Kink confinement in the antiferromagnetic spin- chain in a weak staggered magnetic field
The Heisenberg spin-1/2 chain is considered in the massive antiferromagnetic regime in the presence of a staggered longitudinal magnetic field. The Hamiltonian of the model is characterised by the anisotropy parameter , and by the magnetic field strength . At zero magnetic field, the model is exactly solvable. In the thermodynamic limit, it has two degenerate vacua and the kinks (which are also called spinons) interpolating between these vacua, as elementary excitations. Application of the staggered magnetic field breaks integrability of the model and induces the long-range attractive potential between two adjacent kinks leading to their confinement into the bound states. The energy spectra of the resulting two-kink bound states is perturbatively calculated in the extreme anisotropic (Ising) limit to the first order in the inverse anisotropy constant , and also for generic values of to the first order in the weak magnetic field .
Introduction. – The confinement phenomenon occurs when the constituents of a compound particle cannot be separated from each other and therefore cannot be observed directly. A prominent and important example in high-energy physics is the confinement of quarks in hadrons. It is remarkable, that confinement can also be realized in such condensed matter systems, as quantum quasi-one-dimensional easy-axis ferro- and anti-ferromagnets Coldea et al. (2010); Morris et al. (2014); Grenier et al. (2015); Wang et al. (2015); Bera et al. (2017). The present theoretical understanding Fonseca and Zamolodchikov (2006); Rutkevich (2008) of the confinement in such systems originates from the Wu and McCoy scenario McCoy and Wu (1978), in which the two kinks are treated as quantum particles moving in the line and attracting one another with a linear potential proportional to the external magnetic field.
Very recently Bera et al. (2017), the magnetic excitations energy spectra in the quasi-one-dimensional spin-1/2 antiferromagnetic compound in the confinement regime have been studied by means of the inelastic neutron scattering. The experimentally observed energy spectra were interpreted in Bera et al. (2017) in terms of the one-dimensional spin-1/2 chain Hamiltonian. We write this Hamiltonian directly in the thermodynamic limit in a slightly different form using the more traditional parametrization (see, eg. equation (1.1) in Jimbo et al. (1995))
Here are the Pauli matrices, , is the anisotropy parameter, is the strength of the staggered magnetic field, which mimics Bera et al. (2017) in the Hamiltonian (1) the weak interchain interaction in the array of parallel spin chains in the ordered phase of the compound . The massive antiferromagnetic phase is realised at . The dynamical structure factors and the spectrum of magnetic excitations of the model (1) were numerically studied in Bera et al. (2017) in three different cases: (i) in the extreme anisotropic (Ising) limit , (ii) close to the isotropic point , and (iii) for generic . The resulting energy spectra of the magnetic excitations were presented in the form of graphics in Figures 8 - 15, and in phenomenological formulas like (26), which contain fitting parameters.
The aim of the present paper is to find analytic representations for the energy spectra of the two-kink bound states in the whole Brillouin zone in model (1) expressed solely in terms of the Hamiltonian parameters and . The problem is solved perturbatively in two asymptotical regimes: in the extreme anisotropic limit , and for generic at small .
The Hamiltonian symmetries. – The Hamiltonian (1) commutes with the -projection of the total spin
and with the modified translation operator , where stands for the unit translation, and is the global rotation by around the -axis. Note, that modified translation operator satisfies the relations
and anticommutes with . Clearly, the operator reduces to the translation by two lattice sites, , , for . For short, the operator will be called the ”total spin” in the sequel.
Ising limit. – In the extreme anisotropic limit , it is convenient to rescale the Hamiltonian (1) to the form
where . The ground states and the low-energy excitations of the Hamiltonian (3) can be effectively studied Jimbo et al. (1995); Bera et al. (2017) by means of the Rayleigh-Schrödinger perturbation theory in the small parameter . We shall describe these straightforward calculations to the first order in in order to gain insight into the nature of the two-kink bound states of the Hamiltonian (1) in the case of generic .
The zero order Hamiltonian has two antiferromagnetic ground states
which are degenerate in energy,
The localized one-kink states interpolate between the vacua and to the left, and to the right, respectively, from the bond connecting the sites , . In the state , only two adjacent spins at the sites and have the same orientations. Denote by the space of one-kink states, by the orthogonal projector on this space, and by the restriction of the Hamiltonian on . Denote, further, by the one-kink states with the definite quasi-momentum , which are distinguished by the spin index ,
The modified translations operator acts in the following way on these states, . The states (5) are the eigenstates of the Hamiltonian , with the dispersion law
Let us turn now to the two-kink sector . The basis of localized states in it is formed by the vectors where , . The space is the vector sum of two subspaces, , where is spanned by the basis states , and is spanned by the states . Denote by the orthogonal projector onto the subspace .
Since acts as the unit operator on , and since the Hamiltonian does not mix the subspaces and , one can restrict to the subspace the first-order perturbative analysis of the eigenvalue problem for the Hamiltonian (3). Accordingly, we define the restriction of the Hamiltonian (3) onto the subspace . The reduced two-kink Hamiltonian acts on the basis states of as follows,
Here is the ”string tension”, which determines the linear attractive potential acting between the two kinks.
Let us proceed to the eigenvalue problem in ,
where is the total quasi-momentum of the two-kink excitation, and the discrete quantum numbers are defined in equation (10).
Writing the eigenvector in the form
one obtains the simple discrete Sturm-Liouville problem for the wave function ,
where , and the Dirichlet boundary condition
is imposed. Note, that the state has the total spin for , and for . Both states with have the zero total spin, .
where are the quasi-momenta of two kinks having the dispersion law (7).
It will be useful for the sequel to present the two-kink state in a slightly different form. Let us define the two-kink state characterised by the momenta , and the spins of the individual kinks,
where , , and denotes the two-kink scattering amplitude
One can easily check, that
where , , , and . The vectors (14) are the eigenstates of the operators , and with the eigenvalues , and , respectively.
The two-kink states (14) satisfy the Faddeev-Zamolodchikov commutation relations,
At , equations (11), (12) describe the discrete Sturm-Liouville problem in the half-line with the linear potential. It can be easily solved Gallinar and Mattis (1985) exploiting the appropriate equality for the Bessel functions ,
The resulting energy spectrum reads as
where , and are the solutions of the equation
which are ordered according to their values, , and
Note, that inside the Brillouin zone .
Equations (18)-(20) determine the exact small- asymptotics for the energy spectrum of the two-kink bound states for the Hamiltonian (3) to the first order in . A very similar energy spectrum was found in paper Rutkevich (2008) [see equations (54), (59), (60) there], where the kink confinement in the quantum Ising spin-chain ferromagnet was studied.
In the case of small , two asymptotical expansions can be extracted from the exact spectrum (18)-(20) following the lines described in Rutkevich (2008). For not very large , the low energy expansion in the fractional powers of holds,
where are the zeroes of the Airy function. In the case of large , one can use instead the semiclassical expansion
Here is the solution of two equations
which determine also the parameter .
It turns out, that the small- asymptotic representations (21), (22) for the two-kink bound state energy spectra can be also derived by a means of different, semi-heuristic approach, which was initially developed for the Ising field theory Rutkevich (2005); Fonseca and Zamolodchikov (2006), and then applied to the quantum Ising spin-chain model Rutkevich (2008), and to the Potts field theory Rutkevich (2010). The high accuracy of the analytical predictions obtained by this technique was confirmed later Lencsés and Takács (2014); Kormos et al. (2017) by direct numerical calculations of the kink bound state energy spectra in the confinement regime for all three models mentioned above. In what follows, we shall describe briefly this technique for the case of the extreme anisotropic limit of the antiferromagnetic model, and then apply it to the generic case .
Let us treat the two kinks as classical particles having the spins , respectively, moving along the line , and attracting one another with a linear potential. Their Hamiltonian will be taken in the form
Here are the kink spacial coordinates, is the kink dispersion relation (7), and .
After the canonical transformation
the Hamiltonian (25) takes the form
In order to simplify notations, we have dropped here the spin arguments in the functions , , and .
The canonical equations of motion for the Hamiltonian (27) can be easily solved. The topology of the phase trajectories in the -plane depends on the total energy of the two kinks, as it is clear from Figure 1. The phase trajectories are open for , and closed for . In the latter case, the solution of the canonical equations describes the oscillatory motion of two kinks in the centre of mass frame that drift with a constant average velocity.
There are two different ways to quantize the model (25). For small oscillations, the kinetic energy term (28) can be expanded to the second order in the momentum , with subsequent replacement of the latter by the operator . The resulting Schrödinger equation can be reduced to the Airy equation in the half-line , which, together with the Dirichlet boundary condition at , gives rise to the energy spectrum (21). For the high-amplitude oscillations with energies in the interval , the energy levels can be found Rutkevich (2008) by means of the Bohr-Sommerfeld quantisation rule. The resulting spectrum reads as
General case . – Now let us turn to the general case of the spin-chain model (1) in the antiferromagnetic phase . We shall use the standard parametrisation for the anisotropy constant ,
At zero staggered magnetic field , the model considered on the finite spin chain with sites is solvable by the Bethe Ansatz method Orbach (1958), see also Zvyagin (2010); Dugave et al. (2015) for further references. In the thermodynamic limit , it has two energetically degenerate ground states , , showing a Néel-type order,
The lowest energy excitations are topologically charged, being represented by the kinks interpolating between two different vacua and . The dispersion relation of these excitations was found by Johnson, Krinsky, and McCoy Johnson et al. (1973),
Here and are the complete elliptic integrals of modulus and respectively, such that . The dispersion relation (33) can be parametrized in terms of the Jacobi elliptic functions:
with the rapidity variable .
The number of kinks must be even in the topologically neutral sector. The two-kink basis states reduce to (14) in the limit . These states diagonalize the Hamiltonian and the total spin with the eigenvalues , and , respectively.
The two-kink scattering at can be described by the Faddeev-Zamolodchikov commutation relations:
The scattering amplitude was found by Zabrodin Zabrodin (1992), and the whole two-kink scattering matrix was determined by Davies et al. Davies et al. (1993). The scattering amplitudes can be parametrized by the rapidity variable,
For later use, let us define the following basis in the two-kink subspace with ,
The modified translation operator and the scattering matrix become diagonal in this basis,
Let us define also the corresponding scattering phases , with , by the relation . In the rapidity variables, these scattering phases take the form,
where is given by (39).
Application of a staggered magnetic field breaks integrability of the model and leads at to confinement of kinks into the bound states. The natural way to study their energy spectrum is to apply some perturbative technique in small around the exact solution at . The most systematic, but technically rather hard realization of this idea should exploit the Bethe-Salpeter equation Fonseca and Zamolodchikov (2003, 2006), together with the appropriate form factor perturbative expansion Rutkevich (2009, 2017). Here we shall apply instead the more simple semi-heuristic approach, which was outlined above. In what follows, we shall concentrate on the topological neutral sector spanned by the basis states .
So, let us consider two interacting particles moving in the line, which classical evolution is described by the Hamiltonian
Now the particle kinetic energy is taken in the form (33), and for the string tension we shall use its value at , where the spontaneous magnetisation is given by (32). Quantization of the periodical motion of two particles in the center of mass frame allows one to determine the energy spectrum of their bound states.
Two new features, which modify the analysis, should be taken into account. First, due to the different kink dispersion law (33), the profile of the effective kinetic energy in the centre of mass frame,
now transforms with increasing total momentum , as it is shown in Figures 2 and 3 of ref. Rutkevich (2008). At small total momenta , the kinetic energy takes its minimal value at the origin , and monotonically increases with at . At large enough , the kinetic energy becomes non-monotonic. It has a local maximum at , and two minima located at , where . The transition between these two regimes takes place at the critical value of the total momentum, which is fixed by the condition . As the result, the classical phase portrait of the two particle relative motion changes at , which also affects the quantization of their dynamics.
Fortunately, the kink dispersion law (33) in the antiferromagnetic -model coincides up to a re-parametrization with the kink dispersion law in the ferromagnetic phase in the Ising spin- chain in a transverse magnetic field. This fact allows one to apply the results of the paper Rutkevich (2008), in which the same semi-heuristic approach has been used for the latter model, together with the method based on the Bethe-Salpeter equation.
Second, in contrast to the Ising model, the kinks in the -model are not free particles, but strongly interact at small distances already at . This short-range interaction leads to the nontrivial two-kink scattering, which must be properly taken into account. The problem of kink confinement in the presence of a nontrivial kink-kink scattering has been already studied in the case of the Potts field theory Rutkevich (2010). The dynamics of two kinks confined into a bound state in the semiclassical regime at was described in Rutkevich (2010) in two different ways. At large separations , the two kinks were treated as classical particles, which attract one another with a linear potential and move along the line according the canonical equations of motion. When the two kinks approach one another at some point having the momenta and , they undergo the quantum scattering, which is described by the Faddeev-Zamolodchikov commutation relations analogous to (36). As the result, the semiclassical energy spectrum of the two-kink bound states determined by the Bohr-Sommerfeld quantization rule becomes explicitly dependent on the kink-kink scattering phases. Applying the same strategy, we calculated the energy spectrum of the two-kink bound states in the -model for any to the first order in . Here we shall present the results only. The details of the calculations, which are to much extent similar to those described in papers Rutkevich (2008, 2010), will be published elsewhere.
There are three spectral modes, which will be distinguished by the parameter taking the ’values’ and . The two-fold degenerate mode with corresponds to the kink bound states with . Two other modes correspond to the kink bound-states with . The wave functions of such states can be expanded in the bases (42), which diagonalize the scattering matrix at . These two modes, which degenerate in the Ising limit , split at finite due to the difference in their two-kink scattering phases.
At , the initial terms of the low energy expansion take the form,
where , and are the zeroes of the Airy function. So, the shifts between the energy spectra of three modes distinguished by the parameter are proportional to the magnetic field.
The leading term of the semiclassical expansion for the energy spectra of all three modes at reads
where , are the scattering phases defined by (45), and is the solution of the equation . Note, that the left-hand side of (52) is the Legendre transform of the integral considered as a function of the variable .
At , the low-energy expansion takes the form
where is the location of the minimum of the kinetic energy , and and are the zeroes of the the Airy function and of its derivative, respectively, , , . The semiclassical asymptotics at modifies to the form
where are the positive solutions of the equation
For the energies in the interval , the semiclassical spectrum is described by equation (52).
At , the Taylor expansion of the kinetic energy does not contain the quadratic term. As the result, the low-energy expansion changes to the form
where are the consecutive solutions of the equation (93) in Rutkevich (2008). The numerical values of the first three ones are , , . The low-energy expansion (56) holds for the energies slightly above the lower bound of the spectrum, . For higher energies in the interval , the semiclassical asymptotics (52) can be used.