Publication: Critical behavior of su(1|1) supersymmetric spin chains with long-range interactions
Loading...
Official URL
Full text at PDC
Publication Date
2016-06-01
Advisors (or tutors)
Editors
Journal Title
Journal ISSN
Volume Title
Publisher
American Physical Society
Citations
Exportar
Abstract
We introduce a general class of su(1|1) supersymmetric spin chains with long-range interactions which includes as particular cases the su(1|1) Inozemtsev (elliptic) and Haldane-Shastry chains, as well as the XX model. We show that this class of models can be fermionized with the help of the algebraic properties of the su(1|1) permutation operator and take advantage of this fact to analyze their quantum criticality when a chemical potential term is present in the Hamiltonian. We first study the low-energy excitations and the low-temperature behavior of the free energy, which coincides with that of a (1+1)-dimensional conformal field theory (CFT) with central charge c=1 when the chemical potential lies in the critical interval (0,E(π)), E(p) being the dispersion relation. We also analyze the von Neumann and Rényi ground state entanglement entropies, showing that they exhibit the logarithmic scaling with the size of the block of spins characteristic of a one-boson (1+1)-dimensional CFT. Our results thus show that the models under study are quantum critical when the chemical potential belongs to the critical interval, with central charge c=1. From the analysis of the fermion density at zero temperature, we also conclude that there is a quantum phase transition at both ends of the critical interval. This is further confirmed by the behavior of the fermion density at finite temperature, which is studied analytically (at low temperature), as well as numerically for the su(1|1) elliptic chain.
Description
©2016 American Physical Society.
This work was partially supported by Spain’s MINECO under Grant No. FIS2015-63966-P, and by the Universidad Complutense de Madrid and Banco Santander under Grant No. GR3/14-910556. P.T. has been partly supported by the ICMAT Severo Ochoa Project SEV-2015-0554 (MINECO). J.A.C. would also like to thank the Universidad Complutense de Madrid, the Madrid township and the “Residencia de Estudiantes” for their financial support.
UCM subjects
Unesco subjects
Keywords
Citation
1. D. Porras and J. I. Cirac, Phys. Rev. Lett. 92, 207901 (2004).
2. K. Kim, M.-S. Chang, R. Islam, S. Korenblit, L.-M. Duan, and C. Monroe, Phys. Rev. Lett. 103, 120502 (2009).
3. P. Jurcevic, B. P. Lanyon, P. Hauke, C. Hempel, P. Zoller, R. Blatt, and C. F. Roos, Nature (London) 511, 202 (2014).
4. P. Richerme, Z.-X. Gong, A. Lee, C. Senko, J. Smith, M. Foss-Feig, S. Michalakis, A. V. Gorshkov, and C. Monroe, Nature (London) 511, 198 (2014).
5. P. Schauß, J. Zeiher, T. Fukuhara, S. Hild, M. Chenau, T. Macrì, T. Pohl, I. Bloch, and C. Gross, Science 347, 1455 (2015).
6. F. D. M. Haldane, Phys. Rev. Lett. 60, 635 (1988).
7. B. S. Shastry, Phys. Rev. Lett. 60, 639 (1988).
8. V. I. Inozemtsev, J. Stat. Phys. 59, 1143 (1990).
9. F. D. M. Haldane, Z. N. C. Ha, J. C. Talstra, D. Bernard, and V. Pasquier, Phys. Rev. Lett. 69, 2021 (1992).
10. F. D. M. Haldane, in Correlation Effects in Low-dimensional Electron Systems, Springer Series in Solid-State Sciences, Vol. 118, edited by A. Okiji and N. Kawakami (Springer, Berlin, 1994), pp. 3–20.
11. B. Basu-Mallick and N. Bondyopadhaya, Nucl. Phys. B 757, 280 (2006).
12. B. Basu-Mallick, N. Bondyopadhaya, and D. Sen, Nucl. Phys. B 795, 596 (2008).
13. B. Sutherland, Phys. Rev. B 12, 3795 (1975).
14. P. Schlottmann, Phys. Rev. B 36, 5177 (1987).
15. P. A. Bares and G. Blatter, Phys. Rev. Lett. 64, 2567 (1990).
16. Y. Kuramoto and H. Yokoyama, Phys. Rev. Lett. 67, 1338 (1991).
17. F. Finkel and A. González-López, J. Stat. Mech. (2014) P12014.
18. H. W. J. Blöte, J. L. Cardy, and M. P. Nightingale, Phys. Rev. Lett. 56, 742 (1986).
19. I. Affleck, Phys. Rev. Lett. 56, 746 (1986).
20. C. Holzhey, F. Larsen, and F. Wilczek, Nucl. Phys. B 424, 443 (1994).
21. P. Calabrese and J. Cardy, J. Stat. Mech. (2004) P06002.
22. P. Calabrese and J. Cardy, J. Stat. Mech. (2005) P04010.
23. Here and in what follows, all sums range from 1 to N unless otherwise stated.
24. E. T. Whittaker and G. N. Watson, A Course of Modern Analysis (Cambridge University Press, Cambridge, 1927).
25. F. W. J. Olver, D. W. Lozier, R. F. Boisvert, and C. W. Clark, editor, NIST Handbook of Mathematical Functions (Cambridge University Press, Cambridge, 2010).
26. S. Sachdev, Quantum Phase Transitions, 2nd ed. (Cambridge University Press, Cambridge, 2011).
27. It can be easily checked with the help of Legendre's identity (η1/α)+i+ζ(iπ/α;π,iπ/α)=1/2 that E(p) is not an elliptic function.
28. F. Göhmann and M. Wadati, J. Phys. Soc. Jpn. 64, 3585 (1995).
29. Indeed, log(1+e−x)=O(e−x) for x→∞, so that the error incurred in extending the upper integration limit to +∞ is bounded by C∫∞λβe−xdx=Ce−λβ≪Tj for all j>0, where C is a positive constant.
30. G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev. Lett. 90, 227902 (2003).
31. Although, strictly speaking, Eq. (25) only holds for m≠n, it is obvious that Amm coincides with the m→n limit of the latter equation.
32. M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary ed. (Cambridge University Press, Cambridge, 2010).
33. B.-Q. Jin and V. E. Korepin, J. Stat. Phys. 116, 79 (2004).
34. More precisely, since we have not actually computed the entanglement entropy for λ exactly equal to 0 the previous calculation only shows that the entropy has the same limit as λ→0− and λ→0+.
35. Cf. the previous footnote.
36. M. E. Fisher and R. E. Hartwig, Adv. Chem. Phys. 15, 333 (1968).
37. E. L. Basor, Indiana Math. J. 28, 975 (1979).
38. R. Klabbers, Nucl. Phys. B 907, 77 (2016).
39. F. Castilho Alcaraz, M. Ibáñez Berganza, and G. Sierra, Phys. Rev. Lett. 106, 201601 (2011).
40. M. Ibáñez Berganza, F. Castilho Alcaraz, and G. Sierra, J. Stat. Mech. (2012) P01016.