Vector-spin-chirality bound state driven by the inverse Dzyaloshinskii–Moriya mechanism

We illustrate analytically the formation of vector-spin-chirality bound state due to spin–phonon interaction conditioned by the inverse Dzyaloshinskii–Moriya mechanism. The non-equilibrium dynamics of spin-chirality is effectively mapped into the spin-boson model. For spin-1/2 systems, our study suggests an existence of a gapless first-order phase transition from incoherent to coherent spin fluctuations, which is quantified to an emergence of spin-chirality bound state. The critical strength of spin–phonon interaction is found to be determined by the ratio between the amplitude of spin fluctuations and the Debye frequency of system.


Introduction
The magnetic properties of physical systems dramatically depend on the dimensionality, the frustration, and the thermal/quantum spin fluctuations of the system [1]. One-dimensional (1D) frustrated quantum spin-1/2 chain with competing nearest-neighboring ferromagnetic (J 1 <0) and next-nearest-neighboring antiferromagnetic (J 2 >0) exchange coupling, despite its simple structure, offers a good playground to look for exotic quantum phases in both experiment and theory [2]. In the classical spin approximation, it is well known that the ground state of the J 1 -J 2 spin chain possesses a helimagnetic state with an incommensurate pitch angle Q J J arccos 4 1 2 = -( )in the range of 0>J 1 /J 2 >−4. For the SU(2)-invariant quantum case, such a long-range helical ordering is destroyed by strong quantum/thermal spin fluctuations, whereas the system possesses types of hidden multiple-spin ordering, such as magnetic multipolar phase or spin-nematic state in which magnon bound states are formed from the subtle competition between geometrical balance of ferromagnetic and antiferromagnetic correlations among spins [3][4][5][6][7][8][9][10].
The J 1 -J 2 spin-1/2 chain also provides a minimal model for understanding the multiferroic behavior of the quasi-1D edge-sharing cuprates, such as LiCu 2 O 2 [11,12], LiCuVO 4 [13,14], CuCl 2 [15] and PbCuSO 4 (OH) 2 [16,17], in which the ferroelectricity is found to be of spin origin and inherently related to a vector spin chirality, S S i j ( ) of nearby spin S i [18]. A spin excitation gap is therefore believed to be required for protecting the vector spin chirality ordered state without a magnetic spiral long-range order in multiferroic cuprates. By introducing a small bond alternation in the nearest-neighboring ferromagnetic exchange and an easy-plane anisotropy into the isotropic J 1 -J 2 Hamiltonian, [19] proposed a gaped vector-chiral dimer state in the absence of magnetic field based on density-matrix renormalization-group calculations. Such gapped vector-chiral phase was also obtained for the J 1 -J 2 Heisenberg model with added uniform Dzyaloshinskii-Moriya (DM) interaction by using the numerical Lanczos diagonalization [20,21]. However, it is difficult to handle the DM interaction analytically [22], and the role of DM interaction in the vector-chiral state is needed to be further clarified. In the present study, we revisit a vector spin chiral bond that is coupled with phonons by the spin-phonon interaction of DM type. We reveal analytically that a novel first-order phase transition is induced by the formation of a vector-spinchirality bound state. This phase transition is gapless in energy but causes a strongly dynamical suppression of Any further distribution of this work must maintain attribution to the author(s) and the title of the work, journal citation and DOI. decoherence in the quantum and thermal spin fluctuations as the strength of spin-phonon interaction increasing. c = -l k . However, it should be noted that the displacement u i is not essential to the spin-driven ferroelectricity. u i represents the contribution of the displacement of electronic cloud as well. H DM corresponds to the DM interaction once the static displacement u i á ñ is nonzero and breaks the space inversion symmetry.
On the other hand, considering that the exchange interaction J falls off as a power law with the separation of the magnetic ions, with γ in the range 6-14 [28]. R i is the bare value of the position of the magnetic atom at site i, and R R i j -| | determines the lattice constant (set here to 1). The dynamical exchange striction of transverse displacement [29], intrinsically generates a quadratic coupling, u u i j · between the neighboring transverse displacements, which gives rise to an effective coupling between the neighboring spin chirality i c and j c under the inverse DM mechanism in the J 1 -J 2 spin chain.
With the aforementioned properties in hand, after applying the molecular mean-field approximation to the spin-chirality interactions, we have an effective spin-phonon model that describes the coupling between the local spin-chirality i c and the transverse phonon modes

Spin-chirality bound state
In the presence of the spin-phonon coupling, complete information about the interaction effects (H DM ) can be further encapsulated in the spectral function, g k k 2 k w p l d w w = å -( ) | | ( )in the momentum k-representation of discrete phonon modes after thermal equilibrium average [30]. Here λ k denotes the effective coupling strength between the spin chirality and the kth mode of transverse phonons with frequency ω k . The Hamiltonian H of a single vector-spin-chirality can then be rewritten as the spin-boson model being the phonon creation (annihilation) operator. In the Debye model, there are no phonon modes with the frequency above the Debye frequency ω D . Such an abrupt cutoff can result in a simple form of the spectral density g w ( ) in the continuum limit [30], where α is a dimensionless coupling constant and Θ(x) is the usual step function. For the case of (threedimensional) phonon-related interaction in a solid, s can be 3 or 5 for frequencies well below the Debye frequency [31,32]. However, in general, the frequency behavior of g w ( ) is complicated, especially in the low dimensional systems. In the following discussion, we treat s as a free parameter with no qualitative change in conclusion of the present study.
Under the unitary transformation U exp i 4 ) and the successive the rotating-wave approximation (RWA) [33], we have then a transformed integrable Hamiltonian H (hereafter, transformed quantities are marked by a tilde), Noted that 0 z  = is assumed in H at the first step in analytical discussions for emphasizing the effect of spin fluctuations. As for the case of J 0 z ¹ , we find that J z does modify slightly the tunneling parameter [34], but has no qualitative influence on the non-equilibrium dynamics of spin-chirality discussed in the following section.
being the total excitation number operator, and the eigenstate of H is given by the direct sum of the subspace with definite quantum number N.
As λ k =0, the spin chirality decouples completely from the phonons. The ground state is given by a tensor productor of the zero-excitation modes of two subsystems, i.e., Consequently, we have zero value of ferroelectric signal, and the long-range order of spin chirality is broken down by the spin fluctuations even down to zero temperature. However, 0 y ñ |˜is not always the ground state of the whole system when the spin-phonon coupling is turned on, as shown below.
Considering the single-excitation state with N=1, c c 0 1 1 0 k k k k y ñ =  ñ Ä ñ + å  ñ Ä ñ |˜|˜| |˜| , its eigenenergy E 1 is determined by the following transcendental eigenequation [33], The analysis for algebraic relationship of above equation reveals two inequality constrains, Once such conditions are satisfied, E 1 ( ) always has one and only one real root, which is exact the eigen-energy E 1 of the formed bound state 1 y ñ |˜of the system in the presence of the spin-phonon interaction of DM type. For a excited state with N 2  , its eigenvalue is found to be always greater than E 1 . This suggests that the state with higher-phonon modes cannot be the ground state. The above inequality constraints, equation (11), yields a critical value of the spin-phonon coupling, which is simply determined by the ratio between the strength of spin fluctuations and the Debye frequency of phonons. In the case of strong spin-phonon interaction, i.e., αα c , the ground state of the system is not the zero-excitation mode 0 y ñ |˜but the single-excitation bound state 1 y ñ |˜. Considering that 0 1 0 y y á ñ = |˜and the discontinuity of the first derivative of the ground state energy, one can see that the system undergoes a first-order phase transition, accompanying with a gapless change in the ground state from 0 y ñ |˜to 1 y ñ |˜by increasing the spin-phonon interaction.