Can one trust quantum simulators?

Reports on Progress in Physics

Volumn 75, Issue 8, 2012, Pages

  Hauke, Philipp ^a   Cucchietti, Fernando M ^a,b   Tagliacozzo, Luca ^a   Deutsch, Ivan ^c,d   Lewenstein, MacIej ^a,e  

^a ICFO INSTITUT DE CIENCIES FOTONIQUES   (Spain)

^b BARCELONA SUPERCOMPUTING CENTER   (Spain)

^c Center for Quantum Information and Control (CQuIC)   (United States)

^d UNIVERSITY OF NEW MEXICO   (United States)

^e INSTITUCIÓ CATALANA DE RECERCA I ESTUDIS AVANÇATS ICREA   (Spain)

Author keywords

[No Author keywords available]

Indexed keywords

ALGORITHM; ARTICLE; CHEMICAL MODEL; COMPUTER SIMULATION; MATHEMATICAL COMPUTING; QUANTUM THEORY;

ALGORITHMS; COMPUTER SIMULATION; MODELS, CHEMICAL; NUMERICAL ANALYSIS, COMPUTER-ASSISTED; QUANTUM THEORY;

EID: 84864415066     PISSN: 00344885     EISSN: None     Source Type: Journal    
DOI: 10.1088/0034-4885/75/8/082401     Document Type: Review

References (206)

1. Feynman R P 1982 Int. J. Theor. Phys. 21 467
2. Buluta I and Nori F 2009 Science 326 108
3. Lewenstein M, Sanpera A and Ahufinger V 2012 Ultracold Atoms in Optical Lattices: Simulating Quantum Many-Body Systems (Oxford: Oxford University Press)
4. Breakthrough of the Year The Runners-Up 2010 Science 330 1605
5. Lin Y-J, Compton R L, Jiménez-García K, Porto J V and Spielman I B 2009 Nature 462 628
6. Bakr W S, Peng A, Tai M E, Ma R, Simon J, Gillen J I, Fölling S, Pollet L and Greiner M 2010 Science 329 547
7. Trotzky S, Pollet L, Gerbier F, Schnorrberger U, Bloch I, Prokof'ev N V, Svistunov B and Troyer M 2010 Nature Phys. 6 998
8. Jördens R et al 2010 Phys. Rev. Lett. 104 180401
9. Baumann K, Guerlin C, Brennecke F and Esslinger T 2010 Nature 464 1301
10. Liao Y, Rittner A S C, Paprotta T, Li W, Partridge G B, Hulet R G, Baur S K and Mueller E J 2010 Nature 467 567
11. Simon J, Bakr W S, Ma R, Tai M E, Preiss P M and Greiner M 2011 Nature 472 307
12. Struck J, Ölschläger C, Le Targat R, Soltan-Panahi P, Eckardt A, Lewenstein M, Windpassinger P and Sengstock K 2011 Science 333 996
13. Van Houcke K, Werner F, Kozik E, Prokof'ev N, Svistunov B, Ku M J H, Sommer A T, Cheuk L W, Schirotzek A and Zwierlein M W 2012 Nature Phys. 8 366
14. Trotzky S, Chen Y-A, Flesch A, McCulloch I P, Schollwöck U, Eisert J and Bloch I 2012 Nature Phys. 8 325
15. Cheneau M, Barmettler P, Poletti D, Endres M, Schausz P, Fukuhara T, Christian C G, Bloch I, Kollath C and Kuhr S 2012 Nature 481 484
16. Aidelsburger M, Atala M, Nascimbène S, Trotzky S, Chen Y-A and Bloch I 2011 Phys. Rev. Lett. 107 255301
17. Jimínez-García K, LeBlanc L J, Williams R A, Beeler M C, Perry A R and Spielman I B 2012 Phys. Rev. Lett. 108 225303
18. Bloch I, Dalibard J and Zwerger W 2008 Rev. Mod. Phys. 80 885
19. Bloch I, Dalibard J and Nascimbène S 2012 Nature Phys. 8 267
20. Müller M, Diehl S, Pupillo G and Zoller P 2012 arXiv:1203.6595v1 [quant-ph]
21. Friedenauer A, Schmitz H, Glueckert J T, Porras D and Schaetz T 2008 Nature Phys. 4 757
22. Kim K, Chang M-S, Korenblit S, Islam R, Edwards E E, Freericks J K, Lin G-D, Duan L-M and Monroe C 2010 Nature 465 590
23. Islam R et al 2011 Nature Commun. 2 377
24. Lanyon B P et al 2011 Science 334 57
25. Barreiro J T, Müller M, Schindler P, Nigg D, Monz T, Chwalla M, Hennrich M, Roos C F, Zoller P and Blatt R 2011 Nature 470 486
26. Gerritsma R, Kirchmair G, Zähringer F, Solano E, Blatt R and Roos C F 2010 Nature 463 68
27. Gerritsma R, Lanyon B, Kirchmair G, Zähringer F, Hempel C, Casanova J, García-Ripoll J J, Solano E, Blatt R and Roos C F 2011 Phys. Rev. Lett. 106 060503
28. Johanning M, Varón A F and Wunderlich C 2009 J. Phys. B: At. Mol. Opt. 42 154009
29. Schneider C, Porras D and Schaetz T 2012 Rep. Prog. Phys. 75 024401
30. Blatt R and Roos C F 2012 Nature Phys. 8 277
31. Lanyon B P, Weinhold T J, Langford N K, Barbieri M, James D F V, Gilchrist A and White A G 2007 Phys. Rev. Lett. 99 250505 (Pubitemid 350281274)
32. Lanyon B P, Barbieri M, Almeida M P and White A G 2008 Phys. Rev. Lett. 101 200501
33. Lanyon B P et al 2010 Nature Chem. 2 106
34. Belgiorno F, Cacciatori S L, Clerici M, Gorini V, Ortenzi G, Rizzi L, Rubino E, Sala V G and Faccio D 2010 Phys. Rev. Lett. 105 203901
35. Kitagawa T, Broome M, Fedrizzi A, Rudner M, Berg E, Kassal I, Aspuru -Guzik A, Demler E and White A G 2011 arXiv:1105.5334
36. Ma X, Dakic B, Naylor W, Zeilinger A and Walther P 2011 Nature Phys. 7 399
37. Aspuru-Guzik A and Walther P 2012 Nature Phys. 8 285
38. Greentree A D, Tahan C, Cole J H and Hollenberg L C L 2006 Nature Phys. 2 856 (Pubitemid 44853584)
39. Hartmann M J, Brandão F G S L and Plenio M B 2006 Nature Phys. 2 849 (Pubitemid 44853583)
40. Angelakis D G, Santos M F and Bose S 2007 Phys. Rev. A 76 031805 (Pubitemid 47510818)
41. Wang Z, Chong Y, Joannopoulos J D and Soljačić M 2009 Nature 461 772
42. Koch J, Houck A A, Hur K L and Girvin S M 2010 Phys. Rev. A 82 043811
43. Angelakis D G et al 2012 in preparation
44. Singha A et al 2011 Science 332 1176
45. Somaroo S, Tseng C H, Havel T F, Laflamme R and Cory D G 1999 Phys. Rev. Lett. 82 5381
46. Tseng C H, Somaroo S, Sharf Y, Knill E, Laflamme R, Havel T F and Cory D G 1999 Phys. Rev. A 61 012302
47. Du J, Xu N, Peng X, Wang P, Wu S and Lu D 2010 Phys. Rev. Lett. 104 030502
48. Álvarez G and Suter D 2010 Phys. Rev. Lett. 104 230403
49. Li Z, Yung M-H, Chen H, Lu D, Whitfield J D, Peng X, Aspuru-Guzik A and Du J 2011 Sci. Rep. 1 88
50. Lu D, Xu N, Xu R, Chen H, Gong J, Peng X and Du J 2011 Phys. Rev. Lett. 107 020501
51. Makhlin Y, Schön G and Shnirman A 2001 Rev. Mod. Phys. 73 357
52. Devoret M, Wallraff A and Martinis J 2004 arXiv:cond-mat/0411174
53. You J and Nori F 2005 Phys. Today 58 42
54. Girvin S M, Devoret M H and Schoelkopf R J 2009 Phys. Scr. T137 014012 (Proc. Nobel Sympo. 141: Qubits for Future Quantum.)
55. Pritchett E J, Benjamin C, Galiautdinov A, Geller M R, Sornborger A T, Stancil P C and Martinis J M 2010 arXiv:1008.0701v1 [quant-ph]
56. Reed M D, DiCarlo L, Nigg S E, Sun L, Frunzio L, Girvin S M and Schoelkopf R J 2012 Nature 482 382
57. Zoller P et al 2005 Eur. Phys. J. D 36 203 (Pubitemid 41442672)
58. Kendon V, Nemoto K and Munro W 2010 Phil. Trans. R. Soc. A 368 3609
59. Landauer R 1996 Phys. Lett. A 217 188 (Pubitemid 126372168)
60. Deutsch D 1985 Proc. R. Soc. Lond. A 400 97
61. Knill E and Laflamme R 1997 Phys. Rev. A 55 900
62. Preskill J 1998 Introduction to Quantum Computation (Singapore: World Scientific) chapter Fault Tolerant Quantum Computation
63. Brown K R, Clark R J and Chuang I L 2006 Phys. Rev. Lett. 97 050504 (Pubitemid 44188133)
64. DiVincenzo D P 2000 Fortschr. Phys. 48 771
65. Cirac J I and Zoller P 2012 Nature Phys. 8 264
66. Houck A A, Türeci H E and Koch J 2012 Nature Phys. 8 292
67. Lee P A, Nagaosa N and Wen X-G 2006 Rev. Mod. Phys. 78 17 (Pubitemid 43279669)
68. Lee P A 2008 Rep. Prog. Phys. 71 012501
69. Jaksch D, Bruder C, Cirac J I, Gardiner C W and Zoller P 1998 Phys. Rev. Lett. 81 3108
70. Greiner M, Bloch I, Mandel O, Hänsch T W and Esslinger T 2001 Phys. Rev. Lett. 87 160405
71. Hadzibabic Z 2011 private communication
72. Leibfried D 2010 Nature 463 608
73. Sandvik A 2010 AIP Conf. Proc. 1297 135
74. Dusuel S and Uhrig G S 2004 J. Phys. A: Math. Gen. 37 9275 (Pubitemid 39403620)
75. Schollwöck U 2011 Ann. Phys. 326 96
76. Alba V, Tagliacozzo L and Calabrese P 2011 J. Stat. Mech. P06012
77. Evenbly G and Vidal G 2011 arXiv:1109.5334v1 [quant-ph]
78. Stoudenmire E and White S 2011 Ann. Rev. Condens. Matter Phys. 3 111
79. Tagliacozzo L, Evenbly G and Vidal G 2009 Phys. Rev. B 80 235127
80. Tagliacozzo L and Vidal G 2011 Phys. Rev. B 83 115127
81. Corboz P, White S, Vidal G and Troyer M 2011 Phys. Rev. B 84 041108
82. Wang L, Gu Z, Wen X and Verstraete F 2011 arXiv:1112.3331v2 [cond-mat.str-el]
83. Binney J J, Dowrick N J, Fisher A J and Newman M E J 1992 The Theory Of Critical Phenomena : an Introduction to the Renormalization Group (Oxford: Clarendon)
84. Martin R 2004 Electronic Structure: Basic Theory and Practical Methods (Cambridge: Cambridge University Press)
85. Stoudenmire E, Wagner L O, White S R and Burke K 2011 arXiv:1107.2394v2 [cond-mat.str-el]
86. Georges A, Kotliar G, Krauth W and Rozenberg M 1996 Rev. Mod. Phys. 68 13
87. Misguich G and Lhuillier C 2004 Frustrated Spin Systems vol 229 (Singapore: World Scientific)
88. Das A and Chakrabarti B K 2008 Rev. Mod. Phys. 80 1061
89. Lieb E and Robinson D 1972 Commun. Math. Phys. 28 251
90. Bravyi S, Hastings M B and Verstraete F 2006 Phys. Rev. Lett. 97 050401 (Pubitemid 44156696)
91. Calabrese P and Cardy J 2006 Phys. Rev. Lett. 96 136801
92. Eisert J and Osborne T 2006 Phys. Rev. Lett. 97 150404 (Pubitemid 44570427)
93. Nachtergaele B, Ogata Y and Sims R 2006 J. Stat. Phys. 124 1 (Pubitemid 44300290)
94. Schuch N, Wolf M, Vollbrecht K and Cirac J 2008 New J. Phys. 10 033032
95. Eisert J, Cramer M and Plenio M B 2010 Rev. Mod. Phys. 82 277
96. Lloyd S 1996 Science 273 1073
97. Müller M, Lesanovsky I, Weimer H, Büchler H and Zoller P 2009 Phys. Rev. Lett. 102 170502
98. Weimer H, Müller M, Lesanovsky I, Zoller P and Büchler H P 2010 Nature Phys. 6 382
99. Jane E, Vidal G, Dür W, Zoller P and Cirac J 2003 Quantum Comput. Inform. 3 15
100. Wiebe N, Berry D, Hoyer P and Sanders B 2011 J. Phys. A: Math. Theor. 44 445308
101. Aspuru-Guzik A, Dutoi A D, Love P J and Head-Gordon M 2005 Science 309 1704 (Pubitemid 41285935)
102. Jordan S P, Lee K S and Preskill J 2011 arXiv:1111.3633v2
103. Jordan S P, Lee K S and Preskill J 2011 arXiv:1112.4833v1
104. Shor P W 1995 Phys. Rev. A 52 2493(R)
105. Nielsen M A and Chuang I L 2000 Quantum Computation and Quantum Information (Cambridge: Cambridge University Press)
106. Cory D G, Price M D, Maas W, Knill E, Laflamme R, Zurek W H, Havel T F and Somaroo S S 1998 Phys. Rev. Lett. 81 2152
107. Chiaverini J et al 2004 Nature 432 602 (Pubitemid 39626262)
108. Pittman T B, Jacobs B C and Franson J D 2005 Phys. Rev. A 71 052332 (Pubitemid 41481398)
109. Lassen M, Sabuncu M, Huck A, Niset J, Leuchs G, Cerf N J and Andersen U L 2010 Nature Photon. 4 700
110. Schindler P, Barreiro J T, Monz T, Nebendahl V, Nigg D, Chwalla M, Hennrich M and Blatt R 2011 Science 332 1059
111. Aharonov D and Ta-Shma A 2003 Proc. 35th STOC (San Diego, CA,) (New York: ACM) p 20
112. Berry D, Ahokas G, Cleve R and Sanders B 2007 Commun. Math. Phys. 270 359
113. Childs A and Kothari R 2011 Theory of Quantum Computation, Communication, and Cryptography (5th Conf. TQC, Leeds, 2010) (Berlin: Springer) p 94
114. Brown K, Munro W and Kendon V 2010 Entropy 12 2268
115. Wu L A, Byrd M S and Lidar D A 2002 Phys. Rev. Lett. 89 057904
116. Kitaev A, Shen A and Vyalyi M 2002 Classical and Quantum Computation, Graduate Studies in Mathematics vol 47 (Providence, RI: American Mathematical Society)
117. Clark C, Metodi T, Gasster S and Brown K 2009 Phys. Rev. A 79 062314
118. Abrams D and Lloyd S 1999 Phys. Rev. Lett. 83 5162
119. Schwarz M, Temme K and Verstraete F 2011 arXiv:1104.1410 [quant-ph]
120. Temme K, Osborne T J, Vollbrecht K G, Poulin D and Verstraete F 2011 Nature 471 87
121. Haeberlen U and Waugh J S 1968 Phys. Rev. 175 453
122. Waugh J 2007 Average Hamiltonian Theory (Encyclopedia of Magnetic Resonance) (New York: Wiley)
123. Diehl S, Micheli A, Kantian A, Kraus B, Büchler H P and Zoller P 2008 Nature Phys. 4 878
124. Kraus B, Büchler H P, Diehl S, Kantian A, Micheli A and Zoller P 2008 Phys. Rev. A 78 042307
125. Verstraete F, Wolf M and Cirac I 2009 Nature Phys. 5 633
126. Wu R, Pechen A, Brif C and Rabitz H 2007 J. Phys. A: Math. Theor. 40 5681
127. Wolf M, Eisert J, Cubitt T and Cirac J 2008 Phys. Rev. Lett. 101 150402
128. Cubitt T, Eisert J and Wolf M 2009 arXiv:0908.2128
129. Cucchietti F M, Zhang J-F, Lombardo F C, Villar P I and Laflamme R 2010 Phys. Rev. Lett. 105 240406
130. Temme K, Wolf M and Verstraete F 2009 arXiv:0912.0858
131. Dodd J, Nielsen M, Bremner M and Thew R 2002 Phys. Rev. A 65 040301
132. Wocjan P, Roetteler M, Janzing D and Beth T 2002 Quant. Inf. Comp. 2 133
133. Dür W, Bremner M J and Briegel H J 2008 Phys. Rev. A 78 052325
134. Britton J W, Sawyer B C, Keith A C, Wang C-C J, Freericks J K, Uys H, Biercuk M J and Bollinger J J 2012 Nature 484 489
135. Sowiski T, Dutta O, Hauke P, Tagliacozzo L and Lewenstein M. 2012 Phys. Rev. Lett. 108 115301
136. Dutta O, Eckardt A, Hauke P, Malomed B and Lewenstein M 2011 New J. Phys. 13 023019
137. Schuch N, Wolf M, Verstraete F and Cirac J 2008 Phys. Rev. Lett. 100 030504 (Pubitemid 351161830)
138. Schuch N, Cirac J and Verstraete F 2008 Phys. Rev. Lett. 100 250501
139. Hastings M 2009 Phys. Rev. Lett. 103 050502
140. Quan H, Song Z, Liu X, Zanardi P and Sun C P 2006 Phys. Rev. Lett. 96 140604
141. Zanardi P and Paunkovic N 2006 Phys. Rev. E 74 031123 (Pubitemid 44494310)
142. Georgeot B and Shepelyansky D L 2000 Phys. Rev. E 62 3504
143. Frahm K M, Fleckinger R and Shepelyansky D L 2004 Eur. Phys. J. D 29 139 (Pubitemid 38483586)
144. Gardiner C, Zoller P, Ballagh R and Davis M 1997 Phys. Rev. Lett. 79 1793
145. Gardiner C and Zoller P 1997 Phys. Rev. A 55 2902
146. Jaksch D, Gardiner C and Zoller P 1997 Phys. Rev. A 56 575
147. Gardiner C and Zoller P 1998 Phys. Rev. A 58 536
148. Jaksch D, Gardiner C, Gheri K and Zoller P 1998 Phys. Rev. A 58 1450
149. Gardiner C, Lee M, Ballagh R, Davis M and Zoller P 1998 Phys. Rev. Lett. 81 5266
150. Gardiner C and Zoller P 2000 Phys. Rev. A 61 033601
151. Cirac J, Lewenstein M and Zoller P 1994 Phys. Rev. Lett. 72 2977
152. Cirac J, Lewenstein M and Zoller P 1994 Phys. Rev. A 50 3409
153. Cirac J, Lewenstein M and Zoller P 1995 Phys. Rev. A 51 2899
154. Morigi G, Cirac J I, Lewenstein M and Zoller P 1997 Europhys. Lett. 39 13
155. Santos L and Lewenstein M 1999 Eur Phys. J. D 7 379
156. Santos L and Lewenstein M 1999 Appl. Phys. B 69 363 (Pubitemid 30514153)
157. Castin Y, Cirac J and Lewenstein M 1998 Phys. Rev. Lett. 80 5305
158. Idziaszek Z, Santos L and Lewenstein M 2001 Phys. Rev. A 64 051402
159. Idziaszek Z, Santos L, Baranov M and Lewenstein M 2003 J. Opt. B: Quantum Semiclass. Opt. 5 S190
160. Idziaszek Z, Santos L, Baranov M and Lewenstein M 2003 Phys. Rev. A 67 041403(R)
161. Dziarmaga J and Lewenstein M 2005 Phys. Rev. Lett. 94 090403 (Pubitemid 40620587)
162. Dziarmaga J and Lewenstein M 2005 Phys. Rev. A 72 033606 (Pubitemid 41776568)
163. Lewenstein M, Cirac J and Zoller P 1995 Phys. Rev. A 51 4617
164. Mosk A, Kraft S, Mudrich M, Singer K, Wohlleben W, Grimm R and Weidemüller M 2001 Appl. Phys. B 73 791 (Pubitemid 34043635)
165. Papenbrock T, Salgueiro A and Weidenmüller H 2002 Phys. Rev. A 65 043601
166. Papenbrock T, Salgueiro A and Weidenmüller H 2002 Phys. Rev. A 66 025603
167. Wang S, Nemes M, Salgueiro A and Weidenmüller H 2002 Phys. Rev. A 66 033608
168. Augusiak R, Cucchietti F M, Haake F and Lewenstein M 2010 New J. Phys. 12 025021
169. Mostame S, Rebentrost P, Eisfeld A, Kerman A J, Tsomokos D I and Aspuru-Guzik A 2011 arXiv:1106.1683v3 [quant-ph]
170. Sanchez-Palencia L and Lewenstein M 2010 Nature Phys. 6 87
171. Zuniga-Hansen N, Chi Y-C and Byrd M S 2012 arXiv:1202.6337v2 [quant-ph]
172. Cardy J 1996 Scaling and Renormalization in Statistical Physics (Cambridge Lecture Notes in Physics) (Cambridge: Cambridge University Press) Number 5
173. De Martino A, Thorwart M, Egger R and Graham R 2005 Phys. Rev. Lett. 94 060402 (Pubitemid 43272592)
174. Lieb E, Schultz T and Mattis D 1961 Ann. Phys. 16 407
175. Griffiths R. 1969 Phys. Rev. Lett. 23 17
176. Refael G and Moore J 2004 Phys. Rev. Lett. 93 260602 (Pubitemid 40213716)
177. Fisher M P, Weichmann P B, Grinstein G and Fisher D S 1989 Phys. Rev. B 40 546
178. Freedman R and Hertz J A 1977 Phys. Rev. B 15 2384
179. Lewenstein M, Sanpera A, Ahufinger V, Damski B, Sen A and Sen U 2007 Adv. Phys. 56 243 (Pubitemid 46710973)
180. Uhlmann A 1976 Rep. Math. Phys. 9 273
181. Barber M N 1983 Phase Transitions and Critical Phenomena vol 8 (New York: Academic)
182. Cardy J 2010 Boltzmann Medal Lecture, StatPhys24, Cairns http://www-thphys.physics.ox.ac.uk/people/JohnCardy/seminars/statphys24.pdf
183. Holzhey C, Larsen F and Wilczek F 1994 Nucl. Phys. B 424 443
184. Vidal G, Latorre J I, Rico E and Kitaev A 2003 Phys. Rev. Lett. 90 227902
185. Calabrese P and Cardy J 2004 J. Stat. Mech. P06002
186. Gorin T, Prosen T, Seligman T and Znidaric M 2006 Phys. Rep. 435 33 (Pubitemid 44696158)
187. Prosen T and Znidaric M 2007 Phys. Rev. E 75 015202 (Pubitemid 46134154)
188. Perales A and Vidal G 2008 Phys. Rev. A 78 042337
189. Aharonov D and Ben-Or M 1996 Proc. 37th Conference on Foundations of Computer Science (Burlington, VT,) arXiv:quant-ph/9611029v1
190. Harrow A and Nielsen M 2003 Phys. Rev. A 68 012308
191. Virmani S, Huelga S F and Plenio M B 2005 Phys. Rev. A 71 042328 (Pubitemid 41481474)
192. Vidal G 2003 Phys. Rev. Lett. 91 147902
193. Verstraete F and Cirac J I 2004 arXiv:cond-mat/0407066v1
194. Evenbly G and Vidal G 2011 J. Stat. Phys. 145 891
195. Cozzini M, Giorda P and Zanardi P 2007 Phys. Rev. B 75 014439
196. Cramer M and Eisert J 2006 New J. Phys. 8 71 (Pubitemid 43788089)
197. Nightingale M 1975 Physica A 83 561
198. Jacobson N T, Garnerone S, Haas S and Zanardi P 2009 Phys. Rev. B 79 184427
199. Levitov L, Lee H and Lesovik G 1996 J. Math. Phys. 37 4845
200. Klich I 2003 Quantum Noise in Mesoscopic Systems (Dordrecht: Kluwer)
201. Sorella S and Becca F 2011 http://people.sissa.it/sorella/Simulazioni.pdf
202. Pineda C, Barthel T and Eisert J 2009 arXiv:0905.0669
203. Corboz P and Vidal G 2009 Phys. Rev. B 80 165129
204. Barthel T, Pineda C and Eisert J 2009 Phys. Rev. A 80 042333
205. Shi Q, Li S, Zhao J and Zhou H 2009 arXiv:0907.5520
206. Corboz P, Orús R, Bauer B and Vidal G 2010 Phys. Rev. B 81 165104