the stress tensor as Now we find the transport current which is made of particle flux J N and orbital magnetization. where Using the Galilean symmetry and expanding up to second order in u and E we find (we should note we drop the band index) We separate the terms in particle flux like what we did for stress tensor, where J N Ω,i = eE k mu j kli [dp] Using integrating by part: where Now for J N E in Eq(A22) we have If we consider TRS system (E → E, p → − p and v → − v)then the first term in J E vanish. Using integration by part we find where Now for the orbital magnetization part we have the same separation and expansion, We can expand the ∂ j C li term: where and For M E we use similar approach By considering the parabolic dispersion relation the above equation can be written as the following form where Finally we can write the final expression for the transport current Eq(A19) Using constitutive relations and Eq(A6) we can find a hydrodynamic equation where we used following relation for coefficients g (0) and D (0) are defined by the free electron and phonon's Hamiltonian: where α γ ( k) = αsk y + γλ(k) is the eigenvalue of the H 0 , k = k 2 x + k 2 y and λ(k) = (vk) 2 + ∆ 2 . Also for the eigenvectors we have where Using Eq(B6), Eq(B6) and Eq(B7) we can find where η is a small positive number. By calculating the following expression m∈odd e iωmη 1 iω m − e α γ (k + q) and summing over Matsubara frequencies in Eq(B13) we can find Using analytic continuation, in the limit |ω − µ| ω 0 and T → 0, we can find the real part of the self energy as the following The projection operator for the mentioned model is Using projection operator in Eq(B16) we can find The coefficient of τ y term vanishes because it is an odd function on k x . Now we can write the above equation in a simpler formσ and S α 2 (ω) = Now one can write the effective LagrangianL(k, ω) = H 0 (k) +σ(k, ω) as an expansion of Pauli matricesL(k, ω) = N µ τ µ + C(k, ω)1L (k, ω) = −α vk y + S α 2 (ω) τ x + vk x τ y + (∆ + S 1 (ω))τ z + (αsk y + S 0 (ω))1 , where To find the AMF we use following definition The second term is the correction to the AMF up to second order in electron-phonon coupling .