Hydrodynamic transport and violation of the viscosity-to-entropy ratio bound in nodal-line semimetals

The ratio between the shear viscosity and the entropy $\eta/s$ is considered a universal measure of the strength of interactions in quantum systems. This quantity was conjectured to have a universal lower bound $(1/4\pi)\hbar/k_{B}$, which indicates a very strongly correlated quantum fluid. By solving the quantum kinetic theory for a nodal-line semimetal in the hydrodynamic regime, we show that $\eta/s\propto T$ violates the universal lower bound, scaling towards zero with decreasing temperature $T$ in the perturbative limit. We find that the hydrodynamic scattering time between collisions is nearly temperature independent, up to logarithmic scaling corrections, and can be extremely short for large nodal lines, near the Mott-Ragel-Ioffe limit. Our finding suggests that nodal-line semimetals can be very strongly correlated quantum systems.


I. INTRODUCTION
Hydrodynamics describes the behavior of quantum fluids in the regime where the relaxation of electrons is dominated by collision among the quasiparticles. This theory describes long wavelength deviations from local thermal equilibrium, when transport is dominated by conservation laws [1]. Since the time between collisions is the shortest time scale in the problem, the electrons exchange momentum faster than they can relax to phonons or disorder. That leads to universal behavior in the form of a slow diffusion of densities and to viscous flow. This framework has been successfully applied to a variety of different systems, ranging from strong coupling gauge theories with holographic duals [2], quark-gluon plasma [3], cold-atoms systems [4,5], thin wires [6], and graphene [7][8][9].
The shear viscosity measures the longitudinal resistivity to transverse gradients in the velocity of a fluid. It has been conjectured by Kovtun et al. [10] that quantum systems have a universal lower bound for the ratio between the sheer viscosity and the entropy, The equality was found in an infinitely strongly coupled field theory and has been associated with "perfect fluids," systems that are so strongly interacting that they can display quantum turbulence [7,11]. This ratio is widely believed to be a proxy for the strength of interactions in many classes of quantum systems, including relativistic, non-relativistic systems and Plankian metals [12], which entirely lack quasiparticles. By dimensional analysis, the shear viscosity η ∼ F τ , where F is the free energy and τ is the relaxation time [13,14]. In hydrodynamic relativistic systems, the free energy is mostly entropic, F ∼ sT . In the absence of screening, the scattering time due to Coulomb interactions is τ ∼h/(k B T ), and hence η/s ∼h/k B , with a prefactor of order unity. In general, screened electronic quasiparticles are long lived and typically lead to high viscosity in quantum fluids. In Fermi liquids, the free energy is dominated by the Fermi energy E F at low temperature, whereas τ ∝ T −2 . The ratio η/s ∼h/k B (E F /T ) 3 for T < E F [15], saturating to a constant η/s ∼h/k B at T > E F , above the conjectured universal lower bound.
Violations of the universal bound were found before in some strongly interacting conformal field theories [16][17][18] and holographic gravity models [19,20], and were predicted near a superfluid transition [21]. In quantum materials, it has been recently suggested that anisotropic Dirac fermions found at a topological Lifshitz transition, where two Dirac cones merge [22], violate the proposed lower bound in the non-perturbative regime of interactions [23]. Coulomb interactions, nevertheless, were more recently shown to restore the isotropy of the Dirac cone near the fixed point of that problem [24], effectively reinstating a lower bound. The extent to which the universal lower bound is violated (or not) in that problem requires a closer examination.
In this paper, we show that in nodal systems where the density of states vanishes along a Fermi line, such as in nodal-line semimetals (NLSMs) [25][26][27][28][29][30][31][32][33][34][35][36][37], the ratio between the shear viscosity and the entropy strongly violates the conjectured lower bound, scaling towards zero where k F is the radius of the nodal line, v F is the Fermi velocity of the quasiparticles, and α = e 2 /v F is the fine structure constant. This is the main result of the paper.
In the absence of screening, the scattering rate τ −1 is set by the volume of the phase space available for collisions. Due to the lack of dispersion along the line, as illustrated in Fig. 1, there is no energy cost for the quasiparticles to scatter in that direction, even at zero temperature. From this phase space argument, the scattering time is hence temperature independent, scales inversely with the length of the nodal line and can be extremely short for large nodal lines, possibly close to the Mott-Ragel-Ioffe limit [38]. We find that with additional logarithmic scaling corrections in temperature in the perturbative regime. We confirm that result by calculating the longitudinal conductivity in the collision dominated regime (ω τ −1 ), which is indicative of insulating behavior. We note that in Weyl semimetals, the dc conductivity σ ∝ T 2 τ also scales linearly with temperature (since τ ∝ 1/T , as in graphene [39,40]), although reflecting a completely distinct behavior in the scaling of the scattering rate, and hence in the viscosity-to-entropy ratio. We conclude that the violation of the bound due to an unusually short and nearly temperature-independent scattering time suggests that NLSMs can be extremely correlated quantum systems.
In the following, we outline the structure of the paper. In Sec. II, we derive the quantum kinetic equation. In Sec. III, we calculate the conductivity in the hydrodynamic regime, including a discussion on many-body effects through a renormalization group analysis. In Sec. IV, we calculate the shear viscosity and demonstrate the violation of the viscosity-to entropy ratio bound. Finally, in Sec. V, we discuss experimental implications of this result.

II. QUANTUM KINETIC EQUATION
We adopt the low-energy Hamiltonian of a NLSM that is described by a circular nodal line in the k z = 0 plane. The low-energy quasiparticles are Dirac fermions located in the vicinity of the nodal line, where δk r = k r − k F is the in-plane momentum away from the nodal line, and v F = k F /m is the Fermi velocity in the radial direction and v z along the the z direction. The quasiparticles interact through the threedimensional (3D) Coulomb potential and disperse linearly near the nodal line.
In the hydrodynamic regime, the particles interact with each other more quickly than they lose energy to the lattice. The electronic relaxation is driven by the collision between particles, leading to local thermalization. The out-of-equilibrium distribution function of the quasiparticles f λ (k, x, t) satisfies the Boltzmann equation where λ = ±1 for quasiparticles and quasiholes respectively, and v λ,k = ∇ k ε λ,k is the velocity of the quasiparticles, with being the equilibrium energy spectrum. The term eE = ∂k/∂t is the external force driving the system, with E being the electric field, and I col [f λ ] is the collision integral, which includes all scattering processes between quasiparticles allowed by Fermi's golden rule. For a nonequilibrium state, where f 0 λ = [e ε 0 λ β +1] −1 is the equilibrium Fermi distribution, which solves the Boltzmann equation in the absence of interactions (I col = 0), β = 1/k B T , and δf λ (k, x, t) is the non-equilibrium correction in linear response to an external perturbation such as electric field and strain. In general, I col ≈ δf /τ , where τ is the scattering time between collisions.

III. CONDUCTIVITY
To gain physical intuition in the problem, we derive first the conductivity and the scattering time for NLSMs. If the system is spatially homogenous, the nonequilibrium current carried by the quasiparticles in the presence of an external electric field is with´k ≡ (2π) −3´d3 k. In linear response, where J i = σ ij E j , the conductivity per spin is In leading order and close to equilibrium, the driving force term on the left-hand side of (7) is The non-equilibrium dispersion can be written in the form with g λ (k) being some unknown function to be found from the solution of the kinetic equation, where k ≡ (k, ω). With this ansatz, the non-equilibrium correction of the distribution function assumes the form For convenience, we define χ λ,i ≡ (v λ,k ) i g λ . In the collision dominated regime ω τ −1 , the linearized kinetic equation (2) can be approximately expressed in terms of the collision operator as where with M Col λλ1λ2λ3 being the collision matrix element [41]. For details in the derivation of the collision term and integration, see Appendix A.
The solution of the Boltzmann equation requires inverting the collision operator C, which can be done through the standard procedure [41]. The dominant contribution to the conductivity follows from the eigenfunctions of the collision operator with the lowest eigenvalues. In the collinear approximation, where the momenta of the quasiparticles point in the same direction, the momenta embedded in the definition of the velocities v λ,k factor out in the integrand of C, which is proportional to The zero modes of the collision operator Cχ λ,i = 0 in this restricted phase space are and corresponding to conservation of charge, chirality, and momentum, respectively. In the absence of noncollinear processes, those zero modes would produce infinite conductivity [41]. To account for non-collinear processes, we express the eigenfunctions of full collision operator C that have the lowest eigenvalues in a basis of zero modes of the collinear regime. We note that due to translational symmetry, the momentum zero mode is an exact eigenfunction of (16), as can be readily checked [41,42]. It does not, however, contribute to the conductivity (11) due to particlehole symmetry at the nodal line. For the same reason, the chiral modes do not contribute the the charge transport either. We are then left with the charge zero mode, which provides the only contribution to the conductivity.
We next restore the frequency dependence of the Boltzmann equation, φ λ,i = Cχ λ,i + iωg λ φ λ,i . In order to calculate the function a (e) (ω), we define the inner product (a λ,i , b λ,i ) = λ,i´k a λ,i (k)b λ,i (k) and set the variational functional which is to be minimized, ∂Q/∂a (e) = 0. The momentum integral of the collision operator is performed in the collinear approximation, where all momenta are nearly parallel to each other. As shown in Appendix B, this approximation is justified by the fact that for a large nodal , the weight of collinear processes in the collision phase space is logarithmically divergent, as in the case of two-dimensional (2D) Dirac fermions [7,41]. We also restrict scattering to channels that conserve the number of particles and holes, which are dominant processes in the collinear regime.
Combining the solution of Eq. (21) with Eqs. (11) and (14), we obtain the frequency dependent conductivity in the hydrodynamic regime, where for i = x, y and N is the spin degeneracy [43]. The coefficient c(1) ≈ 1.034 was numerically extracted from the collision integral for N = 2. This value decreases monotonically away from γ = 1. The functions α(T ) and v F (T ) are the fine-structure constant and Fermi velocity, respectively, dressed by interaction effects.

A. Renormalization group analysis
As in graphene [44], Coulomb interactions are marginal and renormalize the velocity of the quasiparticles in the perturbative regime. The velocity grows logarithmically with decreasing temperature, where The electron charge does not run and the fine-structure constant is also renormalized, and decreases logarithmically at low temperature. The renormalization group (RG) results mimic the structure of the calculation in graphene. The details can be found in Appendix C. The combination [αv F ] (T ) does not run, whereas the ratio γ ≡ v z /v F flows toward 1. Hence, in the collisiondominated regime ω τ −1 , σ(0, T ) scales linearly up to logarithmic corrections, suggesting that the system behaves as an insulator, as shown in Fig. 2(a). In that plot, we use k B Λ T = 0.2×v F k F , γ = 1 and α = 0.6. The static charge polarization bubble of a NLSM, Π(q, 0) ∼ −N/(2π) 3 k F q/v F , scales linearly with momentum. Since the Coulomb interaction ∝ q −2 , the quasiparticles are partially screened at momenta q N αk F /2π 2 , where interactions decay as ∝ q −1 . Below the cut-off temperature λ T /Λ T ≈ N αk F /2π 2 Λ ∼ 0.2, the Coulomb interaction is therefore screened by charge polarization effects, although still long ranged, indicating a crossover in the T → 0 limit. In that regime, the velocity is not further renormalized by the screened Coulomb interaction and the RG flow stops.
A previous on-shell Wilson-Yukawa RG analysis has indicated the presence of a screened interacting fixed point in this problem [45,46]. In the vicinity of that fixed point, a strong charge renormalization was found, suggesting a crossover to a Fermi liquid. We point out that the analysis of Ref. [45,46] did not incorporate the nonanalytic structure of the infrared (IR) polarization bubble in the bosonic propagator, which is relevant in the RG sense. For Dirac fermions, it has been recently shown [47] that the incorporation of the IR bubble (which is nonanalytic) in the on-shell propagator of the bosons is necessary and correctly recovers previous numerical results based on conformal bootstrap calculations. The fermionic analysis for NLSMs shown above indicates that the charge is This is the second main result of the paper. In Fermi liquids, the scattering time diverges as τ ∝hε F /(k B T ) 2 , with ε F the Fermi energy. Relativistic systems have a parametrically shorter scattering time (τ ∼h/k B T ), reflecting the absence of screening. The nodal line significantly enlarges the phase space for collisions among the quasiparticles, without producing any screening effects at T > ∼ λ T . That further reduces the scattering time, which increases only logarithmically with decreasing temperature, as shown in Fig. 2(b).

IV. SHEAR VISCOSITY
The shear viscosity η is the dissipative response of fluids to transverse gradients in their velocity field. It is defined after the strain contribution to the stress tensor away from the equilibrium distribution [48] where u = ∂ξ/∂t is the velocity field of the fluid, with ξ i being a strain deformation field. The gradient u ij ≡ ∂u i /∂x j = ∂ξ ij /∂t is the time derivative of the strain tensor ξ ij ≡ ∂ξ i /∂x j . For systems that preserve time-reversal symmetry, the viscosity tensor is symmetric, obeying the Onsager relation η ijk = η k ij [49].
The stress tensor can be derived from the change of the Hamiltonian with respect to the strain tensor, In linear response, the first-order contribution of strain to the Hamiltonian can be shown [50] to appear through a term with the general form From Eq. (28), the deviation of the expectation value of the stress tensor T ij away from equilibrium is from which the shear viscosity in Eq. (27) can be extracted. For details of the derivation, see Appendix D.
Going back to the kinetic equation (7), the second term on the left gives Setting the electric field to zero, the change in the energy spectrum can be parametrized with the ansatz where h λ (k, t) is to be determined by solving the kinetic equation. Hence, the nonequilibrium correction to the distribution function due to strain has the form Defining χ λ,ij ≡ I ij h λ , the kinetic equation in the stationary regime (ω → 0) is The definition of the collision operator follows directly from Eq. (16) under the substitution χ λ,i → χ λ,ij . In the collinear regime, there are three zero modes that are eigenfunctions of the collision operator, Cχ λ,ij = 0, namely, and Those modes correspond to conservation of charge, energy, and number of particles respectively. The particle number zero mode, however, does not contribute to the shear viscosity due to particle-hole symmetry at the nodal line. This mode is orthogonal to the other two and can be completely decoupled. Setting a basis with the charge and energy modes λ,ij (k), with α = 1, 2, one can express χ λ,ij as a linear combination in that basis, If we project the kinetic equation in that basis, namely and then the solution (39) follows from the determination of the a β coefficients C −1 αβ is the inverse of a 2 × 2 matrix that can be evaluated numerically through the momentum integration of the collision operator in the collinear approximation, as shown in Appendix C. Substitution in Eqs. (30) and (34) gives the viscosity tensor where c y (1) ≈ 0.569 and c z (1) ≈ 0.759 for N = 2.
A. Viscosity-entropy ratio The entropy density of a NLSM can be calculated from the entropy of a noninteracting system dressed by interactions with the renormalized observables, where ζ(3) ≈ 1.20 is a zeta function. Allowing the Fermi velocity and the fine structure constant to be renormalized according to the RG prescription, the ratio η/s is In Fig. 3, we plot the temperature dependence of the shear viscosity-entropy ratio in units ofh/k B versus temperature in units of the temperature cutoff. The horizontal line is the conjectured lower bound η/s = (1/4π)h/k B . The ratio has a quasilinear scaling toward zero with decreasing temperature T ∈ [λ T , Λ T ], in violation of the lower bound. The violation reflects the enlarged phase space for collisions at low temperature in unscreened relativistic systems with a nodal line. For T λ T , partial screening effects can lead to the restoration of a non-universal lower bound, below the one previously conjectured [10].

V. DISCUSSION
In the hydrodynamic regime, the usual manifestations of the viscous flow of electrons in constrained geometries include nonlocal negative resistance [8,9,51] and fluid dynamics with vortex lines [52]. The very low viscosity compared to the amount of entropy production, in violation of the conjectured lower bound, is highly suggestive that NLSMs may exhibit quantum turbulence [7,11,52].
In general, observation of hydrodynamics requires quasiparticles with a relatively short scattering time. Signatures of hydrodynamic behavior can be detected in the collision-dominated regime through optical and transport measurements when k B T ε F , ∆, with ε F being the energy of the Fermi surface and ∆ being the gap induced by spin-orbit coupling effects or possible many-body instabilities [53,54], including excitonic phases [55]. NLSMs that combine inversion, time reversal, and mirror glide symmetry have nodal lines that are robust against spinorbit coupling [56].
NLSMs are unique in that the nodal line introduces a length scale that does not generate fully screned interactions, as in Fermi liquids. That length scale substantially enlarges the size of the phase space for collision of thermally excited quasiparticles and is responsible for the unusual temperature scaling of the scattering time in the hydrodynamic regime. Materials such as ZrSiSe [57] have a large nodal line gapped by a small spin-orbit coupling gap of ∆ ≈ 30 meV, with v F Λ ≈ 0.4 eV (Λ T ≈ 4 × 10 3 K) and v F k F ≈ 2eV. In this material, the Fermi velocitȳ hv F ≈ 2eVÅ is three times smaller than in graphene. Experimental control over the value of the fine structure constant can be achieved with experiments on thin films encapsulated by dielectric materials. In ZrSiSe, for a moderate fine structure constant α ≈ 0.6 within the perturbative regime, the scattering length at T ∆/k B , is of the order of the lattice constant, near the Mott-Ragel-Ioffe limit, indicating the presence of very strong correlations. We speculate that hydrodynamic behavior may be observable in a number of different NLSM materials. Following the derivation of Kadanoff [58], the Boltzmann equation has the general form:

t) is the nonequilibrium Fermi distribution, and
is the collision term, with V (k) = 4πe 2 /k 2 is the Coulomb interaction and M is a tensor in the quasiparticle-hole basis. Explicitly, with U k being a unitary transformation that diagonalizes the Hamiltonian. For nodal-line semimetals (NLSMs), or where H = h x + ih y , δk is a relative momentum from the node line. The Hamiltonian can be diagonalized in the quasi-particle and quasi-hole basis with their energy ±v F h. We assign each basis as λ = ±1, and thus ε λ,k = λv F h where λ = +1 corresponds to a excited particle and λ = −1 to a excited hole. The unitary transformation matrix is and thus, the tensor M is The velocity of quasiparticles in the Boltzmann equation is, by definition,

Linearized Boltzmann equation
Starting from the the nonequilibrium correction of the distribution function due to the presence of an external electric field, where χ λ,i ≡ (v λ,k ) i g λ , with g λ being a function to be determined by solving the quantum kinetic equation (7). The left-hand side of that equation is Defining f λi ≡ f λi (k i ), the collision term in the righthand side is with N being the fermionic degeneracy, and where M λλ1 ≡ M λλ1 (k, k 1 ), and so on. The third line of (A13) has two terms with four f functions. One should expand it in eight terms to linear order in δf , with three f 0 and one δf . We can simplify them using which is restricted by the energy conservation. After some straightforward algebra, we find with the collision matrix element equating the left-and the right-hand side of the quantum Boltzmann equation, Eq. (A12) and (A16), we have with C being the collision operator as defined in the main text, Appendix B: Collinear approximation

Collision phase space
Due to the Coulomb potential V (k − k 2 ) and V (k − k 3 ) in the integrand of the collision operator, the integral is governed by small momentum transfer due to collision processes. In the collinear approximation, where the four momenta are nearly aligned to each other around the nodal line, we can define the momenta where we assume that the ⊥ components are small compared to the radius of the nodal line k F . The phase space for collision processes is set by conservation of energy, We now expressing it in terms of the dimensionless variables, and with n = 1, 2, 3. Performing a suitable change of variables k 2 → k − k 2 and k 3 → k 1 − k 3 , where while at the same time after using momentum conservation k + k 1 − k 2 − k 3 = 0.
Since ξ 1 and ξ 2 are much smaller than κ 0 , we can rewrite the argument of the δ function D as and w i (i = 1, 2, 3) are functions of the dimensionles variables x, y, x i , y i . D is a quadratic function of ξ 1 . We can then express the δ function as where D is the first derivative of D, and ξ i=± are the two roots of the quadratic function, namely Hence, It is clear that the phase space has a logarithmic divergence in the ξ 2 variable whenĀ → 0. At the same time, the Coulomb interaction termsV 1 andV 2 defined in Eqs. (B9) and (B10) decay quickly to zero with ξ 2 when it is large. Thus, there are two important regions of the integrand in phase space:Ā → 0 and ξ 2 → 0. This phase space argument justifies the validity of the collinear approximation, with which the conductivity and the shear viscosity were calculated.

a. Calculation of the conductivity
The variational functional of the conductivity is with a (e) (ω) being the variational function corresponding to charge conservation in the collinear regime. For convenience, after multiplying the factor v F β 3 in both sides of Eq. (B20), the first term is To calculate the second term, we consider the dominant processes in the near collinear regime, which conserve the number of particles and holes. We have where α ≡ e 2 /v F , and I(γ) is a dimensionless number. The extra factor of 1 4 on the right-hand side is due to the symmetrization in the four momenta. In terms of the dimensionless variables (B5) and (B6), the combination a (e) 2 I(γ) can be written in the collinear approximation as where D,V 1 , andV 2 are given in Eq. (B8), (B9), and (B10). The W and Y tensors follow from Eqs. (A5) and (A14) with the substitution h x → x, h y → y and so on. The integral is performed enforcing the restriction in momentum space This implies that where In the near collinear approximation, we find c (γ = 1) ≈ 1.034 for N = 2. When the nodal line is spin polarized, with N = 1, c(1) ≈ 0.361. In the two anisotropic limits γ → 0 and γ → ∞, c (γ) is proportional to γ 2 and γ −2 respectively, and scales toward zero. The conductivity is dy 2π x 2 e r r 2 (e r + 1)

Calculation of the viscosity
In the collinear regime, we set a basis with the zero modes reflecting conservation of energy and number of particles {χ (1) λ,ij , χ (2) λ,ij }, with as described in the main text. One can express χ λ,ij as a linear combination in that basis.

Appendix C: Renormalization Group Analysis
We perform the renormalization group (RG) analysis using standard perturbation theory. Since Coulomb interactions are marginal operators in the RG sense, perturbation theory is well controlled in the regime where the fine structure constant α = e 2 /v F 1. In the spirit of perturbation theory, in one loop one needs to extract the leading logarithmic divergences of three diagrams: The Fock diagram for the self-energy, the polarization bubble, and the vertex diagram, as shown in Fig. 4.
The Green's function for a NLSM is given bŷ with v F = k F /m and δk r = k r − k F . The pole of the Green's function gives the energy dispersion The Fock self-energy is given by the diagram At one loop level, the self-energy is frequency independent. In the regime where the radius of the nodal line k F Λ, with Λ the momentum ultraviolet cut-off around the line, one can ignore terms such as q 2 /k F , ≈ v F (δk r + δk r · q r ).
We integrate the bosonic momentum q of the selfenergy in the regime δk q k F , where the leading logarithmic divergence of the diagram is expected.
The perturbative velocity renormalization is v = v 0 1 + α 0 I 1 (γ) ln Λ δk , (C9) v z = v z0 1 + α 0 I 2 (γ) ln Λ δk . (C10) Next, we examine the vertex and the bubble diagrams. In standard perturbation theory for Coulomb interactions, the vertex diagram does not contribute to the charge renormalization due to a Ward identity, which relates the vertex and the quasiparticle residue renormalizations. In one loop, the self-energy is frequency independent, and hence the vertex diagram is zero at this order. The polarization bubble renormalizes the Coulomb interaction and could also renormalize the charge. However, the static polarization bubble of a NLSM is perfectly regular and does not contain logarithmic divergences [45], with a 1 and a 2 of order unity. Therefore, neither diagram contributes to the renormalization of the charge, which does not run in the perturbative regime.
We also point out that since the polarization Π(0, q r , q z ) is linear in q whereas V (q) ∝ 1/q 2 , Π changes the form of the Coulomb propagator due to screening effects at small q, 4πe 2 q 2 − 4πe 2 Π(0, q r q z ) .
For q N α 2π 2 k F = q c , Coulomb interactions are screened (although still long range) and the analysis in the vicinity of the fixed point will change. Our analysis indicates that further away from that fixed point, for q > ∼ q c and N α < 1, where interactions are unscreened and standard perturbation theory applies, only the velocities run. At low momentum, for q q c and N α < 1 where interactions are partially screened, no logarithmic divergences are present and the RG flow stops, whereas the charge remains unrenormalized.