Symmetry Considerations and the Exact Diagonalization of Finite Spin ...

CHINESE JOURNAL OF PHYSICS

VOL. 48, NO. 4

AUGUST 2010

Symmetry Considerations and the Exact Diagonalization of Finite Spin Systems Kunle Adegoke1, ∗ and Helmut B¨ uttner2, † 1 2

Department of Physics, Obafemi Awolowo University, Ile-Ife, Nigeria Physics Institute, University of Bayreuth, 95440, Bayreuth, Germany (Received January 4, 2010)

We have exploited the symmetries of a model spin Hamiltonian to facilitate its block diagonalization for finite system sizes. The simplifications arising from the symmetries are discussed in considerable detail. We also present exact diagonalization results based on a working code written in the programming language of the computer algebra system Waterloo Maple. It should be easy to modify or adapt the code to handle similar models. PACS numbers: 03.65.Aa, 05.30.Rt, 64.70.Tg

I. INTRODUCTION

The Hilbert space corresponding to a system of N spin 21 atoms is 2N dimensional. Exact diagonalization of such systems can be quite formidable, even on the fastest computers, due to the exponential increase in the dimensions of the Hamiltonian matrix with increasing N . When the symmetries of the system are gainfully employed, however, the diagonalization process can be considerably simplified, since a Hamiltonian can be (block) diagonalized in the degenerate-eigenvalued subspaces of its symmetries. This is particularly useful if one is interested only in the ground state of the system, since the ground state will be contained in the most symmetric subspace [1] of the system, and hence only one (the largest) Hamiltonian submatrix will need to be diagonalized. In this paper we will discuss and apply some symmetries which will often be useful in simplifying the diagonalization of the Hamiltonians of spin systems. In this work, the exact diagonalization of a model Hamiltonian will be performed based on three important symmetries: translational invariance, spin reflection symmetry, and the all-spins inversion symmetry. As will be seen in the following sections, reflection and translational invariance are symmetries of the model Hamiltonian for all parameters, whereas the all-spins inversion symmetry is broken by the presence of a longitudinal field. The Model Hamiltonian To illustrate the tremendous simplification which can arise from symmetry considerations we will study a magnetic model that is endowed with frustration. This choice is based on the fact that frustration as a result of competitive interactions in magnetic models has remained a subject of active research [2–4]. The most popular model in which the effects of regular frustration on spin models have been extensively studied is the axial next nearest neighbour Ising (ANNNI) model [5, 6]. The ANNNI model is described by a system of Ising spins with nearest neighbour interactions along all the lattice directions (x,

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y, and z) as well as a competing next nearest neighbour interaction in one axial (e.g., the z) direction. In this work we will explore an Ising system in which frustration is due to the presence of an external transverse field, as well as competitive interactions from next nearest neighbour spins and the influence of an external longitudinal field. Specifically, we will study the one-dimensional ANNNI model in an external transverse magnetic field hx and a uniform longitudinal field hz . The system is described by the Hamiltonian X X X X z z H= Siz Si+1 +j Siz Si+2 − hz Siz − hx Six , (1) i

i

i

i

where j is the next nearest neighbour exchange interaction, Si are the usual spin- 21 operators, and the fields hx and hz are measured in units where the splitting factor and Bohr magneton are unity. In discussing the Hamiltonian (1), it is convenient to choose a system of basis vectors in which Siz is diagonal and write the direct product basis vectors, spanning the 2N -dimensional Hilbert space associated with a Hamiltonian, H, in the form |S1 S2 · · · SN i ≡ |S1 i ⊗ |S2 i ⊗ · · · ⊗ |SN i so that Siz |S1 S2 · · · · · · Si · · · SN i = Si |S1 S2 · · · · · · Si · · · SN i ,

(2)

where Si = ±1/2. The ANNNI model in two perpendicular fields is particularly interesting because it is a rather complete model in the following sense: various special cases of the model have either been exactly solved or their phase diagrams obtained using numerical and approximate techniques. In particular we would like to mention the following cases: 1. The one-dimensional ANNNI model

hz = 0, hx = 0 in Hamiltonian (1) is the well-known and well-studied one-dimensional ANNNI model. The ANNNI model was proposed by Elliot [7] to account for the existence of modulated phases in some rare-earth compounds. The ANNNI model is the simplest non-trivial model that exhibits spatially modulated phases [8–11]. The ground state of the model at zero temperature is well known in all dimensions [12, 13]: the two-fold degenerate antiferromagnetic state for j < 1/2 and the four-fold degenerate antiphase configuration for j > 0.5. The model is infinitely degenerate at j = 1/2, with the degeneracy being of the order of τ N for a system of N spins, τ being the golden ratio. An excellent review of the ANNNI model can be found in Selke [12]. 2. The Ising model in a transverse field

The case hz = 0, j = 0 corresponds to the Ising model in a transverse field. This model belongs to the same universality class of the two-dimensional Ising model [4, 6]. The

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transverse Ising model has been solved analytically by Pfeuty [14], who obtained the ground state energy of the model using a technique developed by Lieb [15] and employed the results of McCoy [16] to investigate the order in the system. The model is gapped at hx < 0.5 with non-zero staggered magnetization, with the ground state being two-fold degenerate in the thermodynamic limit. The transverse Ising model becomes gapless at hx = 1/2, and the order parameter (staggered magnetization) vanishes as a function of hx = −1/2 with the critical exponent 81 . 3. The one-dimensional ANNNI model in a transverse field

When hz = 0 in the Hamiltonian (1), we have the one-dimensional ANNNI model in a transverse field. Again, this is a well-studied model, and one which has continued to arouse interest among researchers. The reason this model has been extensively studied is probably due to the fact that the zero temperature (quantum) critical behaviour of a quantum spin Ising system in d-dimension is usually related to the thermal behaviour of the corresponding classical system in d + 1-dimensions, and vice versa [5, 17]. For spin-1/2, the relation between the quantum d-dimensional transverse Ising model and the (d + 1)-dimensional classical Ising model is most clearly seen by considering the Ising model in an extremely anisotropic limit of the exchange couplings [18, 19]. It is not clear if the transverse ANNNI model can be solved exactly, although there have been several attempts in this direction, such as can be found in the works of Ruján [20] and Sen [21]. Various techniques, such as the self-consistent Hartree-Fock method [22], real space renormalization group (RSRG) calculations [21], and field-theoretic renormalization group calculations, which give direct evidence for the existence of a floating phase with algebraically decaying correlations [23], have been employed to investigate the one-dimensional ANNNI model in a transverse field. So far the most detailed phase diagrams for the transverse ANNNI model have been obtained using numerical or approximate calculations, such as perturbation expansions and finite size scaling [8, 20, 24–26], the strong coupling eigenstate method (SCEM) [27–29], and Monte Carlo methods [10, 30]. 4. the Ising model in two external magnetic fields

When j = 0 in the Hamiltonian (1), the model is the Ising model in two external magnetic fields, longitudinal and transverse. Models incorporating two noncommuting fields are gaining popularity among experimentalists as well as theoreticians as is evident for example in references [4, 6, 31, 32]. Sen [6] investigated the quantum phase transitions in the ferromagnetic transverse Ising model in a spatially modulated longitudinal field, and obtained the phase diagram of the model at zero temperature, using finite size scaling techniques. It was found that a continuous phase transition occurs everywhere except at the multiphase point hx = 0, where a first order transition exists. The values of the critical exponents obtained in reference [6] are identical to those of the transverse Ising model, putting the model in the same universality class as the two-dimensional classical Ising model.


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Ovchinnikov [4] investigated the antiferromagnetic Ising chain in the presence of a transverse magnetic field and a longitudinal magnetic field, and showed that the quantum phase transition existing in the transverse Ising model remains in the presence of the longitudinal field. Using the density matrix renormalization group (DMRG) technique of White [33], they found the critical line in the (hx , hz ) plane where the mass gap disappears and the staggered magnetizations along the X and Z axes vanish. The authors of reference [4] established that the Ising model in non-commuting fields belongs to the universality class of the transverse Ising model. 5. The ANNNI model in a longitudinal field

The case hx = 0 in the Hamiltonian (1) corresponds to the ANNNI model in a longitudinal field. This is a classical model in the sense that all operators involved commute. The longitudinal ANNNI model has interesting properties and it has been shown that there are four possible ground state configurations, the ferromagnetic, antiferromagnetic, antiphase, and the three-fold degenerate ↑↑↓ ground states (Adegoketh [34], Morita and Horiguchi [35]). We note that this is a classical model with competitive interaction from the nearest neighbours, next nearest neighbours, and the longitudinal field. The effect of the transverse field in the general Hamiltonian (1) is therefore to introduce quantum fluctuations in the system. As can be seen in Chapters 4 and 5 of Adegoketh [34], the existing order of the longitudinal ANNNI model is destroyed by quantum fluctuations.

II. SYMMETRIES OF THE MODEL HAMILTONIAN

II-1. The translation invariance symmetry T Consider any basis vector |ui = |S1 S2 · · · SN −1 SN i of the direct product total Sz basis of a system of N spins 1/2. The translation operator T whose action on |ui produces another basis vector |vi of the total Sz basis belonging to the same value of total Sz as |ui is defined by |vi =T |ui

= |S2 S3 · · · SN −1 SN S1 i

(3)

Definition 1. Two vectors |ui and |vi are translationally related if T n |ui = |vi for some integer n ≤ N . Definition 2. A set of m translationally related vectors {|u1 i , |u2 i , · · · , |uk i , · · · , |um i} such that for any member |uk i, the relationship T m |uk i = |uk i holds is called a cycle of period m.

(4)

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Clearly, the Hamiltonian (1) is invariant under spin translation, that is [H, T ] = 0. Ordinarily one can build and diagonalize the matrix of H in the basis defined in Equation (2). However, as noted earlier, even with the fastest computer, a direct diagonalization of a 2N × 2N matrix even for a relatively small system is quite inefficient and impractical and as such should be avoided. We take advantage of the fact that the Siz basis vectors can be sorted into cycles, as long as periodic boundary conditions are imposed. Although total Sz is not a symmetry of the Hamiltonian (1), T is. We are then able to use the translation symmetry together with the reflection symmetry and, in some special cases, the inversion symmetry discussed in later sections, to simplify the diagonalization process. Following the convention of B¨ arwinkel et al. [36] cycles will be called proper cycles if they have period N , otherwise they will be called epicycles. A more detailed discussion of how one can gainfully employ the symmetries of a Hamiltonian can be found in [36–38]. We note however that some important observations that are made here are not discussed in the cited references, nor in fact anywhere else. Since T D |mi = |mi, m = 1, 2, . . . , D, for D vectors that are translationally related, the eigenvalues xk of T are the D Dth roots of unity: 2πik k = 0, 1, . . . , D − 1 , (5) xk = exp D with corresponding eigenvectors D 1 X −m |xk i = √ x |mi k = 0, 1, . . . , D − 1 . D m=1 k

(6)

A recursion relation for the number of cycles to a period

Let X(m) be the number of cycles to a given period m. Clearly X(1) = 2 and X(2) = 1. Now since the number of cycles is independent of system size, in order to determine X(m) it is sufficient to consider system size N = m. This is a straightforward task since, due to the fact that the periods are factors of N we have that X (period × number of cycles) = 2N . (7) The simplest case is when m is prime, since in this case, we have that for N = m there are only cycles of period 1 and cycles of period m. The number of period m cycles is then determined by solving 1 × 2 + m × X(m) = 2m for X(m). Thus if m is prime, the number of period m cycles for any system size is X(m) =

2m − 2 . m


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For a general m we have from Equation (7) the following recursion relation for the number of cycles having period m:

X(m) =

2m −

γ−1 P

λk X (λk )

k=1

,

m

(8)

with X(1) = 2, λk = kth factor of m, and γ = total number of factors of m. The total number of cycles of a chain of N spins is then given by Totalcycles(N ) =

α X

X (βi ),

(9)

i=1

where βi is the ith factor of N and α is the total number of factors of N . An interesting case is when m is a power of 2. Then the factors of m will be 2 1, 2, 2 , . . . , 2m−1 , 2m and the recursion (8) can then be written in closed form as m

m−1

22 − 22 X(2 ) = 2m m

.

Dimensions of the subspaces of the space of eigenstates of T

The orthogonal subspaces of T eigenstates contain eigenvector contributions from proper cycles as well as from epicycles of the total Sz basis vectors of the Hilbert space of a system of N spins. An epicycle of period D ≤ N contributes an eigenvector to the k-subspace only if kD/N is an integer or zero. II-2. The spin reflection symmetry R Definition 3. The spin reflection operator R is defined by its action on an arbitrary direct product state |S1 S2 · · · Si · · · SN −1 SN i as follows: R |S1 S2 · · · Si · · · SN −1 SN i = |SN SN −1 · · · SN −i+1 · · · S2 S1 i

(10)

Clearly, R2 = 1, so that the eigenvalues of R are ±1. It is useful to note that if any two states |ui and |vi are related by reflection their linear combinations always are eigenstates of R2 = 1. Clearly R is a symmetry of the Hamiltonian H given by equation (1). It is also straightforward to see that, although R and T are both symmetries of the Hamiltonian (1), they do not commute in general. R and T do however commute in the k = 0 and k = N/2 subspaces of the eigenstates of T . In the k = 0 and k = N/2 subspaces, R can be diagonalized, and thereafter H can then be diagonalized in the two subspaces of the eigenstates of R (corresponding to eigenvalues +1 and −1). Consider an example of the simplification which arises from the combined use of the symmetries R and T for a system with N = 8:

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The states |u1 i = {|↓↓↑↑↓↑↓↓i} and |u2 i = {|↓↓↑↓↑↑↓↓i} are mirror reflections of each other. Similarly the following pairs are related by reflection: |v1 i = {|↑↑↓↓↑↓↓↓i} and |v2 i = |↑↓↓↑↑↓↓↓i, |w1 i = {|↓↑↓↓↑↑↓↑i} and |w2 i = {|↑↓↓↑↓↑↓↑i}, |x1 i = {|↑↓↑↓↓↓↑↑i} and |x2 i = {|↑↓↓↓↑↓↑↑i}, |y1 i = {|↑↑↓↓↑↑↓↑i} and |y2 i = {|↑↓↓↑↑↑↓↑i}, and |z1 i = {|↑↑↑↓↓↑↓↑i} and |z2 i = {|↓↑↓↓↑↑↑↑i}. Here we can employ the reflection symmetry to further reduce the dimensions of the matices in the k = 0 and k = 8/2 = 4 subspaces in solving the full Hamiltonian of the ANNNI model in the presence of both longitudinal and transverse fields. We take the subspace k = 0 as an example. The following 6 linear combinations are eigenstates of R belonging to eigenvalue −1. They are also eigenstates of T of eigenvalue 1. √ |ui = (− |u1 i + |u2 i) / 16 , √ |vi = (− |v1 i + |v2 i) / 16 , √ |wi = (− |w1 i + |w2 i) / 16 , √ |xi = (− |x1 i + |x2 i) / 16 , √ |yi = (− |y1 i + |y2 i) / 16 , √ |zi = (− |z1 i + |z2 i) / 16 . (11) The 6 new states in (11) constitute 6 eigenstates of the subspace of R of eigenvalue −1. Thus the 36 states of the subspace k = 0 of T now split into a union of 6 eigenstates of R of eigenvalue −1 and 30 eigenstates of R of eigenvalue 1. II-3. All spin inversion operator I Another useful operator which we have employed to advantage is the inversion operator I defined in the notations of the previous section by I |S1 S2 · · · SN −1 SN i = |(−S1 )(−S2 ) · · · (−SN −1 )(−SN )i .

(12)

Again we note that I 2 = 1, so that the eigenvalues of I are ±1. We hasten to emphasize that I is not a symmetry of the general Hamiltonian (1), for a finite hz . Nonetheless I is very useful even when hz 6= 0. I is a symmetry of the Ising model, the ANNNI model, and the transverse ANNNI model, as can be easily proved. When hx = 0, the model (1) reduces to the ANNNI model in a longitudinal field. In this case I is useful in writing down immediately the energy of any state obtained from another state by inversion, since then one merely has to change the sign of hz in the energy of the former state. In other words, the use of I makes it possible to restrict the discussion of the eigenstates of H(j, hx = 0, hz ) to only those with total Sz ≥ 0 or total Sz ≤ 0. On the other hand, when hz = 0, the Hamiltonian (1) reduces to that of the transverse ANNNI model and, in this case I is a symmetry. Combined with the translation symmetry and the reflection symmetry, exact diagonalization of the transverse ANNNI model can be significantly simplified. That I and R are compatible operators is rather obvious.


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III. RESULTS AND DISCUSSION

III-1. The longitudinal ANNNI model The longitudinal ANNNI model described by the Hamiltonian HANNNI =

N X i=1

z Siz Si+1

+j

N X i=1

z Siz Si+2

− hz

N X

Siz

(13)

i=1

is diagonal in the Sz representation and is therefore an exactly solvable system. Although in principle there are 2N possible states for a chain of N spins-1/2, translational invariance and reflection symmetry lead to a considerable, in fact drastic, reduction in the dimension of the Hilbert space. As pointed out in sections II II-1 and II II-2 both R and T are good quantum operators, but they however do not mutually commute, that is [R, T ] 6= 0 except in the subspaces k = 0 and, when N is even, k = N/2 of the eigenstates of T . As shown in the previous sections, T and R are, in fact, symmetries of the full Hamiltonian (1) of the ANNNI model in the presence of both longitudinal and transverse fields hz and hx . The all-spins inversion operator I is however not a symmetry of the Hamiltonian (13) except in the special case of total Sz = 0 or hz = 0. Notwithstanding that I is not a good quantum number it is still useful, because it simplifies the classification of the eigenstates and energies of the model (13), since if the energy of a given Sz configuration is known, the energy of the corresponding −Sz state obtained by inversion can be written down immediately by simply changing the sign of hz . We also remark that I is a symmetry of the Hamiltonian (1) in the special case hz = 0 (that is the transverse ANNNI model). Since the ground state energies are proportional to N , it is also not difficult to generalize the results of finite size exact diagonalization to an infinite system. Furthermore, the states are highly degenerate, so that the number of independent energies among which to search for a minimum is quite few for a given chain length. For example for N = 12 there are only 84 independent energies out of which only 4 can be ground state energies (corresponding to 10 states—the two-fold degenerate antiferro states, the three-fold degenerate ↑↑↓ · · · states, the four-fold degenerate antiphase states, and the non-degenerate ferromagnetic state). Similarly, for N = 20, there are 396 energies out of which only 6 belong to ground state configurations. We have implemented a program consisting of Maple procedures to classify the energy eigenvalues of the Hamiltonian (13) for finite lattice sizes. The values of the dimensions of the reduced Hilbert space can be determined, in advance, for any N by the Maple procedure totalcycles, based on the recursion relation in section II II-1, while another procedure carries out the actual diagonalization and classification of energies. The procedure energy makes it easy to calculate the eigenenergy of an arbitrary state of any N . We used the Sz representation for the spin operators. In the program, spin up is denoted by +1 and spin down by −1. The total Sz direct product basis states are conveniently represented as Maple lists, whose elements consist of a series of 1 and −1. This representation is particularly convenient because the permute command from the combinat package makes the generation of the basis states a boon (for large N however, direct permutation is avoided and the randperm command is used to generate the independent states (cycles) based on

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FIG. 1: T = 0 ground state energy diagram of the longitudinal ANNNI model for N = 12. This is also the diagram in the thermodynamic limit [34, 35].

the advance knowledge of the total number of cycles as returned by totalcycles). Whenever direct permutation is done, a procedure prune then employs translational invariance to reduce the dimension of the Hilbert space to the value returned by totalcycles and another procedure computes the states and the corresponding energies. As examples, we now present the results for the energy classification of the longitudinal ANNNI model for system sizes N = 12 and N = 20, respectively.


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SNo

States

k−values

Sz

Energies

Degen

1

[↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓↑↓]

[0, 10]

0

−5 + 5j

2

2

[↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓↑↑↓↓]

[0, 5, 10, 15]

0

−5j

4

3

[↑↑↓↑↑↓↑↑↓↓↑↑↓↑↑↓↑↑↓↓] [0, 2, 4, . . . , 16, 18] 2 −1 − 3j − 2hz

10

4

[↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓↓↑↑↓↓] [0, 1, . . . , 18, 19]

2 −1 − 3j − 2hz

20

5

[↑↑↓↑↑↓↑↑↓↑↑↓↓↑↑↓↑↑↓↓] [0, 1, . . . , 18, 19]

2 −1 − 3j − 2hz

20

6

[↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓↑↓] [0, 1, . . . , 18, 19]

3 −2 − j − 3hz

20

7

[↑↑↓↑↑↓↑↑↑↓↑↑↓↑↑↓↑↑↑↓] [0, 2, 4, . . . , 16, 18] 4 −1 − j − 4hz

10

8

[↑↑↓↑↑↓↑↑↓↑↑↑↓↑↑↓↑↑↑↓] [0, 1, . . . , 18, 19]

4 −1 − j − 4hz

20

9

[↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓↑↑] [0, 1, . . . , 18, 19]

4 −1 − j − 4hz

20

10 [↑↑↓↑↑↓↑↑↓↑↑↓↑↑↓↑↑↑↓↑] [0, 1, . . . , 18, 19]

4 −1 − j − 4hz

20

11 [↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑↑]

10 5 + 5j − 10hz

1

[0]

TABLE I: Energy classification of 20 spins showing the lowest 6 energies. The complete list of energies is obtained by running the code stateswithk (available from the authors) with N = 20.

1. N=12

Classified by translation invariance, the total number of configurations for spins on a chain of 12 spins is 352. The states are highly degenerate and the number of distinct energies is only 84, out of which only 4 correspond to ground state alignments. The 4 ground state structures are the non-degenerate ferromagnetic state with energy 3 + 3j − 6hz , the antiferromagnetic states with energy −3+3 j, the antiphase states with energy −3 j and the period-3 ↑↑↓ states with energy −1 − j − 2 hz . The resulting ground state energy diagram is shown in Figure 1.

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FIG. 2: T = 0 Ground State Energy diagram of the longitudinal ANNNI model for N = 20. 2. N=20

The reduced Hilbert space of 20 spin-1/2 atoms has 52488 states (as returned by the procedure totalcycles). These states are very highly degenerate, so that they share only 396 energies. Of these energies, only 6 belong to ground state configurations. The 6 lowest energies are tabulated in Table I. The ordering of the energies in the antiferro region is as follows: −5 + 5 j < −4 + 3 j − hz < −3 + j − 2 hz < −2 − j − 3 hz .

(14)

We note that the energies −4 + 3 j − hz and −3 + j − 2 hz , the energies of the first and second excited states, respectively, never make it to the ground state, since there is a lower


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energy, −2 − j − 3 hz , outside the antiferro region. The ground state energy diagram for the longitudinal ANNNI model for N = 20 is plotted in Figure 2. The regions marked R1, R2, and R3 are characteristic of system sizes N fulfilling N = 3n + 2, n an integer. They correspond respectively to ground state energies [34]: N +4 N −8 N −2 R1 EN =− − j− hz , 12 12 6 N −8 N −8 N +4 R2 EN = − − j− hz , 12 12 6 N −8 N + 16 N −8 R3 EN =− − j− hz . 12 12 6 In the particular case of N = 20, we have R1 EN = −2 − j − 3hz ,

R2 EN = −1 − j − 4hz ,

R3 EN = −1 − 3j − 2hz .

We note that when N is a multiple of 3, for example N = 12, the configurations depicted in regions R1, R2, and R3 are absent and are replaced by the 3 fold degenerate ↑↑↓ Q configuration with energy EN = −N/12(1 + j + 2hz ). That is the regions R1, R2, and R3 of Figure 2 merge into the region Q of Figure 1. In fact, in the thermodynamic limit, we see from the above that the ground state energy is R1 EN E R2 E R3 = N = N N N N (1 + j + 2hz ) . =− 12

ε0 =

The ground state energy diagram for N = 12 is in fact identical to the ground state energy diagram of the longitudinal ANNNI model. III-2. The ANNNI model in two perpendicular fields When the longitudinal ANNNI model is placed in a transverse field hx , quantum fluctuations are introduced into the system, in addition to the competitive interactions of the nearest neighbours, next nearest neighbours, and the longitudinal field. The model is now fully quantum and the Hamiltonian (1) is not diagonal in the total Sz direct product basis. As noted in previous sections, the translation invariance T and the spin reflection operator R are symmetries of the Hamiltonian H, with T and R being, however, not compatible. Since a Hamiltonian can always be block diagonalized in the subspaces of the degenerate eigenstates of its symmetries, we have written a Maple code to build the N Hamiltonian blocks in the N subspaces of the eigenstates of the translation invariance operator corresponding to the quantum numbers k = 0, 1, 2, . . . , N − 1.

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FIG. 3: Energy spectrum of the transverse Ising model for N = 12 (j = 0, hz = 0), showing the lowest five energies. Note the vanishing of the gap in the excitation spectrum around hx = 0.2, this should be compared with Figure 4 where the energy gap remains for this value of hx . 1. Energy Spectra

As examples of practical application, the lowest energy levels as functions of the transverse field hx for a system size N = 12 are plotted in Figures 3–6 for selected values of the longitudinal field hz and the exchange interaction j. Figure 3 describes the effect of quantum fluctuations on the antiferromagnetic order. One notices the initial wide energy gap between the first excited state and the ground state j = 0, hz = 0 is in fact the exactly solvable transverse Ising model whose ground state energy is expressed in term of the elliptic integral of the second kind [14]. The N = 12 ground state energy per spin agrees with the exact ground state energy per spin to a very high degree of accuracy. Figure 4 corresponds to the effect of quantum fluctuations on the ferromagnetic order. Since the ferro ground state is non-degenerate at hx = 0 there is no sharp drop in the energy gap, unlike in the antiferro case. Figure 5 shows the effect of the transverse field on the ↑↑↓↓ . . . ground state configuration, while Figure 6 is about the effect of quantum fluctuations on the ↑↑↓ . . . ground state structure.

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FIG. 4: Energy spectrum of the ANNNI model in two fields for N = 12 for j = 0.1, hz = 1.2, showing the lowest five energies. The figure shows the effect of quantum fluctuations on the ferromagnetic order which exists at hx = 0. Note the steady widening of the energy gap ∆ε = ε1 − ε0 for increasing values of hx . This is because the model is ferromagnetic again (but with the spins now aligned in the transverse direction) for high values of the transverse field. 2. Continuous phase transitions in the ANNNI model in two perpendicular fields

In a further application, we have employed the finite size scaling ansatz to determine the critical points (hxC , jC ) for fixed hz for the one dimensional ANNNI model in two perpendicular magnetic fields, described by the Hamiltonian (1). The method introduced by Kramers and Wannier [39] and later developed and generalized by Fisher and Barber [40] has turned out to be a valuable tool in evaluating critical behaviour from numerical results by extrapolating information obtained from a finite system to the thermodynamic limit [41, 42]. The technique gives reliable results for quite different models and different types of critical behaviour [41]. A list of references on numerous successful applications of the finite size scaling technique (at nonzero temperature) to various quite different models is found in reference [41]. The finite size scaling technique is also gaining popularity in the study of quantum phase transitions (that is, phase transitions at zero temperature that are driven by competition and quantum fluctuations alone), as opposed to conventional, thermally driven phase transitions [6, 42].

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507

FIG. 5: Energy spectrum of the ANNNI model in two fields for N = 12 for j = 0.8, hz = 0.2, showing the lowest five energies. Note the vanishing of the gap ∆ε = ε1 − ε0 around hx = 0.1. At hx = 0 the ground state structure is the four-fold degenerate antiphase configuration.

The basic equation for finite size scaling is N ∆EN = N 0 ∆EN 0 ,

(15)

1 0 (j, hx , hz ) . ∆EN = EN (j, hx , hz ) − EN

(16)

with

0 and E 1 respectively are the ground state N and N 0 are different system sizes, EN N energy and the energy of the first excited state of the model as functions of the parameters j, hx , and hz . The energy gap ∆EN is related to the inverse correlation length of the classical model. In considering quantum fluctations introduced by the transverse field hx for two different system sizes N and N 0 , j and hz are kept fixed and one keeps adjusting (refining) hx until Equation (15) is satisfied to the desired accuracy. This value of hx is then the critical value of the transverse field. The correlation length critical exponent is estimated from

ν=

N ∂∆EN /∂hx . N 0 ∂∆EN 0 /∂hx

The partial derivatives are evaluated at the critical value of hx for given j and hz .

(17)

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FIG. 6: Energy spectrum of the ANNNI model in two fields for N = 12 for j = 0.5, hz = 0.5, showing the lowest five energies. Here we observe the effect of quantum fluctuations introduced by the transverse field hx on the three-fold degenerate up-up-down ground state of the longitudinal ANNNI model.

We stress here the point that we have investigated the different phases in relation to the magnetic fields always at temperature T = 0. We took advantage of the translational symmetry of the Hamiltonian under periodic boundary conditions to drastically reduce the dimensions of the Hilbert space of the spin systems in the total Sz basis. The Hamiltonian H was diagonalized in the orthogonal subspaces of the translation operator. A maximum system size of 12 spins was considered. The matrix of H in each subspace was generated by Maple and diagonalized by Matlab. The critical exponents were calculated using (17) and tabulated in Table II. From the table ν ≈ 1, so that the ANNNI model in two orthogonal magnetic fields is still in the same universality class as the zero field classical two-dimensional ANNNI model. The T = 0 phase diagram of the ANNNI model in two fields, obtained from the finite size scaling data points, is presented in Figure 7. Considering the high accuracy which is characteristic of the finite size scaling technique [41], we believe that N = 12 is adequate to produce reliable results. In fact fewer or comparable systems have been treated in the recent past [6, 26]. The convergence and smoothness of the data points indicate that these are clearly second order phase transition lines. This however does not rule out the possibility of the existence of floating phases in the regions as well. Implementing the DMRG algorithm for entanglement entropy calculations, Beccaria et al. [43] obtained clear evidence for the existence of a floating phase for j > 0.5, for the transverse ANNNI model (hz = 0 in our model), extending at least up to j = 5. It however

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j hx c ν 0 0.48518 1.07281 0.1 0.397037 1.064492 0.2 0.30105 1.042743 0.3 0.19032 0.964838 0.5 0.265 0.907053 0.6 0.39016 1.0762272 0.7 0.498 1.191895

TABLE II: The critical field and the corresponding critical exponent for the phase boundary of the ANNNI model in two orthogonal magnetic fields.

remains an open question whether the floating phase extends up to j = ∞. We remark that the existence of floating phases will make the phase diagram richer, but will have no grave consequences for the second order transition lines. Other methods can be used to obtain the tricritical points and floating phases and these can then be inserted in the phase diagram as was done, for example, in the article by Guimar∼aes et al. [26].

↑↓↑↓↑↓

↑↑↓↑↑↓ ↑↑↓↓

FIG. 7: T = 0 phase diagram of the ANNNI model in two perpendicular magnetic fields for hz = 0.2. The convergence and smoothness of the data points indicate that these are clearly second order phase transition lines. Other methods (e.g. those employed in references [44] and [43] ) can be used to obtain the tricritical lines and floating phases, and these can then be inserted in the phase diagram as was done, for example, in the article by Guimar˜ aes et al. [26].


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IV. CONCLUSION

We have presented results for the finite longitudinal and cross-field ANNNI model with special symmetry considerations. In addition we did a finite size scaling analysis for the latter spin system and obtained a phase diagram for periodic states. Our results may serve as a starting point for investigations of floating phases, tricritical lines, and finite temperature effects.

Acknowledgments KA is grateful to the DAAD for a scholarship and thanks the Physics Institute, Universität Bayreuth for hospitality.

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