Quantum renormalization of \(l_1\)-norm and relative entropy of coherence in quantum spin chains. (English) Zbl 1508.82016
Summary: We investigate quantum-phase transitions (QPT) in the Ising transverse field model, the XY-Heisenberg model with staggered Dzyaloshinskii-Moriya (DM) interaction, and the bond alternating Ising model with DM interaction, on a one-dimensional periodic chain using the quantum renormalization group (QRG) method. Based on \(l_1\)-norm and relative entropy, we employ two quantum coherence measures to identify QPT. Our results show that the QPT can be characterized by the coherence measure calculated for each QRG step. It is found that after large number of QRG iterations, the coherence in the ground state develops two saturated values corresponding to two different phases and changes abruptly at a quantum critical point (QCP). The scaling behavior of the system is investigated by computing the derivative of the coherence measure around the QCP that shows a logarithmic dependence on the length of the chain. The critical exponent of the correlation length is also calculated.
MSC:
| 82B26 | Phase transitions (general) in equilibrium statistical mechanics |
| 82B20 | Lattice systems (Ising, dimer, Potts, etc.) and systems on graphs arising in equilibrium statistical mechanics |
| 82B28 | Renormalization group methods in equilibrium statistical mechanics |
References:
| [1] | Sachdev, S., Quantum Phase Transitions (1999), Cambridge: Cambridge University Press, Cambridge |
| [2] | Nishimori, H.; Ortiz, G., Elements of Phase Transitions and Critical Phenomena (2010), Oxford: Oxford University Press, Oxford |
| [3] | Cardy, J., Scaling and Renormalization in Statistical Physics (1996), Cambridge: Cambridge University Press, Cambridge |
| [4] | Stanley, HE, Introduction to Phase Transitions and Critical Phenomena (1987), Oxford: Oxford University Press, Oxford |
| [5] | Domb, C.; Lebowitch, JL, Phase Transitions and Critical Phenomena (2000), New York: Academic Press, New York · Zbl 1063.82001 |
| [6] | Goldenfeld, N., Lectures on Phase Transitions and Renormalization Group (1992), New York: Addison-Wesley, New York |
| [7] | Carr, LD, Understanding Quantum Phase Transitions (2010), Boca Raton: Taylor and Francis, Boca Raton |
| [8] | Osterloh, A.; Amico, L.; Falci, G.; Rosario, F., Scaling of entanglement close to a quantum phase transition, Nature, 416, 608-610 (2002) |
| [9] | Osborne, TJ; Nielsen, MA, Entanglement in a simple quantum phase transition, Phys. Rev. A, 66, 032110 (2002) |
| [10] | Vidal, G.; Latorre, JI; Rico, E.; Kitaev, A., Entanglement in quantum critical phenomena, Phys. Rev. Lett., 90, 227902 (2003) |
| [11] | Vidal, J.; Palacios, G.; Mosseri, R., Entanglement in a second-order quantum phase transition, Phys. Rev. A, 69, 022107 (2004) |
| [12] | Wu, LA; Sarandy, MS; Lidar, DA, Quantum phase transitions and bipartite entanglement, Phys. Rev. Lett., 93, 250404 (2004) |
| [13] | Anfossi, A.; Giorda, P.; Montorsi, A., Entanglement in extended Hubbard models and quantum phase transitions, Phys. Rev. B., 75, 165106 (2007) |
| [14] | Amico, L.; Fazio, R.; Osterloh, A.; Vedral, V., Entanglement in many-body systems, Rev. Mod. Phys., 80, 517-576 (2008) · Zbl 1205.81009 |
| [15] | Najarbashi, G.; Balazadeh, L.; Tavana, A., Thermal entanglement in XXZ Heisenberg model for coupled spin-half and spin-one triangular cell, Int. J. Theor. Phys., 57, 95-111 (2018) · Zbl 1387.82014 |
| [16] | Peschel, I., On the entanglement entropy for an XY spin chain, J. Stat. Mech., 2004, 12005 (2004) · Zbl 1073.82523 |
| [17] | Mukherjee, S.; Nag, T., Dynamics of decoherence of an entangled pair of qubits locally connected to a one-dimensional disordered spin chain, J. Stat. Mech., 2019, 043108 (2019) |
| [18] | Sarandy, MS, Classical correlation and quantum discord in critical systems, Phys. Rev. A, 80, 022108 (2009) |
| [19] | Dillenschneider, R., Quantum discord and quantum phase transition in spin chains, Phys. Rev. B., 78, 224413 (2008) |
| [20] | Zanardi, P.; Paunković, N., Ground state overlap and quantum phase transitions, Phys. Rev. E, 74, 031123 (2006) |
| [21] | Evenbly, G.; Vidal, G., Scaling of entanglement entropy in the (branching) multiscale entanglement renormalization ansatz, Phys. Rev. B., 89, 235113 (2014) |
| [22] | Xu, S.; Song, XK; Ye, L., Measurement-induced disturbance and negativity in mixed-spin XXZ model, Quant. Inf. Process., 13, 1013-1024 (2014) · Zbl 1293.82009 |
| [23] | Song, XK; Wu, T.; Ye, L., The monogamy relation and quantum phase transition in one-dimensional anisotropic XXZ model, Quant. Inf. Process., 12, 3305-3317 (2013) · Zbl 1287.82012 |
| [24] | Balazadeh, L.; Najarbashi, G.; Tavana, Ali, Quantum renormalization of spin squeezing in spin chains, Sci. Rep., 8, 17789 (2018) |
| [25] | Milivojević, M., Maximal thermal entanglement using three-spin interactions, Quant. Inf. Process., 18, 48 (2019) · Zbl 1409.81021 |
| [26] | Wieśniak, Marcin; Vedral, Vlatko; Brukner, Časlav, Magnetic susceptibility as a macroscopic entanglement witness, New Journal of Physics, 7, 258-258 (2005) |
| [27] | Wilson, KG, The renormalization group: critical phenomena and the Kondo problem, Rev. Mod. Phys., 47, 773-840 (1975) |
| [28] | Monthus, C., Block Renormalization for quantum Ising models in dimension d=2: applications to the pure and random ferromagnet, and to the spin-glass, J. Stat. Mech., 2015, 01023 (2015) · Zbl 1457.82363 |
| [29] | Peotta, S.; Mazza, L.; Vicari, E.; Polini, M.; Fazio, R.; Rossini, D., The XYZ chain with Dzyaloshinsky-Moriya interactions: from spin-orbit-coupled lattice bosons to interacting Kitaev chains, J. Stat. Mech., 2014, 09005 (2014) |
| [30] | Langari, A., Phase diagram of the antiferromagnetic XXZ model in the presence of an external magnetic field, Phys. Rev. B., 58, 14467-14475 (1998) |
| [31] | Kargarian, M.; Jafari, R.; Langari, A., Renormalization of concurrence: the application of the quantum renormalization group to quantum-information systems, Phys. Rev. A, 76, 060304 (2007) |
| [32] | Kargarian, M.; Jafari, R.; Langari, A., Renormalization of entanglement in the anisotropic Heisenberg \((XXZ)\) model, Phys. Rev. A, 77, 032346 (2008) |
| [33] | Kargarian, M.; Jafari, R.; Langari, A., Dzyaloshinskii-Moriya interaction and anisotropy effects on the entanglement of the Heisenberg model, Phys. Rev. A, 79, 042319 (2009) |
| [34] | Jafari, R.; Langari, A., Phase diagram of the one-dimensional \(S=\frac{1}{2}XXZ\) model with ferromagnetic nearest-neighbor and antiferromagnetic next-nearest-neighbor interactions, Phys. Rev. B., 76, 014412 (2007) |
| [35] | Langari, A., Quantum renormalization group of \(\rm XYZ\) model in a transverse magnetic field, Phys. Rev. B., 69, 100402 (2004) |
| [36] | Jafari, R.; Kargarian, M.; Langari, A.; Siahatgar, M., Phase diagram and entanglement of the Ising model with Dzyaloshinskii-Moriya interaction, Phys. Rev. B., 78, 214414 (2008) |
| [37] | Zhang, RJ; Xu, S.; Shi, JD; Ma, WC; Ye, L., Exploration of quantum phases transition in the XXZ model with Dzyaloshinskii-Moriya interaction using trance distance discord, Quant. Inf. Process., 14, 4077-4088 (2015) · Zbl 1327.82020 |
| [38] | Zhang, XX; Li, HR, The renormalization of geometric quantum discord in the transverse Ising model, Mod. Phys. Lett. B, 29, 1550002 (2015) |
| [39] | Jafari, R., Quantum renormalization group approach to geometric phases in spin chains, Phys. Lett. A, 377, 3279-3282 (2013) · Zbl 1301.82019 |
| [40] | Langari, A.; Pollmann, F.; Siahatgar, M., Ground-state fidelity of the spin-1 Heisenberg chain with single ion anisotropy: quantum renormalization group and exact diagonalization approaches, J. Phys.: Condens. Matter, 25, 406002 (2013) |
| [41] | Liu, CC; Xu, S.; He, J.; Ye, L., Unveiling \(\pi \) -tangle and quantum phase transition in the one-dimensional anisotropic XY model, Quant. Inf. Process., 14, 2013-2024 (2015) · Zbl 1317.81032 |
| [42] | Langari, A.; Abouie, J.; Asadzadeh, MZ; Rezai, M., Phase diagram of the XXZ ferrimagnetic spin-(1/2, 1) chain in the presence of transverse magnetic field, J. Stat. Mech., 2011, 08001 (2011) |
| [43] | Najarbashi, G.; Balazadeh, L., Quantum renormalization of coherence measure in transverse Ising model, J. Res. Many-body Syst., 9, 192-198 (2019) |
| [44] | Kadanoff, LP, Scaling laws for Ising models near \(T_C=1\), Phys. Physique Fizika., 2, 263-272 (1966) |
| [45] | Kadanoff, LP; Gotze, W.; Hamblen, D.; Hecht, R.; Lewis, EAS; Palciauskas, VV; Rayl, M.; Swift, J.; Aspnes, D.; Kane, J., Static phenomena near critical points: theory and experiment, Rev. Mod. Phys., 39, 395 (1967) |
| [46] | Kadanoff, L.P. (edited by Domb, C., Green, M.S.): Scaling, Universality, and Operator Algebra, in Phase Transitions and Critical Phenomena. Academic, New York (1976) |
| [47] | Demkowicz-Dobrzanski, R.; Maccone, L., Using entanglement against noise in quantum metrology, Phys. Rev. Lett., 113, 250801 (2014) |
| [48] | Giovannetti, V.; Lloyd, S.; Maccone, L., Advances in quantum metrology, Nat. Photonics, 5, 222-229 (2011) |
| [49] | Aberg, J., Catalytic coherence, Phys. Rev. Lett., 113, 150402 (2014) |
| [50] | Narasimhachar, V.; Gour, G., Low-temperature thermodynamics with quantum coherence, Nat. Commun., 6, 7689 (2015) |
| [51] | Lambert, N.; Chen, YN; Cheng, YC; Li, CM; Chen, GY; Nori, F., Quantum biology, Nat. Phys., 9, 10 (2013) |
| [52] | Plenio, MB; Huelga, SF, Dephasing-assisted transport: quantum networks and biomolecules, New J. Phys., 10, 113019 (2008) |
| [53] | Lu, XM; Wang, X.; Sun, CP, Quantum fisher information flow and non-Markovian processes of open systems, Phys. Rev. A, 82, 042103 (2010) |
| [54] | Baumgratz, T.; Cramer, M.; Plenio, MB, Quantifying coherence, Phys. Rev. Lett., 113, 140401 (2014) |
| [55] | Streltsov, A.; Adesso, G.; Plenio, MB, Colloquium: quantum coherence as a resource, Rev. Mod. Phys., 89, 041003 (2017) |
| [56] | Chen, JJ; Cui, J.; Zhang, YR; Fan, H., Coherence susceptibility as a probe of quantum phase transitions, Phys. Rev. A, 94, 022112 (2016) |
| [57] | Dutta, A.; Aeppli, G.; Rosenbaum, TF, Quantum Phase Transitions in Transverse Field Models (2015), Cambridge: Cambridge University Press, Cambridge · Zbl 1321.82001 |
| [58] | Suzuki, S.; Inoue, J.; Chakrabarti, BK, Quantum Ising Phases and Transitions in Transverse Ising Models (2012), Berlin: Springer, Berlin |
| [59] | Martín-Delgado, MA; Sierra, G., Real space renormalization group methods and quantum groups, Phys. Rev. Lett., 76, 1146-1149 (1996) |
| [60] | Pfeuty, P., The one-dimensional Ising model with a transverse field, Ann. Phys., 57, 79-90 (1970) |
| [61] | Langari, A., Quantum renormalization group of XYZ model in a transverse magnetic field, Phys. Rev. B., 69, 100402 (2004) |
| [62] | Ma, FW; Liu, SX; Kong, XM, Quantum entanglement and quantum phase transition in the XY model with staggered Dzyaloshinskii-Moriya interaction, Phys. Rev. A, 84, 042302 (2011) |
| [63] | Amiri, N.; Langari, A., Quantum critical phase diagram of bond alternating Ising model with Dzyaloshinskii-Moriya interaction: signature of ground state fidelity, Phys. Status Solidi B, 250, 537 (2013) |
| [64] | Hao, X., Quantum renormalization of entanglement in an antisymmetric anisotropic and bond-alternating spin system, Phys. Rev. A, 81, 044301 (2010) |
| [65] | Chen, S.; Wang, L.; Hao, Y.; Wang, Y., Intrinsic relation between ground-state fidelity and the characterization of a quantum phase transition, Phys. Rev. A, 77, 032111 (2008) |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
