Fundamentals of the Physics of Solids / 15-Elementary Excitations in Magnetic Systems
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15.5 Low-Dimensional Magnetic Systems |
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Fig. 15.13. The continuum of low-energy excitations in the XY model
a particle–hole pair has to be created in this distribution. The distribution of the occupied states is shown in Fig. 15.14.
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Fig. 15.14. Wave numbers of free spinless fermions in the lowest-energy state of
the Stotz = 1 sector and in a possible excited configuration for N = 16 and N↓ = 7. Occupied sites are indicated by full circles
It can be shown that in the N → ∞ limit the continuum of these excitations coincides with the continuum arising in the subspace Stotz = 0.
This mapping to free spinless fermions is valid for = 0 only. The Jordan– Wigner transformation can be applied to the case when is finite, but then an interaction appears among the spinless fermions. We shall defer the discussion of the e ects of this interaction to the third volume, where we shall derive the above-mentioned result – namely that the picture presented for the isotropic antiferromagnet and the XY model is qualitatively true in the whole planar regime −1 < < 1, where the spin projection along the z direction is expected to vanish classically. In fact, the quantum mechanical ground state
is in the subspace Stotz = 0 and the continuum of low-energy excitations can be interpreted as arising from pairs of spin-1/2 spinons.
15.5.7 The Role of Next-Nearest-Neighbor Interactions
So far it has been assumed that the antiferromagnetic exchange interaction acts between nearest neighbor spins only. A di erent type of disordered spin
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configuration can be found when the interaction between farther neighbors is not negligible, since in addition to quantum fluctuations spin frustration that may arise from the competition between magnetic couplings has to be taken into account, too.
We shall denote the strength of the antiferromagnetic coupling between nearest neighbors by J1, and between next-nearest neighbors by J2. The Hamiltonian of the spin chain is then
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H = J1 Si · Si+1 + J2 |
Si · Si+2 . |
(15.5.126) |
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Both J1 and J2 are assumed to be positive. Spin frustration is best illustrated on a zigzag ladder, as shown in Fig. 15.15. Spins located at even and odd sites of the original chain form two antiferromagnetic chains – the legs of the ladder – that are somewhat displaced relative to each other. Each spin on one leg is coupled antiferromagnetically to two spins on the other leg.
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Fig. 15.15. Spin chain with nearestand next-nearest-neighbor interactions illustrated by a zigzag ladder
Separating the terms within the legs and between them, the system is described by the Hamiltonian
H = J1 |
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· S2i + S2i · S2i+1 |
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(15.5.127) |
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+ J2 |
S2i−1 · S2i+1 + S2i · S2i+2 . |
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The coupling J2 tends to align the spins on the legs antiferromagnetically
– at least on short distances. Assuming that two neighboring spins on a leg are aligned oppositely (as they should be if the antiferromagnetic coupling between them is strong enough), they create opposite e ective fields at the position of the spin lying between them on the other leg, canceling each other’s e ect. If, on the other hand, the coupling J1 is stronger, then a short-range antiferromanetic order may be expected along the zigzag path, which would give rise to a parallel alignment of neighboring spins on the same leg. When J1 and J2 are of comparable strength, spins become frustrated, and a new type of singlet phase appears.
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The ground state of this model is exactly known in the Majumdar–Ghosh point,12 where J2 = 12 J1. In this special case the Hamiltonian can be written as
H = 34 J1 P3/2(S2i−1 +S2i +S2i+1)+P3/2(S2i +S2i+1 +S2i+2) −3J1N/8 ,
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(15.5.128) |
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where the operator |
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P3/2(S2i−1 + S2i + S2i+1) = 31 |
(S2i−1 + S2i + S2i+1)2 − 21 |
21 + 1 |
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(15.5.129) |
projects to the subspace where the three S = 1/2 spins add up to a total spin of 3/2. It is immediately seen from this form that the state in which singlet pairs are formed between nearest neighbors along the zigzag path is an exact ground state as three consecutive spins can never be combined into a total spin of 3/2. As shown in Fig. 15.16, there are two ways to form singlet pairs, so the system has a doubly degenerate ground state. Since the singlet pairs are independent of each other there is no correlation between the spins beyond the nearest-neighbor distance.
Fig. 15.16. The two equivalent ground states of the antiferromagnetic spin chain in the zigzag ladder representation for J2 = 12 J1 . The spins connected by solid lines form singlet pairs
Breaking any of the singlet pairs gives rise to an excited state. A relatively low-energy excitation is expected when the two spins of the broken singlet are not bound into a triplet but move farther away, and the singlet bonds between them are rearranged, as shown in Fig. 15.17. The configuration of singlet bonds corresponds to di erent ground states at the two ends and in the intermediate region. Since the energy of the singlet pairs is not modified by the rearrangement, the energy of the configuration is independent of the length of the rearranged region, i.e., on the distance between the free spins. Therefore they can propagate freely. Consequently the continuum of low-energy excitations can be interpreted as arising from a pair of spinons or domain walls, the creation of which requires a finite amount of energy that is equal to the binding energy of the singlets.
Such a behavior is observed not only in the point J2 = 12 J1 but whenever the antiferromagnetic second-neighbor coupling is stronger than a critical value (J2/J1)c = 0.241.
12 C. K. Majumdar and D. K. Ghosh, 1969.
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Fig. 15.17. Freely moving spinons in the spin-1/2 antiferromagnetic zigzag ladder
In the previous discussion we have seen two possible types of behavior for the S = 1/2 spin chains. The ground state was either a unique singlet state with a gapless excitation spectrum, or the ground state was doubly degenerate, dimerized, and the excitation spectrum was separated from the ground state by a finite gap. According to the Lieb–Schultz–Mattis theorem,13 these two types of behavior are generic in one-dimensional spin models with half-odd-integer spin. More precisely, the authors established that in half- odd-integer spin models that are invariant under translation by the lattice constant and have full rotational symmetry in spin space, the ground state and the excitation continuum may be separated by a gap only if the ground state is degenerate and breaks the translational symmetry spontaneously.
15.5.8 Excitations in the Spin-One Heisenberg Chain
In contrast to the spin-1/2 chain, the ground state and the excitation spectrum of antiferromagnetic Heisenberg models with S > 1/2 cannot be determined exactly. States with more than two reversed spins cannot be given in the Bethe ansatz form, and scattering processes involving several magnons cannot be described in terms of the phase shifts occurring in two-magnon scattering. To obtain exactly solvable models, terms involving higher powers of the product of two spins, or the products of several spins – with suitably chosen coupling constants – must be included in the Hamiltonian. In the S = 1 case, the model defined by the Hamiltonian
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Si · Si+1 − (Si · Si+1)2 , |
(15.5.130) |
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while in the S = 3/2 case, the system with the Hamiltonian |
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H = −J |
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Si · Si+1 − 278 (Si · Si+1)2 − 2716 (Si · Si+1)3 |
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are exactly solvable by straightforward generalization of the Bethe ansatz. Both the ground state and the excitation spectrum can be determined, and the behavior of the system is found to be similar to that of the spin-1/2 isotropic Heisenberg model. When the coupling is ferromagnetic, the ground state is fully aligned, and low-lying excitations are spin-1 magnons. Antiferromagnetic coupling, on the other hand, leads to a singlet ground state with a gapless
13 E. Lieb, T. D. Schultz, and D. C. Mattis, 1962.
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continuum of excitations that consists of singlet and triplet pairs of spin-1/2 spinons.
Another exactly solvable version of the spin-1 model is obtained by reversing the sign of the biquadratic term:
H = −J |
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Si · Si+1 + (Si · Si+1)2 . |
(15.5.132) |
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This model possesses not only SU(2) but also SU(3) symmetry. Here, too, the continuum of excitations appears above the ground state without gap, however, the soft modes appear at k = 0 and ±2π/3a in this model, and not at k = 0 and π/a as in the spin-1/2 antiferromagnetic Heisenberg chain.
These models should nevertheless be considered as exceptions. In general, isotropic magnetic chains with integer spin do not have a gapless branch of excitations. The Lieb–Schultz–Mattis theorem is not applicable to these models, and, barring certain exceptions, even if the ground state is nondegenerate, the continuum of excited states in integer-spin antiferromagnetic chains is separated from the ground state by a forbidden region, the Haldane gap.14
To understand the nature of this state better, consider the most general Hamiltonian describing an interacting S = 1 chain with an SU(2)-invariant isotropic nearest-neighbor interaction. It follows from the properties of spin-1 operators discussed in Appendix F that, besides the usual exchange interaction, a biquadratic term (Si · Si+1)2 also appears in the Hamiltonian, but no higher powers. The most general form satisfying these requirements is
H = J |
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Si · Si+1 + β (Si · Si+1)2 . |
(15.5.133) |
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This Hamiltonian can be written in an alternative form by taking into account that the combination of two neighboring S = 1 spins gives rise to total spins Spair = 0, 1 and 2. The operators that project onto the respective subspaces are
P0(i, i + 1) = 1 − 23 (Si + Si+1)2 + 121 (Si + Si+1)4
= −31 + 31 (Si · Si+1)2 , |
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P1(i, i + 1) = 43 (Si + Si+1)2 − 81 (Si + Si+1)4 |
(15.5.134) |
= 1 − 21 (Si · Si+1) − 21 (Si · Si+1)2 |
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P2(i, i + 1) = −121 (Si + Si+1)2 + 241 (Si + Si+1)4 = 13 + 12 (Si · Si+1) + 16 (Si · Si+1)2 .
The most general isotropic model is constructed by taking an arbitrary linear combination,
14 F. D. M. Haldane, 1983.
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N2S
H = |
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fj Pj (i, i + 1) . |
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i=1 j=0
The usual antiferromagnetic Heisenberg model is recovered by choosing fj = j(j + 1).
Let us now consider the special case when only those states contribute to the energy for which every pair of neighboring spins is in the Spair = 2 state. With an appropriately chosen additive constant the Hamiltonian may
be written as |
H = J |
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2P2(i, i + 1) − 32 |
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(15.5.136) |
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or, in terms of the spin operators, as |
31 (Si · Si+1)2 . |
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H = J |
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Si · Si+1 + |
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If we succeed in constructing a state without any pairs of neighbors being in the Spair = 2 state, then the energy of the state in question will simply be E0 = −23 N J. Such a state can be put together as follows. First, at each lattice site, the S = 1 spin operator is built up of two spin-1/2 operators, σi and τi, allowing only the symmetric combination. Then, a singlet state is formed between lattice sites i and i + 1, using one of the 1/2 spins at each lattice site. The other 1/2 spin at lattice site i (i + 1) is paired with one of the spins at lattice site i − 1 (i + 2) to give another singlet state. Thus, with the exception of the two end points, every lattice point of the chain is bound to its left and right neighbors by a singlet bond. The bond arrangement is shown schematically in Fig. 15.18.
Fig. 15.18. Singlet bonds in the ground state of the model defined by (15.5.137)
When the projection operator P2 acts on any such configuration, the result is indeed zero. By choosing the linear combination of all possible states with equal weights, symmetrization is taken care of. The resulting state turns out to be more than just an eigenstate of the Hamiltonian (15.5.137): it is the ground state. Since the state comprises singlets between nearest neighbors, and is therefore reminiscent of valence bonds, such models are also called valence bond solids (VBS).
To create an excited state one or more short-range valence bonds have to be broken. Since this requires a finite amount of energy, the ground state is separated by a gap from the excitation spectrum in these models. Consequently the spin–spin correlation function decays exponentially, with a correlation length that is not much larger than atomic dimensions. Another interesting point that distinguishes this spin-1 model from the spin-1/2 Heisenberg
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model is that the low-lying excitations do not form a continuum. The excitations are not pairs of deconfined spin-1/2 spinons but spin-1 magnons with a well-defined dispersion relation.
If the parameter β in (15.5.133) is not exactly β = 1/3, then for −1 < β < 1 (note that this interval contains the isotropic antiferromagnetic Heisenberg model, which corresponds to β = 0) the ground state and the excitation spectrum are similar to those discussed above – with the exception that singlet bonds are not necessarily formed between neighboring sites, however their average range is short. Once again, the spin–spin correlation functions decay exponentially. This is the Haldane phase of the spin-1 Heisenberg chain.
15.5.9 Spin Ladders
Having overviewed the properties of spin chains, we can now generalize these considerations to two or more coupled spin chains. In the simplest case, where two chains are coupled by an exchange coupling J , the system is described by the Hamiltonian
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H = J (S1,i · S1,i+1 + S2,i · S2,i+1) + J |
S1,i · S2,i . (15.5.138) |
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When the couplings between the spins are drawn as straight lines, they make up a ladder: J is the coupling along the legs, while J is the coupling between spins on the same rung. This is why the system is called a spin ladder. When several chains are coupled, multileg ladders are obtained.
If the rung coupling J is ferromagnetic and strong enough, the spins on the same rung are bound into a triplet (S = 1), and the spin ladder behaves as an S = 1 spin chain. When J is antiferromagnetic, and periodic boundary conditions are imposed, then the ground state is a nondegenerate singlet state (the analogue of the VBS state discussed above), in which spins on neighboring or not too distant rungs form singlets. Such a configuration is shown in Fig. 15.19(a). Note that if the ladder were cut anywhere between two neighboring rungs an odd number of singlet bonds would be broken.
Figure 15.19(b) shows an excited state in which one singlet bond is broken. Breaking a bond requires a finite amount of energy, thus the excitation spectrum starts with a finite gap, the Haldane gap. It can also be understood from the figure why the two “unbound” spins do not propagate freely, and why the excitation spectrum is not a continuum of spinon pairs (unlike for the zigzag ladder). If this ladder were cut between the two “unbound” spins, an even number of singlet bonds would be broken. The energy of this configuration increases when the two spins are moved farther apart, therefore they do not become truly unbound. Using the language of particle physics, the force increases linearly with distance, and confines the two spinons into a magnon.
A similar, nondegenerate singlet ground state is found when J is antiferromagnetic: for su ciently strong coupling spins on the same rung form
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Fig. 15.19. (a) A possible configuration of singlet bonds in the ground state of a ferromagnetically coupled two-leg spin ladder. (b) An excited state obtained by breaking a singlet bond
singlets with high probability. Such a configuration is shown in Fig. 15.20. Note that in this case the number of singlet bonds between neighboring rungs is always even.
Fig. 15.20. A possible configuration of singlet bonds in the ground state of an antiferromagnetic two-leg spin ladder
The low-lying excitations are magnons again, and the spectrum has a finite energy gap, for the same reason as above: if a singlet bond is broken, the number of singlet bonds between the two “free” spins changes parity, and the energy increases linearly with the distance between the spins, confining the spinons into magnons.
Thus, whether the coupling between the legs is ferromagnetic or antiferromagnetic, two-leg spin ladders behave like spin-1 chains. It can be shown by a generalization of the Lieb–Schultz–Mattis theorem that in odd-leg spin ladders made up of an odd number of S = 1/2 spin chains either the ground state is degenerate or the excitation spectrum is gapless. Such spin ladders behave like chains of half-integer spins. On the other hand, the behavior of even-leg spin ladders is analogous to that of chains of integer spins.
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15.5.10 Physical Realizations of Spin Chains and Spin Ladders
Ideal, free-standing spin chains and spin ladders (such as the ones studied theoretically in the previous subsections) do not exist in nature. However, there are a number of materials that have truly three-dimensional crystalline structure but quantum mechanical exchange is strongly anisotropic in them: much stronger in one direction than in others. These materials can, therefore, be considered as if they consisted of weakly coupled chains of spins. At low temperatures, where the thermal energy is comparable to or less than the exchange energy between the chains, a three-dimensional ordered magnetic structure may arise. Above this temperature, the chains behave independently as far as magnetic properties are concerned. Here we give just a small selection of materials that exhibit such properties.
A typical example is CPC – CuCl2·2N(C5H5) –, in which the chains formed by spin-1/2 Cu2+ ions are separated by large pyridine molecules. The exchange interaction between the chains is about 300 times weaker than within the chains, and antiferromagnetic ordering takes place at only TN = 1.14 K. Copper pyrazine dinitrate (CuPzN) – Cu(C4H4N2)(NO3)2 – is an even better candidate for the ideal isotropic antiferromagnetic Heisenberg chain. The coupling between the chains is four orders of magnitude weaker than within the chains, and no magnetic ordering has been observed down to 0.1 K. The continuum of low-lying excitations of the spin-1/2 Heisenberg chain, which has been interpreted in terms of spinon pairs, can be observed very well in these materials.
An almost ideal representative of integer-spin antiferromagnetic Heisenberg chains is Y2BaNiO5. The spin-1 nickel ions surrounded by oxygens form chains that are extremely weakly coupled. In agreement with theoretical predictions, the excitation spectrum has a finite Haldane gap ( = 8.6 meV).
Several materials have been produced synthetically in which spins form twoor multileg ladders. In the homologous series Srn−1CunO2n−1 spin-1/2 Cu2+ ions are arranged in such a way that the copper chains lie in magnetically isolated layers; within the layers n chains form an n-leg ladder, but adjacent ladders are displaced by half a unit, and thus the coupling between ladders is frustrated, and has a negligible e ect. Figure 15.21 shows the susceptibility for the twoand three-leg members of this series.
It can be seen that by subtracting the Curie component – which arises from the end spins of the chains or from paramagnetic impurities –, the susceptibility becomes exponentially small at low temperatures for the two-leg ladder, while it remains finite for the three-leg ladder. This is in good agreement with the result explained above: there are no gapless magnetic excitations in the two-leg ladder, while the excitations spectrum of the three-leg ladder is gapless.
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Fig. 15.21. Temperature dependence of the susceptibility of the two-leg (SrCu2O3) and three-leg (Sr2Cu3O5) members of the series. Empty circles indicate raw data, while full circles are corrected data obtained by subtracting the Curie component [M. Azuma et al., Phys. Rev. Lett. 73, 3463 (1994)]
15.6 Spin Liquids
It was shown in the previous sections that quantum fluctuations become relevant in low-dimensional systems. In one dimension they will hinder magnetic ordering, with the sole exception of the ferromagnetic phase. The ground state of antiferromagnetic spin chains and spin ladders is a spin singlet in which the continuous SU(2) symmetry of the Hamiltonian is not broken. Such a state might be called a spin liquid.
When considering the excitation spectrum and the decay of the correlation functions in these systems, various fundamentally di erent types of behavior are found. As we shall see in Chapter 32, owing to the gapless character of the excitation continuum the decay of the correlation function is power-law- like in the spin-1/2 antiferromagnetic Heisenberg model, just like in a critical system. Such systems are called algebraic or critical spin liquids. When the next-nearest exchange is su ciently strong and also antiferromagnetic, the competition of the couplings leads to frustration, resulting in an exponentially decaying correlation. Although the situation is similar to the decay of correlations in liquids, the spin system is not homogeneous: the symmetry of the Hamiltonian under translation through the lattice constant is broken. Correlations decay exponentially in the Haldane phase, too, however, a certain order is exhibited by the singlet valence bonds: a hidden order in the spin components.
Thus, none of the above examples is a true spin liquid in the strict sense – namely, that the magnetically disordered singlet ground state should break no symmetry of the Hamiltonian, that there should be no hidden order parameters, and that the spin–spin correlation functions should decay exponentially. They indicate, however, that if a spin liquid phase exists, it may arise from quantum fluctuations, frustration caused by competing interactions, or geometric frustration resulting from the topology of the arrangement of spins.