Abstract
We study the effects of quenched one-dimensional disorder on the holographic Weyl semimetal quantum phase transition (QPT), with a particular focus on the quantum critical region. We observe the smearing of the sharp QPT linked to the appearance of rare regions at the horizon where locally the order parameter is non-zero. We discuss the role of the disorder correlation and we compare our results to expectations from condensed matter theory at weak coupling. We analyze also the interplay of finite temperature and disorder. Within the quantum critical region we find indications for the presence of log-oscillatory structures in the order parameter hinting at the existence of an IR fixed point with discrete scale invariance.
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References
S. Sachdev, Quantum phase transitions, second ed., Cambridge University Press, Cambridge U.K., (2011).
M. Vojta, Quantum phase transitions, Rept. Prog. Phys. 66 (2003) 2069 [cond-mat/0309604].
J.A.N. Bruin, H. Sakai, R.S. Perry and A.P. Mackenzie, Similarity of scattering rates in metals showing t-linear resistivity, Science 339 (2013) 804.
T. Vojta, Computing quantum phase transitions, arXiv:0709.0964.
T. Vojta, Topical review: rare region effects at classical, quantum and nonequilibrium phase transitions, J. Phys. A 39 (2006) R143 [cond-mat/0602312].
T. Vojta, Phases and phase transitions in disordered quantum systems, AIP Conf. Ser. 1550 (2013) 188 [arXiv:1301.7746].
A.B. Harris, Effect of random defects on the critical behaviour of Ising models, J. Phys. C 7 (1974) 1671.
T. Vojta and J.A. Hoyos, Criticality and quenched disorder: rare regions vs. Harris criterion, arXiv:1309.0753.
T. Vojta and R. Sknepnek, Critical points and quenched disorder: from Harris criterion to rare regions and smearing, Phys. Stat. Solidi B 241 (2004) 2118 [cond-mat/0405070].
J.A. Hoyos and T. Vojta, Theory of smeared quantum phase transitions, Phys. Rev. Lett. 100 (2008) 240601 [arXiv:0802.2303].
T. Vojta, Disorder-induced rounding of certain quantum phase transitions, Phys. Rev. Lett. 90 (2003) 107202.
R.B. Griffiths, Nonanalytic behavior above the critical point in a random Ising ferromagnet, Phys. Rev. Lett. 23 (1969) 17.
M. Randeria, J.P. Sethna and R.G. Palmer, Low-frequency relaxation in Ising spin-glasses, Phys. Rev. Lett. 54 (1985) 1321.
T. Vojta, Quantum Griffiths effects and smeared phase transitions in metals: theory and experiment, J. Low Temp. Phys. 161 (2010) 299 [arXiv:1005.2707].
T. Vojta, Smearing of the phase transition in Ising systems with planar defects, J. Phys. A 36 (2003) 10921.
T. Vojta, Broadening of a nonequilibrium phase transition by extended structural defects, Phys. Rev. E 70 (2004) 026108.
L. Demkó et al., Disorder promotes ferromagnetism: rounding of the quantum phase transition in sr1−xcaxruo3, Phys. Rev. Lett. 108 (2012) 185701.
D. Nozadze, C. Svoboda, F. Hrahsheh and T. Vojta, Modification of smeared phase transitions by spatial disorder correlations, AIP Conf. Ser. 1550 (2013) 263 [arXiv:1212.5962].
C. Svoboda, D. Nozadze, F. Hrahsheh and T. Vojta, Disorder correlations at smeared phase transitions, EPL (Europhys. Lett.) 97 (2012) 20007 [arXiv:1109.4290].
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, arXiv:1612.07324 [INSPIRE].
M. Ammon and J. Erdmenger, Gauge/gravity duality, Cambridge University Press, Cambridge U.K., (2015).
J. Zaanen, Y. Liu, Y. Sun and K. Schalm, Holographic duality in condensed matter physics, Cambridge University Press, Cambridge U.K., (2015).
E. D’Hoker and P. Kraus, Magnetic brane solutions in AdS, JHEP 10 (2009) 088 [arXiv:0908.3875] [INSPIRE].
E. D’Hoker and P. Kraus, Magnetic field induced quantum criticality via new asymptotically AdS 5 solutions, Class. Quant. Grav. 27 (2010) 215022 [arXiv:1006.2573] [INSPIRE].
N. Iqbal, H. Liu, M. Mezei and Q. Si, Quantum phase transitions in holographic models of magnetism and superconductors, Phys. Rev. D 82 (2010) 045002 [arXiv:1003.0010] [INSPIRE].
K. Landsteiner, Y. Liu and Y.-W. Sun, Quantum phase transition between a topological and a trivial semimetal from holography, Phys. Rev. Lett. 116 (2016) 081602 [arXiv:1511.05505] [INSPIRE].
E. Gubankova, M. Cubrovic and J. Zaanen, Exciton-driven quantum phase transitions in holography, Phys. Rev. D 92 (2015) 086004 [arXiv:1412.2373] [INSPIRE].
N. Iqbal, H. Liu and M. Mezei, Quantum phase transitions in semilocal quantum liquids, Phys. Rev. D 91 (2015) 025024 [arXiv:1108.0425] [INSPIRE].
A. Donos and S.A. Hartnoll, Interaction-driven localization in holography, Nature Phys. 9 (2013) 649 [arXiv:1212.2998] [INSPIRE].
M. Baggioli and O. Pujolàs, On holographic disorder-driven metal-insulator transitions, JHEP 01 (2017) 040 [arXiv:1601.07897] [INSPIRE].
M. Baggioli and O. Pujolàs, On effective holographic Mott insulators, JHEP 12 (2016) 107 [arXiv:1604.08915] [INSPIRE].
V. Dobrosavljevic, Introduction to metal-insulator transitions, arXiv:1112.6166.
K. Landsteiner and Y. Liu, The holographic Weyl semi-metal, Phys. Lett. B 753 (2016) 453 [arXiv:1505.04772] [INSPIRE].
S.-Y. Xu et al., Discovery of a Weyl fermion semimetal and topological Fermi arcs, Science 349 (2015) 613.
Z.K. Liu et al., Discovery of a three-dimensional topological Dirac semimetal, Na 3 Bi, Science 343 (2014) 864.
P. Hosur and X. Qi, Recent developments in transport phenomena in Weyl semimetals, Comptes Rendus Physique 14 (2013) 857 [arXiv:1309.4464] [INSPIRE].
L. Lu et al., Experimental observation of Weyl points, Science 349 (2015) 622 [arXiv:1502.03438] [INSPIRE].
B.Q. Lv et al., Experimental discovery of Weyl semimetal TaAs, Phys. Rev. X 5 (2015) 031013 [arXiv:1502.04684] [INSPIRE].
D.E. Kharzeev, J. Liao, S.A. Voloshin and G. Wang, Chiral magnetic and vortical effects in high-energy nuclear collisions — a status report, Prog. Part. Nucl. Phys. 88 (2016) 1 [arXiv:1511.04050] [INSPIRE].
K. Landsteiner, Notes on anomaly induced transport, Acta Phys. Polon. B 47 (2016) 2617 [arXiv:1610.04413] [INSPIRE].
J. Gooth et al., Experimental signatures of the mixed axial-gravitational anomaly in the Weyl semimetal NbP, Nature 547 (2017) 324 [arXiv:1703.10682] [INSPIRE].
Q. Li et al., Observation of the chiral magnetic effect in ZrTe 5, Nature Phys. 12 (2016) 550 [arXiv:1412.6543] [INSPIRE].
X. Huang et al., Observation of the chiral-anomaly-induced negative magnetoresistance in 3d Weyl semimetal TaAs, Phys. Rev. X 5 (2015) 031023.
H. Li et al., Negative magnetoresistance in Dirac semimetal Cd 3 As 2, Nature Commun. 7 (2016) 10301.
A.G. Grushin, Consequences of a condensed matter realization of Lorentz violating QED in Weyl semi-metals, Phys. Rev. D 86 (2012) 045001 [arXiv:1205.3722] [INSPIRE].
H.B. Nielsen and M. Ninomiya, Adler-Bell-Jackiw anomaly and Weyl fermions in crystal, Phys. Lett. B 130 (1983) 389 [INSPIRE].
R. Jackiw, When radiative corrections are finite but undetermined, Int. J. Mod. Phys. B 14 (2000) 2011 [hep-th/9903044] [INSPIRE].
A. Lucas, R.A. Davison and S. Sachdev, Hydrodynamic theory of thermoelectric transport and negative magnetoresistance in Weyl semimetals, Proc. Nat. Acad. Sci. 113 (2016) 9463 [arXiv:1604.08598] [INSPIRE].
A. Altland and D. Bagrets, Effective field theory of the disordered Weyl semimetal, Phys. Rev. Lett. 114 (2015) 257201 [arXiv:1501.06537] [INSPIRE].
A. Altland and D. Bagrets, Theory of the strongly disordered Weyl semimetal, Phys. Rev. B 93 (2016) 075113 [arXiv:1511.01876] [INSPIRE].
C.-Z. Chen, J. Song, H. Jiang, Q.-F. Sun, Z. Wang and X.C. Xie, Disorder and metal-insulator transitions in Weyl semimetals, Phys. Rev. Lett. 115 (2015) 246603 [arXiv:1507.00128].
Y.X. Zhao and Z.D. Wang, Disordered Weyl semimetals and their topological family, Phys. Rev. Lett. 114 (2015) 206602 [arXiv:1412.7678] [INSPIRE].
B. Roy, R.-J. Slager and V. Juricic, Global phase diagram of a dirty Weyl liquid and emergent superuniversality, arXiv:1610.08973 [INSPIRE].
T. Louvet, D. Carpentier and A.A. Fedorenko, New quantum transition in Weyl semimetals with correlated disorder, Phys. Rev. B 95 (2017) 014204 [arXiv:1609.08368] [INSPIRE].
B. Roy and S. Das Sarma, Quantum phases of interacting electrons in three-dimensional dirty Dirac semimetals, Phys. Rev. B 94 (2016) 115137 [arXiv:1511.06367] [INSPIRE].
B. Sbierski, G. Pohl, E.J. Bergholtz and P.W. Brouwer, Quantum transport of disordered Weyl semimetals at the nodal point, Phys. Rev. Lett. 113 (2014) 026602 [arXiv:1402.6653].
K. Landsteiner, E. Lopez and G. Milans del Bosch, Quenching the chiral magnetic effect via the gravitational anomaly and holography, Phys. Rev. Lett. 120 (2018) 071602 [arXiv:1709.08384] [INSPIRE].
G. Grignani, A. Marini, F. Pena-Benitez and S. Speziali, AC conductivity for a holographic Weyl semimetal, JHEP 03 (2017) 125 [arXiv:1612.00486] [INSPIRE].
K. Landsteiner, Y. Liu and Y.-W. Sun, Odd viscosity in the quantum critical region of a holographic Weyl semimetal, Phys. Rev. Lett. 117 (2016) 081604 [arXiv:1604.01346] [INSPIRE].
C. Copetti, J. Fernández-Pendás and K. Landsteiner, Axial Hall effect and universality of holographic Weyl semi-metals, JHEP 02 (2017) 138 [arXiv:1611.08125] [INSPIRE].
M. Rogatko and K.I. Wysokinski, Holographic calculation of the magneto-transport coefficients in Dirac semimetals, JHEP 01 (2018) 078 [arXiv:1712.01608] [INSPIRE].
V.P.J. Jacobs, P. Betzios, U. Gürsoy and H.T.C. Stoof, Electromagnetic response of interacting Weyl semimetals, Phys. Rev. B 93 (2016) 195104 [arXiv:1512.04883] [INSPIRE].
M. Ammon, M. Heinrich, A. Jiménez-Alba and S. Moeckel, Surface states in holographic Weyl semimetals, Phys. Rev. Lett. 118 (2017) 201601 [arXiv:1612.00836] [INSPIRE].
Y. Liu and Y.-W. Sun, Topological nodal line semimetals in holography, arXiv:1801.09357 [INSPIRE].
O. Aharony, Z. Komargodski and S. Yankielowicz, Disorder in large-N theories, JHEP 04 (2016) 013 [arXiv:1509.02547] [INSPIRE].
D. Vegh, Holography without translational symmetry, arXiv:1301.0537 [INSPIRE].
T. Andrade and B. Withers, A simple holographic model of momentum relaxation, JHEP 05 (2014) 101 [arXiv:1311.5157] [INSPIRE].
M. Baggioli and O. Pujolàs, Electron-phonon interactions, metal-insulator transitions and holographic massive gravity, Phys. Rev. Lett. 114 (2015) 251602 [arXiv:1411.1003] [INSPIRE].
L. Alberte, M. Baggioli, A. Khmelnitsky and O. Pujolàs, Solid holography and massive gravity, JHEP 02 (2016) 114 [arXiv:1510.09089] [INSPIRE].
M. Baggioli and D.K. Brattan, Drag phenomena from holographic massive gravity, Class. Quant. Grav. 34 (2017) 015008 [arXiv:1504.07635] [INSPIRE].
A. Lucas and S. Sachdev, Conductivity of weakly disordered strange metals: from conformal to hyperscaling-violating regimes, Nucl. Phys. B 892 (2015) 239 [arXiv:1411.3331] [INSPIRE].
A. Lucas, S. Sachdev and K. Schalm, Scale-invariant hyperscaling-violating holographic theories and the resistivity of strange metals with random-field disorder, Phys. Rev. D 89 (2014) 066018 [arXiv:1401.7993] [INSPIRE].
A. Lucas, Hydrodynamic transport in strongly coupled disordered quantum field theories, New J. Phys. 17 (2015) 113007 [arXiv:1506.02662] [INSPIRE].
A. Donos and J.P. Gauntlett, The thermoelectric properties of inhomogeneous holographic lattices, JHEP 01 (2015) 035 [arXiv:1409.6875] [INSPIRE].
A.M. Garcıa-García and B. Loureiro, Marginal and irrelevant disorder in Einstein-Maxwell backgrounds, Phys. Rev. D 93 (2016) 065025 [arXiv:1512.00194] [INSPIRE].
T. Andrade, A.M. Garcıa-García and B. Loureiro, Coherence effects in disordered geometries with a field-theory dual, arXiv:1711.10953 [INSPIRE].
S.A. Hartnoll and J.E. Santos, Disordered horizons: holography of randomly disordered fixed points, Phys. Rev. Lett. 112 (2014) 231601 [arXiv:1402.0872] [INSPIRE].
S.A. Hartnoll and J.E. Santos, Cold planar horizons are floppy, Phys. Rev. D 89 (2014) 126002 [arXiv:1403.4612] [INSPIRE].
D. Sornette, Discrete scale invariance and complex dimensions, Phys. Rept. 297 (1998) 239 [cond-mat/9707012] [INSPIRE].
S.A. Hartnoll, D.M. Ramirez and J.E. Santos, Emergent scale invariance of disordered horizons, JHEP 09 (2015) 160 [arXiv:1504.03324] [INSPIRE].
S.A. Hartnoll, D.M. Ramirez and J.E. Santos, Thermal conductivity at a disordered quantum critical point, JHEP 04 (2016) 022 [arXiv:1508.04435] [INSPIRE].
K. Balasubramanian, Gravity duals of cyclic RG flows, with strings attached, arXiv:1301.6653 [INSPIRE].
M. Flory, Discrete scale invariance in holography revisited, Fortsch. Phys. (2018) [arXiv:1711.03113] [INSPIRE].
S. Ryu, T. Takayanagi and T. Ugajin, Holographic conductivity in disordered systems, JHEP 04 (2011) 115 [arXiv:1103.6068] [INSPIRE].
M. Fujita, Y. Hikida, S. Ryu and T. Takayanagi, Disordered systems and the replica method in AdS/CFT, JHEP 12 (2008) 065 [arXiv:0810.5394] [INSPIRE].
D. Arean, L.A. Pando Zayas, I.S. Landea and A. Scardicchio, Holographic disorder driven superconductor-metal transition, Phys. Rev. D 94 (2016) 106003 [arXiv:1507.02280] [INSPIRE].
D. Areán, A. Farahi, L.A. Pando Zayas, I. Salazar Landea and A. Scardicchio, Holographic p-wave superconductor with disorder, JHEP 07 (2015) 046 [arXiv:1407.7526] [INSPIRE].
D. Arean, A. Farahi, L.A. Pando Zayas, I.S. Landea and A. Scardicchio, Holographic superconductor with disorder, Phys. Rev. D 89 (2014) 106003 [arXiv:1308.1920] [INSPIRE].
A. Cortijo, Y. Ferreirós, K. Landsteiner and M.A.H. Vozmediano, Hall viscosity from elastic gauge fields in Dirac crystals, 2D Mater. 1 (2016) 011002 [arXiv:1506.05136] [INSPIRE].
A. Cortijo, Y. Ferreirós, K. Landsteiner and M.A.H. Vozmediano, Elastic gauge fields in Weyl semimetals, Phys. Rev. Lett. 115 (2015) 177202 [arXiv:1603.02674] [INSPIRE].
M. Shinozuka and G. Deodatis, Simulation of stochastic processes by spectral representation, Appl. Mech. Rev. 44 (1991) 191.
F. Hrahsheh, D. Nozadze and T. Vojta, Composition-tuned smeared phase transitions, Phys. Rev. B 83 (2011) 224402 [arXiv:1103.5439].
E. D’Hoker and P. Kraus, Quantum criticality via magnetic branes, Lect. Notes Phys. 871 (2013) 469 [arXiv:1208.1925] [INSPIRE].
M. Frigo and S.G. Johnson, The design and implementation of FFTW3, Proc. IEEE 93 (2005) 216.
X.S. Li, An overview of SuperLU: algorithms, implementation, and user interface, ACM Trans. Math. Softw. 31 (2005) 302.
O. Tange, Gnu parallel — the command-line power tool, ;login: the USENIX Magazine 36 (2011) 42.
J.P. Boyd, The asymptotic Chebyshev coefficients for functions with logarithmic endpoint singularities: mappings and singular basis functions, Appl. Math. Comput. 29 (1989) 49.
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Ammon, M., Baggioli, M., Jiménez-Alba, A. et al. A smeared quantum phase transition in disordered holography. J. High Energ. Phys. 2018, 68 (2018). https://doi.org/10.1007/JHEP04(2018)068
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DOI: https://doi.org/10.1007/JHEP04(2018)068