Hall and spin Hall viscosity in topological insulators
Taro Kimura***E-mail address: firstname.lastname@example.org
Department of Basic Science, University of Tokyo, Tokyo 153-8902, Japan
Mathematical Physics Laboratory, RIKEN Nishina Center, Saitama 351-0198, Japan
We study responses to metric perturbation in topological insulator models. In this paper we introduce a novel quantity, Hall viscosity to particle density ratio, which is analogous to the viscosity to entropy ratio suggested by AdS/CFT correspondence. This quantity corresponds to the filling fraction which characterizes the quantum Hall states, and hence it could be discussed in Haldane’s zero-field quantum Hall system. We also consider dissipationless viscosity in the time reversal invariant quantum spin Hall system, which is related to a spin current.
Recent interest in topological aspects of electron states has led to a large number of experimental and theoretical works. In particular the study of the quantum Hall effect (QHE)  has shown the idea of the adiabatic geometric phase is quite useful to understand their topological properties. One of the most remarkable phenomena in such a topological state is dissipationless transport: a dissipative longitudinal current vanishes, and only a dissipationless transverse current, called Hall current, is observed in the QH phase. In a time reversal broken system, we can consider not only such a dissipationless charge transport, but a dissipationless momentum transport. The latter one is related to a dissipationless viscosity coefficient [2, 3], which is called “asymmetric viscosity,”  “Hall viscosity,” [4, 5] and also “Lorentz shear modulus.”[6, 7] To obtain the Hall viscosity, we can use the Kubo formula with a correlation function of a stress tensor as well as the dissipative shear viscosity. It means the viscosity indicates a response to a metric perturbation. On the other hand, as the case of the Hall conductivity, the Hall viscosity is also given by the Berry curvature on a certain parameter space. In the case of viscosity, it turns out to be the moduli space of the torus , while we should consider the Brillouin zone for the conductivity . It reflects that the metric of the system can be parametrized by the modulus of the torus.
In earlier works, mainly the Hall viscosity of the lowest Landau level (LLL) has been computed for non-interacting  and interacting electron states, such as the Laughlin state and generalized conformal blocks [4, 6, 7]. They correspond to integer and fractional QH states, respectively. The adiabatic property for the higher LL state has been also considered in Ref. , but the connection with the Hall viscosity has not been explicitly referred. On the other hand, all the previous results are restricted to the non-relativistic systems although there are various applications of the relativistic Dirac fermion in the context of condensed-matter physics. Thus, in this paper, we will discuss the Hall viscosity of the two dimensional Dirac fermion, and consider applications to graphene  and topological insulators , which have been remarkable topics in recent years. More recently the field theoretical [12, 13, 14] and the holographic approaches  have been investigated.
To discuss topological insulator models, we will start with Haldane’s zero-field QH model . For the QH system, the filling fraction, which is defined as a ratio of density of states to magnetic flux, is useful to characterize its topological property. In the topological phase of the Haldane model, however, density of states itself vanishes because no net external magnetic field is applied to this model. Although, in this sense, the filling fraction becomes indeterminate, we can obtain a finite Hall conductivity through an alternative method, called Widom-Středa formula [17, 18].111Remark this formula is also applied to the thermal Hall conductivity  and also the Hall vistocity . Similarly the Hall viscosity of this model also goes to zero even in the topological phase because viscosity is the hydrodynamical quantity, and should be proportional to the density or the magnetic field. It means the naive definition of the Hall viscosity cannot capture the topological property of the Haldane model. Thus we have to consider a novel viscosity, which is able to characterize the topological phase of the Haldane model. We will show a hint to resolve this difficulty can be found in string theory.
Recent progress in string theory has provided us with a remarkable method to approach the strongly correlated system, which is called AdS/CFT correspondence . Indeed there are a large number of applications ranging from particle physics to condensed-matter physics. The shear viscosity for a certain strongly coupled system was calculated in Ref. , and universal behavior of the viscosity to entropy density ratio, , was proposed in Ref. . It suggests that a ratio of viscosity is more fundamental quantity than viscosity itself. Thus, in this paper, we will consider a ratio for the Hall viscosity and discuss some analogous properties.
Furthermore, it has been recently shown that the topological phase can be observed even in the time reversal invariant systems , which is characterized by topological invariant . Its discovery has a significant influence on both of experimental and theoretical sides of condensed-matter physics. The effective theory of such a system is well described by the Dirac fermion . Thus we can also generalize the discussion of the dissipationless viscosity to the time reversal invariant system. As discussed in Ref. , the Hall viscosity itself is observed only in the time reversal broken system. However we can introduce another kind of viscosity in such a system, which is analogous to the spin Hall conductivity. We will show it characterizes a stress induced by a spin current while the Hall viscosity is related to a current of particles.
This paper is organized in the following way. We begin in section 2 with the theory of Dirac fermion with metric perturbation as preliminaries to discuss topological insulators. Then we will also discuss its application to graphene system. In section 3 we consider two dimensional topological insulator models with/without time reversal symmetry. We then introduce a novel quantity, a ratio for the Hall viscosity, to discuss these models, and some analogous properties with a similar quantity, which is proposed in the context of AdS/CFT correspondence. We also discuss a spin Hall analogue of the viscosity ratio in the time reversal invariant system. Section 4 is devoted to summary and discussions.
2 Hall viscosity and Dirac fermion
Let us start with how to calculate the Hall viscosity. As the case of the usual dissipative shear viscosity, the Hall viscosity is obtained by response to metric perturbation, which is related to the shear stress mode. We now introduce the metric perturbation as follows:
Here stands for the inverse of , and is a complex form of the perturbation parameter, which is interpreted as a modulus of the torus. We obtain the Euclidean metric by setting , and . Then the stress tensor is given by response to the metric perturbation, . According to the Kubo formula, one obtains the usual shear viscosity from a correlation function of . Not only the dissipative shear and bulk viscosity, but another transport coefficient can be observed, in the time reversal broken system , which we call Hall viscosity.
The Hall viscosity is also given by correlators of stress tensors, such as . However it is shown in Ref.  that the quantity is related to the Berry curvature on the moduli space of the metric perturbation parameters ,
To recover the physical unit of the viscosity, it is required to multiply the factor of , where is the system size. This is just a contribution of an eigenstate to the viscosity, where denotes a label of the particle state, e.g. the Landau level index. In general, we should consider a whole contribution of all the eigenstates under the Fermi energy. It is quite similar way to obtain the Hall conductivity, which is given by the Berry curvature on the momentum space.
Here we remark the difference between the Hall viscosity and conductivity. The Hall conductivity is quantized because it is given by the integral of the Berry curvature over the Brillouin zone . On the other hand, the Hall viscosity is directly related to the curvature on the specific point. Thus it is not expected to be quantized since it is not given by the integral of the curvature while its integral is quantized .
2.1 Dirac fermion
As a preliminary to discuss the topological insulator models, we now consider the Hall viscosity of the two dimensional Dirac fermion model with external magnetic field.
The unperturbed two dimensional Dirac Hamiltonian is given by
where is the Fermi velocity. To consider the modification of (2.3) under the metric perturbation (2.1), we should introduce modified Pauli matrices, satisfying . The metric (2.1) can be written with the “zweibein,” , where is the Euclidean metric. In a matrix form, it becomes
This helps us to obtain the modified version of the Pauli matrices: they are given by a linear combination of the original ones,
Hence the Dirac Hamiltonian with the metric (2.1) yields
This is just the Dirac operator in a flat, but non-standard metric spacetime.222See e.g. Ref. .
To discuss the Landau quantization for this model, let us take into account the magnetic field. Thus we introduce a modified momentum as with the Landau gauge . Then the Hamiltonian becomes
The cyclotron frequency is given by . Here the creation and annihilation operators are defined as
We can see they satisfy the ordinary bosonic relation, .
We then search for eigenstates of the Hamiltonian (2.6). The Dirac zero mode, corresponding to the LLL state, is obtained by the equation of . Here labels the degenerated LLL state, , where the magnetic field is quantized as . Then the higher LL states are also given by
Hence the eigenvalues and the eigenstates of the Hamiltonian yield
where the normalization condition for the LL state is given by . As discussed in Ref. , the LLL wavefunction can be written in a form of where consists of the theta function depending on only either of ,
with letting be the complex coordinate of the two dimension.
The property of the theta function yields the normalization constant . It depends on both of and because it is also written as . In general, the Berry curvature on the moduli space is related to the holomorphic structure of the eigenstate. Actually the LLL state splits into the (anti-)holomorphic and non-holomorphic parts, corresponding to the theta function and the normalization factor. It is then shown that the contribution to the curvature comes only from the normalization factor [4, 6, 7]. According to the Berry curvature of the LLs given in Ref. , we obtain the curvature of the eigenstate of (2.7),
Here the curvature is the two form defined as with the Berry connection . Therefore the Hall viscosity is obtained from this curvature on the moduli space by taking into account the degeneracy of each LL state. The LL state is -fold degenerated, but the zero mode turns out to be half filled because of the parity anomaly . Thus contributions to the Hall viscosity coming from the curvature (2.15) and (2.16) is given by
where is the density of particles and the magnetic length is defined as . We remark the result (2.17) depends on the level index , while the contribution to the Hall conductivity does not. It has been shown this level dependence indeed reproduces the classical result , where is the cyclotron frequency [2, 4].
2.2 Application to graphene
The most desirable application of the relativistic -dimensional Dirac fermion theory in condensed-matter systems would be the graphene system [28, 10]. The low energy behavior of graphene is well described by two Dirac fermions at two valleys, and points, and each valley has up and down spin degrees of freedom. Hence we often treat it as the four flavor, totally eight component, Dirac fermion system. Indeed the anomalous QHE of graphene is well explained by the four flavor theory .
We shall discuss the Hall viscosity of the graphene system. First of all, it should be noted, as discussed in Ref. , it would be ambiguous for the system where translational symmetry is violated at short length scales, since it is the transport coefficient related to momentum transport. Therefore we have to consider the system such that it is well described by the continuum effective theory, i.e. the magnetic length, which characterizes the extent of the wavefunction, is much larger than the lattice spacing of the system. Furthermore, in the case of graphene, we should also deal with the low energy region where the Dirac fermion description is suitable.
We then introduce the effective Hamiltonian for the monolayer graphene ,
where the cyclotron frequency for this system is defined as , and is the hopping amplitude between the nearest neighbor sites. Here the annihilation/creation operators are defined as , satisfying where and refer to the and valleys, respectively. This is just the model discussed in section 2.1. Therefore we can apply the result of the Dirac fermion to this system by considering the metric perturbation. However, since the Dirac zero mode is shared by both of and valleys due to the anomaly as discussed before, we must not doubly count the contribution of the zero mode. Hence, by considering valley and spin degrees of freedom, the Hall viscosity of the graphene system yields for . The sign corresponds to whether it is associated with particle or hole states, as the anomalous Hall conductivity of the graphene . On the other hand, in the non-interacting and non-relativistic system, it becomes for , since the contribution of each LL is given by for , as shown in Refs. [4, 9].
3 Hall and spin Hall viscosity ratio
In this section we then apply the argument of the Hall viscosity to two dimensional topological insulators . All the topological insulator models have a band gap, which is generated by magnetic field in a time reversal symmetry broken system, and spin-orbit interaction in a time reversal symmetric system, respectively. Although the origin of the gap is different, topological aspects of such systems are well described by the effective Dirac fermion model discussed in section 2.
3.1 Hall viscosity ratio in topological phase
First we consider the model proposed by Haldane . The remarkable property of this model is that the topological phase can be observed even though no net external magnetic field is applied while the time reversal symmetry is broken. Indeed it is shown that anomalous contribution of the Dirac zero mode, relating to the parity anomaly, plays an important role in this model.
Now let us discuss how to obtain the Hall conductivity for the model without external magnetic field. It is given by a derivative of charge density with respect to magnetic flux, , which is called Widom-Středa formula [17, 18] and available even in the zero field limit. It can be interpreted as l’Hopital’s rule to evaluate an indeterminate form of the filling fraction in the limit of . Therefore we can obtain the Hall conductivity of this model as follows: we calculate the Hall conductivity with Widom-Středa formula after taking into account infinitesimal magnetic field as a probe. We then obtain the Hall conductivity without external magnetic field by taking the limit of . Indeed the charge density becomes when we apply magnetic field in a certain phase of the Haldane model. As a result, the Hall conductivity is given by , thus it turns out to be QH phase. Then we can safely take the limit of , since there is no dependence on the magnetic field.
We then consider the Hall viscosity in the topological phase of the Haldane model. According to the result of the Dirac fermion model discussed in section 2, it yields in the topological phase with infinitesimal magnetic flux. We now remark that not charge, but particle number density is used in the expression of the Hall viscosity (2.17). However, taking the zero field limit, the Hall viscosity goes to zero as the particle density vanishes. This reflects that viscosity is the hydrodynamical quantity which should depend on the density. In the Haldane model the incompressible fluid state of the LLs cannot be observed, and thus the naive definition of the Hall viscosity cannot capture the topological property of the model.
To resolve the difficulty in considering viscosity for a system without the LLs, we now introduce a novel quantity, a ratio of the Hall viscosity to particle density,
This is similar to kinematic viscosity, which is defined as the ratio of viscosity to mass density. The reason why we prefer the quantity (3.1) to kinematic viscosity is that it is not well-defined for the massless Dirac fermion. Actually the topological phase generally has an energy gap, which corresponds to a kind of mass term. In this sense it should be enough to apply the kinematic viscosity to characterize the topological phase. However, since the unit of viscosity is times density of particles, the ratio of viscosity to density simply has the dimension of . Therefore the viscosity ratio (3.1) should be independent of the characteristic scale of the system, and a much universal quantity rather than the kinematic viscosity, which includes a system dependent non-universal mass scale.
With emphasis on its similarity to the Widom-Středa formula, we can introduce another expression,
This is regarded as a generalized form of the expression (3.1). At least, in the linear response region, these two expressions are consistent. By using l’Hopital’s rule as the case of the Hall conductivity, we can see it takes non-zero value,
This expression is available even in the zero field limit, thus it can be interpreted as alternative formula for the Hall viscosity, which is associated with Widom-Středa formula. In general, it stands for with effective one-particle spin because the Hall viscosity can be represented as .
We now comment universal behavior of the Hall viscosity for the LLL state. The Hall viscosity of the LLL state, which is discussed in Refs. [3, 6], and of the Dirac zero mode shown in (2.17) are given by , yielding . Furthermore it is shown in Refs. [4, 7] that the Hall viscosity of Laughlin state becomes with . Although the value of the Hall viscosity itself is different from the above, dependence on the magnetic field is universal, . On the other hand, this universality is not observed in the excited state. Indeed the Hall viscosity for more generalized conformal blocks is obtained in Ref. , , with a conformal weight . Similarly, the -th LL state gives as discussed in the previous section. These results give a larger value , compared with the LLL state. It is because an excited state has a higher spin.
We also remark similarities between the Hall viscosity ratio and another quantity, the viscosity to entropy density ratio, , which is well studied by using AdS/CFT correspondence. Remark AdS/CFT is just a tool to analyse a strongly correlated system, thus the ratio is well-defined without using the gravity theory. The usual dissipative shear viscosity in a certain strongly coupled system is obtained in Ref. , and its universal behavior, , called KSS bound, is discussed in Ref. . In the context of AdS/CFT, the Hawking temperature of the black hole is interpreted as the temperature of the dual field theory at boundary. Although the temperature goes to zero in the extremal black hole limit, the residual entropy is observed in some cases. Actually it is discussed in Refs. [30, 31] that this universality is still available even in such an extremal case, corresponding to the zero temperature limit. This extremal limit is analogous to the zero magnetic field limit in our model because the ratio can be indeterminate in the limit. It suggests such a quantity could be available even for the extremal case.
It is shown in Ref.  that the KSS bound can be violated in generalized holographic model, e.g. Gauss-Bonnet gravity, which includes higher order curvature terms. In the case of Hall viscosity, the shift of its value, coming from the effect of the geometry, is pointed out . These results suggest the geometrical effect should not be neglected for arguments on viscosity. Thus the universal behavior could be seen only in the flat space.
These universal behaviors are not discussed for viscosity itself, but the viscosity ratios. The reason why is the natural unit of viscosity should be times density as discussed in Ref. . This density dependence suggests viscosity relates to the characteristic scale of the system, e.g. the magnetic length. Thus it is preferred that we consider the ratio of quantities rather than viscosity itself. We have shown there are some similar properties in both of the viscosity ratios. We have used this similarity as a hint to introduce the viscosity density in the topological phase. The connection between these quantities is not yet obvious, but we hope such a similarity is helpful on the both sides of physics.
3.2 Spin Hall viscosity with time reversal symmetry
It is well known that the Hall conductivity is observed if and only if time reversal symmetry is broken. As the case of conductivity, it is pointed out in Ref.  that the Hall viscosity can be also observed in a system without time reversal symmetry. On the other hand, spin Hall conductivity induced by spin-orbit interaction is found in time reversal symmetric systems [33, 34], where there is no external magnetic field. We then consider application of the idea of the Hall viscosity to the time reversal system, and spin Hall analogue of viscosity.
The most fundamental model for the two dimensional topological insulator with time reversal symmetry is the Kane-Mele model, which is proposed in Refs. [24, 25]. In this model the spin-orbit interaction is taken into account on the honeycomb lattice instead of magnetic field, and it corresponds to the next nearest neighbor (NNN) hopping term of the Haldane model. Actually either of up or down spin sector of the Kane-Mele model is equivalent to the Haldane model. However, although the time reversal symmetry of the Haldane model is explicitly broken, the Kane-Mele model preserves the time reversal symmetry by considering both of up and down spin sectors. In this sense, the Kane-Mele model can be interpreted as the doubled Haldane model where the signs of the NNN hopping amplitude is opposite for up and down spins. Hence, if we do not introduce any other spin-mixing terms, a spin current is derived from the quantized Hall current of the Haldane model, , and thus it yields a spin Hall conductivity . In this sense, we then consider spin Hall analogue of the Hall viscosity ratio
which we will call spin Hall viscosity ratio. As the case of the Haldane model, the quantity defined in (3.4) is just for the one-particle state since it is divided by the particle density. We now remark that it is not derived from hydrodynamical relations, however such a spin-related viscosity would be well-defined unless spin-mixing terms, which do not preserve -component of spin, are not there. In this sense, as the spin Hall conductivity, this quantity would not be directly related to a generic topological invariant, which characterize the topological phase. Then we comment many-body interaction effect on this quantity. Although some generalizations of topological insulators for correlated electron systems have been considered [35, 36], an explicit consequence is not yet obvious. If certain aspects of highly correlated topological insulators are understood, we would be able to discuss the spin Hall viscosity for such systems.
We then study a physical meaning of the spin Hall viscosity. In the presence of magnetic field, the Hall viscosity gives a relation between the electric current and the stress tensor, e.g. , where stands for the current in -direction . In the case of the spin Hall system, it corresponds to where represents -component of a spin current. This means the spin current induces the stress in the Kane-Mele model as shown in Fig. 1 while the charge current does not. It is dissipationless because the stress is perpendicular to the current as the cases of Hall and spin Hall conductivity. Finally, we comment a possibility of experimental detection. In such a topological phase, as discussed before, some non-trivial responses are observed. Thus we could extract an effect of the dissipationless viscosity by totally utilizing them. For example, when we apply stress on carriers of the system, a current is generated through a certain direction due to the viscosity. Then this current induces an electric field due to the conductivity, therefore it can be detected by measuring its voltage.
In this paper we have discussed the Hall and spin Hall viscosity in some topological insulator models. In section 2 we have studied the metric perturbation for the Dirac fermion model. The Berry curvature on the moduli space, which gives the Hall viscosity of the Dirac fermion, is computed, and its application to the graphene system is also discussed. In section 3 we have proposed the novel quantity, the Hall viscosity to particle density ratio, to study the viscosity of Haldane’s zero-field model for QHE. It is analogous to the ratio of viscosity to entropy density, which is often referred in the context of AdS/CFT correspondence, and some similar properties are discussed. We have also studied the spin Hall viscosity of the Kane-Mele model. Although the Hall viscosity is observed only in the time reversal broken systems, the spin Hall viscosity can be found even in the time reversal invariant systems. We have pointed out that the spin current induces the dissipationless stress in the topological phase of the Kane-Mele model.
We now comment some open problems on this topic. One of them is the relation between the Hall viscosity and the edge state. It is well known that the edge state plays an important roll on the transport phenomena in the QH and QSH systems: the Hall conductivity is actually obtained by different ways, due to the bulk/edge correspondence . In this sense, the Hall viscosity should be computed from the edge state. Next is a generalization to the three dimensional topological insulators, which are also characterized by invariant [38, 39, 40]. Three dimensional anomalous viscosity induced by the external field was indeed considered in Ref. , and its classical computation was also given. Since the surface state of the three dimensional model can be described by Dirac fermions, the results discussed in this paper would be related to it. Furthermore the four dimensional generalization would be also discussed. The anomalous contribution in the four dimensional hydrodynamics is recently considered in Ref. . It is related to the triangle anomaly, and corresponds to the non-linear response coefficient. Indeed, in the four dimensional QH system , the non-linear coefficient is quantized while the linear part is not. Therefore it is expected that the four dimensional anomalous contribution can be found in the four dimensional QH system.
The author would like to thank S. Hikami, H. Katsura, S. Murakami, H. Shimada, A. Shitade and T. Yoshimoto for valuable comments and discussions. The author is supported by Grant-in-Aid for the Japan Society for Promotion of Science (JSPS) Postdoctoral Fellows (No. 23-593).
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