Skip to main content

Advertisement

SpringerLink
Log in

Quantum phase transition in skewed ladders: an entanglement entropy and fidelity study

  • Regular Article - Solid State and Materials
  • Published:
The European Physical Journal B Aims and scope Submit manuscript

Abstract

Entanglement entropy (EE) of a state is a measure of correlation or entanglement between two parts of a composite system and it may show appreciable change when the ground state (GS) undergoes a qualitative change in a quantum phase transition (QPT). Therefore, the EE has been extensively used to characterise the QPT in various correlated Hamiltonians. Similarly fidelity also shows sharp changes at a QPT. We characterized the QPT of frustrated antiferromagnetic Heisenberg spin-1/2 systems on 3/4, 3/5 and 5/7 skewed ladders using the EE and fidelity analysis. It is noted that all the non-magnetic to magnetic QPT boundary in these systems can be accurately determined using the EE and fidelity, and the EE exhibits a discontinuous change, whereas fidelity shows a sharp dip at the transition points. It is also noted that in case of the degenerate GS, the unsymmetrized calculations show wild fluctuations in the EE and fidelity even without actual phase transition, however, this problem is resolved by calculating the EE and the fidelity in the lowest energy state of the symmetry subspaces, to which the degenerate states belong.

Graphic abstract

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8

Similar content being viewed by others

Data Availibility Statement

This manuscript has no associated data or the data will not be deposited. [Authors’ comment: The data that support the findings of this study are available from the corresponding author upon reasonable request.]

References

  1. C.K. Majumdar, D.K. Ghosh, On next-nearest-neighbor interaction in linear chain. ii. J. Math. Phys. 10(8), 1399–1402 (1969). https://doi.org/10.1063/1.1664979

    Article  ADS  MathSciNet  Google Scholar 

  2. A.V. Chubukov, Chiral, nematic, and dimer states in quantum spin chains. Phys. Rev. B 44, 4693–4696 (1991). https://doi.org/10.1103/PhysRevB.44.4693

    Article  ADS  Google Scholar 

  3. R. Chitra, S. Pati, H.R. Krishnamurthy, D. Sen, S. Ramasesha, Density-matrix renormalization-group studies of the spin-1/2 heisenberg system with dimerization and frustration. Phys. Rev. B 52, 6581–6587 (1995). https://doi.org/10.1103/PhysRevB.52.6581

    Article  ADS  Google Scholar 

  4. S.R. White, I. Affleck, Dimerization and incommensurate spiral spin correlations in the zigzag spin chain: analogies to the kondo lattice. Phys. Rev. B 54, 9862–9869 (1996). https://doi.org/10.1103/PhysRevB.54.9862

    Article  ADS  Google Scholar 

  5. C. Itoi, S. Qin, Strongly reduced gap in the zigzag spin chain with a ferromagnetic interchain coupling. Phys. Rev. B 63, 224–423 (2001). https://doi.org/10.1103/PhysRevB.63.224423

    Article  Google Scholar 

  6. S. Mahdavifar, Numerical study of the frustrated ferromagnetic spin-1/2 chain. J. Phys. Condens. Matter 20(33), 335–430 (2008). https://doi.org/10.1088/0953-8984/20/33/335230

  7. J. Sirker, Thermodynamics of multiferroic spin chains. Phys. Rev. B 81, 014–419 (2010). https://doi.org/10.1103/PhysRevB.81.014419

    Article  Google Scholar 

  8. M. Kumar, A. Parvej, Z.G. Soos, Level crossing, spin structure factor and quantum phases of the frustrated spin-1/2 chain with first and second neighbor exchange. J. Phys. Conden. Matter 27(31), 316001 (2015). https://doi.org/10.1088/0953-8984/27/31/316001

  9. Z.G. Soos, A. Parvej, M. Kumar, Numerical study of incommensurate and decoupled phases of spin-1/2 chains with isotropic exchange \(J_1, J_2\) between first and second neighbors. J. Phys. Condensed Matter. 28(17), 175603 (2016). https://doi.org/10.1088/0953-8984/28/17/175603

  10. M. Kumar, S. Ramasesha, Z.G. Soos, Bond-order wave phase, spin solitons, and thermodynamics of a frustrated linear spin-\(\frac{1}{2}\) heisenberg antiferromagnet. Phys. Rev. B 81, 054413 (2010). https://doi.org/10.1103/PhysRevB.81.054413

  11. T. Hamada, J.i. Kane, S.i. Nakagawa, Y. Natsume, Exact solution of ground state for uniformly distributed rvb in one-dimensional spin-1/2 heisenberg systems with frustration. J. Phys. Soc. Jpn. 57(6), 1891–1894 (1988). https://doi.org/10.1143/JPSJ.57.1891

  12. M. Kumar, B.J. Topham, R. Yu, Q.B.D. Ha, Z.G. Soos, Magnetic susceptibility of alkali-tetracyanoquinodimethane salts and extended hubbard models with bond order and charge density wave phases. J. Chem. Phys. 134(23), 234304 (2011). https://doi.org/10.1063/1.3598952

  13. M. Kumar, S. Ramasesha, Z.G. Soos, Tuning the bond-order wave phase in the half-filled extended hubbard model. Phys. Rev. B 79, 035102 (2009). https://doi.org/10.1103/PhysRevB.79.035102

  14. J.E. Hirsch, Charge-density-wave to spin-density-wave transition in the extended hubbard model. Phys. Rev. Lett. 53, 2327–2330 (1984). https://doi.org/10.1103/PhysRevLett.53.2327

    Article  ADS  Google Scholar 

  15. J.E. Hirsch, Phase diagram of the one-dimensional molecular-crystal model with coulomb interactions: half-filled-band sector. Phys. Rev. B 31, 6022–6031 (1985). https://doi.org/10.1103/PhysRevB.31.6022

    Article  ADS  Google Scholar 

  16. D. Dey, M. Kumar, Z.G. Soos, Boundary-induced spin-density waves in linear heisenberg antiferromagnetic spin chains with \(s \ge 1\). Phys. Rev. B 94, 144417 (2016). https://doi.org/10.1103/PhysRevB.94.144417

  17. S.R. White, D.A. Huse, Numerical renormalization-group study of low-lying eigenstates of the antiferromagnetic s=1 heisenberg chain. Phys. Rev. B 48, 3844–3852 (1993). https://doi.org/10.1103/PhysRevB.48.3844

    Article  ADS  Google Scholar 

  18. E.S. So/rensen, I. Affleck, Equal-time correlations in haldane-gap antiferromagnets. Phys. Rev. B 49, 15771–15788 (1994). https://doi.org/10.1103/PhysRevB.49.15771

  19. T. Hikihara, L. Kecke, T. Momoi, A. Furusaki, Vector chiral and multipolar orders in the spin-\(\frac{1}{2}\) frustrated ferromagnetic chain in magnetic field. Phys. Rev. B 78, 144404 (2008). https://doi.org/10.1103/PhysRevB.78.144404

    Article  ADS  Google Scholar 

  20. J. Sudan, A. Lüscher, A.M. Läuchli, Emergent multipolar spin correlations in a fluctuating spiral: The frustrated ferromagnetic spin-\(\frac{1}{2}\) heisenberg chain in a magnetic field. Phys. Rev. B 80, 140402 (2009). https://doi.org/10.1103/PhysRevB.80.140402

  21. A. Parvej, M. Kumar, Degeneracies and exotic phases in an isotropic frustrated spin-1/2 chain. J. Magnet. Magn. Mater. 401, 96–101 (2016). https://doi.org/10.1016/j.jmmm.2015.10.017

    Article  ADS  Google Scholar 

  22. I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets. Phys. Rev. Lett. 59, 799–802 (1987). https://doi.org/10.1103/PhysRevLett.59.799

    Article  ADS  Google Scholar 

  23. I. Affleck, T. Kennedy, E.H. Lieb, H. Tasaki, Rigorous results on valence-bond ground states in antiferromagnets. Commun. Math. Phys. 115, 477–528 (1988). https://doi.org/10.1007/BF01218021

    Article  ADS  Google Scholar 

  24. U. Schollwöck, O. Golinelli, T. Jolicœur, S = 2 antiferromagnetic quantum spin chain. Phys. Rev. B 54, 4038–4051 (1996). https://doi.org/10.1103/PhysRevB.54.4038

    Article  ADS  Google Scholar 

  25. Z.C. Gu, X.G. Wen, Tensor-entanglement-filtering renormalization approach and symmetry-protected topological order. Phys. Rev. B 80, 155131 (2009). https://doi.org/10.1103/PhysRevB.80.155131

  26. F. Pollmann, E. Berg, A.M. Turner, M. Oshikawa, Symmetry protection of topological phases in one-dimensional quantum spin systems. Phys. Rev. B 85, 075125 (2012). https://doi.org/10.1103/PhysRevB.85.075125

  27. F. Haldane, Continuum dynamics of the 1-d heisenberg antiferromagnet: identification with the o(3) nonlinear sigma model. Phys. Lett. A 93(9), 464–468 (1983). https://doi.org/10.1016/0375-9601(83)90631-X

    Article  ADS  MathSciNet  Google Scholar 

  28. F.D.M. Haldane, Model for a quantum hall effect without landau levels: condensed-matter realization of the “parity anomaly.” Phys. Rev. Lett. 61, 2015–2018 (1988). https://doi.org/10.1103/PhysRevLett.61.2015

  29. H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T. Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, H. Ohta, Experimental observation of the \(1/3\) magnetization plateau in the diamond-chain compound \({\rm cu\rm _{3}({\rm co}}_{3}{)}_{2}(\rm OH{)}_{2}\). Phys. Rev. Lett. 94, 227201 (2005). https://doi.org/10.1103/PhysRevLett.94.227201

  30. H. Kikuchi, Y. Fujii, M. Chiba, S. Mitsudo, T. Idehara, T. Tonegawa, K. Okamoto, T. Sakai, T. Kuwai, H. Ohta, Kikuchi et al. reply. Phys. Rev. Lett. 97, 089702 (2006). https://doi.org/10.1103/PhysRevLett.97.089702

  31. B. Gu, G. Su, Comment on “experimental observation of the 1/3 magnetization plateau in the diamond-chain compound \({\rm cu}_{3}({\rm co}_{3})2({\rm OH})2\). Phys. Rev. Lett. 97, 563089 (2006)

  32. M. Hase, M. Kohno, H. Kitazawa, N. Tsujii, O. Suzuki, K. Ozawa, G. Kido, M. Imai, X. Hu, \(13\) magnetization plateau observed in the spin-\(12\) trimer chain compound \({\rm cu}_{3} ({\rm P}_{2}{\rm O}_{6} {\rm H})_{2}\). Phys. Rev. B 73, 104419 (2006). https://doi.org/10.1103/PhysRevB.73.104419

  33. A. Maignan, V. Hardy, S. Hébert, M. Drillon, M.R. Lees, O. Petrenko, D.M.K. Paul, D. Khomskii, Quantum tunneling of the magnetization in the ising chain compound ca3co2o6. J. Mater. Chem. 14, 1231–1234 (2004). https://doi.org/10.1039/B316717H

    Article  Google Scholar 

  34. V. Hardy, D. Flahaut, M.R. Lees, O.A. Petrenko, Magnetic quantum tunneling in \({\rm ca}_{3}{\rm co}_{2}{\rm o}_{6}\) studied by ac susceptibility: temperature and magnetic-field dependence of the spin-relaxation time. Phys. Rev. B 70, 214439 (2004). https://doi.org/10.1103/PhysRevB.70.214439

  35. V. Hardy, C. Martin, G. Martinet, G. André, Magnetism of the geometrically frustrated spin-chain compound \({\rm sr}_{3} {\rm Ho} {\rm Cr}{\rm o}_{6}\): magnetic and heat capacity measurements and neutron powder diffraction. Phys. Rev. B 74, 064413 (2006). https://doi.org/10.1103/PhysRevB.74.064413

  36. S. Ishiwata, D. Wang, T. Saito, M. Takano, High-pressure synthesis and structure of srco6o11: pillared kagomé lattice system with a 1/3 magnetization plateau. Chem. Mater. 17(11), 2789–2791 (2005). https://doi.org/10.1021/cm050657p

    Article  Google Scholar 

  37. X.X. Wang, J.J. Li, Y.G. Shi, Y. Tsujimoto, Y.F. Guo, S.B. Zhang, Y. Matsushita, M. Tanaka, Y. Katsuya, K. Kobayashi, K. Yamaura, E. Takayama-Muromachi, Structure and magnetism of the postlayered perovskite \({\rm sr}_{3}{\rm co}_{2}{o}_{6}\): a possible frustrated spin-chain material. Phys. Rev. B 83, 100410 (2011). https://doi.org/10.1103/PhysRevB.83.100410

  38. X. Yao, 1/3 magnetization plateau induced by magnetic field in monoclinic cov2o6. J. Phys. Chem. A 116(9), 2278–2282 (2012). https://doi.org/10.1021/jp209830b. (PMID: 22364513)

    Article  Google Scholar 

  39. M. Lenertz, J. Alaria, D. Stoeffler, S. Colis, A. Dinia, Magnetic properties of low-dimensional and cov2o6. J. Phys. Chem. C 115(34), 17190–17196 (2011). https://doi.org/10.1021/jp2053772

    Article  Google Scholar 

  40. Z. He, J.I. Yamaura, Y. Ueda, W. Cheng, Cov2o6 single crystals grown in a closed crucible: unusual magnetic behaviors with large anisotropy and 1/3 magnetization plateau. J. Am. Chem. Soc. 131(22), 7554–7555 (2009). https://doi.org/10.1021/ja902623b. (PMID: 19489641)

    Article  Google Scholar 

  41. W. Shiramura, K.i. Takatsu, B. Kurniawan, H. Tanaka, H. Uekusa, Y. Ohashi, K. Takizawa, H. Mitamura, T. Goto, Magnetization plateaus in nh 4cucl 3. J. Phys. Soci. Jpn. 67(5), 1548–1551 (1998). https://doi.org/10.1143/JPSJ.67.1548

  42. E. Dagotto, Complexity in strongly correlated electronic systems. Science 309(5732), 257–262 (2005). https://doi.org/10.1126/science.1107559

    Article  ADS  Google Scholar 

  43. G. Castilla, S. Chakravarty, V.J. Emery, Quantum magnetism of cuge\({\rm o }_{3}\). Phys. Rev. Lett. 75, 1823–1826 (1995). https://doi.org/10.1103/PhysRevLett.75.1823

    Article  ADS  Google Scholar 

  44. G. Giri, D. Dey, M. Kumar, S. Ramasesha, Z.G. Soos, Quantum phases of frustrated two-leg spin-\(\frac{1}{2}\) ladders with skewed rungs. Phys. Rev. B 95, 224408 (2017). https://doi.org/10.1103/PhysRevB.95.224408

  45. D. Dey, S. Das, M. Kumar, S. Ramasesha, Magnetization plateaus of spin-\(\frac{1}{2}\) system on a \(5/7\) skewed ladder. Phys. Rev. B 101, 195110 (2020). https://doi.org/10.1103/PhysRevB.101.195110

    Article  ADS  Google Scholar 

  46. S. Das, D. Dey, M. Kumar, S. Ramasesha, Quantum phases of a frustrated spin-1 system: the 5/7 skewed ladder. Phys. Rev. B 104, 125–138 (2021). https://doi.org/10.1103/PhysRevB.104.125138

    Article  Google Scholar 

  47. S. Das, D. Dey, S. Ramasesha, M. Kumar, Quantum phases of spin-1 system on 3/4 and 3/5 skewed ladders. J. Appl. Phys. 129(22), 223–902 (2021). https://doi.org/10.1063/5.0048811

  48. M. Oshikawa, M. Yamanaka, I. Affleck, magnetization plateaus in spin chains: “haldane gap’’ for half-integer spins. Phys. Rev. Lett. 78, 1984–1987 (1997). https://doi.org/10.1103/PhysRevLett.78.1984

    Article  ADS  Google Scholar 

  49. Y.C. Li, Y.H. Zhu, Z.G. Yuan, Entanglement entropy and the Berezinskii–Kosterlitz–Thouless phase transition in the j1–j2 heisenberg chain. Phys. Lett. A 380(9), 1066–1070 (2016). https://doi.org/10.1016/j.physleta.2016.01.004

    Article  ADS  MathSciNet  Google Scholar 

  50. D.W. Luo, J.B. Xu, Quantum phase transition by employing trace distance along with the density matrix renormalization group. Ann. Phys. 354, 298–305 (2015). https://doi.org/10.1016/j.aop.2014.12.023

  51. S.J. Gu, Fidelity approach to quantum phase transitions. Int. J. Mod. Phys. B 24(23), 4371–4458 (2010). https://doi.org/10.1142/S0217979210056335

    Article  ADS  MathSciNet  MATH  Google Scholar 

  52. D. Petz, Entropy, von Neumann and the von Neumann Entropy (Springer Netherlands, Dordrecht, 2001), pp. 83–96. https://doi.org/10.1007/978-94-017-2012-0_7

  53. A. Rényi, On measures of entropy and information. In: Proceedings of the 4th Berkeley Symposium on Mathematics, Statistics and Probability, 1, 547–561 (1961)

  54. V.M.L.D.P. Goli, S. Sahoo, S. Ramasesha, D. Sen, Quantum phases of dimerized and frustrated heisenberg spin chains withs= 1/2, 1 and 3/2: an entanglement entropy and fidelity study. J. Phys. Conden. Matter 25(12), 125–603 (2013). https://doi.org/10.1088/0953-8984/25/12/125603

  55. A.B. Kallin, I. González, M.B. Hastings, R.G. Melko, Valence bond and von neumann entanglement entropy in heisenberg ladders. Phys. Rev. Lett. 103, 117–203 (2009). https://doi.org/10.1103/PhysRevLett.103.117203

    Article  Google Scholar 

  56. M. Thesberg, E.S. Sørensen, General quantum fidelity susceptibilities for the \({J}_{1}-{J}_{2}\) chain. Phys. Rev. B 84, 224–435 (2011). https://doi.org/10.1103/PhysRevB.84.224435

    Article  Google Scholar 

  57. S. Chen, L. Wang, S.J. Gu, Y. Wang, Fidelity and quantum phase transition for the heisenberg chain with next-nearest-neighbor interaction. Phys. Rev. E 76, 061–108 (2007). https://doi.org/10.1103/PhysRevE.76.061108

    Article  Google Scholar 

Download references

Acknowledgements

S. Ramasesha acknowledges the Indian National Science Academy and DST-SERB for supporting this work. Manoranjan Kumar acknowledges the SERB for financial support through Project File No. CRG/2020/000754.

Author information

Authors and Affiliations

  1. Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore, 560012, India

    Sambunath Das, Dayasindhu Dey & S. Ramasesha

  2. S. N. Bose National Centre for Basic Sciences, Block-JD, Sector-III, Salt lake, Kolkata, 700106, India

    Sambunath Das & Manoranjan Kumar

  3. UGC-DAE Consortium for Scientific Research, University Campus, Khandwa Road, Indore, 452001, India

    Dayasindhu Dey

Authors
  1. Sambunath Das
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Dayasindhu Dey
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. S. Ramasesha
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Manoranjan Kumar
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

The project was conceived by S. Ramasesha and Manoranjan Kumar. Sambunath Das and Dayasindhu Dey have performed the numerical calculations. All the authors were involved in designing the calculations and interpretation of results. The work was jointly written up by all the authors.

Corresponding author

Correspondence to Manoranjan Kumar.

Rights and permissions

Springer Nature or its licensor holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Das, S., Dey, D., Ramasesha, S. et al. Quantum phase transition in skewed ladders: an entanglement entropy and fidelity study. Eur. Phys. J. B 95, 147 (2022). https://doi.org/10.1140/epjb/s10051-022-00411-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjb/s10051-022-00411-z

Search

Navigation