Holographic DC conductivity for backreacted nonlinear electrodynamics with momentum dissipation
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Abstract
We consider a holographic model with the charge current dual to a general nonlinear electrodynamics (NLED) field. Taking into account the backreaction of the NLED field on the geometry and introducing axionic scalars to generate momentum dissipation, we obtain expressions for DC conductivities with a finite magnetic field. The properties of the inplane resistance are examined in several NLED models. For Maxwell–Chern–Simons electrodynamics, negative magnetoresistance and Mottlike behavior could appear in some parameter space region. Depending on the sign of the parameters, we expect the NLED models to mimic some type of weak or strong interactions between electrons. In the latter case, negative magnetoresistance and Mottlike behavior can be realized at low temperatures. Moreover, the Mott insulator to metal transition induced by a magnetic field is also observed at low temperatures.
1 Introduction
Gauge/gravity duality [1, 2, 3, 4] has provided powerful tools for exploring the behavior of strongly coupled quantum phases of matter, and some remarkable progresses have been made [5, 6, 7, 8, 9]. Conductivity is an important transport quantity in condensed matter, and the gauge/gravity duality provides a framework to compute it for strongly interacting field theories.
Studying the behavior of the conductivity in the presence of external magnetic fields can help us to better understand the transport properties of materials. For normal metals, the resistance is a monotonically increasing function of the magnetic field [10], which appears as positive magnetoresistance. However, negative magnetoresistance has been observed in several experiments [11, 12, 13]. On the other hand, the behavior of negative magnetoresistance was found in strongly coupled holographic chiral anomalous systems [14, 15, 16, 17]. In [18], it showed that negative magnetoresistance could also arise in nonanomalous relativistic fluids due to the distinctive gradient expansion. Note that the transport phenomena in the presence of Weyl corrections have also been discussed in [19, 20, 21]. Recently, the magnetotransport of a strongly interacting system in \(2+1\) dimensions was examined in a holographic DiracBornInfeld model in [22, 23]. Negative magnetoresistance was found for a family of dynoic solutions in [23]. The DC conductivity in the probe DBI case with the vanishing magnetic field was also discussed in [24].
Mott insulators can be parent materials of high \(T_{c}\) cuprate superconductors. A Mott insulator has an insulating ground state driven by Coulomb repulsion. Mottlike behavior is that strong interactions between electrons would prevent the charge carriers to efficiently transport charges. Constructing a holographic model describing Mott insulators is still a challenging task. In [25, 26, 27, 28], dynamically generating a Mott gap has been proposed in holographic models by considering fermions with dipole coupling. A holographic construction of the largeN Bose–Hubbard model was presented in [29], and the model admitted Mott insulator ground states in the limit of large Coulomb repulsion. Some other holographic models dual to Mott insulators include [30, 31, 32]. Recently, a holographic model using a particular type of NLED, namely iDBI, was proposed in [33] to mimic interactions between electrons by selfinteractions of the NLED field. It showed that Mottlike behavior appeared for large enough selfinteraction strength.
In this paper, we extend the analysis of the magnetotransport in a holographic DiracBornInfeld model in [23] to a general NLED model. As in [23], our analysis is performed in a full backreacted fashion. To break translational symmetry, we follow the method in [34] to add axionic scalars, which depend on the spatial directions linearly.
The rest of this article is organized as follows. In Sect. 2, we set up our holographic model. The expressions for the DC conductivities with a finite magnetic field are obtained in Sect. 3. Some limiting cases, including high temperature limit, are then discussed. In Sect. 4, the dependence of the inplane resistance on the temperature, the charge density and the magnetic field are investigated for Maxwell, Maxwell–Chern–Simons, BornInfeld, square and logarithmic electrodynamics. In Sect. 5, we summarize our results and conclude with a brief discussion.
2 Holographic setup
3 DC conductivity
Via gauge/gravity duality, the black brane solution (8) describes an equilibrium state at finite temperature T, which is given by Eq. (14). The NLED field is a U\(\left( 1\right) \) gauge field and dual to a conserved current \(\mathcal {J}^{\mu }\) in the boundary theory. In this section, we calculate the DC conductivities for \(\mathcal {J}^{\mu }\) using the method developed in [35, 36].
3.1 Derivation of DC conductivity
3.2 Various limiting cases
In Sect. 4, we will use Eqs. (30) to discuss the properties of the DC conductivities in some NLED models. Before focusing on a specific model, we now consider some limiting cases of the general formulae for \(\sigma _{ij}\) or \(R_{ij}\).
3.2.1 Weak and strong dissipation limits
3.2.2 Vanishing magnetic field and charge density
3.2.3 High temperature limit
 Green Region: In this region, one has that \(\partial R_{xx} /\partial \left h\right <0\). To describe how the electrical resistance responds to an externallyapplied magnetic field, one can define magnetoresistance asSo the green region has negative magnetoresistance at given temperature and charge density.$$\begin{aligned} MR=\frac{R_{xx}\left( h\right) R_{xx}\left( 0\right) }{R_{xx}\left( 0\right) }\text {.} \end{aligned}$$(41)

Yellow Region: In this region, one has that \(\partial R_{xx} /\partial \left \rho \right >0\). This is Mottlike behavior, which can be explained by the electronic traffic jam: strong enough ee interactions prevent the available mobile charge carriers to efficiently transport charges. In particular, when \(h=0\), Eq. (40) gives that \(\partial R_{xx}/\partial \left \rho \right >0\) as long as \(\theta ^{2}>1\).
4 Examples
In this section, we will use Eqs. (14), (23) and (30) to study the dependence of the inplane resistance \(R_{xx}\) on the temperature T, the charge density \(\rho \) and the magnetic field h in Maxwell, Maxwell–Chern–Simons, BornInfeld, square and logarithmic electrodynamics. The behavior of \(R_{xx}\) in the high temperature limit has already been discussed in Sect. 3. So we will focus on the behavior of \(R_{xx}\) around \(T=0\) in this section.
4.1 Maxwell electrodynamics
4.2 Maxwell–Chern–Simons electrodynamics
4.3 BornInfeld electrodynamics
4.4 Square electrodynamics
4.5 Logarithmic electrodynamics
5 Discussion and conclusion
The dependence of the inplane resistance \(R_{xx}\) on \(\rho /\alpha ^{2}\) and \(h/\alpha ^{2}\) at \(T=0\). Here \(\alpha \) is a parameter responsible for generating momentum dissipation. Note that \(\partial _{\left h\right }R_{xx}<0\) means negative magnetoresistance while \(\partial _{\left \rho \right }R_{xx}<0\) means Mottlike behavior
Lagrangian  Parameter  \(\rho \) dependence of \(R_{xx}\)  h dependence of \(R_{xx}\)  

Maxwell  s  \(\partial _{\left \rho \right }R_{xx}<0\). See Fig. 2a.  \(\partial _{\left h\right }R_{xx}>0\). See Fig. 2a.  
Maxwell–Chern–Simons  \(s+\theta p\)  \(\theta \ne 0\)  There exists parameter space region for \(\partial _{\left \rho \right }R_{xx}>0\). See Fig. 5.  There exists parameter space region for \(\partial _{\left h\right }R_{xx}<0\). See Eq. (47). 
BornInfeld  \(\frac{1\sqrt{12asa^{2}p^{2}}}{a}\)  \(a>0\)  \(\partial _{\left \rho \right }R_{xx}<0\). See Fig. 6a.  \(\partial _{\left h\right }R_{xx}>0\). See Fig. 6a. 
\(a=0.4\)  \(\partial _{\left \rho \right }R_{xx}>0\). See Fig. 7a.  \(\partial _{\left h\right }R_{xx}<0\). See Fig. 7b.  
\(a=1\)  \(\partial _{\left \rho \right }R_{xx}>0\) for small values of \(h/\alpha ^{2}\) and \(\rho /\alpha ^{2}\). See Fig. 7c.  \(\partial _{\left h\right }R_{xx}<0\). See Fig. 7d.  
Square  \(\frac{1\sqrt{12as}}{a}\)  \(a>0\)  \(\partial _{\left \rho \right }R_{xx}<0\).  \(\partial _{\left h\right }R_{xx}>0.\) 
\(a=0.4\)  \(\partial _{\left \rho \right }R_{xx}>0\) for small values of \(h/\alpha ^{2}\). For larger values of \(h/\alpha ^{2}\), \(\partial _{\left \rho \right }R_{xx}>0\) only for large enough values of \(\rho /\alpha ^{2}\). See Fig. 10b.  \(\partial _{\left h\right }R_{xx}<0\). See Fig. 10a.  
\(a=1\)  \(\partial _{\left \rho \right }R_{xx}>0\). See Fig. 10d.  \(\partial _{\left h\right }R_{xx}<0\). See Fig. 10c.  
Logarithmic  \(\frac{\log \left( 1as\frac{a^{2}p^{2}}{2}\right) }{a}\)  \(a>0\)  \(\partial _{\left \rho \right }R_{xx}<0.\)  \(\partial _{\left h\right }R_{xx}>0.\) 
\(a=1\)  \(\partial _{\left \rho \right }R_{xx}>0\) for small values of \(h/\alpha ^{2}\). See Fig. 11b.  \(\partial _{\left h\right }R_{xx}<0\). See Fig. 11a. 
It is interesting to note that BornInfeld electrodynamics with \(a=1\) could describe Mott insulator to metal transition (IMT) induced by a magnetic field. In fact, Fig. 8d shows that, at low temperatures, the system has insulating/metallic behavior for a weak/strong magnetic field. On the other hand, Fig. 7c displays that, for a weak magnetic field at low temperatures, the system usually has Mottlike behavior and hence is a MottInsulator. When the magnetic field grows strong enough, Mottlike behavior disappears, and meanwhile, the system exhibits metallic behavior. A magnetic fieldinduced IMT for a Mott system, namely a bilayer ruthenate, Tidoped Ca\(_{3}\)Ru\(_{2}\)O\(_{7}\), was presented in [40]. Our analysis for IMT is rather qualitative, and it deserves future more detailed studies.
For a negative enough a, we found that, at low temperatures, the resistance \(R_{xx}\) always decreases with increasing magnetic field, which appears as negative magnetoresistance. On the other hand, the behavior of \(R_{xx}\) in the \(a>0\) NLED models is similar to that in Maxwell electrodynamics, in which one always have positiveresistance. It seems that, decreasing a from a positive value to a negative one, which corresponds to increasing the strength of the interactions between electrons, would lead the magnetoresistance to change from positive to negative. It showed in [41] that the magnetoresistance could change from positive to negative by gradually introducing artificial disorder through Ga\(^{+}\) ion irradiation to pristine graphene. To relate our results to the experiments, we need to better understand how a is related to external control parameters.
In this paper, we found the \(\sigma _{ij}/R_{ij}\) expressions for a general NLED field. Our analyses for the properties of the resistance in NLED models are preliminary. One can use these expressions to find or construct a NLED model to realize some interesting experimental results, such as the scaling relationship between applied magnetic field and temperature observed in the magnetoresistance of the pnictide superconductor.
Footnotes
 1.
In fact, the NLED Lagrangian \(\mathcal {L}\left( s,p\right) =\frac{1}{a}\left( 1\sqrt{12as}\right) \), instead of Eq. (50), was used in [33]. However, for the \(h=0\) case, these two Lagrangian would give the same result for \(\sigma _{xx}\) since they have the same value of \(\mathcal {L}^{\left( 1,0\right) }\left( s,0\right) \).
 2.
In [33], the role of a has been discussed for iDBI model.
Notes
Acknowledgements
We are grateful to Zheng Sun for useful discussions and valuable comments. This work is supported in part by NSFC (Grant No. 11005016, 11175039 and 11375121).
References
 1.T. Banks, W. Fischler, S.H. Shenker, L. Susskind, M theory as a matrix model: A conjecture. Phys. Rev. D 55, 5112 (1997). https://doi.org/10.1103/PhysRevD.55.5112. arXiv:hepth/9610043 ADSMathSciNetCrossRefzbMATHGoogle Scholar
 2.J.M. Maldacena, The Large N limit of superconformal field theories and supergravity. Int. J. Theor. Phys. 38, 1113 (1999)MathSciNetCrossRefGoogle Scholar
 3.J.M. Maldacena, The large N limit of superconformal field theories and supergravity. Adv. Theor. Math. Phys. 2, 231 (1998). https://doi.org/10.1023/A:1026654312961. arXiv:hepth/9711200 ADSMathSciNetCrossRefzbMATHGoogle Scholar
 4.E. Witten, Antide Sitter space and holography. Adv. Theor. Math. Phys. 2, 253 (1998). https://doi.org/10.4310/ATMP.1998.v2.n2.a2. arXiv:hepth/9802150 ADSMathSciNetCrossRefzbMATHGoogle Scholar
 5.S.S. Gubser, Breaking an Abelian gauge symmetry near a black hole horizon. Phys. Rev. D 78, 065034 (2008). https://doi.org/10.1103/PhysRevD.78.065034. arXiv:0801.2977 [hepth]ADSCrossRefGoogle Scholar
 6.S.A. Hartnoll, C.P. Herzog, G.T. Horowitz, Building a holographic superconductor. Phys. Rev. Lett. 101, 031601 (2008). https://doi.org/10.1103/PhysRevLett.101.031601. arXiv:0803.3295 [hepth]ADSCrossRefGoogle Scholar
 7.S.S. Lee, A nonFermi liquid from a charged black hole: A critical Fermi ball. Phys. Rev. D 79, 086006 (2009). https://doi.org/10.1103/PhysRevD.79.086006. arXiv:0809.3402 [hepth]ADSCrossRefGoogle Scholar
 8.H. Liu, J. McGreevy, D. Vegh, NonFermi liquids from holography. Phys. Rev. D 83, 065029 (2011). https://doi.org/10.1103/PhysRevD.83.065029. arXiv:0903.2477 [hepth]ADSCrossRefGoogle Scholar
 9.M. Cubrovic, J. Zaanen, K. Schalm, String theory, quantum phase transitions and the emergent Fermiliquid. Science 325, 439 (2009). https://doi.org/10.1126/science.1174962. arXiv:0904.1993 [hepth]ADSMathSciNetCrossRefzbMATHGoogle Scholar
 10.G.H. Wannier, Theorem on the magnetoconductivity of metals. Phys. Rev. B 5, 3836 (1972)ADSCrossRefGoogle Scholar
 11.H. Negishi, H. Yamada, K. Yuri, M. Sasaki, M. Inoue, Negative magnetoresistance in crystals of the paramagnetic intercalation compound \(\text{ Mn }_{x}\text{ TiS }_{2}\). Phys. Rev. B 56, 11144 (1997)ADSCrossRefGoogle Scholar
 12.C.Z. Li, Giant negative magnetoresistance induced by the chiral anomaly in individual Cd3As2 nanowires. Nat. Commun. 6, 10137 (2015). arXiv:1504.07398 [condmat.strel]ADSCrossRefGoogle Scholar
 13.H.J. Kim, K.S. Kim, J.F. Wang, M. Sasaki, N. Satoh, A. Ohnishi, M. Kitaura, M. Yang, L. Li, Dirac vs. Weyl in topological insulators: AdlerBellJackiw anomaly in transport phenomena. Phys. Rev. Lett. 111, 246603 (2013). arXiv:1307.6990 [condmat.strel]ADSCrossRefGoogle Scholar
 14.A. JimenezAlba, K. Landsteiner, L. Melgar, Anomalous magnetoresponse and the Stückelberg axion in holography. Phys. Rev. D 90, 126004 (2014). arXiv:1407.8162 [hepth]ADSCrossRefGoogle Scholar
 15.A. JimenezAlba, K. Landsteiner, Y. Liu, Y.W. Sun, Anomalous magnetoconductivity and relaxation times in holography. JHEP 1507, 117 (2015). https://doi.org/10.1007/JHEP07(2015)117. arXiv:1504.06566 [hepth]ADSCrossRefGoogle Scholar
 16.K. Landsteiner, Y. Liu, Y.W. Sun, Negative magnetoresistivity in chiral fluids and holography. JHEP 1503, 127 (2015). https://doi.org/10.1007/JHEP03(2015)127. arXiv:1410.6399 [hepth]MathSciNetCrossRefzbMATHGoogle Scholar
 17.Y.W. Sun, Q. Yang, Negative magnetoresistivity in holography. JHEP 1609, 122 (2016). https://doi.org/10.1007/JHEP09(2016)122. arXiv:1603.02624 [hepth]ADSMathSciNetCrossRefzbMATHGoogle Scholar
 18.A. Baumgartner, A. Karch, A. Lucas, Magnetoresistance in relativistic hydrodynamics without anomalies. JHEP 1706, 054 (2017). https://doi.org/10.1007/JHEP06(2017)054. arXiv:1704.01592 [hepth]ADSMathSciNetCrossRefzbMATHGoogle Scholar
 19.A. Mokhtari, S.A. Hosseini Mansoori, K. Bitaghsir Fadafan, Diffusivities bounds in the presence of Weyl corrections, arXiv:1710.03738 [hepth]
 20.C.S. Chu, R.X. Miao, Anomaly Induced Transport in Boundary Quantum Field Theories, arXiv:1803.03068 [hepth]
 21.C.S. Chu, R.X. Miao, Anomalous Transport in Holographic Boundary Conformal Field Theories, arXiv:1804.01648 [hepth]
 22.E. Kiritsis, L. Li, Quantum criticality and DBI magnetoresistance. J. Phys. A 50(11), 115402 (2017). https://doi.org/10.1088/17518121/aa59c6. arXiv:1608.02598 [condmat.strel]ADSMathSciNetCrossRefzbMATHGoogle Scholar
 23.S. Cremonini, A. Hoover, L. Li, Backreacted DBI magnetotransport with momentum dissipation. JHEP 1710, 133 (2017). https://doi.org/10.1007/JHEP10(2017)133. arXiv:1707.01505 [hepth]ADSMathSciNetCrossRefGoogle Scholar
 24.C. Charmousis, B. Gouteraux, B.S. Kim, E. Kiritsis, R. Meyer, Effective holographic theories for lowtemperature condensed matter systems. JHEP 1011, 151 (2010). https://doi.org/10.1007/JHEP11(2010)151. arXiv:1005.4690 [hepth]ADSCrossRefzbMATHGoogle Scholar
 25.M. Edalati, R.G. Leigh, P.W. Phillips, Dynamically generated Mott gap from holography. Phys. Rev. Lett. 106, 091602 (2011). https://doi.org/10.1103/PhysRevLett.106.091602. arXiv:1010.3238 [hepth]ADSCrossRefGoogle Scholar
 26.M. Edalati, R.G. Leigh, K.W. Lo, P.W. Phillips, Dynamical gap and cupratelike physics from holography. Phys. Rev. D 83, 046012 (2011). https://doi.org/10.1103/PhysRevD.83.046012. arXiv:1012.3751 [hepth]ADSCrossRefGoogle Scholar
 27.J.P. Wu, H.B. Zeng, Dynamic gap from holographic fermions in charged dilaton black branes. JHEP 1204, 068 (2012). https://doi.org/10.1007/JHEP04(2012)068. arXiv:1201.2485 [hepth]ADSCrossRefGoogle Scholar
 28.Y. Ling, P. Liu, C. Niu, J.P. Wu, Z.Y. Xian, Holographic fermionic system with dipole coupling on Qlattice. JHEP 1412, 149 (2014). https://doi.org/10.1007/JHEP12(2014)149. arXiv:1410.7323 [hepth]ADSCrossRefGoogle Scholar
 29.M. Fujita, S. Harrison, A. Karch, R. Meyer, N.M. Paquette, Towards a holographic BoseHubbard model. JHEP 1504, 068 (2015). https://doi.org/10.1007/JHEP04(2015)068. arXiv:1411.7899 [hepth]ADSMathSciNetCrossRefzbMATHGoogle Scholar
 30.Y. Ling, P. Liu, C. Niu, J.P. Wu, Building a doped Mott system by holography. Phys. Rev. D 92(8), 086003 (2015). https://doi.org/10.1103/PhysRevD.92.086003. arXiv:1507.02514 [hepth]ADSCrossRefGoogle Scholar
 31.T. Nishioka, S. Ryu, T. Takayanagi, Holographic superconductor/insulator transition at zero temperature. JHEP 1003, 131 (2010). https://doi.org/10.1007/JHEP03(2010)131. arXiv:0911.0962 [hepth]ADSCrossRefzbMATHGoogle Scholar
 32.E. Kiritsis, J. Ren, On holographic insulators and supersolids. JHEP 1509, 168 (2015). https://doi.org/10.1007/JHEP09(2015)168. arXiv:1503.03481 [hepth]ADSCrossRefGoogle Scholar
 33.M. Baggioli, O. Pujolas, On effective holographic Mott insulators. JHEP 1612, 107 (2016). https://doi.org/10.1007/JHEP12(2016)107. arXiv:1604.08915 [hepth]ADSCrossRefGoogle Scholar
 34.T. Andrade, B. Withers, A simple holographic model of momentum relaxation. JHEP 1405, 101 (2014). https://doi.org/10.1007/JHEP05(2014)101. arXiv:1311.5157 [hepth]ADSCrossRefGoogle Scholar
 35.A. Donos, J.P. Gauntlett, Novel metals and insulators from holography. JHEP 1406, 007 (2014). https://doi.org/10.1007/JHEP06(2014)007. arXiv:1401.5077 [hepth]ADSCrossRefGoogle Scholar
 36.M. Blake, A. Donos, Quantum critical transport and the Hall angle. Phys. Rev. Lett. 114(2), 021601 (2015). https://doi.org/10.1103/PhysRevLett.114.021601. arXiv:1406.1659 [hepth]ADSCrossRefGoogle Scholar
 37.X. Guo, P. Wang, H. Yang, Membrane Paradigm and Holographic DC Conductivity for Nonlinear Electrodynamics, arXiv:1711.03298 [hepth]
 38.S.A. Hartnoll, P. Kovtun, Hall conductivity from dyonic black holes. Phys. Rev. D 76, 066001 (2007). https://doi.org/10.1103/PhysRevD.76.066001. arXiv:0704.1160 [hepth]ADSCrossRefGoogle Scholar
 39.R.A. Davison, B. Goutéraux, Dissecting holographic conductivities. JHEP 1509, 090 (2015). https://doi.org/10.1007/JHEP09(2015)090. arXiv:1505.05092 [hepth]ADSMathSciNetCrossRefzbMATHGoogle Scholar
 40.M. Zhu, J. Peng, T. Zou, K. Prokes, S.D. Mahanti, T. Hong, Z.Q. Mao, G.Q. Liu, X. Ke, Colossal magnetoresistance in a Mott insulator via magnetic fielddriven insulatormetal transition. Phys. Rev. Leet. 116, 216401 (2016)ADSCrossRefGoogle Scholar
 41.Y.B. Zhou, B.H. Han, Z.M. Liao, H.C. Wu, D.P. Yu, From positive to negative magnetoresistance in graphene with increasing disorder. Appl. Phys. Lett. 98, 222502 (2011)ADSCrossRefGoogle Scholar
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