Abstract
Topological semimetals are a class of many-body systems exhibiting novel macroscopic quantum phenomena at the interplay between high energy and condensed matter physics. They display a topological quantum phase transition (TQPT) which evades the standard Landau paradigm. In the case of Weyl semimetals, the anomalous Hall effect is a good non-local order parameter for the TQPT, as it is proportional to the separation between the Weyl nodes in momentum space. On the contrary, for nodal line semimetals (NLSM), the quest for an order parameter is still open. By taking advantage of a recently proposed holographic model for strongly-coupled NLSM, we explicitly show that entanglement entropy (EE) provides an optimal probe for nodal topology. We propose a generalized c-function, constructed from the EE, as an order parameter for the TQPT. Moreover, we find that the derivative of the renormalized EE with respect to the external coupling driving the TQPT diverges at the critical point, signaling the rise of non-local quantum correlations. Finally, we show that these quantum information quantities are able to characterize not only the critical point but also features of the quantum critical region at finite temperature.
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References
L.D. Landau, On the theory of phase transitions, Zh. Eksp. Teor. Fiz. 7 (1937) 19 [INSPIRE].
P. Toledano and J.-C. Toledano, Landau Theory Of Phase Transitions, The: Application To Structural, Incommensurate, Magnetic And Liquid Crystal Systems, World Scientific Publishing Company (1987).
P.C. Hohenberg and A.P. Krekhov, An introduction to the Ginzburg-Landau theory of phase transitions and nonequilibrium patterns, Phys. Rept. 572 (2015) 1.
X.-G. Wen, Colloquium: Zoo of quantum-topological phases of matter, Rev. Mod. Phys. 89 (2017) 041004 [arXiv:1610.03911] [INSPIRE].
S. Sachdev, Quantum Phase Transitions, Cambridge University Press (2011) [https://doi.org/10.1017/cbo9780511973765].
T. Senthil et al., Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm, Phys. Rev. B 70 (2004) 144407.
J. McGreevy, Generalized Symmetries in Condensed Matter, arXiv:2204.03045 [https://doi.org/10.1146/annurev-conmatphys-040721-021029] [INSPIRE].
P. Calabrese and J.L. Cardy, Entanglement entropy and quantum field theory, J. Stat. Mech. 0406 (2004) P06002 [hep-th/0405152] [INSPIRE].
A. Kitaev and J. Preskill, Topological entanglement entropy, Phys. Rev. Lett. 96 (2006) 110404 [hep-th/0510092] [INSPIRE].
M. Levin and X.-G. Wen, Detecting Topological Order in a Ground State Wave Function, Phys. Rev. Lett. 96 (2006) 110405 [cond-mat/0510613] [INSPIRE].
H.C. Jiang, Z. Wang and L. Balents, Identifying topological order by entanglement entropy, Nature Phys. 8 (2012) 902 [arXiv:1205.4289] [INSPIRE].
E. Fradkin and J.E. Moore, Entanglement entropy of 2D conformal quantum critical points: hearing the shape of a quantum drum, Phys. Rev. Lett. 97 (2006) 050404 [cond-mat/0605683] [INSPIRE].
A. Osterloh, L. Amico, G. Falci and R. Fazio, Scaling of entanglement close to a quantum phase transition, Nature 416 (2002) 608.
J. Vidal, G. Palacios and R. Mosseri, Entanglement in a second-order quantum phase transition, Phys. Rev. A 69 (2004) 022107.
S.-J. Gu, S.-S. Deng, Y.-Q. Li and H.-Q. Lin, Entanglement and Quantum Phase Transition in the Extended Hubbard Model, Phys. Rev. Lett. 93 (2004) 086402.
T.J. Osborne and M.A. Nielsen, Entanglement in a simple quantum phase transition, Phys. Rev. A 66 (2002) 032110 [quant-ph/0202162] [INSPIRE].
A.A. Burkov, Topological semimetals, Nature Mater. 15 (2016) 1145.
L.-K. Lim, J.-N. Fuchs and G. Montambaux, Bloch-Zener Oscillations across a Merging Transition of Dirac Points, Phys. Rev. Lett. 108 (2012) 175303.
M. Tarnowski et al., Observation of Topological Bloch-State Defects and Their Merging Transition, Phys. Rev. Lett. 118 (2017) 240403.
B. Roy and M.S. Foster, Quantum Multicriticality near the Dirac-Semimetal to Band-Insulator Critical Point in Two Dimensions: A Controlled Ascent from One Dimension, Phys. Rev. X 8 (2018) 011049 [arXiv:1705.10798] [INSPIRE].
L. Tarruell et al., Creating, moving and merging Dirac points with a Fermi gas in a tunable honeycomb lattice, Nature 483N7389 (2012) 302 [arXiv:1111.5020] [INSPIRE].
J. Kim et al., Observation of tunable band gap and anisotropic dirac semimetal state in black phosphorus, Science 349 (2015) 723.
Y. Shao et al., Electronic correlations in nodal-line semimetals, Nature Phys. 16 (2020) 636 [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].
S.W. Kim, G. Jose and B. Uchoa, Hydrodynamic transport and violation of the viscosity-to-entropy ratio bound in nodal-line semimetals, Phys. Rev. Res. 3 (2021) 033003 [arXiv:2009.01271] [INSPIRE].
J.-R. Wang, G.-Z. Liu and C.-J. Zhang, Breakdown of Fermi liquid theory in topological multi-Weyl semimetals, Phys. Rev. B 98 (2018) 205113 [arXiv:1612.01729] [INSPIRE].
J. Zaanen, Y.-W. Sun, Y. Liu and K. Schalm, Holographic duality in condensed matter physics, Cambridge University Press (2015).
S.A. Hartnoll, A. Lucas and S. Sachdev, Holographic quantum matter, MIT press (2018).
M. Baggioli and B. Goutéraux, Colloquium: Hydrodynamics and holography of charge density wave phases, Rev. Mod. Phys. 95 (2023) 011001 [arXiv:2203.03298] [INSPIRE].
K. Landsteiner and Y. Liu, The holographic Weyl semi-metal, Phys. Lett. B 753 (2016) 453 [arXiv:1505.04772] [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].
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].
K. Landsteiner, Y. Liu and Y.-W. Sun, Holographic topological semimetals, Sci. China Phys. Mech. Astron. 63 (2020) 250001 [arXiv:1911.07978] [INSPIRE].
B. Yan and C. Felser, Topological Materials: Weyl Semimetals, Ann. Rev. Condensed Matter Phys. 8 (2017) 337 [arXiv:1611.04182] [INSPIRE].
S. Jia, S.-Y. Xu and M.Z. Hasan, Weyl Semimetals, Fermi Arcs and Chiral Anomalies (A Short Review), Nature Mater. 15 (2016) 1140 [arXiv:1612.00416] [INSPIRE].
K. Landsteiner, Notes on Anomaly Induced Transport, Acta Phys. Polon. B 47 (2016) 2617 [arXiv:1610.04413] [INSPIRE].
D. Colladay and V.A. Kostelecky, Lorentz violating extension of the standard model, Phys. Rev. D 58 (1998) 116002 [hep-ph/9809521] [INSPIRE].
M. Baggioli and D. Giataganas, Detecting Topological Quantum Phase Transitions via the c-Function, Phys. Rev. D 103 (2021) 026009 [arXiv:2007.07273] [INSPIRE].
M. Baggioli, B. Padhi, P.W. Phillips and C. Setty, Conjecture on the Butterfly Velocity across a Quantum Phase Transition, JHEP 07 (2018) 049 [arXiv:1805.01470] [INSPIRE].
C. Fang, H. Weng, X. Dai and Z. Fang, Topological nodal line semimetals, Chin. Phys. B 25 (2016) 117106.
L.S. Xie et al., A new form of ca3p2 with a ring of Dirac nodes, APL Materials 3 (2015) 083602.
Y.-H. Chan, C.-K. Chiu, M.Y. Chou and A.P. Schnyder, ca3p2 and other topological semimetals with line nodes and drumhead surface states, Phys. Rev. B 93 (2016) 205132.
G. Bian et al., Topological nodal-line fermions in spin-orbit metal PbTaSe2, Nature Commun. 7 (2016) 10556.
L.M. Schoop et al., Dirac cone protected by non-symmorphic symmetry and three-dimensional Dirac line node in ZrSiS, Nature Commun. 7 (2016) 11696.
M. Moore, P. Surówka, V. Juričić and B. Roy, Shear viscosity as a probe of nodal topology, Phys. Rev. B 101 (2020) 161111 [arXiv:1912.07611] [INSPIRE].
Y. Liu and Y.-W. Sun, Topological nodal line semimetals in holography, JHEP 12 (2018) 072 [arXiv:1801.09357] [INSPIRE].
R. Rodgers, E. Mauri, U. Gürsoy and H.T.C. Stoof, Thermodynamics and transport of holographic nodal line semimetals, JHEP 11 (2021) 191 [arXiv:2109.07187] [INSPIRE].
Y.H. Kwan et al., Quantum oscillations probe the Fermi surface topology of the nodal-line semimetal CaAgAs, Phys. Rev. Res. 2 (2020) 012055.
M. Pretko, Nodal Line Entanglement Entropy: Generalized Widom Formula from Entanglement Hamiltonians, Phys. Rev. B 95 (2017) 235111 [arXiv:1609.07502] [INSPIRE].
M. Srednicki, Entropy and area, Phys. Rev. Lett. 71 (1993) 666 [hep-th/9303048] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
D. Gioev and I. Klich, Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture, Phys. Rev. Lett. 96 (2006) 100503 [quant-ph/0504151] [INSPIRE].
B. Swingle, Entanglement Entropy and the Fermi Surface, Phys. Rev. Lett. 105 (2010) 050502 [arXiv:0908.1724] [INSPIRE].
N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi Surfaces and Entanglement Entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].
L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B 85 (2012) 035121 [arXiv:1112.0573] [INSPIRE].
Y. Liu and X.-M. Wu, An improved holographic nodal line semimetal, JHEP 05 (2021) 141 [arXiv:2012.12602] [INSPIRE].
A.A. Burkov, M.D. Hook and L. Balents, Topological nodal semimetals, Phys. Rev. B 84 (2011) 235126.
J.J. Bisognano and E.H. Wichmann, On the Duality Condition for Quantum Fields, J. Math. Phys. 17 (1976) 303 [INSPIRE].
E. Bianchi and R.C. Myers, On the Architecture of Spacetime Geometry, Class. Quant. Grav. 31 (2014) 214002 [arXiv:1212.5183] [INSPIRE].
C. Hoyos and P. Koroteev, On the Null Energy Condition and Causality in Lifshitz Holography, Phys. Rev. D 82 (2010) 084002 [Erratum ibid. 82 (2010) 109905] [arXiv:1007.1428] [INSPIRE].
L.-L. Gao, Y. Liu and H.-D. Lyu, Black hole interiors in holographic topological semimetals, JHEP 03 (2023) 034 [arXiv:2301.01468] [INSPIRE].
R.C. Myers and A. Singh, Comments on Holographic Entanglement Entropy and RG Flows, JHEP 04 (2012) 122 [arXiv:1202.2068] [INSPIRE].
H. Liu and M. Mezei, Probing renormalization group flows using entanglement entropy, JHEP 01 (2014) 098 [arXiv:1309.6935] [INSPIRE].
V.E. Hubeny, Extremal surfaces as bulk probes in AdS/CFT, JHEP 07 (2012) 093 [arXiv:1203.1044] [INSPIRE].
C.-S. Chu and D. Giataganas, c-Theorem for Anisotropic RG Flows from Holographic Entanglement Entropy, Phys. Rev. D 101 (2020) 046007 [arXiv:1906.09620] [INSPIRE].
A. Osterloh, L. Amico, G. Falci and R. Fazio, Scaling of Entanglement close to a Quantum Phase Transitions, quant-ph/0202029 [https://doi.org/10.1038/416608a].
Y. Huh, E.-G. Moon and Y.B. Kim, Long-range Coulomb interaction in nodal-ring semimetals, Phys. Rev. B 93 (2016) 035138.
T. Faulkner and J. Polchinski, Semi-Holographic Fermi Liquids, JHEP 06 (2011) 012 [arXiv:1001.5049] [INSPIRE].
Acknowledgments
We thank Karl Landsteiner and Ya-Wen Sun for useful comments on a preliminary draft of this work. M.B. and X.-M.W. acknowledge the support of the Shanghai Municipal Science and Technology Major Project (Grant No.2019SHZDZX01). M.B. acknowledges the sponsorship from the Yangyang Development Fund. Y.L. is supported by the National Natural Science Foundation of China grant No.11875083.
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Baggioli, M., Liu, Y. & Wu, XM. Entanglement entropy as an order parameter for strongly coupled nodal line semimetals. J. High Energ. Phys. 2023, 221 (2023). https://doi.org/10.1007/JHEP05(2023)221
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DOI: https://doi.org/10.1007/JHEP05(2023)221