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Quantum antiferromagnets in one and two dimensions
Journal of Magnetism North-Holland
and Magnetic
Materials
104-107
(1992) 761-765
Invited paper
Quantum Bertrand
antiferromagnets
in one and two dimensions
I. Halperin
Physics Department, Harcard Unicersity, Cambridge, MA 02138, USA
We review some theoretical developments of the last few years on the low-temperature behavior of quantum Heisenberg anfiferromagnets in one and two dimensions. For simple two-dimensional systems without frustration, it is now clear that the ground state has Ntel order, for any value of the quantum spin. Theoretical predictions for the behavior as T --) 0 of the diverging correlation length, the spin-wave frequencies and damping rates, and the form of the static and dynamic spin-correlation functions, have been obtained from a combination of renormalization group and scaling analyses, spin-wave hydrodynamics, and computer simulations of classical systems. Results are in good agreement with neutron experiments on La,CuO, and K,NiF,. For the S = 1 antiferromagnetic chain, with a Haldane gap, a recent development is the realization that there should be two effective S = $ degrees of freedom localized at the ends of a finite chain. These have been observed in recent spin resonance experiments on NENP with dilute concentrations of Cu or Zn substituted for the Ni.
1. Introduction In recent years, there has been much interest in quantum antiferromagnets, particularly one- and twodimensional systems with Heisenberg coupling between spins. The 2D problem, particularly the spin-i system on a square lattice, with nearest neighbor coupling, has received particular attention, beginning in 1987, because of its relevance to the spin correlations in the copper-oxide layers in the insulating parent compounds of the new high-temperature superconductors. Interest in 1D spin systems has been stimulated by the mathematical connections that exist between such systems and problems in conformal field theory, as well as by new experimental developments in quasi-1D spin systems, and by the hope that belter understanding of 1D systems might also lead to insights into exotic 2D systems and high-temperature superconductivity. I will review here two theoretical problems which have direct relevance to experimental systems: the behavior of a 2D Heisenberg antiferromagnet at low temperatures, and the behavior of a spin-l chain with free ends. 2. Two-dimensional
Heisenberg
antiferromagnets
According to the rigorous Hohenberg-MerminWagner theorem, a 2D spin system with Heisenberg symmetry (i.e., invariance under the SU(2) group of rotations in spin space) and finite-range coupling between spins cannot have long-range antiferromagnetic order at any temperature different from zero. However, at zero temperature long-range order is not forbidden by the theorem. In fact, long-range order has been rigorously established for spin S 2 1, on a square lattice with nearest-neighbor coupling [1,2]. For the case of S = i, there is no rigorous proof, but there are strong theoretical arguments that long-range order ex0312~8853/92/$05.00
0 1992 - Elsevier
Science
Publishers
ists [3-51. Singh and Huse [4] estimated the sublattice magnetization for S = i, using a perturbation expansion with the Ising model as a starting Hamiltonian, as N,, = 0.302 + 0.007, which is 40% below the Ntel value. Similar estimates of N,, are obtained [5] from the spin-wave series expansion in S- ‘. It is known from the spin-wave theory that N,, can be further decreased if competing interactions are added to the model, such as antiferromagnetic coupling between second-neighbor spins. It has been suggested that for suitable choices of interaction constants it is possible to obtain a groundstate where there is neither broken rotational symmetry in spin-space nor broken translational symmetry in real space. One class of possible states, the chiral spin liquid states have broken time-reversal symmetry and broken reflection symmetry, leading to non-zero expectation values of the form (S, CS, x S,)) where i, j and k are three nearby lattice sites that are noncollinear [6]. Two-spin correlations fall off exponentially with separation in these states. There are good arguments that a chiral spin liquid state can occur as the groundstate of a generalized Heisenberg Hamiltonian, with carefully chosen interaction strengths, but it is not at all clear that such a state can occur for a realistic Hamiltonian, with say just two or three short-range competing interactions. Competing states which have been proposed for frustrated S = i systems include helical spin orderings, and dimerized states, which are rotationally invariant spin singlets but have broken translational symmetry [7]. For a 2D quantum Heisenberg model that has a groundstate with conventional antiferromagnetic order at T= 0, the most interesting theoretical questions concern the behavior at T # 0, where long-range order is forbidden. According to our current theoretical understanding, the two-point correlation function for the
B.V. All rights reserved
staggcrcd spin should decay cxponcntially at large scparations. with a decay length [ that may hc written in the following form at low temperatures [8]: A,(hc/27V,) [=
“XP(2TPJT)
F+Q/2Tip,)
TO(T/27;&)3]
(1)
.
whcrc A, and A; arc universal constants. c is the zero-tcmperaturc spin-wave velocity, and p, is the zero-temperature spin stiffness constant, defined such that a gradient in the direction R of the antiferromagnctic Ao
J’(k[),
(2)
B,5
S( k = 0) = [(2np,/T)
‘N,f
+ L?: + O( T/2lTp,)]?
’
(j)
where k is the wavevector, measured from the antiferromagnetic Bragg peak, f is a function of k<. and B,$ and B.4 arc dimensionless constants, while N,, is the quantum-rcnormalizcd staggered magnetization at T = 0. The function .f’ and the constant B,$ should he universal and arc the same for the quantum and classical models. Fits to Monte Carlo data on classical models give B,S = I25 to 180, with B,;, = I. An approximate form for the scaling function ,f(s), which has the correct limits for x + 0 and x + 3~. is [IO]
where (1 = k.$ and v = w/W,,. (The scaling form applies for T + 0, with q, v fixed.) The scaling function 4 is again universal, and identical for the quantum and classical models. A combination of renormalization group analysis, spin-wave hydrodynamics, and spinwave scattering theory suggests that for large k[, Sk, co) has a well-defined spin-wave peak with maximum at frequency We - Wll,, [+ln(l + y’)]“’ and a width 11 of the approximate form [I I] ‘h -hwL
7i 2 In[cS + l/2 2
[8+
ln( 1 fq’)] 1/2In(l
+ ln(2n)
+y’)]’
’
(7)
with 6 = 1.7. (Note that the value of oh is shifted significantly from the T = 0 value WY= ck, and the ratio I’Joi decrcascs only logarithmically with increasing k[. For kt --j 0, WC expect an overdamped relaxation of the staggered magnetization, so that S( k. w) has a peak centcrcd at w = 0, with a width y,,W,,. A fit to simulations of the classical model [IO] gave the value y,, = 0.85 i 0.15, which is close to an cstimatc ot y. = 0.96 by Grcmpel [12], hascd on a coupled-mode approximation. An approximate analytic form for 4(rl, V) has been fit to the simulations over the cntirc range of intcrcst of r, [ IO,1 I]. For the nearest-neighbor Hcisenberg model, the zero-tempcraturc constants 0, and c may be rclatcd to the coupling constant J and the lattice constant (I hy means of spin-wave theory or other approximate methods [Xl. Singh and Huse [4] studied the S = k model using series cxpanaions and found he = l.h7Ju, and of i2% and (27TTTpJ~K)= O.h8N ’ with uncertainties * 0% respectively. If we now write the pre-exponential factor .4,(hc/2~p,), which appears in the numerator of eq. (I) in the form (‘,a. the ahovc results imply that C, = 0.50 + 0.03 for S = +. For S = 1, the spin-wave expansion gives a value C i = 0.19. with an estimated uncertainty of order 5%. The theoretical results prcscnted above for the correlation length and S(k = 0) at low temperatures. have hcen compared with Monte Carlo calculations on the S = i antiferromagnct [ 13,141 which arc most accurate at high tempcraturcs. The agrecmcnt is satisfactory. and rcasonahlc interpolations, valid over the entire tcmpcraturc range, are therefore poasihle.
1 + 2~rB,~ ’ ln( I +x’) f(X)
=
(4)
I + x?
The dynamic structure factor S(k, w) is predicted to have the dynamic scaling form [8] S(k.
w) =W,j ‘S(k)c$(q,
o,, = c<- ‘( T/hp,)‘?
I’),
(5) (6)
For a 3D stack of Heiscnberg layers, with a weak coupling J’ between the layers, one expects that there will hc a true phase transition, wtth 3D longrange order below the N6el temperature T, whose value is given approximately by the criterion ([/a)‘J’ = T,, where 5 is the value of the tempcraturc-dependent correlation length given by cq. (I) for the pure 2D system [Xl. At tempcraturcs sufficiently far above T,
6. I. Halperin / Quantum untiferromagnets in one and two dimensions
such that ([/a)‘J’ < T, the spin correlation should be well described by the 2D Heisenberg model. (Analogous reasoning applies if deviations from the isotropic Heisenberg spin symmetry are responsible for the onset of long-range order.) The theoretical results described above for the case S = $ have been compared with quasi-elastic and inelastic neutron scattering experiments on La,CuO, and Cu(DCO,), .4D,O, and the results have been judged quite satisfactory [14-161. Also, Birgcneau [17] has reanalyzed twenty-year-old quasi-elastic scattering data for KzNiFd, and found that they are in excellent agreement with the predictions for S = 1. Other properties of quantum antifcrromagnets which have been cxplorcd theoretically and experimentally include Raman scattering, ESR and NMR [ 181.
3. The S = 1 antiferromagnetic
chain
In contrast to the 2D case, a spin-wave analysis suggests that the ID quantum Heisenberg antiferromagnet cannot have long-range order even at T= 0. It has long been known that for the case of S = $ with ncarcst-neighbor coupling, the groundstate has spin correlations which fall off as a power law with distance, and the excitation spectrum has no gap at low energies. In the early 1980s Haldane suggested that similar rcsuits apply for any half-integer spin, but not for integer spins [19]. In the integer case there should be an energy gap for excitations, and the groundstate should have spin-correlations which fall off exponentially with distance, similar to the classical 2D Heisenberg model at finite temperatures. In the last few years, theoretical and experimental developments have given strong evidence for the validity of Haldane’s ideas, and have filled in quantitative details for the integer case of greatest interest, viz. S = 1. The most recent developments have included theoretical predictions of the appearance of S = 4 spin degrees of freedom at the ends of a finite S = 1 chain, and the experimental verification of this via ESR measurements in doped samples of the compound NENP. It is convenient to consider a more general Hamiltonian of the form [I] A?%“=
~[Js,~s,+,+pJ(s,~s,+,)z+D(s:)z]. (8)
The ordinary Heisenberg model has p = D = 0. A nonzero value of the parameter D introduces uniaxial single ion anisotropy, which breaks the Heisenberg symmetry; it is necessary to include a term of this type when comparing results with actual experimental systems. The parameter p is probably not important in real materials, but it is useful as a conceptual generalization which preserves the Heisenberg symmetry.
763
A groundstate with a Haldane gap is expected for a range of values of the parameters p and D/J which includes the Heisenberg point. The lowest lying excitation modes arc spin waves near the boundary of the Brillouin zone, whose spectrum may bc written as [lY-211 E,,(k)
= (c’k’
+ E;,
(9)
where k is the wave vector measured from the zone boundary rr/a, the paramctcr (Y takes on the values f 1 or 0, corresponding to the value of S, for the excitation, the constant c may be interpreted as a spin wave velocity, and E,,, is the energy gap for mode cy. We may define the Haldane gap A as the minimum of length 5 is then the gaps E,,,, and the correlation given by .$ = c/A. In the isotropic cast Es,_ is independent of N, so that the three excitations branches are degenerate. For D f 0, the excitations arc split, according to the approximate formula E g.,, = 4
+ YD,
(11)
where A,, is the energy gap at D = 0, and y is a numerical factor of order unity. Note that the minimum gap A is decreased by the anisotropy, regardless of its sign. For the Hcisenberg model (p = D = 01, numerical studies of finite S = 1 chains lead to the estimates 3 ,, = 0.41 J, and 5 = 7a [20,21]. A simple intuitive interpretation of the Haldane gap phenomenon is given by the valence bond solid model (VBS) of Aflleck, Kennedy, Lieb and Tasaki [l]. They use a representation in which there are two S = $ variables at each lattice site, and the requirement that there be a total spin S = 1 at each site is achieved by requiring that the wave function bc symmetric under the interchange of the two spin variables at any site. The VBS state, which is the exact groundstate of eq. (8) for p = + and D = 0, is formed by taking a symmetric linear superposition of states where one of the two S = $ variables at each site is paired in a spin-singled state (valence bond) with one of the two S = t variables at the following site. For the Heisenberg model, with p = 0, the VBS state is only an approximation to the groundstatc, but we may expect that it has the correct qualitative features. The exact groundstate should be obtainable as an adiabatic perturbation from the VBS state, as long as the energy gap to the lowest excited state dots not vanish at any point in the interval f 2 p 2 0. Using the VBS ideas, it is easy to XC that there should be a free S = 4 variable at each end of a long but finite S = 1 chain [21,22]. These variables are strictly localized at the ends of the chain for the case p = 4
764
B.I. Halperin
/ Quantum
antiferromagnets
where the VBS state is exact. For j3 < i, the free S = 4 variable may hop virtually to other lattice sites, within a distance of order .$ of the end, but it cannot escape from this region as long as the gap remains open and { is much smaller than the chain length. For a chain of finite length L, the spin variables at the two ends interact with eachother by virtual exchange of spin waves. This leads to an interaction proportional to (- l)Le-‘./’ for large L. The four almost degenerate groundstates are thus weakly split, into states which may be characterized in the isotropic case as a singlet and a triplet, but the magnitude of the splitting is negligible when the chain length is sufficiently large. If the anisotropy parameter D in eq. (8) is positive and sufficiently large, the groundstate will be (approximately) the state with eigenvalue S,* = 0 on every site. In that case there will be an energy gap of order D for all excited states, and there will be no low-lying S = i degrees of freedom at the chain ends. This shows that the Haldane-gap state is fundamentally different than the state with large anisotropy. The best studied physical example of an S = 1 antiferromagnetic chain system is the compound Ni abbreviated NENP. Meas(C2H8N,)2N0,(C10,), urements of the susceptibility and the high-field magnetization, and neutron studies of the spin-wave spectrum, show that there is no long-range antiferromagnetic order down to the lowest measured temperatures, and that the spin system exhibits all the properties expected for a set of weakly-coupled Haldane chains [23.24]. The data has been fit using a coupling constant J = 47.5 K, and an anisotropy constant D = 10 K. The Haldane gap measured for the lower branch of the spin-wave spectrum is d = E,,, , = 14 K, while the upper branch has a gap E,,, = 30 K. The weighted averwith the age value of = 19 K is in good agreement calculated value of the Haldane gap for an isotropic spin chain with J = 47.5 K. Experimental demonstration of the existence of S = i spins at the ends of a finite chain was first provided by the ESR experiments of Hagiwara et al. 1251, on NENP samples where approximately 0.7 at% of the Ni was replaced by Cu. At low temperatures they observed a set of lines in the ESR data, whose dependence on frequency and field direction could be explained by assuming that at each Cu site there is an S = 4 variable s associated with the Cu*+ ion, which is weakly coupled to two of the S = i variables S, and S, which appear because the Ni chain has been cut. The coupling has the form zctr=2s.K.(S,
+S,),
(12)
where the tensor K is mildly anisotropic and very weak compared to the coupling J between Ni spins in NENP. More recent measurements by the same group, on
in one and two dimensions
samples with = 0.1 at% Cu, which will be reported elsewhere at this conference, have seen additional splitting of the ESR lines due to the hyperfine interaction with the Cu nuclei, giving additional confirmation to this interpretation [26]. The principal values of the tensor K obtained in these experiments are K, = 0.96 K, for the direction parallel to the chain axis, and K, = 1.19 K,, = 1.14 K, in the perpendicular plane 1261. We note that the ESR experiments cannot be readily interpreted in terms of a copper S = i variable coupled to a single S = 1 spin localized on a Ni site on one or the other side of the Cu. In such a case, the motion of the Ni spin would be dominated by the large single-ion anisotropy, and the ESR spectrum would be greatly altered. Further confirmation of these ideas was provided by Glarum et al. [27], who studied NENP samples in which small concentrations of the nonmagnetic divalent ions Zn, or Hg were substituted for Ni. They observed ESR signals characteristic of uncoupled S = 4 spins, as one would expect at the ends of the broken Ni chains in this situation. The ESR signals in the measurements of Hagiwara et al. and Glarum et al. are observed to drop rapidly with increasing temperature, at temperatures which are still well below the value of the Haldane gap. This can be explained by the fact that thermally excited spin-wave excitations are highly mobile, and even a very small density of such excitations on a long chain segment should be sufficient to wipe out the narrow resonance lines from the S = 4 variables at the ends
[281. The author is grateful for very helpful discussions with I. Affleck, S. Chakravarty, K. Katsumata, D.R. Nelson, S. Ty;, P. Mitra, R. Birgeneau and S. Geschwind. This work was supported in part by NSF Grant DMRXS-17291. References [I] I. Affleck, T. Kennedy, [2] [3] [4] [5] [6]
[J]
[x]
E.H. Lieb and H. Tasaki, Commun. Math. Phys. 115 (1988) 417. J. Neves and F. Perez, Phys. Lett. Al44 (1986) 331. T. Kennedy, E.H. Lieb and B.S. Shastry, J. Stat. Phys. 53 (1988) 1019. R.R.P. Singh and D.A. Huse, Phys. Rev. B 40 (1989) 1247. G.E. Castilla and S. Chakravarty, preprint. X.G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413. V. Kalmeyer and R.B. Laughlin, Phys. Rev. Lett. 59 (1987) 2095. Cf. N. Read and S. Sachdev, Phys. Rev. Lett. 62 (1989) 1694. P. Chandra, P. Coleman, and A. Larkin, Phys. Rev. Lett. 64 (1990) 88. A. Angelucci, preprint. S. Chakravarty, B. Halperin and D. Nelson. Phys. Rev. B 39 (1989) 2344.
B.I. Halperin
/ Quantum
antiferromagnets
191 P. Hasenfratz and F. Niedermayer, preprint. [lo] S. TyE, B.I. Halperin and S. Chakravarty, Phys. Rev. Lett. 62 (19891 835. [ll] S. Ty: and B.I. Halperin, Phys. Rev. B 42 (1990) 2096. [12] D.R. Grempel, Phys. Rev. Lett. 61 (1988) 1041. [13] H.-Q. Ding and M. S. Makivic, Phys. Rev. Lett. 64 (1990) 1449. [14] E. Manousakis, Rev. Mod. Phys. 63 (1991) 1. [15] K. Yamada et al., Phys. Rev. B 40 (1989) 4557. [16] S.J. Clarke, A. Harrison, T.E. Mason, G.J. McIntyre and D. Visser, preprint. [17] R. Birgeneau, Phys. Rev. B 41 (1990) 2514. [18] Cf. R.R.P. Singh et al., Phys. Rev. Lett. 62 (1989) 2736. S. Chakravarty and R. Orbach, Phys. Rev. Lett. 64 (1990) 224. 1191 F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [20] M. Takahashi, Phys. Rev. Lett. 62 (1989) 2313. G. Gomez-Santos, J. Appl. Phys. 67 (1990) 5610. I. Affleck,
[21] [22] [23] [24] [25] 1261
[27] 1281
in one and two dimensions
765
Phys. Rev. B 43 (1991) 3215. R. Botet, R. Julien and M. Kolb, Phys. Rev. B 28 (1983) 3914. T. Kennedy, J. Phys.: Condens. Mater. 2 (1990) 5737. I. Affleck and B.I. Halperin, unpublished. J.P. Renard et al., Europhys. Lett. 3 (1987) 945. J. Appl. Phys. 63 (1988) 3538. K. Katsumata et al., Phys. Rev. Lett. 63 (1989) 86. M. Hagiwara, K. Katsumata, I. Affleck, B.I. Halperin and J.P. Renard, Phys. Rev. Lett. 65 (1990) 3181. M. Hagiwara, K. Katsumata, J.P. Renard, I. Affleck and B.I. Halperin, J. Magn. Magn. Mater. 104-107 (1992) 839. S.H. Glarum, S. Geschwind, K.M. Lee, M.L. Kaplan and J. Michel, Phys. Rev. Lett. 67 (1991) 1614. P. Mitra, B.I. Halperin and I. Affleck, preprint, submitted to Phys. Rev. B.
and Magnetic
Materials
104-107
(1992) 761-765
Invited paper
Quantum Bertrand
antiferromagnets
in one and two dimensions
I. Halperin
Physics Department, Harcard Unicersity, Cambridge, MA 02138, USA
We review some theoretical developments of the last few years on the low-temperature behavior of quantum Heisenberg anfiferromagnets in one and two dimensions. For simple two-dimensional systems without frustration, it is now clear that the ground state has Ntel order, for any value of the quantum spin. Theoretical predictions for the behavior as T --) 0 of the diverging correlation length, the spin-wave frequencies and damping rates, and the form of the static and dynamic spin-correlation functions, have been obtained from a combination of renormalization group and scaling analyses, spin-wave hydrodynamics, and computer simulations of classical systems. Results are in good agreement with neutron experiments on La,CuO, and K,NiF,. For the S = 1 antiferromagnetic chain, with a Haldane gap, a recent development is the realization that there should be two effective S = $ degrees of freedom localized at the ends of a finite chain. These have been observed in recent spin resonance experiments on NENP with dilute concentrations of Cu or Zn substituted for the Ni.
1. Introduction In recent years, there has been much interest in quantum antiferromagnets, particularly one- and twodimensional systems with Heisenberg coupling between spins. The 2D problem, particularly the spin-i system on a square lattice, with nearest neighbor coupling, has received particular attention, beginning in 1987, because of its relevance to the spin correlations in the copper-oxide layers in the insulating parent compounds of the new high-temperature superconductors. Interest in 1D spin systems has been stimulated by the mathematical connections that exist between such systems and problems in conformal field theory, as well as by new experimental developments in quasi-1D spin systems, and by the hope that belter understanding of 1D systems might also lead to insights into exotic 2D systems and high-temperature superconductivity. I will review here two theoretical problems which have direct relevance to experimental systems: the behavior of a 2D Heisenberg antiferromagnet at low temperatures, and the behavior of a spin-l chain with free ends. 2. Two-dimensional
Heisenberg
antiferromagnets
According to the rigorous Hohenberg-MerminWagner theorem, a 2D spin system with Heisenberg symmetry (i.e., invariance under the SU(2) group of rotations in spin space) and finite-range coupling between spins cannot have long-range antiferromagnetic order at any temperature different from zero. However, at zero temperature long-range order is not forbidden by the theorem. In fact, long-range order has been rigorously established for spin S 2 1, on a square lattice with nearest-neighbor coupling [1,2]. For the case of S = i, there is no rigorous proof, but there are strong theoretical arguments that long-range order ex0312~8853/92/$05.00
0 1992 - Elsevier
Science
Publishers
ists [3-51. Singh and Huse [4] estimated the sublattice magnetization for S = i, using a perturbation expansion with the Ising model as a starting Hamiltonian, as N,, = 0.302 + 0.007, which is 40% below the Ntel value. Similar estimates of N,, are obtained [5] from the spin-wave series expansion in S- ‘. It is known from the spin-wave theory that N,, can be further decreased if competing interactions are added to the model, such as antiferromagnetic coupling between second-neighbor spins. It has been suggested that for suitable choices of interaction constants it is possible to obtain a groundstate where there is neither broken rotational symmetry in spin-space nor broken translational symmetry in real space. One class of possible states, the chiral spin liquid states have broken time-reversal symmetry and broken reflection symmetry, leading to non-zero expectation values of the form (S, CS, x S,)) where i, j and k are three nearby lattice sites that are noncollinear [6]. Two-spin correlations fall off exponentially with separation in these states. There are good arguments that a chiral spin liquid state can occur as the groundstate of a generalized Heisenberg Hamiltonian, with carefully chosen interaction strengths, but it is not at all clear that such a state can occur for a realistic Hamiltonian, with say just two or three short-range competing interactions. Competing states which have been proposed for frustrated S = i systems include helical spin orderings, and dimerized states, which are rotationally invariant spin singlets but have broken translational symmetry [7]. For a 2D quantum Heisenberg model that has a groundstate with conventional antiferromagnetic order at T= 0, the most interesting theoretical questions concern the behavior at T # 0, where long-range order is forbidden. According to our current theoretical understanding, the two-point correlation function for the
B.V. All rights reserved
staggcrcd spin should decay cxponcntially at large scparations. with a decay length [ that may hc written in the following form at low temperatures [8]: A,(hc/27V,) [=
“XP(2TPJT)
F+Q/2Tip,)
TO(T/27;&)3]
(1)
.
whcrc A, and A; arc universal constants. c is the zero-tcmperaturc spin-wave velocity, and p, is the zero-temperature spin stiffness constant, defined such that a gradient in the direction R of the antiferromagnctic Ao
J’(k[),
(2)
B,5
S( k = 0) = [(2np,/T)
‘N,f
+ L?: + O( T/2lTp,)]?
’
(j)
where k is the wavevector, measured from the antiferromagnetic Bragg peak, f is a function of k<. and B,$ and B.4 arc dimensionless constants, while N,, is the quantum-rcnormalizcd staggered magnetization at T = 0. The function .f’ and the constant B,$ should he universal and arc the same for the quantum and classical models. Fits to Monte Carlo data on classical models give B,S = I25 to 180, with B,;, = I. An approximate form for the scaling function ,f(s), which has the correct limits for x + 0 and x + 3~. is [IO]
where (1 = k.$ and v = w/W,,. (The scaling form applies for T + 0, with q, v fixed.) The scaling function 4 is again universal, and identical for the quantum and classical models. A combination of renormalization group analysis, spin-wave hydrodynamics, and spinwave scattering theory suggests that for large k[, Sk, co) has a well-defined spin-wave peak with maximum at frequency We - Wll,, [+ln(l + y’)]“’ and a width 11 of the approximate form [I I] ‘h -hwL
7i 2 In[cS + l/2 2
[8+
ln( 1 fq’)] 1/2In(l
+ ln(2n)
+y’)]’
’
(7)
with 6 = 1.7. (Note that the value of oh is shifted significantly from the T = 0 value WY= ck, and the ratio I’Joi decrcascs only logarithmically with increasing k[. For kt --j 0, WC expect an overdamped relaxation of the staggered magnetization, so that S( k. w) has a peak centcrcd at w = 0, with a width y,,W,,. A fit to simulations of the classical model [IO] gave the value y,, = 0.85 i 0.15, which is close to an cstimatc ot y. = 0.96 by Grcmpel [12], hascd on a coupled-mode approximation. An approximate analytic form for 4(rl, V) has been fit to the simulations over the cntirc range of intcrcst of r, [ IO,1 I]. For the nearest-neighbor Hcisenberg model, the zero-tempcraturc constants 0, and c may be rclatcd to the coupling constant J and the lattice constant (I hy means of spin-wave theory or other approximate methods [Xl. Singh and Huse [4] studied the S = k model using series cxpanaions and found he = l.h7Ju, and of i2% and (27TTTpJ~K)= O.h8N ’ with uncertainties * 0% respectively. If we now write the pre-exponential factor .4,(hc/2~p,), which appears in the numerator of eq. (I) in the form (‘,a. the ahovc results imply that C, = 0.50 + 0.03 for S = +. For S = 1, the spin-wave expansion gives a value C i = 0.19. with an estimated uncertainty of order 5%. The theoretical results prcscnted above for the correlation length and S(k = 0) at low temperatures. have hcen compared with Monte Carlo calculations on the S = i antiferromagnct [ 13,141 which arc most accurate at high tempcraturcs. The agrecmcnt is satisfactory. and rcasonahlc interpolations, valid over the entire tcmpcraturc range, are therefore poasihle.
1 + 2~rB,~ ’ ln( I +x’) f(X)
=
(4)
I + x?
The dynamic structure factor S(k, w) is predicted to have the dynamic scaling form [8] S(k.
w) =W,j ‘S(k)c$(q,
o,, = c<- ‘( T/hp,)‘?
I’),
(5) (6)
For a 3D stack of Heiscnberg layers, with a weak coupling J’ between the layers, one expects that there will hc a true phase transition, wtth 3D longrange order below the N6el temperature T, whose value is given approximately by the criterion ([/a)‘J’ = T,, where 5 is the value of the tempcraturc-dependent correlation length given by cq. (I) for the pure 2D system [Xl. At tempcraturcs sufficiently far above T,
6. I. Halperin / Quantum untiferromagnets in one and two dimensions
such that ([/a)‘J’ < T, the spin correlation should be well described by the 2D Heisenberg model. (Analogous reasoning applies if deviations from the isotropic Heisenberg spin symmetry are responsible for the onset of long-range order.) The theoretical results described above for the case S = $ have been compared with quasi-elastic and inelastic neutron scattering experiments on La,CuO, and Cu(DCO,), .4D,O, and the results have been judged quite satisfactory [14-161. Also, Birgcneau [17] has reanalyzed twenty-year-old quasi-elastic scattering data for KzNiFd, and found that they are in excellent agreement with the predictions for S = 1. Other properties of quantum antifcrromagnets which have been cxplorcd theoretically and experimentally include Raman scattering, ESR and NMR [ 181.
3. The S = 1 antiferromagnetic
chain
In contrast to the 2D case, a spin-wave analysis suggests that the ID quantum Heisenberg antiferromagnet cannot have long-range order even at T= 0. It has long been known that for the case of S = $ with ncarcst-neighbor coupling, the groundstate has spin correlations which fall off as a power law with distance, and the excitation spectrum has no gap at low energies. In the early 1980s Haldane suggested that similar rcsuits apply for any half-integer spin, but not for integer spins [19]. In the integer case there should be an energy gap for excitations, and the groundstate should have spin-correlations which fall off exponentially with distance, similar to the classical 2D Heisenberg model at finite temperatures. In the last few years, theoretical and experimental developments have given strong evidence for the validity of Haldane’s ideas, and have filled in quantitative details for the integer case of greatest interest, viz. S = 1. The most recent developments have included theoretical predictions of the appearance of S = 4 spin degrees of freedom at the ends of a finite S = 1 chain, and the experimental verification of this via ESR measurements in doped samples of the compound NENP. It is convenient to consider a more general Hamiltonian of the form [I] A?%“=
~[Js,~s,+,+pJ(s,~s,+,)z+D(s:)z]. (8)
The ordinary Heisenberg model has p = D = 0. A nonzero value of the parameter D introduces uniaxial single ion anisotropy, which breaks the Heisenberg symmetry; it is necessary to include a term of this type when comparing results with actual experimental systems. The parameter p is probably not important in real materials, but it is useful as a conceptual generalization which preserves the Heisenberg symmetry.
763
A groundstate with a Haldane gap is expected for a range of values of the parameters p and D/J which includes the Heisenberg point. The lowest lying excitation modes arc spin waves near the boundary of the Brillouin zone, whose spectrum may bc written as [lY-211 E,,(k)
= (c’k’
+ E;,
(9)
where k is the wave vector measured from the zone boundary rr/a, the paramctcr (Y takes on the values f 1 or 0, corresponding to the value of S, for the excitation, the constant c may be interpreted as a spin wave velocity, and E,,, is the energy gap for mode cy. We may define the Haldane gap A as the minimum of length 5 is then the gaps E,,,, and the correlation given by .$ = c/A. In the isotropic cast Es,_ is independent of N, so that the three excitations branches are degenerate. For D f 0, the excitations arc split, according to the approximate formula E g.,, = 4
+ YD,
(11)
where A,, is the energy gap at D = 0, and y is a numerical factor of order unity. Note that the minimum gap A is decreased by the anisotropy, regardless of its sign. For the Hcisenberg model (p = D = 01, numerical studies of finite S = 1 chains lead to the estimates 3 ,, = 0.41 J, and 5 = 7a [20,21]. A simple intuitive interpretation of the Haldane gap phenomenon is given by the valence bond solid model (VBS) of Aflleck, Kennedy, Lieb and Tasaki [l]. They use a representation in which there are two S = $ variables at each lattice site, and the requirement that there be a total spin S = 1 at each site is achieved by requiring that the wave function bc symmetric under the interchange of the two spin variables at any site. The VBS state, which is the exact groundstate of eq. (8) for p = + and D = 0, is formed by taking a symmetric linear superposition of states where one of the two S = $ variables at each site is paired in a spin-singled state (valence bond) with one of the two S = t variables at the following site. For the Heisenberg model, with p = 0, the VBS state is only an approximation to the groundstatc, but we may expect that it has the correct qualitative features. The exact groundstate should be obtainable as an adiabatic perturbation from the VBS state, as long as the energy gap to the lowest excited state dots not vanish at any point in the interval f 2 p 2 0. Using the VBS ideas, it is easy to XC that there should be a free S = 4 variable at each end of a long but finite S = 1 chain [21,22]. These variables are strictly localized at the ends of the chain for the case p = 4
764
B.I. Halperin
/ Quantum
antiferromagnets
where the VBS state is exact. For j3 < i, the free S = 4 variable may hop virtually to other lattice sites, within a distance of order .$ of the end, but it cannot escape from this region as long as the gap remains open and { is much smaller than the chain length. For a chain of finite length L, the spin variables at the two ends interact with eachother by virtual exchange of spin waves. This leads to an interaction proportional to (- l)Le-‘./’ for large L. The four almost degenerate groundstates are thus weakly split, into states which may be characterized in the isotropic case as a singlet and a triplet, but the magnitude of the splitting is negligible when the chain length is sufficiently large. If the anisotropy parameter D in eq. (8) is positive and sufficiently large, the groundstate will be (approximately) the state with eigenvalue S,* = 0 on every site. In that case there will be an energy gap of order D for all excited states, and there will be no low-lying S = i degrees of freedom at the chain ends. This shows that the Haldane-gap state is fundamentally different than the state with large anisotropy. The best studied physical example of an S = 1 antiferromagnetic chain system is the compound Ni abbreviated NENP. Meas(C2H8N,)2N0,(C10,), urements of the susceptibility and the high-field magnetization, and neutron studies of the spin-wave spectrum, show that there is no long-range antiferromagnetic order down to the lowest measured temperatures, and that the spin system exhibits all the properties expected for a set of weakly-coupled Haldane chains [23.24]. The data has been fit using a coupling constant J = 47.5 K, and an anisotropy constant D = 10 K. The Haldane gap measured for the lower branch of the spin-wave spectrum is d = E,,, , = 14 K, while the upper branch has a gap E,,, = 30 K. The weighted averwith the age value of = 19 K is in good agreement calculated value of the Haldane gap for an isotropic spin chain with J = 47.5 K. Experimental demonstration of the existence of S = i spins at the ends of a finite chain was first provided by the ESR experiments of Hagiwara et al. 1251, on NENP samples where approximately 0.7 at% of the Ni was replaced by Cu. At low temperatures they observed a set of lines in the ESR data, whose dependence on frequency and field direction could be explained by assuming that at each Cu site there is an S = 4 variable s associated with the Cu*+ ion, which is weakly coupled to two of the S = i variables S, and S, which appear because the Ni chain has been cut. The coupling has the form zctr=2s.K.(S,
+S,),
(12)
where the tensor K is mildly anisotropic and very weak compared to the coupling J between Ni spins in NENP. More recent measurements by the same group, on
in one and two dimensions
samples with = 0.1 at% Cu, which will be reported elsewhere at this conference, have seen additional splitting of the ESR lines due to the hyperfine interaction with the Cu nuclei, giving additional confirmation to this interpretation [26]. The principal values of the tensor K obtained in these experiments are K, = 0.96 K, for the direction parallel to the chain axis, and K, = 1.19 K,, = 1.14 K, in the perpendicular plane 1261. We note that the ESR experiments cannot be readily interpreted in terms of a copper S = i variable coupled to a single S = 1 spin localized on a Ni site on one or the other side of the Cu. In such a case, the motion of the Ni spin would be dominated by the large single-ion anisotropy, and the ESR spectrum would be greatly altered. Further confirmation of these ideas was provided by Glarum et al. [27], who studied NENP samples in which small concentrations of the nonmagnetic divalent ions Zn, or Hg were substituted for Ni. They observed ESR signals characteristic of uncoupled S = 4 spins, as one would expect at the ends of the broken Ni chains in this situation. The ESR signals in the measurements of Hagiwara et al. and Glarum et al. are observed to drop rapidly with increasing temperature, at temperatures which are still well below the value of the Haldane gap. This can be explained by the fact that thermally excited spin-wave excitations are highly mobile, and even a very small density of such excitations on a long chain segment should be sufficient to wipe out the narrow resonance lines from the S = 4 variables at the ends
[281. The author is grateful for very helpful discussions with I. Affleck, S. Chakravarty, K. Katsumata, D.R. Nelson, S. Ty;, P. Mitra, R. Birgeneau and S. Geschwind. This work was supported in part by NSF Grant DMRXS-17291. References [I] I. Affleck, T. Kennedy, [2] [3] [4] [5] [6]
[J]
[x]
E.H. Lieb and H. Tasaki, Commun. Math. Phys. 115 (1988) 417. J. Neves and F. Perez, Phys. Lett. Al44 (1986) 331. T. Kennedy, E.H. Lieb and B.S. Shastry, J. Stat. Phys. 53 (1988) 1019. R.R.P. Singh and D.A. Huse, Phys. Rev. B 40 (1989) 1247. G.E. Castilla and S. Chakravarty, preprint. X.G. Wen, F. Wilczek and A. Zee, Phys. Rev. B 39 (1989) 11413. V. Kalmeyer and R.B. Laughlin, Phys. Rev. Lett. 59 (1987) 2095. Cf. N. Read and S. Sachdev, Phys. Rev. Lett. 62 (1989) 1694. P. Chandra, P. Coleman, and A. Larkin, Phys. Rev. Lett. 64 (1990) 88. A. Angelucci, preprint. S. Chakravarty, B. Halperin and D. Nelson. Phys. Rev. B 39 (1989) 2344.
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/ Quantum
antiferromagnets
191 P. Hasenfratz and F. Niedermayer, preprint. [lo] S. TyE, B.I. Halperin and S. Chakravarty, Phys. Rev. Lett. 62 (19891 835. [ll] S. Ty: and B.I. Halperin, Phys. Rev. B 42 (1990) 2096. [12] D.R. Grempel, Phys. Rev. Lett. 61 (1988) 1041. [13] H.-Q. Ding and M. S. Makivic, Phys. Rev. Lett. 64 (1990) 1449. [14] E. Manousakis, Rev. Mod. Phys. 63 (1991) 1. [15] K. Yamada et al., Phys. Rev. B 40 (1989) 4557. [16] S.J. Clarke, A. Harrison, T.E. Mason, G.J. McIntyre and D. Visser, preprint. [17] R. Birgeneau, Phys. Rev. B 41 (1990) 2514. [18] Cf. R.R.P. Singh et al., Phys. Rev. Lett. 62 (1989) 2736. S. Chakravarty and R. Orbach, Phys. Rev. Lett. 64 (1990) 224. 1191 F.D.M. Haldane, Phys. Rev. Lett. 50 (1983) 1153. [20] M. Takahashi, Phys. Rev. Lett. 62 (1989) 2313. G. Gomez-Santos, J. Appl. Phys. 67 (1990) 5610. I. Affleck,
[21] [22] [23] [24] [25] 1261
[27] 1281
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Phys. Rev. B 43 (1991) 3215. R. Botet, R. Julien and M. Kolb, Phys. Rev. B 28 (1983) 3914. T. Kennedy, J. Phys.: Condens. Mater. 2 (1990) 5737. I. Affleck and B.I. Halperin, unpublished. J.P. Renard et al., Europhys. Lett. 3 (1987) 945. J. Appl. Phys. 63 (1988) 3538. K. Katsumata et al., Phys. Rev. Lett. 63 (1989) 86. M. Hagiwara, K. Katsumata, I. Affleck, B.I. Halperin and J.P. Renard, Phys. Rev. Lett. 65 (1990) 3181. M. Hagiwara, K. Katsumata, J.P. Renard, I. Affleck and B.I. Halperin, J. Magn. Magn. Mater. 104-107 (1992) 839. S.H. Glarum, S. Geschwind, K.M. Lee, M.L. Kaplan and J. Michel, Phys. Rev. Lett. 67 (1991) 1614. P. Mitra, B.I. Halperin and I. Affleck, preprint, submitted to Phys. Rev. B.