Non-equilibrium Chiral Magnetic/Vortical Effects in Viscous Fluids
We utilize the chiral kinetic theory (CKT) in a relaxation-time (RT) approximation to investigate the nonlinear anomalous responses of chiral fluids with viscous effects. Unlike the cases in equilibrium, it is found that the chiral magnetic effect (CME) and chiral vortical effect (CVE) are modified by the shear and bulk strengths. Particularly, the shear strength could result in charged Hall currents for CME and CVE, which propagate perpendicular to applied magnetic fields and vorticity. These quantum corrections stemming from side jumps and anomalies are dissipative and pertinent to interactions. Moreover, we show that the non-equilibrium corrections for the energy-density current should vanish based on the matching condition obtained from the CKT with the classical RT approximation. Although the non-equilibrium effects upon charge currents are dissipative, the second law of thermodynamics is still satisfied.
Introduction.—The anomalous transport for Weyl fermions related to quantum anomalies such as chiral magnetic/vortical effects (CME/CVE), from which charged currents are induced by magnetic/vortical fields, has recently aroused great interest in the studies of relativistic heavy ion collisions (HIC) and Weyl semimetals Vilenkin (1979); Kharzeev et al. (2008); Fukushima et al. (2008); Li et al. (2014). Such effects and relevant phenomena associated with chiral imbalance have been investigated from various approaches including field-theory calculations based on Kubo formula Fukushima et al. (2008); Kharzeev and Warringa (2009); Landsteiner et al. (2011a), kinetic theory Gao et al. (2012); Son and Yamamoto (2012); Stephanov and Yin (2012); Son and Yamamoto (2013); Chen et al. (2013); Manuel and Torres-Rincon (2014a, b); Kharzeev et al. (2017); Huang et al. (2017); Gao et al. (2017); Ebihara et al. (2017), relativistic hydrodynamics Son and Surowka (2009); Neiman and Oz (2011); Sadofyev and Isachenkov (2011); Pu et al. (2011); Kharzeev and Yee (2011), lattice simulations Abramczyk et al. (2009); Buividovich et al. (2009a, b, 2010); Yamamoto (2011); Müller et al. (2016); Mace et al. (2017), and gauge/gravity duality Erdmenger et al. (2009); Torabian and Yee (2009); Banerjee et al. (2011); Landsteiner et al. (2011b). Particularly, recent progress in chiral kinetic theory (CKT) with the manifestation of Lorenz symmetry related to side jumps and the incorporation of collisions has improved our understandings on anomalous transport out of equilibrium Chen et al. (2014, 2015); Hidaka et al. (2017).
It is generally believed that CME in equilibrium is protected by the chiral anomaly and unaffected by interactions. On the other hand, CVE in equilibrium could be protected by mixed-axial-gravitational anomaly with only background fields Landsteiner et al. (2011b, a), whereas the modification from interactions could emerge in the presence of dynamical gauge fields Hou et al. (2012); Golkar and Son (2015). See Refs. Chowdhury and David (2015); Golkar and Sethi (2016); Chowdhury and David (2016) for some following works. In addition, there are studies for the axial currents induced by electric fields and chiral imbalance such as the chiral electric separation effect (CESE) or chiral Hall effect (CHE), which are pertinent to interactions Huang and Liao (2013); Pu et al. (2014, 2015), while these effects are not directly connected to the helicity of Weyl fermions and quantum anomalies. Nonetheless, an interplay between the chiral anomaly, magnetization current stemming from side jumps, and magnetic-moment coupling could result in non-equilibrium corrections involving interactions upon anomalous transport. For example, the alternative-current (AC) conductivity for CME with nonzero frequency is modified in the presence of time-dependent magnetic fields Kharzeev and Warringa (2009); Satow and Yee (2014); Kharzeev et al. (2017). Also, the nonlinear responses on anomalous transport triggered by fluctuations near local equilibrium in inviscid chiral fluids have been analyzed by some of the authors in this letter Hidaka et al. (2018), where novel anomalous Hall currents led by electric fields and temperature/chemical-potential gradients are found 111See Chen et al. (2016); Gorbar et al. (2016) for related studies in open systems, where the energy-momentum conservation led by hydrodynamics is not considered.. However, the non-equilibrium corrections on charge currents from magnetic fields or vorticity may appear in the viscous case 222The viscous effects in the absence of background fields are incorporated in the study presented in Ref. Gorbar et al. (2017b), while the authors focus on the hydrodynamic dispersion relation modified by vorticity and the charge current is not computed therein.. Experimentally, in HIC, due to the viscous corrections in the quark gluon plasma, the non-equilibrium effects upon the CME conductivity may impact the charge separation associated with the signal for CME. Moreover, based on recent observations of hydrodynamic transport in graphene Crossno et al. (2016); Ghahari et al. (2016), there exists mounting interest in the possible realization of chiral fluids in Weyl semimetals (see e.g., Refs. Lucas et al. (2016); Gorbar et al. (2017a) for relevant studies in theory). The theoretical investigation of relativistic viscous hydrodynamics of chiral fluids is thus imperative.
In this letter, we follow the approach in Ref. Hidaka et al. (2018) to further study the viscous corrections on anomalous transport contributed by the aforementioned quantum effects associated with the helicity of Weyl fermions. It is found that the CME and CVE conductivities of charge currents are modified by shear and bulk strengths. Nevertheless, the non-equilibrium corrections on the energy-density currents should vanish according to the matching condition in the classical relaxation-time (RT) approximation. Although the non-equilibrium quantum corrections are dissipative, the second law of thermodynamics is still satisfied.
Wigner functions and CKT.—We begin with a brief introduction to the Wigner-function formalism of CKT which will be exploited to study the non-equilibrium transport. As derived in Ref. Hidaka et al. (2017) by solving Dirac equations up to from the Wigner-function approach, the perturbative solution for the less propagators of right-handed Weyl fermions reads
where represents the sign of and
corresponds to the spin tensor depending on a frame vector Chen et al. (2015). The frame vector can be understood as the zeroth component of a vierbein for the local transformation of introduced on the local tangent space to depending on the global spacetime coordinates, where represent Pauli matrices for , such that . That is, we define with being the spacetime indices. In flat spacetime, we may set such that the spin connection vanishes. Now, the global spacetime coordinate transformation corresponds to the frame transformation .
Here we denote , where , with being less/greater self-energies and and being the distribution functions of incoming and outgoing particles, respectively. The second term in Eq. (1) associated with as the side-jump term only contributes for the non-equilibrium cases or for a rotating system in global equilibrium, which results in magnetization currents and CVE. On the other hand, the third term in Eq. (1) yields CME in equilibrium.
Given the Wigner functions, one can directly evaluate the charge current and energy-momentum tensor through
The distribution function in has to be solved from the CKT led by the Dirac equation. In light of the study in Ref. Hidaka et al. (2018), we focus on the fluctuations slightly away from the local equilibrium distribution function defined in a comoving frame with being a fluid velocity.
In such a frame, the CKT takes the form Hidaka et al. (2018),
For simplicity, we apply the RT approximation for the collisional kernel,
where denotes a vector with and represents the relaxation time charactering the inverse strength of interactions, which will be treated as a constant. In addition, we define for the electric and magnetic fields in the comoving frame. We will then follow the computations in Ref. Hidaka et al. (2018) to perturbatively solve for the non-equilibrium distribution function incorporating viscous corrections from the CKT in Eq. (Non-equilibrium Chiral Magnetic/Vortical Effects in Viscous Fluids) based on the and derivatives expansions. Here the local equilibrium distribution function takes the form with for and being the local temperature and chemical potential, respectively. Also, the vorticity is defined as .
Hydrodynamics and matching conditions.— Furthermore, following the energy-momentum conservation, we should also implement the anomalous hydrodynamic EOM led by
These two equations provide the physical constrains for CKT, which dictate the dynamics of thermodynamic parameters , , and in . However, by utilizing the Dirac equation as the origin of the CKT, it is shown in Ref. Hidaka et al. (2018) that the divergence of currents manifests the chiral anomaly,
where we denotes . Here the second term above should vanish as required by Eq. (6) for realistic collisions. In the classical RT approximation by setting , the vanishing second term results in the matching condition such that there exist no non-equilibrium corrections on the charge density Hidaka et al. (2018). That is, , where with being the local equilibrium current.
Moreover, performing similar computations as in the case for , the divergence of the energy-momentum gives rise to
In analogous to Eq. (7), the collisional part should vanish in light of Eq. (6), whereas this condition yields for the classical RT approximation. It turns out that the matching conditions for the classical RT approximation giving in the traditional kinetic theory also hold in the presence of corrections. However, the matching conditions do not constrain the possible corrections on and from .
Non-equilibrium responses and charge currents.— Following Ref. Hidaka et al. (2018), the perturbative solution for the non-equilibrium distribution function is given by
We may make the decomposition, , where the superindices and correspond to the classical and quantum corrections, respectively. We then further separate the part for ideal fluids, and the viscous correction, , where the subindices and denote the inviscid and viscous parts. The explicit expression of can be found in Ref. Hidaka et al. (2018). In light of Ref. Hidaka et al. (2018), we decompose the quantum corrections of the non-equilibrium distribution function into three parts as , where is led by the perturbative solution out of equilibrium solved from CKT and is attributed to the corrections of the temporal derivatives () upon , , and from hydrodynamic EOM obtained from Eq. (6). Finally, comes from the corrections in the collisional kernel, while this term depends on the assumption of in the RT approximation, which does not play a significant role in our analysis.
For convenience, hereafter we denote and . By implementing CKT with the RT approximation, in the local rest frame, we find
where we define as a projection operator with the Minkowski metric , as the bulk strength, as the shear strength, and as the transverse component of an arbitrary vector .
Next, we shall consider the viscous corrections on the hydrodynamic EOM. By solving Eq. (6) with the current and energy-momentum tensor in local equilibrium, it is found that and with , while the viscous correction does not lead to corrections for and . We also include their contributions for the computation of currents. In addition, we find that the viscous correction gives rise to the correction upon and accordingly the hydrodynamic EOM results in
where and with being pressure. On the other hand, the corrections in collisions give
From Eqs. (1) and (Non-equilibrium Chiral Magnetic/Vortical Effects in Viscous Fluids), the quantum corrections of the non-equilibrium current reads
When not applying the hydrodynamic EOM, only contributes and one finds
where . The results suggest that the viscous corrections upon CME and CVE conductivities should exist even for an open system in which the back-reaction on environments is neglected and the energy-momentum conservation is violated such as the case in Weyl semimetals when the scattering between quasi-particles and impurities dominate the interactions among quasi-particles.
Now, for the right-handed chiral fluid as a closed systems with energy-momentum conservation, by implementing the hydrodynamic EOM, the coefficients become
where . Note that there exist two terms in , and , which also contribute to viscous corrections with vorticity when the hydrodynamic EOM are applied in computations. In general, by redefining , one can shift the above corrections on CME/CVE conductivities to the transport coefficients of energy density currents. Regarding the validity of our findings, due to the gradient expansion, the results should be legitimate for , which imparts an upper bound for . In addition, given that the CKT itself is subject to weakly coupled systems, cannot be too small. It is more enlightening to simplify the expressions of and in distinct limits. In the high-temperature limit , the coefficients in Eq. (15) reduce to
On the contrary, in the low-temperature limit , one obtains
So far, we have only considered the contributions for right-handed fermions. The quantum corrections for left-handed fermions will yield the same results but with the change of an overall sign for each term at . Since and are even under the parity () transformation, it is anticipated that the transport coefficients of the non-equilibrium corrections on CME/CVE have the same parity as those in equilibrium, which can be more apparently observed from the simplified expressions shown in two limits above. Therefore, the bulk and shear strengths not only affect the vector currents () induced by CME/CVE with nonzero axial-charge chemical potentials () but also the axial currents () from the chiral separation effect(CSE) 333Although CSE is known as a dual effect for CME in the vector/axial bases Fukushima et al. (2008), it is automatically included in the CME currents in the right/left-handed bases. and CVE with nonzero vector-charge chemical potentials (). It is worthwhile to note that such second-order quantum corrections on currents have different symmetry properties compared to the second-order classical effects also pertinent to magnetic fields such as the Hall-diffusion currents in Ref. Gorbar et al. (2016). Because there is no sign flipping for right/left-handed fermions in the classical case, the corresponding axial current can only be generated when (or ), which is analogous to CESE Huang and Liao (2013); Pu et al. (2014).
Entropy production.— In contrast to the anomalous transport in equilibrium, the non-equilibrium quantum corrections of the charge current are dissipative. This is foreseen by the time-reversal symmetry (). Since the charge current , , and are -odd and is -even, from the classical ohmic current led by the RT approximation , one finds that is also -odd, whereas the CME/CVE conductivities in equilibrium are -even and thus non-dissipative. Nonetheless, because and are both -odd, the corresponding transport coefficients of the viscous corrections upon chiral magnetic/vortical currents are proportional to multiplied by -even functions of and , which accordingly yield dissipation. The same arguments can be applied to the non-equilibrium transport found in inviscid cases Hidaka et al. (2018). Notably, in light of symmetry, one finds or for viscous corrections. Consequently, the corrections in practical collisions can only affect the prefactors of transport coefficients without altering their structures.
Albeit the non-equilibrium quantum transport results in dissipation, its entropy production is suppressed by the dissipation from classical effects as shown below. We may introduce the entropy density current in an usual form as for relativistic hydrodynamics,
where the non-dissipative corrections proportional to and originating from the non-dissipative charge and energy-density currents. The coefficients and should be determined by the transport coefficients of CME and CVE in equilibrium. See e.g., Ref. Son and Surowka (2009) for details. The explicit form of is not important in our discussion.
The constitutive relation now can be written as , where denotes the dissipative corrections characterized by viscous effects and corresponds to the non-dissipative corrections led by anomalous effects in equilibrium. In the classical RT approximation, based on the matching conditions such that , we find
In Eq. (21), the classical contributions result in positive entropy production at , while the corresponding quantum corrections are at . Although the non-equilibrium quantum corrections here could be either positive or negative, they are always suppressed by the classical contributions and the second law of thermodynamics is satisfied.
Acknowledgements.Y. H. was partially supported by Japan Society of Promotion of Science (JSPS), Grants-in-Aid for Scientific Research (KAKENHI) Grants No. 15H03652, 16K17716, and 17H06462. Y. H. was also partially supported by RIKEN iTHES Project and iTHEMS Program. D. Y. was supported by the RIKEN Foreign Postdoctoral Researcher program. Supplementary materials : Divergence of the energy-momentum tensor.— Here we present some critical steps for the derivation of Eq. (8). Following the trick in Ref. Gorbar et al. (2017b), we find
Performing straightforward computations, one should obtain
which results in
where the first term on the right-hand side of the first equality in fact vanishes. On the other hand, we find
By performing the integration by part and dropping the divergent and vanishing surface terms, we obtain
Combining Eqs. (Non-equilibrium Chiral Magnetic/Vortical Effects in Viscous Fluids) and (26), we acquire Eq. (8).
- Vilenkin (1979) A. Vilenkin, Phys. Rev. D20, 1807 (1979).
- Kharzeev et al. (2008) D. E. Kharzeev, L. D. McLerran, and H. J. Warringa, Nucl. Phys. A803, 227 (2008), arXiv:0711.0950 [hep-ph] .
- Fukushima et al. (2008) K. Fukushima, D. E. Kharzeev, and H. J. Warringa, Phys. Rev. D78, 074033 (2008), arXiv:0808.3382 [hep-ph] .
- Li et al. (2014) Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic, et al., (2014), arXiv:1412.6543 [cond-mat.str-el] .
- Kharzeev and Warringa (2009) D. E. Kharzeev and H. J. Warringa, Phys. Rev. D80, 034028 (2009), arXiv:0907.5007 [hep-ph] .
- Landsteiner et al. (2011a) K. Landsteiner, E. Megias, and F. Pena-Benitez, Phys. Rev. Lett. 107, 021601 (2011a), arXiv:1103.5006 [hep-ph] .
- Gao et al. (2012) J.-H. Gao, Z.-T. Liang, S. Pu, Q. Wang, and X.-N. Wang, Phys. Rev. Lett. 109, 232301 (2012), arXiv:1203.0725 [hep-ph] .
- Son and Yamamoto (2012) D. T. Son and N. Yamamoto, Phys. Rev. Lett. 109, 181602 (2012), arXiv:1203.2697 [cond-mat.mes-hall] .
- Stephanov and Yin (2012) M. Stephanov and Y. Yin, Phys. Rev. Lett. 109, 162001 (2012), arXiv:1207.0747 [hep-th] .
- Son and Yamamoto (2013) D. T. Son and N. Yamamoto, Phys. Rev. D87, 085016 (2013), arXiv:1210.8158 [hep-th] .
- Chen et al. (2013) J.-W. Chen, S. Pu, Q. Wang, and X.-N. Wang, Phys. Rev. Lett. 110, 262301 (2013), arXiv:1210.8312 [hep-th] .
- Manuel and Torres-Rincon (2014a) C. Manuel and J. M. Torres-Rincon, Phys. Rev. D89, 096002 (2014a), arXiv:1312.1158 [hep-ph] .
- Manuel and Torres-Rincon (2014b) C. Manuel and J. M. Torres-Rincon, Phys. Rev. D90, 076007 (2014b), arXiv:1404.6409 [hep-ph] .
- Kharzeev et al. (2017) D. E. Kharzeev, M. A. Stephanov, and H.-U. Yee, Phys. Rev. D95, 051901 (2017), arXiv:1612.01674 [hep-ph] .
- Huang et al. (2017) A. Huang, Y. Jiang, S. Shi, J. Liao, and P. Zhuang, (2017), arXiv:1703.08856 [hep-ph] .
- Gao et al. (2017) J.-h. Gao, S. Pu, and Q. Wang, Phys. Rev. D96, 016002 (2017), arXiv:1704.00244 [nucl-th] .
- Ebihara et al. (2017) S. Ebihara, K. Fukushima, and S. Pu, Phys. Rev. D96, 016016 (2017), arXiv:1705.08611 [hep-ph] .
- Son and Surowka (2009) D. T. Son and P. Surowka, Phys. Rev. Lett. 103, 191601 (2009), arXiv:0906.5044 [hep-th] .
- Neiman and Oz (2011) Y. Neiman and Y. Oz, JHEP 03, 023 (2011), arXiv:1011.5107 [hep-th] .
- Sadofyev and Isachenkov (2011) A. V. Sadofyev and M. V. Isachenkov, Phys. Lett. B697, 404 (2011), arXiv:1010.1550 [hep-th] .
- Pu et al. (2011) S. Pu, J.-h. Gao, and Q. Wang, Phys. Rev. D83, 094017 (2011), arXiv:1008.2418 [nucl-th] .
- Kharzeev and Yee (2011) D. E. Kharzeev and H.-U. Yee, Phys. Rev. D84, 045025 (2011), arXiv:1105.6360 [hep-th] .
- Abramczyk et al. (2009) M. Abramczyk, T. Blum, G. Petropoulos, and R. Zhou, PoS LAT2009, 181 (2009), arXiv:0911.1348 [hep-lat] .
- Buividovich et al. (2009a) P. Buividovich, M. Chernodub, E. Luschevskaya, and M. Polikarpov, Phys. Rev. D80, 054503 (2009a), arXiv:0907.0494 [hep-lat] .
- Buividovich et al. (2009b) P. Buividovich, E. Luschevskaya, M. Polikarpov, and M. Chernodub, JETP Lett. 90, 412 (2009b).
- Buividovich et al. (2010) P. Buividovich, M. Chernodub, D. Kharzeev, T. Kalaydzhyan, E. Luschevskaya, et al., Phys. Rev. Lett. 105, 132001 (2010), arXiv:1003.2180 [hep-lat] .
- Yamamoto (2011) A. Yamamoto, Phys. Rev. Lett. 107, 031601 (2011), arXiv:1105.0385 [hep-lat] .
- Müller et al. (2016) N. Müller, S. Schlichting, and S. Sharma, Phys. Rev. Lett. 117, 142301 (2016), arXiv:1606.00342 [hep-ph] .
- Mace et al. (2017) M. Mace, N. Mueller, S. Schlichting, and S. Sharma, Phys. Rev. D95, 036023 (2017), arXiv:1612.02477 [hep-lat] .
- Erdmenger et al. (2009) J. Erdmenger, M. Haack, M. Kaminski, and A. Yarom, JHEP 01, 055 (2009), arXiv:0809.2488 [hep-th] .
- Torabian and Yee (2009) M. Torabian and H.-U. Yee, JHEP 08, 020 (2009), arXiv:0903.4894 [hep-th] .
- Banerjee et al. (2011) N. Banerjee, J. Bhattacharya, S. Bhattacharyya, S. Dutta, R. Loganayagam, et al., JHEP 1101, 094 (2011), arXiv:0809.2596 [hep-th] .
- Landsteiner et al. (2011b) K. Landsteiner, E. Megias, L. Melgar, and F. Pena-Benitez, JHEP 1109, 121 (2011b), arXiv:1107.0368 [hep-th] .
- Chen et al. (2014) J.-Y. Chen, D. T. Son, M. A. Stephanov, H.-U. Yee, and Y. Yin, Phys. Rev. Lett. 113, 182302 (2014), arXiv:1404.5963 [hep-th] .
- Chen et al. (2015) J.-Y. Chen, D. T. Son, and M. A. Stephanov, Phys. Rev. Lett. 115, 021601 (2015), arXiv:1502.06966 [hep-th] .
- Hidaka et al. (2017) Y. Hidaka, S. Pu, and D.-L. Yang, Phys. Rev. D95, 091901 (2017), arXiv:1612.04630 [hep-th] .
- Hou et al. (2012) D.-F. Hou, H. Liu, and H.-c. Ren, Phys. Rev. D86, 121703 (2012), arXiv:1210.0969 [hep-th] .
- Golkar and Son (2015) S. Golkar and D. T. Son, JHEP 02, 169 (2015), arXiv:1207.5806 [hep-th] .
- Chowdhury and David (2015) S. D. Chowdhury and J. R. David, JHEP 11, 048 (2015), arXiv:1508.01608 [hep-th] .
- Golkar and Sethi (2016) S. Golkar and S. Sethi, JHEP 05, 105 (2016), arXiv:1512.02607 [hep-th] .
- Chowdhury and David (2016) S. D. Chowdhury and J. R. David, JHEP 12, 116 (2016), arXiv:1604.05003 [hep-th] .
- Huang and Liao (2013) X.-G. Huang and J. Liao, Phys. Rev. Lett. 110, 232302 (2013), arXiv:1303.7192 [nucl-th] .
- Pu et al. (2014) S. Pu, S.-Y. Wu, and D.-L. Yang, Phys. Rev. D89, 085024 (2014), arXiv:1401.6972 [hep-th] .
- Pu et al. (2015) S. Pu, S.-Y. Wu, and D.-L. Yang, Phys. Rev. D91, 025011 (2015), arXiv:1407.3168 [hep-th] .
- Satow and Yee (2014) D. Satow and H.-U. Yee, Phys. Rev. D90, 014027 (2014), arXiv:1406.1150 [hep-ph] .
- Hidaka et al. (2018) Y. Hidaka, S. Pu, and D.-L. Yang, Phys. Rev. D97, 016004 (2018), arXiv:1710.00278 [hep-th] .
- (47) See Chen et al. (2016); Gorbar et al. (2016) for related studies in open systems, where the energy-momentum conservation led by hydrodynamics is not considered.
- (48) The viscous effects in the absence of background fields are incorporated in the study presented in Ref. Gorbar et al. (2017b), while the authors focus on the hydrodynamic dispersion relation modified by vorticity and the charge current is not computed therein.
- Crossno et al. (2016) J. Crossno, J. K. Shi, K. Wang, X. Liu, A. Harzheim, A. Lucas, S. Sachdev, P. Kim, T. Taniguchi, K. Watanabe, T. A. Ohki, and K. C. Fong, Science 351, 1058 (2016), http://science.sciencemag.org/content/351/6277/1058.full.pdf .
- Ghahari et al. (2016) F. Ghahari, H.-Y. Xie, T. Taniguchi, K. Watanabe, M. S. Foster, and P. Kim, Phys. Rev. Lett. 116, 136802 (2016).
- Lucas et al. (2016) A. Lucas, R. A. Davison, and S. Sachdev, Proc. Nat. Acad. Sci. 113, 9463 (2016), arXiv:1604.08598 [cond-mat.str-el] .
- Gorbar et al. (2017a) E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and P. O. Sukhachov, (2017a), arXiv:1712.01289 [cond-mat.str-el] .
- (53) Although CSE is known as a dual effect for CME in the vector/axial bases Fukushima et al. (2008), it is automatically included in the CME currents in the right/left-handed bases.
- Gorbar et al. (2016) E. V. Gorbar, I. A. Shovkovy, S. Vilchinskii, I. Rudenok, A. Boyarsky, and O. Ruchayskiy, Phys. Rev. D93, 105028 (2016), arXiv:1603.03442 [hep-th] .
- Gorbar et al. (2017b) E. V. Gorbar, D. O. Rybalka, and I. A. Shovkovy, Phys. Rev. D95, 096010 (2017b), arXiv:1702.07791 [hep-th] .
- Chen et al. (2016) J.-W. Chen, T. Ishii, S. Pu, and N. Yamamoto, Phys. Rev. D93, 125023 (2016), arXiv:1603.03620 [hep-th] .