19.06.2023
Novel states of matter in low — dimensional systems
We study states with unconventional types of ordering,which emerge in lowdimensional quantum magnets or systems of ultracold atoms in optical lattices. Among such novel states are quantum chiral states – optically active spin liquids that break the right/left symmetry, states with quadrupolar magnetic order (spin nematics), states with spontaneous circulating currents, various states with socalled topological order, etc.
The term “quantum magnets” refers to systems whose spin dynamics islargely determined by quantum effects and cannot be adequately described using quasiclassical equations of magnetization dynamics. Lowdimensional quantum magnets are highly correlated systems in which quantum fluctuations play a significant role. Nonlinearity and topology play an important role in the physics of lowdimensional quantum magnets: elementary excitations are often domain walls in the magnetic or socalled topological order,and instantons, which can be considered as a special kind of nonlinear processes in spacetime, are responsible for subtle effects such as the difference between the physics of spin chains with integer and halfinteger spin, famouslydiscovered by F. D. M. Haldane. We have proposed a number of new approaches to the analysis of the physical properties of quantum magnets, some of the examples are presented below.We have developed the theory of chiral phases in onedimensional frustrated quantum magnets with easyplane anisotropy, and predicted the appearance of chiral phases under the action of an external magnetic field. Chiral phases in frustrated spin chains are characterized by a nonzero value of a specific order parameter – vector chirality (spin current) (here, two neighboring spins of the chain), the occurrence of which is associated with a spontaneous violation of parity. We show that the behavior of such frustrated onedimensional quantum systems with easyplane anisotropy corresponds to the physics of twodimensional classical planar helimagnets at finite temperature, and we establish the existence of two types of chiral phases, one of which has a gap in the spectrum, and the other is gapless. The phase diagram predicted by us was later fully confirmed by the results of numerical simulations.
N2H5CuCl3 : spin ½ zigzag chain
We have developed a method for constructing variational wave functions of onedimensional magnets in the form of socalled matrix product states, which allows, along with the degree of quantum “entanglement” and topological order, to also control the full spin of the wave function and its projection. This method has been successfully applied for the analysis of elementary excitations of lowdimensional magnets, In particular, we investigated the dynamics and phase diagram of an isotropic frustrated chain of spin 1, and it was shown that at a certain critical value of frustration, a quantum phase transition of the first kind occurs from the Haldane state to a new phase, which can be imagined as a “double” Haldane chain , where the “strings” corresponding to the nonlocal string order in each of the two chains are arbitrarily intertwined. The mechanism of such a topological transition is the formation of pairs of solitons in the string order, which can be imagined as string loops; at the transition, the connectivity (number of strings) jumps from 1 to 2.
Schematic representation of the ground state of a frustrated chain of spin 1: (a) singlestring state corresponding to the Haldane phase under weak frustration; (b)–(d) states with two strings contained in the wave function of the “double” Haldane phase under strong frustration.
By means of the density matrix renormalization group technique, we have studied quantum chiral and spin nematic states in quasionedimensional magnetic material LiCuVO_{4 ,} induced by strong magnetic field. Our theoretical predictions have been confirmed by highfield magnetization measurements and NMR studies.
LiCuVO_{4 }: frustrated spin chain with the anisotropy Δ≈0.8–0.9
We have proposed a new method to determine the ground state of lowdimensional frustrated magnetsinhigh magnetic fields close to saturation.Theme thodisbasedontreatingmagnonsas multi component Bose gas, and calculating renormalized interactions from the BetheSalpeter equation. Our results for the ground state phase diagram off rustrated spinS chain sareinanexc ellent agreement with the numeric al data obtained by density matrix renormalization group (DMRG) technique
Typical magnon spectrum: 2 or more equivalent dispersion minima at incommensurate wave vectors; magnetic field acts asthe “chemical potential” for magnons 

Renormalized interaction Γ_{q}(k,k’) from the BetheSalpeter equation 

Phase diagram of a frustrated S=1 anisotropic chain, at magnetic fields right below the saturation. 
Contacts:
Prof. Dr. Oleksiy Kolezhuk, kolezhuk@knu.ua, kolezhuk@gmail.com
Selected publications:
[1] H.J. Mikeska, A.K. Kolezhuk, Onedimensionalmagnetism //In: Quantum Magnetism, Lecture Notes in Physics Vol. 645, pp. 183 (Springer, 2004).
[2] A.K.Kolezhuk,Quantumchiralphasesinfrustratedeasyplanespinchains,Phys.Rev. B62,R6057R6060 (2000).
[3] A.Kolezhuk,T.Vekua,Fieldinducedchiralphaseinisotropicfrustratedspinchains,Phys.Rev.B72,094424 (2005).
[4] B.A. Ivanov,A.K. Kolezhuk, EffectivefieldtheoryfortheS=1quantumnematic,Phys.Rev.B68,052401(2003).
[5] A. K. Kolezhuk, F. HeidrichMeisner, S. Greschner, and T. Vekua, Frustrated spin chains in strong magnetic field: Dilute twocomponent Bose gas regime, Phys. Rev. B 85, 064420 (2012)
[6] K. Rodriguez, A. Arguelles, A. K. Kolezhuk, L. Santos, and T. Vekua, Fieldinduced phase transitions of repulsive spin1 bosons in optical lattices,Phys. Rev. Lett. 106, 105302 (2011)
[7] A.K. Kolezhuk,Effective manybody interactions in onedimensional dilute Bose gases, Low Temperature Physics46, pp. 947–950 (2020).
[8] T. Zavertanyi, A.Kolezhuk, Quantum antiferromagnets near SU(N) symmetry, AIP Advances 11, 035231 (2021).