Energy Flux Effect in the One-Dimensional Spin-1/2 XX Model of Magnetoelectric. Lagrange Multiplier Method


  • O.R. Baran Institute for Condensed Matter Physics, Nat. Acad. of Sci. of Ukraine



one-dimensional XX model, magnetoelectric, nonequilibrium steady states, energy flux, Lagrange multiplier method


The Lagrange multiplier method is applied to study a nonequilibrium steady state with energy flux in the one-dimensional spin-1/2 XX model of a magnetoelectric with the Katsura–Nagaosa–Balatsky mechanism at sufficiently low temperatures. With the help of the Jordan–Wigner transformation, the problem is reduced to that with the Hamiltonian for spinless noninteracting fermions and can be solved exactly. A number of phase diagrams are plotted, and the dependences of the magnetization, electric polarization, and various susceptibilities on the magnetic and electric fields, as well as on the energy flux, are calculated.


C.-Y. Hsieh, J. Liu, C. Duan, J. Cao. A Nonequilibrium variational polaron theory to study quantum heat transport. J. Phys. Chem. C 123, 17196 (2019).

S. Saryal, H.M. Friedman, D. Segal, B.K. Agarwalla. Thermodynamic uncertainty relation in thermal transport. Phys. Rev. E 100, 042101 (2019).

K.W. Becker, H. Fehske, V.N. Phan. Projector-based renormalization approach to electron-hole-photon systems in their nonequilibrium steady state. Phys. Rev. B 99, 035304 (2019).

H. Ness. Nonequilibrium density matrix in quantum open systems: Generalization for simultaneous heat and charge steady-state transport. Phys. Rev. E 90, 062119 (2014).

D.S. Kosov. Lagrange multiplier based transport theory for quantum wires. J. Chem. Phys. 120, 7165 (2004).

G. Rastelli, W. Belzig. Ground state cooling of nanomechanical resonators by electron transport. Eur. Phys. J. Spec. Top. 227, 1885 (2019).

D. Karevski, T. Platini. Quantum nonequilibrium steady states induced by repeated interactions. Phys. Rev. Lett. 102, 207207 (2009).

T. Antal, Z. R'acz, L. Sasv'ari. Nonequilibrium steady state in a quantum system: One-dimensional transverse Ising model with energy current. Phys. Rev. Lett. 78, 167 (1997).

T. Antal, Z. R'acz, A. R'akos, G.M. Sch¨utz. Isotropic transverse XY chain with energy and magnetization currents. Phys. Rev. E 57, 5184 (1998).

Z. R'acz. Presence of energy flow in quantum spin chains: An experimental signature. J. Stat. Phys. 101, 273 (2000).

V. Eisler, Z. R'acz, F. van Wijland. Magnetization distribution in the transverse Ising chain with energy flow. Phys. Rev. E 67, 056129 (2003).

V. Eisler, Z. Zimbor'as. Entanglement in the XX spin chain with an energy current. Phys. Rev. A 71, 042318 (2005).

J. Hide. A steady state entanglement witness [].

D. Karevski, R.J. Harris. Defect production in quench from a current-carrying nonequilibrium state. J. Stat. Mech.: Theory Exp. 033204 (2016).

C. Mej'ıa-Monasterio, T. Prosen, G. Casati. Fourier's law in a quantum spin chain and the onset of quantum chaos. Europhys. Lett. 72, 520 (2005).

W.H. Aschbacher, C.-A. Pillet. Nonequilibrium steady states of the XY chain. J. Stat. Phys. 112, 1153 (2003).

Y. Ogata. Nonequilibrium properties in the transverse XX chain. Phys. Rev. E 66, 016135 (2002).

D.L. Gonz'alez-Cabrera, Z. R'acz, F. van Wijland. Casimir effect in the nonequilibrium steady state of a quantum spin chain. Phys. Rev. A 81, 052512 (2010).

M.O. Lavrentovich. Steady-state properties of coupled hot and cold Ising chains. J. Phys. A 45, 085002 (2012).

X. Xu, K. Choo, V. Balachandran, D. Poletti. Transport and energetic properties of a ring of interacting spins coupled to heat baths. Entropy 21, 228 (2019).

T. Antal, Z. R'acz, A. R'akos, G.M. Sch¨utz. Transport in the XX chain at zero temperature: Emergence of flat magnetization profiles. Phys. Rev. E 59, 4912 (1999).

M. Brenes, E. Mascarenhas, M. Rigol, J. Goold. Hightemperature coherent transport in the XXZ chain in the presence of an impurity. Phys. Rev. B 98, 235128 (2018).

J.L. Lancaster, J.P. Godoy. Persistence of power-law correlations in nonequilibrium steady states of gapped quantum spin chains. Phys. Rev. Res. 1, 033104 (2019).

V. Popkov, T. Prosen, L. Zadnik. Exact nonequilibrium steady state of open XXZ/XY Z spin-1/2 chain with Dirichlet boundary conditions. Phys. Rev. Lett. 124, 160403 (2020).

M. Rigol, V. Dunjko, V. Yurovsky, M. Olshanii. Relaxation in a completely integrable many-body quantum system: An ab initio study of the dynamics of the highly excited states of 1D lattice hard-core bosons. Phys. Rev. Lett. 98, 050405 (2007).

E. Ilievski, J. De Nardis, B. Wouters, J.-S. Caux, F.H.L. Essler, T. Prosen. Complete generalized gibbs ensembles in an interacting theory. Phys. Rev. Lett. 115, 157201 (2015).

D. Liu, Y. Zhang, Y. Liu, G.-L. Long. Entanglement in the ground state of an isotropic three-qubit transverse XY chain with energy current. Chin. Phys. Lett. 24, 8 (2007).

B.-Q. Liu, B. Shao, J. Zou. Entanglement of two qubits coupled to an XY spin chain: Role of energy current. Phys. Rev. A 80, 062322 (2009).

Z.-M. Wang, B. Shao, P. Chang, J. Zou. Quantum state transfer in a Heisenberg XY chain with energy current. Physica A 387, 2197 (2008).

Y.-C. Qiu, Q.-Q. Wu, W.-L. You. Energy dynamics in a generalized compass chain. J. Phys.: Condens. Matter 28, 496001 (2016).

M. Fiebig. Revival of the magnetoelectric effect. J. Phys. D 38, R123 (2005).

M. Fiebig, T. Lottermoser, D. Meier, M. Trassin. The evolution of multiferroics. Nat. Rev. Mater. 1, 16046 (2016).

K.F. Wang, J.-M. Liu, Z.F. Ren. Multiferroicity: the coupling between magnetic and polarization orders. Adv. Phys. 58, 321 (2009).

Y. Tokura, Sh. Seki, N. Nagaosa. Multiferroics of spin origin. Rep. Prog. Phys. 77, 076501 (2014).

D.I. Khomskii. Transition Metal Compounds (Cambridge University Press, 2014).

I.V. Solovyev, T.V. Kolodiazhnyi. Experimental and firstprinciples studies of magnetism and magnetoelectric effect in Co4Nb2O9 and Co4Ta2O9. Phys. Rev. B 94, 094427 (2016).

Multiferroic Materials: Properties, Techniques, and Applications. Edited by J. Wang (CRC Press, 2017).

E.A. Eliseev, A.N. Morozovska, M.D. Glinchuk, B.Y. Zaulychny, V.V. Skorokhod, R. Blinc. Surface-induced piezomagnetic, piezoelectric, and linear magnetoelectric effects in nanosystems. Phys. Rev. B 82, 085408 (2010).

M.D. Glinchuk, E.A. Eliseev, Y. Gu, L.-Q. Chen, V. Gopalan, A.N. Morozovska. Electric-field induced ferromagnetic phase in paraelectric antiferromagnets. Phys. Rev. B 89, 014112 (2014).

M.D. Glinchuk, V.V. Khist. Renovation of interest in the magnetoelectric effect in nanoferroics. Ukr. J. Phys. 13, 1006 (2018).

I.E. Dzyaloshinskii. On the magneto-electrical effect in antiferromagnets. Sov. Phys. JETP 10, 628 (1960).

D.N. Astrov. The magnetoelectric effect in antiferromagnetics. Sov. Phys. JETP 11, 708 (1960).

H. Katsura, N. Nagaosa, A.V. Balatsky. Spin current and magnetoelectric effect in noncollinear magnets. Phys. Rev. Lett. 95, 057205 (2005).

I.A. Sergienko, E. Dagotto. Role of the Dzyaloshinskii-Moriya interaction in multiferroic perovskites. Phys. Rev. B 73, 094434 (2006).

S. Seki, T. Kurumaji, S. Ishiwata, H. Matsui, H. Murakawa, Y. Tokunaga, Y. Kaneko, T. Hasegawa, Y. Tokura. Cupric chloride CuCl2 as an S = 1/2 chain multiferroic. Phys. Rev. B 82, 064424 (2010).

S. Seki, Y. Yamasaki, M. Soda, M. Matsuura, K. Hirota, Y. Tokura. Correlation between spin helicity and an electric polarization vector in quantum-spin chain magnet LiCu2O2. Phys. Rev. Lett. 100, 127201 (2008).

F. Schrettle, S. Krohns, P. Lunkenheimer, J. Hemberger, N. B¨uttgen, H.-A. Krug von Nidda, A.V. Prokofiev, A. Loidl. Switching the ferroelectric polarization in the S = 1/2 chain cuprate LiCuVO4 by external magnetic fields. Phys. Rev. B 77, 144101 (2008).

Y. Wang, J. Li, D. Viehland. Magnetoelectrics for magnetic sensor applications: status, challenges and perspectives. Materials Today 17, 269 (2014).

N. Ortega, A. Kumar, J.F. Scott, R.S. Katiyar. Multifunctional magnetoelectric materials for device applications. J. Phys.: Condens. Matter 27, 504002 (2015).

F. Matsukura, Y. Tokura, H. Ohno. Control of magnetism by electric fields. Nature Nanotechnol. 10, 209 (2015).

I. K'ezsm'arki, U. Nagel, S. Bord'acs, R.S. Fishman, J.H. Lee, H.T. Yi, S-W. Cheong, T. R˜o˜om. Optical diode effect in the room-temperature multiferroic BiFeO3. Phys. Rev. Lett. 115, 127203 (2015).

M. Sato, Sh. Takayoshi, T. Oka. Laser-driven multiferroics and ultrafast spin current generation. Phys. Rev. Lett. 117, 147202 (2016).

D.M. Juraschek, M. Fechner, A.V. Balatsky, N.A. Spaldin. Dynamical multiferroicity. Phys. Rev. Materials 1, 014401 (2017).

M. Azimi, M. Sekania, S.K. Mishra, L. Chotorlishvili, Z. Toklikishvili, J. Berakdar. Pulse and quench induced dynamical phase transition in a chiral multiferroic spin chain. Phys. Rev. B 94, 064423 (2016).

M. Brockmann, A. Kl¨umper, V. Ohanyan. Exact description of magnetoelectric effect in the spin-1/2 XXZ chain with Dzyaloshinskii-Moriya interaction. Phys. Rev. B 87, 054407 (2013).

O. Menchyshyn, V. Ohanyan, T. Verkholyak, T. Krokhmalskii, O. Derzhko. Magnetism-driven ferroelectricity in

spin-1/2 XY chains. Phys. Rev. B 92, 184427 (2015).

O. Baran, V. Ohanyan, T. Verkholyak. Spin-1/2 XY chain magnetoelectric: Effect of zigzag geometry. Phys. Rev. B 98, 064415 (2018).

V. Ohanyan. Influence of XY anisotropy on a magnetoelectric effect in spin-1/2 XY chain in a transverse magnetic field. Condens. Matter Phys. 23, 43704 (2020).

J. Strecka, L. G'alisov'a, T. Verkholyak. Enhanced magnetoelectric effect near a field-driven zero-temperature quantum phase transition of the spin-1/2 Heisenberg-Ising ladder. Phys. Rev. E 101, 012103 (2020).

W.-L. You, G.-H. Liu, P. Horsch, A.M. Ole's. Exact treatment of magnetism-driven ferroelectricity in the onedimensional compass model. Phys. Rev. B 90, 094413 (2014).

H. Cencarikov'a, J. Strecka. Enhanced magnetoelectric effect of the exactly solved spin-electron model on a doubly decorated square lattice in the vicinity of a continuous phase transition. Phys. Rev. E 98, 062129 (2018).

K. Saito, S. Takesue, S. Miyashita. Thermal conduction in a quantum system. Phys. Rev. E 54, 2404 (1996).

X. Zotos, F. Naef, P. Prelov˘sek. Transport and conservation laws. Phys. Rev. B 55, 11029 (1997).

A. Kl¨umper, K. Sakai. The thermal conductivity of the spin-1/2 XXZ chain at arbitrary temperature. J. Phys. A: Math. Gen. 35, 2173 (2002).

O.R. Baran. Energy current effect in the one-dimensional spin-1/2 XX model of the magnetoelectric. Lagrange

multiplier method. Preprint ICMP-20-06U (Lviv, 2020) [].

I.E. Dzialoshinskii. Thermodynamic theory of "weak" ferromagnetism in antiferromagnetic substances. Sov. Phys. JETP 5, 1259 (1957).

T. Moriya. Anisotropic superexchange interaction and weak ferromagnetism. Phys. Rev. 120, 91 (1960).

V.M. Kontorovich, V.M. Tsukernik. Spiral structure in a one-dimensional chain of spins. Sov. Phys. JETP 25, 960 (1967).

V.N. Krivoruchko. Magnon bound-states in an anisotropic chain of spin with the Dzyaloshinskii interaction. Fiz. Nizk. Temp. 12, 872 (1986).

A.A. Zvyagin. The ground-state structure of a spin chain with the Dzyaloshinsky type interaction. Fiz. Nizk. Temp. 15, 977 (1989).

O.V. Derzhko, A.Ph. Moina. Statistical mechanics of onedimensional s = 1/2 anisotropic XY model in transverse field with Dzyaloshinskii-Moriya interaction. Condens. Matter Phys. No 3, 3 (1994).

O. Derzhko, A. Moina. 1D S = 1/2 anisotropic XY model in transverse field with Dzyaloshinskii-Moriya interaction. Ferroelectrics 153, 49 (1994).

O. Derzhko, T. Verkholyak. Effects of DzyaloshinskiiMoriya interaction in the dynamics of s = 1/2 XX chain. Czech. J. Phys. 54, D531 (2004).

O. Derzhko, T. Verkholyak, T. Krokhmalskii, H. B¨uttner. Dynamic probes of quantum spin chains with the Dzyaloshinskii-Moriya interaction. Phys. Rev. B 73, 214407 (2006).

O. Derzhko, T. Verkholyak. Dynamic structure factors of the spin-1/2 XX chain with Dzyaloshinskii-Moriya interaction. J. Phys. Soc. Jpn. 75, 104711 (2006).

N. Avalishvili, B. Beradze, G.I. Japaridze. Magnetic phase diagram of a spin S=1/2 antiferromagnetic two-leg ladder with modulated along legs Dzyaloshinskii-Moriya interaction. Eur. Phys. J. B 92, 262 (2019).

F.K. Fumani, B. Beradze, S. Nemati, S. Mahdavifar, G.I. Japaridze. Quantum correlations in the spin-1/2 Heisenberg XXZ chain with modulated Dzyaloshinskii-Moriya interaction. J. Magn. Magn. Mater. 518, 167411 (2021).

F. Heidrich-Meisner, A. Honecker, D.C. Cabra, W. Brenig. Zero-frequency transport properties of one-dimensional spin-1/2 systems. Phys. Rev. B 68, 134436 (2003).

M. Michel, O. Hess, H. Wichterich, J. Gemmer. Transport in open spin chains: A Monte Carlo wave-function approach. Phys. Rev. B 77, 104303 (2008).

L.-A. Wu, D. Segal. Energy flux operator, current conservation and the formal Fourier's law. J. Phys. A. 42, 025302 (2009).

R. Steinigeweg, J. Gemmer, W. Brenig. Spin and energy currents in integrable and nonintegrable spin-1/2 chains: A typicality approach to real-time autocorrelations. Phys. Rev. B 91, 104404 (2015).

A.M. Tsvelik. Incommensurate phases of quantum onedimensional magnetics. Phys. Rev. B 42, 779 (1990).

A.A. Zvyagin, A. Kl¨umper. Quantum phase transitions and thermodynamics of quantum antiferromagnets with next-nearest-neighbor couplings. Phys. Rev. B 68, 144426 (2003).

H. Frahm. Integrable spin-1/2 XXZ Heisenberg chain with competing interactions. J. Phys. A 25, 1417 (1992).

I. Titvinidze, G.I. Japaridze. Phase diagram of the spin S = 1/2 extended XY model. Eur. Phys. J. B 32, 383 (2003).

T. Krokhmalskii, O. Derzhko, J. Stolze, T. Verkholyak. Dynamic properties of the spin-1/2 XY chain with threesite interactions. Phys. Rev. B 77, 174404 (2008).

M. Topilko, T. Krokhmalskii, O. Derzhko, V. Ohanyan. Magnetocaloric effect in spin-1/2 XX chains with threespin interactions. Eur. Phys. J. B 85, 278 (2012).

E. Lieb, T. Schultz, D. Mattis. Two soluble models of an antiferromagnetic chain. Ann. Phys. (N.Y.) 16, 407 (1961).

O. Derzhko. Jordan-Wigner fermionization for spin-1/2 systems in two dimensions: A brief review. J. Phys. Studies 5 No. 1, 49 (2001).

T. Krokhmalskii, T. Verkholyak, O. Baran, V. Ohanyan, O. Derzhko. Spin-1/2 XX chain in a transverse field with regularly alternating g factors: Static and dynamic properties. Phys. Rev. B 102, 144403 (2020).

M. Fabrizio. Superconductivity from doping a spin-liquid insulator: A simple one-dimensional example. Phys. Rev. B 54, 10054 (1996).

A.A. Zvyagin. Quantum phase transitions in low-dimensional quantum spin systems with incommensurate magnetic structures. Phys. Rev. B 72, 064419 (2005).

R.K.P. Zia, E.L. Praestgaard, O.G. Mouritsen. Getting more from pushing less: Negative specific heat and conductivity in nonequilibrium steady states. Am. J. Phys. 70, 384 (2002).

E. Boksenbojm, C. Maes, K. Netoˇcn'y, J. Peˇsek. Heat capacity in nonequilibrium steady states. Europhys. Lett. 96, 40001 (2011).



How to Cite

Baran, O. (2021). Energy Flux Effect in the One-Dimensional Spin-1/2 XX Model of Magnetoelectric. Lagrange Multiplier Method. Ukrainian Journal of Physics, 66(10), 890.



Physics of magnetic phenomena and physics of ferroics