ANNNI Model in Transverse Field

Abstract

Competing interactions between spins or frustration give rise to intriguing many-body states with spatially modulated spin structures. The simplest model with the regular frustration is the classical axial next nearest neighbour Ising (ANNNI) model. The classical ANNNI model is described by a system of Ising spins with nearest neighbour interactions along all the lattice directions (x, y and z) as well as a competing next nearest neighbour interaction in one axial direction (z for instance). Chapter 4 discusses detailed results on quantum ANNNI models in a transverse field with a brief introduction to the classical model. Analytic studies on the basis of an interacting fermion representation of the model, real-space and field theoretic renormalisation group techniques, the numerical exact diagonalisation method, and Monte Carlo simulations have revealed a variety of ground-state phases of the quantum ANNNI chain. These are mentioned in Chap. 4. Studies of higher dimensional quantum ANNNI models are also presented there. Appendices of this chapter include details of approximate methods used in the study of the quantum ANNNI chain.

Appendix 4.A

4.1.1 4.A.1 Hartree-Fock Method: Mathematical Details

The general diagonal form of the Hamiltonian in terms of free fermions can be written as

The Hamiltonian H (4.3.15) is to be approximately written in the form of H _g such that J ₁, J ₂ etc. are expressed in terms of $J_{1}'$, $J_{2}'$ etc. The renormalised quantities $J_{1}'$, $J_{2}'$ etc. are obtained by minimising the free energy functional

$$F/N = F_{0}/N - \bigl(\langle H_{g} \rangle- \langle H \rangle\bigr). $$

Applying Wick’s theorem, F is given by

Variation of F with respect to $\varGamma', J_{1}'$ etc. yields under stationary condition the following equation given in matrix form

where the elements of M are the derivatives of $\langle c_{i}c_{k} \rangle, \langle c_{i}^{\dag}c_{k} \rangle$ etc., with k=i,i+1 or i+2, with respect to the parameters $\varGamma',J_{1}'$ etc.

The above equation yields the self-consistent expressions (also $J_{11}' =J_{1}'$ and $J_{22}' = J_{2}'$) given by (4.3.24a), (4.3.24b), (4.3.24c). The correlations $\langle c_{i}^{\dag}c_{j} \rangle$ and 〈c _i c _j 〉 at zero temperature are expressed as [360, 423]

where

$$\lambda_{k} = 4\bigl[ \bigl(\varGamma' \bigr)^{2} + \bigl(J_{1}'\bigr)^{2}/4 + \bigl(J_{2}'\bigr)^{2}/4 + \varGamma' J_{1}' \cos k - \varGamma'J_{2}' \cos2k - J_{1}' J_{2}' \cos k/2\bigr]. $$

In the region κ′<0.5, we put $\varGamma'/J_{1}'=(1-\kappa')$ to get

$$\varGamma/J_{1} = 2\bigl[ I_{1}\bigl(J_{1}'+J_{2}' \bigr)+J_{2}'I_{3}\bigr]/ \bigl(J_{1}'I_{1}-4J_{2}'I_{2} \bigr) $$

where

$$J_{1} = J_{1}' - 4J_{2}'I_{2}/I_{1}, \qquad J_{2} = 2\pi J_{2}'/I_{1} $$

and the integrals I ₁, I ₂ and I ₃ are given by

with

$$\alpha= \bigl[-1/\bigl\{\kappa'\bigl(1-\kappa'\bigr) \bigr\}^{1/2}\bigr] \bigl[ \sin^{-1}2\bigl\{ -4 \kappa'\bigl(1-\kappa'\bigr)+1/2\bigr\} -\pi/2\bigr]/2. $$

In the other region κ>0.5, $\varGamma'/J_{1}' = |\kappa'|$, and

$$J_{2}/J_{1} = -2\pi\kappa'/ \bigl(I_{1}'+4I_{2}' \kappa'\bigr) $$

where

therefore J ₂/J ₁=−1/2 for all values of κ′ and Γ/J ₁=0 identically for κ=0.5.

4.1.2 4.A.2 Large S Analysis: Diagonalisation of the Hamiltonian in Spin Wave Analysis

In the spin wave analysis, in each phase, the angles θ _i are determined by minimising the energy given by (4.4.1). Here, new spin variables $\tilde{S}$ are introduced

which obey the same commutation relations of the old ones. Since

$$\bigl(\tilde{S}_{i}^{x}\bigr)^{2}+ \bigl( \tilde{S}_{i}^{y}\bigr)^{2} + \bigl( \tilde{S}_{i}^{z}\bigr)^{2} = S_{c}^{2} $$

where

$$S_{c} = \bigl[ S(S + 1) \bigr]^{1/2} = S + 1/2 + O(1/S). $$

One can expand $\tilde{S}_{i}^{z}$ in terms of $\tilde{S}_{i}^{x}$ and $\tilde{S}_{i}^{y}$:

Introducing canonically conjugate variables $q_{i} = \tilde{S}_{i}^{x}/\sqrt{S}$ and $p_{i} = \tilde{S}_{i}^{y}/\sqrt{S}$ which satisfy [q _i ,p _j ]=iδ _ij to O(1) in S. The Hamiltonian in phase p is then given by

$$H = E_{0}(p) + \frac{1}{2} \sum_{i} \bigl[ f_{i} \bigl(p_{i}^{2} + q_{i}^{2}\bigr) + g_{i}q_{i}q_{i+1}+ h_{i}q_{i}q_{i+2}\bigr] $$

where

$$f_{i} = \cos\theta_{i} \bigl[ \cos\theta_{i+1} + \cos\theta_{i-1}- \kappa (\cos\theta_{i+2} + \cos \theta_{i-2})\bigr] + \varGamma\sin\theta_{i} $$

$$g_{i} = -2\sin\theta_{i}\sin\theta_{i+1}, \qquad h_{i} = 2\kappa\sin\theta_{i}\sin \theta_{i+2}. $$

Now the Hamiltonian can be diagonalised in the momentum space. One then defines the variables

Provided (f _i ,g _i ,h _i ) are independent of i (which is true for the three elementary phases), a diagonal form of the Hamiltonian is obtained:

$$H_{Q}(p) = \frac{1}{2} \sum_{k} (A_{k}p_{k}p_{-k} + B_{k}q_{k}q_{-k}) $$

where A _k and B _k are real and even functions of k.

4.1.3 4.A.3 Perturbative Analysis

In this Appendix we shall present a first order perturbation theory [67] where the unperturbed Hamiltonian is H _cl and the perturbation is H _p . We start by noting that at κ=0.5 the ground state of H _cl is a state with high degeneracy and any spin configuration that has no spin-domain of length unity can be the ground state. The number of domain walls is immaterial and can be anything between 0 and N/2, N being the total number of spins. (Of course, for periodic boundary there can be only an even number of walls.) Let us denote the set of all such configurations as . Also, let the population of this set be ν which incidentally is of the order of g ^N, where $g = (\sqrt{5} + 1)/2$ [247]. Now, the first-order correction to the eigenvalue [347] are the eigenvalues of the ν×ν matrix P, whose elements are

$$P_{\alpha\beta} \equiv\langle\alpha| H_p | \beta\rangle $$

where |α〉 and |β〉 are configurations within . We shall now transform P to a block diagonal structure. Note that if and only if the j-th spin lies at the boundary of a domain, and the domain too has length larger than 2. Also, in such a case, $S^{x}_{j}$ operating on |β〉 will translate the wall at the left (right) of the j-th site by one lattice spacing to the right (left). This indicates that P _αβ ≠0 if and only if |α〉 and |β〉 have equal number of domain walls. Thus, we can break up into subsets , where contains all possible spin distributions with W walls (W=2,4,…,N/2). Now, the ν×ν matrix P gets block-diagonalised into matrices P(W) of size ν _W ×ν _W , where

$$ P_{\alpha\beta}(W) \equiv\langle\alpha| H_p | \beta\rangle $$

(4.A.1)

where ν _W is the population of and . Now, it can be seen that the longitudinal term in H _p only contributes a diagonal term rΓ(N−4W) to P _αβ (W), so that one can write

$$ P(W) = M(W) + r\varGamma(N-4W)\mathbf{1} $$

(4.A.2)

where 1 is the ν _W ×ν _W unit matrix, M _αβ (W)≡〈α|H _q |β〉 and $H_{q} = - \varGamma\sum_{j} S^{x}_{j}$ is the transverse part of H _p . Thus the non-trivial problem is to solve the eigenproblem of M(W).

To solve the eigenproblem of matrices M(W), let us construct from each member |α〉 of a configuration |α′〉 by removing one spin from each spin domain. The total number of spins in |α′〉 will obviously be N−W=N′, say. Such a transformation was also used by Villain and Bak [412] for the case of two-dimensional ANNNI model. It is crucial to observe that the set composed of the states |α′〉 is then nothing but the set of all possible distributions of N′ spins with W walls, with no restriction on the domains of length unity. Hence $\langle\alpha^{\prime}|\sum_{j=1}^{N^{\prime}} S_{j}^{x}|\beta^{\prime} \rangle$ is non-zero only when $\langle\alpha|\sum_{j=1}^{N} S_{j}^{x}|\beta \rangle$ is non-zero and

$$M^{\prime}_{\alpha\beta}(W) \equiv\bigl\langle\alpha^{\prime} \bigr| H_q \bigl| \beta^{\prime} \bigr\rangle= M_{\alpha\beta}(W) $$

The eigenproblem of M′(W) becomes simple once we observe that is nothing but the set of degenerate eigenstates of the usual classical Ising Hamiltonian

$$H_0^{\prime} = - J \sum_{j=1}^{N^{\prime}} S^z_j S^z_{j+1} $$

corresponding to the eigenvalue

$$ E_W = - J \bigl(N^{\prime} - 2W\bigr). $$

(4.A.3)

Thus, if we perturb $H_{0}^{\prime}$ by H _q , then the first-order perturbation matrix will assume a block diagonal form made up of the matrices M′(W) for all possible values of W.

To solve the perturbation problem for $H_{0}^{\prime} + H_{q}$, we note that this Hamiltonian is nothing but the standard transverse Ising Hamiltonian

$$H_{TI} = - \sum_{j=1}^{N^{\prime}} \bigl[ J S^z_j S^z_{j+1} + \varGamma S^x_j \bigr]. $$

The exact solution for this Hamiltonian is known (Chap. 2). The exact expression for the energy eigenstates are

$$ E= 2 \varGamma\sum_k \xi_k \varLambda_k $$

(4.A.4)

where ξ _k may be 0, ±1 and k runs over N′/2 equispaced values in the interval 0 to π. Also, Λ _k stands for √(λ ²+2λcosk+1), where λ is the ratio J/Γ. For Γ=0, the energy E must be the same as E _W of (4.A.3), so that

$$ 2 \sum_{k=0}^{\pi} \xi_k = - \bigl(N^{\prime} - 2W\bigr). $$

(4.A.5)

and the first order perturbation correction to this energy is

$$\biggl( \frac{\partial E}{\partial\varGamma} \biggr)_{\varGamma=0} = 2 \sum _{k=0}^{\pi} \xi_k \cos k. $$

They are therefore also the eigenvalues of the M′(W) matrix. Thus the eigenvalues of the matrix P(W) of (4.A.2) are

$$ E_P = r\varGamma(N-4W) + 2\varGamma\sum _{k=0}^{\pi} \xi_k \cos k. $$

(4.A.6)

Keeping N fixed we have to find, for which value of W and for which distribution of ξ _k , E _P is minimum subject to the constraint (4.A.5). For a given value of ∑ξ _k , this minimisation is achieved if −1 (+1) values of ξ _k accumulate near large positive (negative) values of cosk. Let the desired distribution be

(4.A.7)

Equation (4.A.5) now gives

$$ N/N^{\prime} = (4\pi- \theta- \phi)/2\pi $$

(4.A.8)

and one obtains,

$$E_P = - \frac{N\varGamma}{4\pi- \theta- \phi} \bigl[ r(4\pi- 3\theta- 3\phi) +2(\sin \theta+ \sin\phi) \bigr]. $$

Minimising E _P with respect to θ and ϕ, we find that θ=ϕ=θ ₀ (say) where

$$ 2\pi r = \sin\phi_0 + (2\pi- \phi_0)\cos \phi_0. $$

(4.A.9)

The minimum value of E _P is given by,

$$ E^{(1)} = - N\varGamma[ 3r - 2\cos \phi_0 ]. $$

(4.A.10)

This is the final expression for the (exact) first order perturbation correction to ground state energy. It is easily seen that for r<−0.5, that is, for Γ/J<(1−2κ), one has ϕ ₀=π and W=0 (ferromagnetic phase), while for r>1, that is Γ/J<(κ−0.5), one has ϕ ₀=0 and W=N/2 (antiphase). As r varies from −0.5 to 1, ϕ ₀ gradually changes from π to 0 according to (4.A.9). It can also be seen that Also d ² E ⁽¹⁾/dΓ ² diverges at r=−0.5 and 1, indicating two critical lines there. It can be shown [67] that except for these two values of r, E ⁽¹⁾ and all its higher derivatives remain finite. We can now conclude that the phase diagram is either like Fig. 4.8(a) or like Fig. 4.8(b). We shall now show that an analysis of longitudinal susceptibility points to the possibility of the former.

Let us call the eigenstate of $H_{0}^{\prime} + H_{q}$ corresponding to θ=ϕ=ϕ ₀ as |ψ′〉. This state will be composed of the spin-distributions that belong to and can be written as

$$\bigl|\psi^{\prime}\bigr\rangle= \sum_{j^{\prime}} a_{j^{\prime}} \bigl|j^{\prime }\bigr\rangle $$

where |j′〉 runs over all the states in . Let us construct from each state |j′〉 another state |j〉 by augmenting each domain by a single spin. Thus for |j′〉=|+++−−++++〉, |j〉 will be |++++−−−+++++〉. Then we combine these states with the same coefficients to get a state ∑ _j a _j |j〉 where a _j =a _j′. The procedure for constructing M′ from M indicates that ∑ _j a _j |j〉 is an eigenstate of M(W) and hence of P(W) (see (4.A.2)) and this eigenstate is nothing but the zero-th order eigenfunction |ψ ⁽⁰⁾〉 for the perturbed ground state of H _cl +H _p . One should observe that although the spin-spin correlation may not be equal for |j〉 and |j′〉, the longitudinal magnetisation M _z must be the same for them as equal number of positive and negative spins have been added while transforming |j′〉 to |j〉. Thus, the longitudinal susceptibility

$$\chi_z \propto\bigl\langle M_z^2 \bigr \rangle- \langle M_z\rangle^2 $$

of |ψ ⁽⁰⁾〉 must be the same as that of |ψ′〉. The spin-spin correlation

$$C^z(n) \equiv\bigl\langle S_i^z S_{i+n}^z \bigr\rangle $$

for |ψ′〉 may be calculated by evaluating the corresponding Toeplitz determinants [68]. In the case of Γ<J, for the entire range 0<ϕ ₀<π, the correlation is

$$C^z(n) = A \frac{1}{\sqrt {n}}\cos\bigl[n(\pi- \phi_0) \bigr] $$

where A is a constant. This is clearly a floating phase since C ^z(n) decays algebraically with n. The susceptibility χ _z is hence infinity for both the states |ψ′〉 and |ψ ⁽⁰⁾〉. This leads us to the conclusion that the zero-th order eigenstate is in floating phase and hence, at least for small values of Γ, the ground state of transverse ANNNI chain must be a floating phase for all values of r between −0.5 and 1. Of course, for large values of Γ the perturbation corrections may cancel the divergence of susceptibility and lead to a paramagnetic state.

One signature of floating phase or diverging correlation length is vanishing mass-gap. Analysis of mass gap corroborates the fact that Fig. 4.8(b) rather than Fig. 4.8(a) holds true. Let us now study the first order (in Γ) correction to the mass-gap. Clearly the first excited state is the smallest possible value of E _P other than the ground state E ⁽¹⁾. To find the lowest excitation over the ground state, we note that such excitation is possible either (i) by keeping ∑ξ _k fixed and rearranging the ξ _k values; or (ii) by altering θ and ϕ and thus altering ∑ξ _k . For (i) the lowest excitation will correspond to an interchange of +1 and −1 at k=ϕ ₀, which will lead to a mass gap (for the whole system)

$$\varDelta^{(1)} = \frac{8\pi\varGamma\lambda\sin\phi_0}{ N^{\prime}\varLambda_{\phi_0}}. $$

For (ii) this quantity will be

$$\varDelta^{(1)} = \frac{1}{2} \biggl(\frac{\partial^2 E_P}{\partial\theta^2} \biggr)_{\theta=\phi_0}(\delta\theta)^2 $$

where δθ is the smallest possible deviation in θ at ϕ ₀. As the smallest possible change in W, and hence in N′ is 2, we get from (4.A.8)

$$\delta\theta= \frac{2(2\pi- \theta)^2}{\pi N} \sim\frac{1}{N}. $$

This shows that for both the mechanisms (i) and (ii), the mass gap Δ ⁽¹⁾ vanishes as N→∞ for all values of ϕ ₀ between 0 and π. This shows that for all values of r between −0.5 and 1 there must be floating phase for small Γ.