Helical multiferroics for electric field controlled quantum information processing
M. Azimi, L. Chotorlishvili, S. K. Mishra, S. Greschner, T. Vekua, J. Berakdar
aa r X i v : . [ c ond - m a t . s t r- e l ] N ov Helical multiferroics for electric field controlled quantum information processing
M. Azimi , L. Chotorlishvili , S. K. Mishra , S. Greschner , T. Vekua , and J. Berakdar Institut f¨ur Physik, Martin-Luther-Universit¨at Halle-Wittenberg,06120 Halle, Germany Institut f¨ur Theoretische Physik, Leibniz Universit¨at Hannover, 30167 Hannover, Germany (Dated: August 10, 2018)Magnetoelectric coupling in helical multiferroics allows to steer spin order with electric fields. Here we showtheoretically that in a helical multiferroic chain quantum information processing as well as quantum phases arehighly sensitive to electric ( E ) field. Applying E -field, the quantum state transfer fidelity can be increased andmade directionally dependent. We also show that E field transforms the spin-density-wave/nematic or multipo-lar phases of frustrated ferromagnetic spin − chain in chiral phase with a strong magnetoelectric coupling. Wefind sharp reorganization of the entanglement spectrum as well as a large enhancement of fidelity susceptibilityat Ising quantum phase transition from nematic to chiral states driven by electric field. These findings point to anew tool for quantum information with low power consumption. PACS numbers:
Introduction .- Multiferroics (MF) are materials that showsimultaneously multiple spontaneous ferroic ordering [1]. In-trinsic coupling between the order parameters, e.g. ferromag-netism (FM), ferroelectricity (FE), and/or ferroelasticity (foran overview we refer to [2–21]), allows for multifunctionalityof devices with qualitatively new conceptions [22–26]. Partic-ulary advantageous is the high sensitivity of some MF com-pounds to external fields [27–30, 32]. This allows to steer, forinstance magnetic order with moderate electric fields openingthus the door for magnetoelectric spintronics and spin-basedinformation processing with ultra low power consumption anddissipation [23–26]. These prospects are fueled by advancesin synthesis and nano fabrication that render feasible versatileMF nano and quantum structures with enhanced multiferroiccoupling [2–6, 31]. From a fundamental point of view MF arealso fascinating as their properties often emerge from an inter-play of competing exchange and electronic correlation, crystalsymmetry, and coupled spin-charge dynamics. For example,the perovskite multiferroics RMnO with R = Tb, Dy, Gd andEu − x Y x exhibit an incommensurate chiral spin order [36]coupled to a finite FE polarization. The underlying physics isgoverned by competing exchange and Dzyaloshiskii-Moriya(DM) [33] interactions: The spin-orbital coupling associatedwith d ( p ) -orbitals of magnetic(oxygen) ions triggers the FEpolarization [34, 37] P ∝ ˆ e ij × ( S i × S j ) ; ˆ e ij is a unit vectorconnecting the sites i and j at which the effective spins S i/j reside (e.g., along [110] direction for TbMnO ). P is thuslinked with the spin order chirality κ = ( S i × S j ) , offeringthus a tool for electrical control of κ because, as shown exper-imentally [35], P can be steered with an external electric field E (with | E | ∼ E [40–45], i.e. phenomena rooted in ME can be driven,and possibly controlled by moderate external fields. Notewor-thy, the chiral behaviour of TbMnO persists with miniatur-ization down to 6nm [31]. Furthermore, the feasibility wasdemonstrated of multiferroic spin / chain of LiCu O [38]and field-switchable LiCuVO [39].These facts combined with the robust topological nature of the intrinsic chirality are the key elements of the presentproposal to utilize chiral MF for E -field controlled, spin-based quantum information processing. Starting from an es-tablished model [37] for chiral MF with the aim to inspectelectrically driven quantum information processing and quan-tum phases in a multiferroic chain, we find that electric field E ∼ kV/cm increases strongly the quantum state trans-fer fidelity making it direction dependent. The system can besteered electrically between spin-density-wave, nematic, mul-tipolar and chiral phases. We find an E- field modifies drasti-cally the entanglement spectrum and an enhances the fidelitysusceptibility at Ising quantum phase transition from nematicto chiral states. Theoretical framework.-
We employ an effective modelwith frustrated spin interaction for the description of a one-dimensional MF chain along the x axis [37]. The chain issubjected to an electric ( E , applied along the y axis) and amagnetic ( B along the z axis) fields. The Hamiltonian reads ˆ H = J X i =1 S i . S i +1 + J X i =1 S i . S i +2 − B X i =1 S zi − E · b P . (1)The exchange interaction constant between nearest neighborspins is chosen FM J < , while next-nearest interactionis antiferromagnetic J > . Below, we use units in which J = 1 (typical values, e.g., for LiCu O are J ≈ − ± meV; and J ≈ ± meV [46–49]). Eq. (1) is an effec-tive Hamiltonian based on the conditions that E ( B ) fieldsare weak such that their direct coupling to electronic orbitalmotion is negligible. The classical E field couples (witha strength g ME ) to the induced polarization, i.e., E · b P = Eg ME P i ( S i × S i +1 ) z . While S i will be treated fully quan-tum mechanically, displacements will not be quantized [45]. κ = h κ i i = h ( S i × S i +1 ) z i is known as z component of vec-tor chirality (VC), which for brevity we call chirality.The frustrated J − J spin − chain was studied extensivelyboth theoretically [50–56] ( exhibiting its rich ground statephase diagram hosting multipolar and chiral phases) and ex-perimentally [57–59]. However, neither the control of quan-tum information processing via external driving fields nor theeffect of electric field on the ground state properties have beenaddressed yet. The present study is a contribution to fill thesegaps.We note that the electric field coupling term resembles aDzyaloshinskii-Morija (DM) anisotropy, with a coupling con-stant d = g ME E . Experiments indicate the presence of asmall DM anisotropy in MF cuprates made of frustrated spinchains [57, 58]; previous theories considered it negligible,however. Here we show that even a tiny DM anisotropy mod-ifies considerably the spin / chain characteristics. In par-ticular, nematic spin-density-wave (SDW) state of magnon aswell as multipolar phases transform into a chiral Luttinger liq-uid with non-zero spin current in the ground state.First we focus analytically on a minimal system of four spinsfor different strengths of magnetic and electric (driving) fieldsfor establishing an efficient protocol to field-control the en-tanglement. We also inspect quantum state transfer fidelity(QSTF) through MF chain and its E -field dependence.For strong B -fields, i.e. B is larger than | J | , , and d = g ME E , the ground state is fully polarized, namely | F i = | ↑↑↑↑i . The corresponding energy is E F = J + 1 − B .The pair entanglement between any two arbitrary spins andthe chirality vanish. Decreasing the magnetic field so that B < B < d + J + 2 , where B = p ( J − + 8 d / J − d ) / , the ground state is | ψ i = i | ↓↑↑↑i + − | ↑↓↑↑i + − i | ↑↑↓↑i + | ↑↑↑↓i with the corresponding energy E = − − B − d . The chirality jumps to κ = h ψ | κ i | ψ i = 1 .We observe a finite entanglement, as quantified by the pairconcurrence between spins on n and m sites [60] C nm = max (0 , √ R −√ R −√ R −√ R ) , where R n are the eigen-values of the matrix R = ρ Rnm ( σ y N σ y )( ρ Rnm ) ∗ ( σ y N σ y ) ,and ρ Rnm is the reduced density matrix of the four spins sys-tem obtained from the density matrix ˆ ρ after tracing over twospins. One can contrast the amount of the entanglement storedin the pair correlations, quantified by the so-called two-tangle τ , with the multi-spin entanglement of the whole spin chain,encapsulated in the one-tangle, τ = 4det ρ [60] ( ρ is thesingle spin reduced density matrix). Two-tangle is calcu-lated as τ = P m C nm . For the state | ψ i we find the ratio τ = τ τ = 1 , thus half of the entanglement generated by de-creasing the magnetic field (or increasing the electric field) in | ψ i is stored in the collective multi-spin correlations and halfin the pair correlations. It is instructive to study the effect of E and B fields on quantum-transfer fidelity, QSTF, [62] betweendifferent states, F ( E, B, t ) = | f j,s ( E, B, t ) | cos γ | f j,s ( E, B, t ) | ,γ = arg { f j,s ( E, B, t ) } . (2) f j,s ( E, B, t ) = h j | exp( − i ˆ Ht ) | s i is the transition amplitudebetween the states | j i and | s i .Time dependencies of QSTF obtained analytically betweenthe initial state | i = | ↓↑↑↑i and final states | i = | ↑↓↑↑↑i and | i = | ↑↑↓↑i are depicted in the Fig.1. The resultsevidence that E -field increases QSTF, particularly from | i to | i . By inspecting (2) we infer that the oscillating behavior of F in Fig.1. is related to the interference effect betweendifferent quantum states E n (cid:0) E (cid:1) / ¯ h . Note that electric field E enters in the energy levels through the DM coupling leadingto a shift of state energies and the transition strength. For theexplicit expression of Fidelity see supporting materials.For confirmation we performed numerical calculations forsystems with a large number of spins (not shown) and ob-served similar behavior of QSTF on E . We note that E -fieldbreaks the parity symmetry of the MF spin chain. Hence,when E -field is present, clockwise and anticlockwise QSTFbetween the states | j i −→ | s i and | s i −→ | j i differ consid-erably (cf. Fig.1, which might be used for information trans-fer control via magnetic chirality [61]. Further decreasing the F d=0.5d=0.5d=0 |2>|1>|2>|1> |2>|1> d=0.5d=0 FIG. 1: (Color online) Time and E -field dependence of the QSTF ina four spin chain, as quantified by F (cf. eq.(2)). d = E g ME . QSTFare depicted on the left panel for the states: | i → | i , | i → | i .Right panel shows QSTF for | i → | i = | i → | i . We set di-mensionless units − J = J = 1 , B = 1 / . Time is measured in ¯ h/J . In material parameters, e.g. for LiCu O (cf. Ref. [38]), and ¯ h/J = 0 . ps ] . d = 0 . corresponds to E = 10 [ kV /cm ] assum-ing in a cell of size a F E ≈ nm ] a polarization of P = P a F E with P = 5 · − [ C/m ] (which is within the range measuredin Ref.[30]. As we choose S N +1 = S , for N = 4 the transition | i → | i shows no directional dependence for the fidelity. magnetic field below B , the ground state becomes | ψ i = β (cid:0) | ↑↑↓↓i − iλ | ↓↑↓↑i − | ↑↓↓↑i − | ↓↑↑↓i + iλ | ↑↓↑↓i + | ↓↓↑↑i (cid:1) ,λ = ( J / − p J / − J / d / / d and β = 1 / √ λ . In this case for chirality we have κ = h ψ | κ i | ψ i = 8 λβ . and we plot its electric field dependencein Fig. 2 (a). The ratio between one-tangle τ and two-tangle τ in the ground state | ψ i reads τ = τ τ = ( − λ λ ) < , for < d ≤ − J / . Therefore, in this case the entanglementgenerated by the electric field is stored basically in many spincorrelations rather then in two spin correlations.Response sensitivity with changing the driving field ampli-tude is quantified by the fidelity susceptibility (FS) [63]. FSwith respect to magnetic field vanishes as the magnetizationis conserved in our model. FS with E-field changes is finite.E.g., for | ψ i state we obtain: χ dF = (cid:0) αβ/d (cid:1) and depict it inFig. 2 (b). As we see even small amplitude of the electric fieldleads to the substantial reduction of the FS. Physical reason of d κ (a) −4 −2 0 2 4084 d F d χ * −2 (b) FIG. 2: a) Electric field dependence of chirality for the following val-ues of the parameters − J = J = 1 , B = 1 / . We see that electricfield generates chirality. Qualitatively similar dependence holds evenin thermodynamic limit. Electric field control of the magnetic chiral-ity in the ferroaxial MF system RbF e ( MoO ) was addressed inRef.[64]. b) Electric field fidelity susceptibility. As we see, due tothe transition to the chiral phase, even a weak electric field leads to asubstantial reduction of the FS. the observed effect is transition to the chiral phase. We willstudy FS for long chains later, especially its behavior near thenematic to chiral quantum phase transition (QPT).Hence depending on the driving fields, quantum informa-tion characteristics such as many particle entanglement andQSTF differ considerably. For macroscopic number of sitesdriving fields lead to different quantum phases and QPTs infrustrated FM chain. For MF chain we can expect thus a simi-lar behavior that can possibly be controlled by E field. Hence,we study below E -field steered quantum phases and their tran-sitions in a macroscopic MF chain. We focus on the thermo-dynamic limit. Before addressing the many-body physics it isinstructive to start with the two-magnon problem: For d = 0 and weak J < a bound state of two magnons forms belowthe scattering continuum. The bound state branch has a min-imum for the total momentum K = π , for antiferromagnetic J disfavors two-magnons occupying sites of the same parity.We solved analytically the two-magnon problem for d = 0 (for L → ∞ ). The solution of two-magnon problem [67]is shown in Fig. 3. One can clearly see that with including d = 0 , the bound state minimum of the two-magnon stateshifts from K = π to K = π − K , where K ∼ d . Thebinding energy decreases as well gradually and after the crit-ical value of d > d c ( J ) (e.g. for J = − , d c ≃ . ) thetwo-magnon scattering state minimum becomes energeticallylower. Hence, bound states disappear from the ground state.When the density of magnons is increased with decreas-ing the magnetic field we expect that the two-magnon boundstates quasi-condense in the minimum of the two-magnon dis-persion at K = π − K . Hence, the ground state will en-ter the nematic-chiral state for an arbitrary small d = 0 .However, when d > d c , the nematicity (magnon pair quasi-condensate) disappears via QPT, and the low energy behav-ior is dominated by a single-particle picture with h S − i S + j i quasi long-range ordered as shown in right panel of Fig. 4.Hence, we anticipate an E -field driven phase transition fromthe ’molecular’ (2-magnon bound state) quasi-condensate tothe ’atomic’ (single-particle) quasi-condensate. This expecta-tion is fully confirmed by the effective field theory descriptionwithin bosonization techniques[67] where the competition be- −π π E π ~d FIG. 3: Two particle spectrum, with scattering states and boundstate branch for d = 0 . and J = − . Parity asymmetry is dueto DM interaction. Inset shows a zoom of the two-body dispersionaround the momentum π indicating a shift of the minimum from π in the direction of the two-magnon scattering state minimum. tween ferromagnetic J (that binds magnons and produces ne-matic order) and electric field (promoting chirality) is resolvedvia an Ising QPT with changing d .We have checked our analytical results with large scale nu-merical calculations using the density matrix renormalizationgroup (DMRG) method [65, 66] on chains up to L = 240 sites.
10 100 r −5 −4 −3 −2 −1 c o rr e l a ti on s
10 100 rd = 0.06 d = 0.38
FIG. 4: (Color online) Various correlation functions for J = − and M = 0 . in nematic (left) and chiral (right) phases for L = 160 sites. In-plane spin-spin correlation functions (cid:10) S + i S − i + r (cid:11) are indi-cated by × and show exponential decay in nematic phase and alge-braic quasi-long-range order in the chiral phase; + indicates chiralitycorrelations h κ i κ i + r i , pair correlation (cid:10) S + i S + i +1 S − i + r S − i + r +1 (cid:11) indi-cated by ◦ and the density correlations h S zi S zi + r i indicated by opensquares decay algebraically in both phases with pronounced oscilla-tions in nematic phase. For small values of | J | < ∼ and for d = 0 the leadingcorrelation function is h S zi S zj i for low magnetic fields B = 0 and the system is in the SDW dominated regime. With in-creasing the magnetic field SDW phase crosses over into thenematic state[53], with leading correlation function given by h S − i S − i +1 S + j S + j +1 i (see left pannel of Fig. 4). In both regimesthe in-plane single-spin correlation function h S − i S + j i decaysexponentially. We have studied various correlation functionsfor different values of electric field. In Fig. 4 we compare thebehavior of the correlation functions in nematic ( d < d c ) and B d B VC VC FF N T J =−3 J =−1 MM Ising I−st order
FIG. 5: (Color online) Phase diagram as a function of electric andmagnetic fields. MM indicates metamagnetic behavior (macroscopicjump) in the magnetization when descending from a saturation value.T indicates multipolar state with three-body bound states. We deter-mined phase boundary between nematic (N) and chiral (VC) statesby looking at magnetization step size with B for finite systems. ∆ M = 2 in N, whereas ∆ M = 1 in VC. Similarly, we observephase boundary between T (with magnetization step ∆ M = 3 ) anda VC. d ~d (a) d χ F / L L = 80L = 120L = 160L = 240 (b) d FIG. 6: (Color online) a) Entanglement spectrum for L=160 sites b)Scaling of DM FS near the nematic to chiral QPT for J = − and M = 0 . . chiral ( d > d c ) phases. In Fig. 5 we depict the phase diagramas a function of driving fields E and B at J = − (a) and J = − (b). To witness the transition from the nematic tothe chiral state induced by E we studied the behavior of theentanglement spectrum (Fig. 6 (a)) and DM FS (Fig. 6 (b)).In the chiral phase of a J − J chain and for d = 0 the com-plete entanglement spectrum is doubly degenerate due to thespontaneously broken parity symmetry, however in the pres-ence of d the degeneracy is lifted. Linear in L scaling of thepeak of DM FS relative to the overall background shown ininset of Fig. 6 b) confirms the Ising nature of QPT.We have studied as well the effect of DM anisotropy onmultipolar phases of the J − J chain for − < J < − . involving bound states with more than 2 magnons. The min-imum of the multi-body bound state dispersion which is at K = π for d = 0 (in both phases T and Q) shifts from π for d = 0 . In fact, ∼ DM anisotropy in J is suffi-cient to remove the three-body and the four-body multipolarphases from the ground state phase diagram below the satura-tion magnetization. Instead, in the presence of a tiny d = 0 theground state magnetization experiences a macroscopic jump −4 −2 0 J d ∆∆∆∆ MMMM =1 =2 =4 =3 N meta−magnetic VC T FM Q IN FIG. 7: (Color online) Phases under saturation magnetization. N, Tand Q stand for multipolar phases with two, three and four-magnonbound states, respectively and IN stands for incommensurate nematicphase. Filled circles indicate that for these parameters the systemexperiences macroscopic magnetization jump when descending froma fully polarized ground state into VC state by lowering B (indicatedby MM in Fig. 5), larger circle meaning greater jump. to the fully saturated value when increasing the magnetic fieldas depicted in Fig. 7. Note, for d = 0 the metamagnetic re-gion is squeezed in the close right-side vicinity of J = − point. In the presence of DM anisotropy the metamagneticjump is observed in much broader region, starting at J ≃ − and extending even in the region J < − [67]. In summary , based on the spin current model for a helicalmultiferroic spin- chain in external B and E -fields we findthat both quantum information processing as well as groundstate phases are extremely sensitive to an electric field that af-fects the magnetoelectric coupling. E -field increases stronglythe quantum state transfer fidelity and makes it directional de-pendent (transfer in clockwise direction differs from that inanticlockwise direction). A tiny magnetoelectric coupling issufficient to change the spin-density-wave/nematic or multi-polar phases in favor of the chiral phase. We analyzed QPT in-duced by ME coupling and find in particular a sharp change ofthe entanglement spectrum and a large enhancement of the fi-delity susceptibility at Ising QPT from nematic to chiral states.Our findings serve as the basis for E field controlled quantuminformation processing in helical multiferoics.MA, LC, SKM and JB acknowledge gratefully finan-cial support by the Deutsche Forschungsgemeinschaft (DFG)through SFB 762, and contract BE 2161/5-1. SG and TV aresupported by QUEST (Center for Quantum Engineering andSpace-Time Research) and DFG Research Training Group(Graduiertenkolleg) 1729. [1] H. Schmid, Int. J. Magn. 4, 337 (1973), Ferroelectrics , 317-338 (1994), J. Phys. Cond. Matt. , 434201, (2008).[2] W. Eerenstein, N. D. Mathur, and J. F. Scott, Nature , 759(2006).[3] Y. Tokura and S. Seki, Adv. Mater. , 1554 (2010). [4] C. A. F. Vaz, J. Hoffman, Ch. H. Ahn, and R. Ramesh, Adv.Mater. , 2900 (2010).[5] F. Zavaliche, T. Zhao, H. Zheng, F. Straub, M. P. Cruz, P. 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Phys. , 259 (2005).[67] see Supplementary materials for details. DETAILS OF BOSONIZATION
Here we provide details of effective field theory description,bosonization applied to microscopic Hamiltonian ˆ H = J X i =1 S i . S i +1 + J X i =1 S i . S i +2 − B X i =1 S zi − d X i =1 ( S i × S i +1 ) z . (3)To develop bosonization description it is convenient to con-sider the limit of strong frustration J ≫ | J | and weak DManisotropy d ≪ J . In this case the system may be viewed astwo antiferromagnetic spin- chains weakly coupled by thezigzag interchain coupling J [1] with DM anisotropy d .Low-energy properties of a single spin- chain in a uniformmagnetic field is described by the standard Gaussian theory[2] known also as the Tomonaga-Luttinger liquid: H = v Z dx n K ( ∂ x φ ) + K ( ∂ x θ ) o . (4)Here φ is a real scalar bosonic field and θ is its dual field, ∂ t φ = v∂ x θ , with the commutation relations [ φ ( x ) , θ ( y )] = i Θ( y − x ) , where Θ( x ) is the Heaviside function. K is Lut-tinger liquid parameter and v is spin-wave velocity.The exact functional dependences v ( J , B ) and K ( J , B ) for isolated chains are known (see [3] and references therein)from the numerical solution of the Bethe ansatz integral equa-tions [4]. In particular, K increases monotonously with themagnetic field, whereas v decreases: K ( B = 0) = , v ( B = 0) = J π/ and K → , v → for B → B sat ,where saturation value B sat = 2 J .Long wave-length fluctuations of spin-1/2 chain are cap-tured by the following representation of the lattice spin oper-ators [2]: S zn → √ π ∂ x φ + aπ sin (cid:8) k F x + √ πφ (cid:9) + M (5) S − n → ( − n e − iθ √ π (cid:8) c + b sin (cid:0) k F x + √ πφ (cid:1)(cid:9) , Here M ( B ) is the ground state magnetization per spin whichdetermines the Fermi wave vector k F = ( − M ) π and a , b ,and c are non-universal numerical constants.For J = d = 0 , two decoupled chains are described bytwo copies of Gaussian models of the form (4) with pair ofdual bosonic fields [ φ , θ ] and [ φ , θ ] . Treating interchaincouplings J and DM anisotropy d perturbatively and intro-ducing the symmetric and antisymmetric combinations of thefields describing the individual chains, φ ± = ( φ ± φ ) / √ K and θ ± = ( θ ± θ ) p K/ , the effective Hamiltonian densitydescribing low-energy properties of (3) takes the followingform: H eff = H +0 + H − + H int , H ± = v ± ∂ x θ ± ) + ( ∂ x φ ± ) ] , H int = g cos (cid:0) k F + p πK − φ − (cid:1) − ( g ∂ x θ + + g ) sin (cid:0)p π/K − θ − (cid:1) . (6) The Fermi velocities v ± ∝ J and coupling constants are g ∝ J cos k F [5], g ∝ J and g ∝ d , with proportional-ity coefficients involving short-distance cut-off. The LuttingerLiquid parameter of antisymmetric sector is given by K − = K ( h ) n J K ( B ) / (cid:0) πv ( B ) (cid:1)o . (7)The inter-sector coupling in Eq. (6) contains a term withcoupling constant g that represents an infrared limit of theproduct of z-components of in-chain and inter-chain vectorchiralities [6], ( κ z i − , i +1 + κ z i, i +2 ) κ z i, i +1 → ∂ x θ + sin s πK − θ − , (8)where κ zi,j ≡ ( S i × S j ) z .The Hamiltonian (6) provides with the effective field theorydescribing the low-energy behavior of a strongly frustratedspin- zigzag chain with DM anisotropy for a nonzero mag-netization M . For small values of magnetization the Luttingerliquid parameter K − ≃ , and the inter-sector g term has ahigher scaling dimension than the strongly relevant g and g terms in the antisymmetric sector. In this case the system is ina phase with relevant competing couplings in antisymmetricsector. In contrast to that, at B = 0 all terms generated by the J zigzag coupling are marginal and only DM coupling g isa relevant perturbation.The competition between cos p πK − φ − (nematicity) and cos p π/K − θ − (chirality) terms is resolved with an Isingphase transition in the antisymmetric sector with changing d/J [7]. EFFECT OF DM ANISOTROPY IN FERROMAGNETICREGION J < − J We now discuss the effect of DM interaction on ferromag-netic region J < − J . For d = 0 , due to SU(2) symme-try the magnon gas behaves as non-interacting bosons. Deepinside ferromagnetic region DM interaction introduces repul-sion (repulsion increases monotonously with increasing d ) be-tween magnons and below the fully polarized state chiral Lut-tinger liquid phase is realized for any d = 0 [8]. However,in close left-side vicinity of J = − (hence J < − ) non-monotonous effect of DM on the effective interaction betweenmagnons is observed. First, for small values d → DManisotropy introduces repulsion between magnons, howeverwith increasing d repulsion transforms into attraction and withfurther increasing d interaction between magnons becomes re-pulsive once again as shown in Fig. 1. Effective couplingconstant of the magnon gas we extracted from the followingrelation [9, 10], g = − h ma D (9)where m is mass of magnon and a D is one-dimensionalscattering length, which we calculated analytically from thelow energy scattering phase shift δ ( k ) , a D = lim k → δ ( k ) k , (10)where k is a relative momentum of scattering magnons. − − FIG. 8: Two-magnon scattering length in units of the lattice con-stant for J = − . , showing non-trivial sequence of resonancesinduced by changing just a single parameter d . For attractive regime g < , a D > , scattering length ex-tracted from scattering problem coincides with the correlationlength of the bound state of magnons. We depict in Fig. 1 scat-tering length from which one can observe due to Eq. (9) thateffective interaction changes sign twice via resonance-like be-havior when changing d . For the values of d which correspondto the positive scattering length (and hence g < ), the exter-nal magnetic field induces a metamagnetic transition (macro-scopic jump of the magnetization) from chiral Luttinger liquidto the fully polarized state (resulting in first order phase transi-tion). For the parameters corresponding to negative scatteringlength (and hence g > ) magnetization will change smoothlyall the way from M = 0 till M = 1 / , in particular lead-ing to usual commensurate-incommensurate phase transitionfrom chiral Luttinger liquid to fully polarized state when in-creasing the magnetic field strength. FIDELITY
Transition amplitudes and energy levels entering in the ex-pression for fidelity Eq. (2), used for plotting Fig. 1: f , = 14 (exp[ − i℘ t ] − exp[ − i℘ t ]) − i − i℘ t ] − exp[ − i℘ t ]) ,f , = −
14 (exp[ − i℘ t ] − exp[ − i℘ t ]) − i − i℘ t ] − exp[ − i℘ t ]) , (11) f , = f , = −
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