Abstract
We consider an interacting quantum field theory on a curved two-dimensional manifold that we construct by geometrically deforming a flat hexagonal lattice by the insertion of a defect. Depending on how the deformation is done, the resulting geometry acquires a locally nonvanishing curvature that can be either positive or negative. Fields propagating on this background are forced to satisfy boundary conditions modulated by the geometry and that can be assimilated by a nondynamical gauge field. We present an explicit example where curvature and boundary conditions compete in altering the way symmetry breaking takes place, resulting in a surprising behavior of the order parameter in the vicinity of the defect. The effect described here is expected to be generic and of relevance in a variety of situations.
| Original language | English |
|---|---|
| Article number | 221601 |
| Journal | Physical review letters |
| Volume | 121 |
| Issue number | 22 |
| DOIs | |
| Publication status | Published - 2018 Nov 27 |
ASJC Scopus subject areas
- Physics and Astronomy(all)
Access to Document
Other files and links
Fingerprint Dive into the research topics of 'Symmetry Breaking and Lattice Kirigami'. Together they form a unique fingerprint.
Cite this
- APA
- Standard
- Harvard
- Vancouver
- Author
- BIBTEX
- RIS
Symmetry Breaking and Lattice Kirigami. / Castro, Eduardo V.; Flachi, Antonino; Ribeiro, Pedro; Vitagliano, Vincenzo.
In: Physical review letters, Vol. 121, No. 22, 221601, 27.11.2018.Research output: Contribution to journal › Article › peer-review
}
TY - JOUR
T1 - Symmetry Breaking and Lattice Kirigami
AU - Castro, Eduardo V.
AU - Flachi, Antonino
AU - Ribeiro, Pedro
AU - Vitagliano, Vincenzo
N1 - Funding Information: In this Letter we have looked at symmetry breaking in the Hubbard model on a curved 2D manifold constructed by taking the continuum limit of a flat hexagonal lattice deformed by insertion of a defect. The two important features of the story turn out to be the increasing (or decreasing) curvature near the defect and the boundary conditions along the cut. The latter can be assimilated by a nondynamical gauge field. As an example, we have considered the staggered magnetization, the order parameter associated with the discrete sublattice symmetry. Numerical results have shown a surprising increase of the order parameter as the locally curved region is approached. This behavior has been explained by the competition between the effect of curvature and of the emergent gauge field (i.e., boundary conditions) modulated by the conical structure. What we have seen here should be generic and should be expected to occur for different QFT phases and different lattice structures as long as the same geometrical traits are maintained. Here, we have ignored fluctuations of the order parameter; thus, the system is always in the ordered state. However, the way R ˜ and the emergent gauge field appear in the determinant suggests an analogous suppression to what we find here of the onset of a metallic phase. An intriguing possibility is to consider a multidefect configuration and look for arrangements where the competition between the geometry-induced gauge fields and curvature can be adjusted to produce specific order parameter profiles. The effect of curvature on the spontaneous breaking of sublattice symmetry could be used to shed light on the question regarding the semimetal-insulator phase transition in graphene: even though graphene is predicted to be very close to the transition point [40,58] , no experimental signature of the insulating behavior has been found in flat graphene so far [59] . Another promising route in graphene would be the combination of curvature with adatom adsorption in order to enhance symmetry breaking, in particular of magnetic order [60] . Cosmic strings in cosmology [61] or their condensed matter analogue (in liquid crystals, for example [62,63] ) offer a straightforward application of our results. In this case the effect discussed here should locally trigger fermion condensation and offer a mechanism for inducing a superconducting phase at the string. Finally, it is tempting to relate the present ideas to gravity at the Planck scale, where spacetime may be discrete and the presence of defects could cause local changes in the lattice geometry and topology similar to those described here, triggering a form of graviton condensation in the vicinity of these spacetime glitches. We acknowledge the support of the Japanese Ministry of Education, Culture, Sports, Science Program for the Strategic Research Foundation at Private Universities “Topological Science” (Grant No. S1511006) and of JSPS KAKENHI Grant No. 18K03626 for A. F.; the JSPS (Grant No. P17763) for V. V.; the FCT-Portugal (Grant No. UID/CTM/04540/2013) for E. V. C. and P. R.; and the Investigador FCT program (Contract No. IF/00347/2014) for P. R. A. F. is grateful to A. Beekman, K. Fukushima, T. Fujimori, M. Nitta, and R. Yoshii for discussions on various aspects of symmetry breaking and geometry, and to A. Beekman for his feedback on the manuscript. [1] 1 N. D. Birrel and P. C. W. Davies , Quantum Fields in Curved Space ( Cambridge University Press , Cambridge, England, 1982 ). [2] 2 L. Parker and D. Toms , Quantum Field Theory in Curved Spacetime ( Cambridge University Press , Cambridge, England, 2009 ). [3] 3 L. Parker , Phys. Rev. 183 , 1057 ( 1969 ). PHRVAO 0031-899X 10.1103/PhysRev.183.1057 [4] 4a S. W. Hawking , Commun. Math. Phys. 43 , 199 ( 1975 ); CMPHAY 0010-3616 10.1007/BF02345020 4b S. W. Hawking Commun. Math. Phys. 46 , 206 ( 1976 ). CMPHAY 0010-3616 10.1007/BF01608497 [5] 5 S. R. Coleman and E. J. Weinberg , Phys. Rev. D 7 , 1888 ( 1973 ). PRVDAQ 0556-2821 10.1103/PhysRevD.7.1888 [6] 6 G. W. Gibbons , J. Phys. A 11 , 1341 ( 1978 ). JPHAC5 0305-4470 10.1088/0305-4470/11/7/021 [7] 7 G. M. Shore , Ann. Phys. (N.Y.) 128 , 376 ( 1980 ). APNYA6 0003-4916 10.1016/0003-4916(80)90326-7 [8] 8 G. Denardo and E. Spallucci , Nuovo Cimento A 71 , 397 ( 1982 ). NCIAAT 0369-3546 10.1007/BF02898934 [9] 9 T. Inagaki , T. Muta , and S. D. Odintsov , Prog. Theor. Phys. Suppl. 127 , 93 ( 1997 ). PTPSEP 0375-9687 10.1143/PTPS.127.93 [10] 10a C. J. Isham , Proc. R. Soc. A 362 , 383 ( 1978 ); PRLAAZ 1364-5021 10.1098/rspa.1978.0140 10b C. J. Isham J. Phys. A 14 , 2943 ( 1981 ). JPHAC5 0305-4470 10.1088/0305-4470/14/11/017 [11] 11a D. J. Toms , Phys. Rev. D 21 , 2805 ( 1980 ); PRVDAQ 0556-2821 10.1103/PhysRevD.21.2805 11b D. J. Toms Phys. Rev. D 25 , 2536 ( 1982 ); PRVDAQ 0556-2821 10.1103/PhysRevD.25.2536 11c D. J. Toms Phys. Rev. D 26 , 958 ( 1982 ). PRVDAQ 0556-2821 10.1103/PhysRevD.26.958 [12] 12 L. H. Ford and D. J. Toms , Phys. Rev. D 25 , 1510 ( 1982 ). PRVDAQ 0556-2821 10.1103/PhysRevD.25.1510 [13] 13 G. Kennedy , Phys. Rev. D 23 , 2884 ( 1981 ). PRVDAQ 0556-2821 10.1103/PhysRevD.23.2884 [14] 14 S. W. Hawking , Commun. Math. Phys. 80 , 421 ( 1981 ). CMPHAY 0010-3616 10.1007/BF01208279 [15] 15 I. G. Moss , Phys. Rev. D 32 , 1333 ( 1985 ). 10.1103/PhysRevD.32.1333 [16] 16 A. Flachi and T. Tanaka , Phys. Rev. D 84 , 061503 ( 2011 ). PRVDAQ 1550-7998 10.1103/PhysRevD.84.061503 [17] 17 A. Flachi , Int. J. Mod. Phys. D 24 , 1542017 ( 2015 ). IMPDEO 0218-2718 10.1142/S0218271815420171 [18] 18 A. H. Castro Neto , F. Guinea , N. M. R. Peres , K. S. Novoselov , and A. K. Geim , Rev. Mod. Phys. 81 , 109 ( 2009 ). RMPHAT 0034-6861 10.1103/RevModPhys.81.109 [19] 19 M. A. H. Vozmediano , M. I. Katsnelson , and F. Guinea , Phys. Rep. 496 , 109 ( 2010 ). PRPLCM 0370-1573 10.1016/j.physrep.2010.07.003 [20] 20 B. Amorim , Phys. Rep. 617 , 1 ( 2016 ). PRPLCM 0370-1573 10.1016/j.physrep.2015.12.006 [21] 21 A. Cortijo , F. Guinea , and M. A. H. Vozmediano , J. Phys. A 45 , 383001 ( 2012 ). JPAMB5 1751-8113 10.1088/1751-8113/45/38/383001 [22] 22 Y. A. Sitenko and N. D. Vlasii , Nucl. Phys. B787 , 241 ( 2007 ). NUPBBO 0550-3213 10.1016/j.nuclphysb.2007.06.001 [23] 23 F. de Juan , A. Cortijo , and M. A. H. Vozmediano , Nucl. Phys. B828 , 625 ( 2010 ). NUPBBO 0550-3213 10.1016/j.nuclphysb.2009.11.012 [24] 24 A. Cortijo and M. A. H. Vozmediano , Eur. Phys. J. 148 , 83 ( 2007 ). 10.1140/epjst/e2007-00228-2 [25] 25 A. Iorio and G. Lambiase , Phys. Rev. D 90 , 025006 ( 2014 ). PRVDAQ 1550-7998 10.1103/PhysRevD.90.025006 [26] 26 J. Gooth , Nature (London) 547 , 324 ( 2017 ). NATUAS 0028-0836 10.1038/nature23005 [27] 27a A. Gromov and A. G. Abanov , Phys. Rev. Lett. 113 , 266802 ( 2014 ); PRLTAO 0031-9007 10.1103/PhysRevLett.113.266802 27b A. Gromov and A. G. Abanov Phys. Rev. Lett. 114 , 016802 ( 2015 ). PRLTAO 0031-9007 10.1103/PhysRevLett.114.016802 [28] 28 A. Flachi and K. Fukushima , Phys. Rev. Lett. 113 , 091102 ( 2014 ). PRLTAO 0031-9007 10.1103/PhysRevLett.113.091102 [29] 29 A. Flachi , K. Fukushima , and V. Vitagliano , Phys. Rev. Lett. 114 , 181601 ( 2015 ). PRLTAO 0031-9007 10.1103/PhysRevLett.114.181601 [30] 30 H. L. Chen , K. Fukushima , X. G. Huang , and K. Mameda , Phys. Rev. D 93 , 104052 ( 2016 ). PRVDAQ 2470-0010 10.1103/PhysRevD.93.104052 [31] 31 S. Ebihara , K. Fukushima , and K. Mameda , Phys. Lett. B 764 , 94 ( 2017 ). PYLBAJ 0370-2693 10.1016/j.physletb.2016.11.010 [32] 32 Y. Jiang and J. Liao , Phys. Rev. Lett. 117 , 192302 ( 2016 ). PRLTAO 0031-9007 10.1103/PhysRevLett.117.192302 [33] 33a M. N. Chernodub and S. Gongyo , Phys. Rev. D 96 , 096014 ( 2017 ); PRVDAQ 2470-0010 10.1103/PhysRevD.96.096014 33b M. N. Chernodub and S. Gongyo J. High Energy Phys. 01 ( 2017 ) 136 ; JHEPFG 1029-8479 10.1007/JHEP01(2017)136 33c M. N. Chernodub and S. Gongyo Phys. Rev. D 95 , 096006 ( 2017 ). PRVDAQ 2470-0010 10.1103/PhysRevD.95.096006 [34] 34 A. Flachi and K. Fukushima , Phys. Rev. D 98 , 096011 ( 2018 ). PRVDAQ 1550-7998 10.1103/PhysRevD.98.096011 [35] 35 A. Flachi and K. Fukushima , Int. J. Mod. Phys. D 26 , 1743007 ( 2017 ). IMPDEO 0218-2718 10.1142/S0218271817430076 [36] 36 X.-G. Huang , K. Nishimura , and N. Yamamoto , J. High Energy Phys. 02 ( 2018 ) 069 . JHEPFG 1029-8479 10.1007/JHEP02(2018)069 [37] 37 A. Yamamoto and Y. Hirono , Phys. Rev. Lett. 111 , 081601 ( 2013 ). PRLTAO 0031-9007 10.1103/PhysRevLett.111.081601 [38] 38 K. H. Villegas and J. P. Esguerra , Mod. Phys. Lett. A 30 , 1550020 ( 2015 ). MPLAEQ 0217-7323 10.1142/S0217732315500200 [39] 39 A. Yamamoto , Phys. Rev. D 90 , 054510 ( 2014 ). PRVDAQ 1550-7998 10.1103/PhysRevD.90.054510 [40] 40 V. N. Kotov , B. Uchoa , V. M. Pereira , F. Guinea , and A. H. Castro Neto , Rev. Mod. Phys. 84 , 1067 ( 2012 ). RMPHAT 0034-6861 10.1103/RevModPhys.84.1067 [41] 41 J. L. Lado and J. Fernández-Rossier , Phys. Rev. Lett. 113 , 027203 ( 2014 ). PRLTAO 0031-9007 10.1103/PhysRevLett.113.027203 [42] 42 Z. Wang , D.-K. Ki , J. Y. Khoo , D. Mauro , H. Berger , L. S. Levitov , and A. F. Morpurgo , Phys. Rev. X 6 , 041020 ( 2016 ). PRXHAE 2160-3308 10.1103/PhysRevX.6.041020 [43] 43 I. F. Herbut , Phys. Rev. Lett. 97 , 146401 ( 2006 ). PRLTAO 0031-9007 10.1103/PhysRevLett.97.146401 [44] 44 See https://en.wikipedia.org/wiki/Kirigami . [45] 45 M. Rickhaus , M. Mayor , and M. Juriček , Chem. Soc. Rev. 46 , 1643 ( 2017 ). CSRVBR 0306-0012 10.1039/C6CS00623J [46] 46 A. Sommerfeld , Proc. London Math. Soc. 28 , 395 ( 1897 ). PLMTAL 0024-6115 [47] 47 D. V. Fursaev and S. N. Solodukhin , Phys. Rev. D 52 , 2133 ( 1995 ). PRVDAQ 0556-2821 10.1103/PhysRevD.52.2133 [48] 48 D. V. Fursaev and D. V. Vassilevich , Operators, Geometry and Quanta ( Springer , New York, 2011 ). [49] 49 D. V. Fursaev , Classical Quantum Gravity 11 , 1431 ( 1994 ). CQGRDG 0264-9381 10.1088/0264-9381/11/6/008 [50] 50 G. Cognola , K. Kirsten , and L. Vanzo , Phys. Rev. D 49 , 1029 ( 1994 ). PRVDAQ 0556-2821 10.1103/PhysRevD.49.1029 [51] 51 R. Seeley , Int. J. Mod. Phys. A 18 , 2197 ( 2003 ). IMPAEF 0217-751X 10.1142/S0217751X03015659 [52] 52 See Supplemental Material at http://link.aps.org/supplemental/10.1103/PhysRevLett.121.221601 for to details of the computation of the effective action. [53] 53a A. Flachi and T. Tanaka , J. High Energy Phys. 02 ( 2011 ) 026 ; JHEPFG 1029-8479 10.1007/JHEP02(2011)026 53b A. Flachi , J. High Energy Phys. 01 ( 2012 ) 023 . JHEPFG 1029-8479 10.1007/JHEP01(2012)023 [54] 54 I. L. Buchbinder , S. D. Odintsov , and I. L. Shapiro , Effective Action in Quantum Gravity ( CRC Press , Boca Raton, 1992 ). [55] 55 E. Elizalde , Ten Physical Applications of Spectral Zeta Functions , Lecture Notes in Physics Vol. 855 ( Springer , New York, 2012 ). [56] 56 I. G. Avramidi , Heat Kernel and Quantum Gravity , Lecture Notes in Physics Monographs Vol. 64 ( Springer , 2000 ). [57] 57 K. Kirsten , Spectral Functions in Mathematics and Physics ( Chapman and Hall , London, 2001 ). [58] 58 M. V. Ulybyshev , P. V. Buividovich , M. I. Katsnelson , and M. I. Polikarpov , Phys. Rev. Lett. 111 , 056801 ( 2013 ). PRLTAO 0031-9007 10.1103/PhysRevLett.111.056801 [59] 59 A. S. Mayorov , D. C. Elias , I. S. Mukhin , S. V. Morozov , L. A. Ponomarenko , K. S. Novoselov , A. K. Geim , and R. V. Gorbachev , Nano Lett. 12 , 4629 ( 2012 ). NALEFD 1530-6984 10.1021/nl301922d [60] 60 H. Gonzalez-Herrero , J. M. Gomez-Rodriguez , P. Mallet , M. Moaied , J. J. Palacios , C. Salgado , M. M. Ugeda , J.-Y. Veuillen , F. Yndurain , and I. Brihuega , Science 352 , 437 ( 2016 ). SCIEAS 0036-8075 10.1126/science.aad8038 [61] 61 A. Vilenkin and E. P. S. Shellard , Cosmic Strings and Other Topological Defects ( Cambridge University Press , Cambridge, England, 2000 ). [62] 62 D. R. Nelson , Defects and Geometry in Condensed Matter Physics ( Cambridge University Press , Cambridge, England, 2002 ). [63] 63 V. Koning and V. Vitelli , in Fluids, Colloids and Soft Materials: An Introduction to Soft Matter Physics , edited by A. Fernandez-Nieves and A. M. Puertas ( Wiley , New York, 2016 ). [64] 64a L. Parker and D. J. Toms , Phys. Rev. D 31 , 953 ( 1985 ); PRVDAQ 0556-2821 10.1103/PhysRevD.31.953 64b I. Jack and L. Parker , Phys. Rev. D 31 , 2439 ( 1985 ). PRVDAQ 0556-2821 10.1103/PhysRevD.31.2439 Publisher Copyright: © 2018 authors. Published by the American Physical Society.
PY - 2018/11/27
Y1 - 2018/11/27
N2 - We consider an interacting quantum field theory on a curved two-dimensional manifold that we construct by geometrically deforming a flat hexagonal lattice by the insertion of a defect. Depending on how the deformation is done, the resulting geometry acquires a locally nonvanishing curvature that can be either positive or negative. Fields propagating on this background are forced to satisfy boundary conditions modulated by the geometry and that can be assimilated by a nondynamical gauge field. We present an explicit example where curvature and boundary conditions compete in altering the way symmetry breaking takes place, resulting in a surprising behavior of the order parameter in the vicinity of the defect. The effect described here is expected to be generic and of relevance in a variety of situations.
AB - We consider an interacting quantum field theory on a curved two-dimensional manifold that we construct by geometrically deforming a flat hexagonal lattice by the insertion of a defect. Depending on how the deformation is done, the resulting geometry acquires a locally nonvanishing curvature that can be either positive or negative. Fields propagating on this background are forced to satisfy boundary conditions modulated by the geometry and that can be assimilated by a nondynamical gauge field. We present an explicit example where curvature and boundary conditions compete in altering the way symmetry breaking takes place, resulting in a surprising behavior of the order parameter in the vicinity of the defect. The effect described here is expected to be generic and of relevance in a variety of situations.
UR - http://www.scopus.com/inward/record.url?scp=85057847797&partnerID=8YFLogxK
UR - http://www.scopus.com/inward/citedby.url?scp=85057847797&partnerID=8YFLogxK
U2 - 10.1103/PhysRevLett.121.221601
DO - 10.1103/PhysRevLett.121.221601
M3 - Article
C2 - 30547615
AN - SCOPUS:85057847797
VL - 121
JO - Physical Review Letters
JF - Physical Review Letters
SN - 0031-9007
IS - 22
M1 - 221601
ER -