Journal ArticleDOI

# The dilute quantum Heisenberg antiferromagnet

15 Apr 1991-Journal of Applied Physics (American Institute of Physics)-Vol. 69, Iss: 8, pp 5191-5193

Abstract: Series expansions are used to treat the randomly diluted spin‐1/2 Heisenberg antiferromagnet at zero temperature A series is obtained at zero temperature in powers of the concentration for many correlation functions and for the correlation length from which static, dynamic, and crossover exponents are estimated The correlation length exponent is found to be 077±010 in two dimensions The critical concentration for the appearance of long‐range order is indistinguishable from the percolation threshold
Topics: Heisenberg model (55%), Critical phenomena (55%), Percolation threshold (52%), Absolute zero (52%)

## Summary (1 min read)

Jump to: [1 (9b a"] and [WpbJApl]

### 1 (9b a"

• Finally, the authors consider the fourth-order staggered susceptibility.
• These quantities have the asymptotic behavior.

### WpbJApl

• The aim is to determine pQ and the above critical exponents.
• Critical exponents were obtained from the series by PadC analyses or more general methods as described previously.'.
• Since the authors believe quantum randomness and classical randomness are in different universality classes, they assume the percolation fixed point to be unstable in the g direction.
• For d=2 and 3 the authors used several of the above series including those for c2 and e to get estimates for the correlation length exponent v.

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University of Pennsylvania University of Pennsylvania
ScholarlyCommons ScholarlyCommons
Department of Physics Papers Department of Physics
4-15-1991
The Dilute Quantum Heisenberg Antiferromagnet The Dilute Quantum Heisenberg Antiferromagnet
C. C. Wan
University of Pennsylvania
A. Brooks Harris
University of Pennsylvania
, harris@sas.upenn.edu
Follow this and additional works at: https://repository.upenn.edu/physics_papers
Part of the Quantum Physics Commons
Recommended Citation Recommended Citation
Wan, C., Harris, A., & Adler, J. (1991). The Dilute Quantum Heisenberg Antiferromagnet.
Journal of Applied
Physics,
69
(8), 5191-5193. http://dx.doi.org/10.1063/1.348123
This paper is posted at ScholarlyCommons. https://repository.upenn.edu/physics_papers/436

The Dilute Quantum Heisenberg Antiferromagnet The Dilute Quantum Heisenberg Antiferromagnet
Abstract Abstract
Series expansions are used to treat the randomly diluted spin1/2 Heisenberg antiferromagnet at zero
temperature. A series is obtained at zero temperature in powers of the concentration for many correlation
functions and for the correlation length from which static, dynamic, and crossover exponents are
estimated. The correlation length exponent is found to be 0.77±0.10 in two dimensions. The critical
concentration for the appearance of longrange order is indistinguishable from the percolation threshold.
Disciplines Disciplines
Physics | Quantum Physics
This journal article is available at ScholarlyCommons: https://repository.upenn.edu/physics_papers/436

The dilute quantum Heisenberg antiferromagnet
C. C. Wan and A. B. Harris
:.
Department
of
Physics, University
of
Pennsylvania, Philadelphia, Pennsylvania 19104
Department
of
Physics, Technion, Haifa 32000, Israel
Series expansions are used to treat .the randomly diluted spin-i Heisenberg antiferromagnet at
zero temperature. A series is obtained at zero temperature in powers of the concentration
for many correlation functions and for the correlation length from which static, dynamic,
I
and crossover exponents are estimated. The correlation length exponent is found to be
0.77*0.10 in two dimensions. The critical concentration for the appearance of long-range E.
order is indistinguishable from the percolation threshold.
Series expansions have often been useful in determin-
ing numerical values of critical exponents,. especially in
cases where existing theories have been unable to elucidate
the nature of the critical phenomena. Here we pursue a
program to obtain numerical data on the zero-temperature
transition that occurs as a function of concentration
p
for
the spin-i Heisenberg antiferromagnet in the presence of
quenched random dilution. The Hamiltonian for this sys-
tem is
R=J & Ei,Si*Sj,
(1)
where S is a quantum spin 4,
(i,j)
indicates thatthe sum is
over pairs of nearest neighbors on a d-dimensional hyper-
cubic lattice, and eii is a random variable which assumes
the value unity with probability
p
if the bond (i,j) is oc-
cupied and the value 0 with probability 1 -
p
if the bondis
unoccupied.
To highlight the differences between the spin-i model
and the classical model (viz., for the S-+.GC limit),Jet us
note the behavior of the staggered structure factor
C(p),
which is defined to be
C(p) =fV- 2 [ (SiSj)]~&Y~ i
iJ
(2)
where N is the total number of sites in the system, [ 1,
indicates an average over all configurations:(of the QS),
( ) denotes a zero-tetnperature average over all state(s) of
the ground manifold, and k is the wave vector associated
with the antiferromagnetic order. For a classccal system a
local gauge transformation in which one reverses the
sign of Si on one sublattice converts
C(p)
into the zero-
wave-vector correlation function for a dilute ferromagnet.
Then all spins are parallel in the groundstates and C(p)
reduces to the mean-square size of connected clusters,
which is the percolation susceptibility. Thus for a classical
system the threshold concentration at which long-range
order sets in coincides with the percolat& threshold. One
may define a critical exponent associated,with C(p) via
-
C@)-IApl -Y,
(3)
i,
)AIso at: School of Physics tid AStronbmy, Tel Aviv University, Tel
Aviv 69978, Israel.
.
where Ap =
pc
-
p
and the critical exponents reduce to
that of percolation. For the dilute quantum system Ap
=
pQ
-
p,
wherep, is the threshold concentration for the
appearance of long-range order
(pe
)
p,)
and y is not ex-
pected to be identical to the percolation value.
We can similarly define other critical exponents. For
instance, if we consider sm+nonzero temperature,
T
then
we may introduce a crossover exponent*2$~ by writing WT) -l&l -WT/]Apld). -c,= -. (4) Also we may define the correlation. length 6 and-its asso- ciated exponent Y by . - pkS c--$[ (si*Sj)]a&““‘/ x- [ (Si*Sj)l,&~ ., .: :(Sa)
.lip, - kv,
ki
(5b)
which we implemented for k= 1 and k=2. For such a
zero-temperature transition one expects3
_. _
,
,
r=(l -rlh
(6)
where n is defined by I[(Si*Sj)]J - rb- d- 7: The differ2
ence between -. Eq. (6) and the usual formula,
y = (2 - 7) Y, is due to the additional time dimension used
to describe zero-temperature quantum systems4 Likewise,
we write the singular part of the configurationally averaged
ground-state energy
E(p) as
.E(pj~+p[~-~, I
(7)
with3 (Y 4 2 - (d + z) Y, where z, the so-called dynamic ex-
ponent, is defined so thatthe characteristic energy scale is
of order 6 -:
The staggered susceptibility xs;- which is
defined to have-the dimensions of (energy) L ,,consists of
two types of terms: I+--
i
xs=WT) + (NJ),
-. (8)
where
L
I.
A=
1
2
(S,(kPS,t
-k)) ,
1
(9s)
a
cl
.~.
.*
B
?=2 2 (S,(k) (N-- 9) (2 -
E#) - S,
1
a
(,-k)) ,
1
(9b)
a
5191 J.Appl.Phys.69(8),15 April 1991
0021~8979/91/085191-03$03.00 @ 1991 American Institute of Physics 5191 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 165.123.108.243 On: Thu, 09 Jul 2015 15:23:28 where a: labels Cartesian components, x is the unit oper- ator, and 9 is a projection operator that is unity only in the ground manifold whose energy is Ea. Finally, we con- sider the fourth-order staggered susceptibility. It is given by more complicated expressions than Eqs. (8) and (9), so we confine our attention to its zero-temperature limit and write where h, is the staggered field. These quantities have the asymptotic behavior WpbJApl -y-=, (114 d(p)--ApI+-=++, (lib) The aim is to determine pQ and the above critical ex- ponents. For this purpose we constructed series expansions of these quantities in powers ofp up to orderpi3. To obtain such an expansion for any quantity X we must calculate X for all possible isolated clusters I consisting of up to 13 bonds. In this task we are aided by the Lieb-Mattis theo- rem6 which says that for any cluster the total spin Stat of the true ground state is identical to the value of S, in the NCel state. Also, when S,,#O, the sum over the degenerate ground-state manifold can be simplified using the Wigner- Eckart theorem. Then [Xl,,= 2 p(r)x(r) = ; pbrXc(r>, r (12) where P( I) is the occurrence probability of the cluster l?, nb(I) is the number of bonds in J?, and X,(F) is the cumulant value of X (which is obtained recursively by subtracting the contributions from cumulants of all subdi- agrams of r). For quantities that only depend on the to- pology (and not on the exact shape) of the cluster, the sum over I? can be restricted to topologically inequivalent dia- grams, since we have the weak embedding constants for general hypercubic lattice in d dimensions. (For the series involving the correlation length this simplification could not be made, and those results were confined to d=2 and 3.) Critical exponents were obtained from the series by PadC analyses or more general methods as described pre- viously. For instance, for d=2 the the coefficients in the series for the correlation correlation length, starting with the constant term and going up to order p, are 0.0000, 1 .oooo, 0.6667, 14.7718, - 28.9345, 396.6181, - 2507.5602, 21 322.9721, - 163 559.2768, 1276 223.5022, - 9 657 938.4516, and 71906 968.9306. From the original series, we can obtain further series either by dividing one series by another or by term by term divi- sion of two series. The latter procedure has the advantage I I that it yields a series whose divergence occurs at the known value p = 1 in contrast to the former series whose threshold remains at the unknown value pQ In principle, the first question should be to determine pp and in particular to see whetherpQ = pc. To answer this 5192 J. Appl. Phys., Vol. 69, No. 8,15 April 1991 FIG. 1. Fixed-point structure -of the randomly dilute quantum antiferro- magnet in the p-g plane, where g is the coupling constant in the nonlinear sigma model. At the dilute quantum fixed point Q, a fixed distribution of coupling of finite width (i.e., inequiv- alent to a pure system ) is maintained. This point, although on the critical surface, does not Iie in the p-g plane. question we show in Fig. 1 the phase diagram we expect for this system. There we incorporate the result of Chakra- varty, Halperin, and Nelson that there is a critical value for the coupling constant in the pure system for the non- linear sigma model. Also it seems certain that the condi- tion for randomness to be relevant,3 viz., a: + v > 0, holds so that the pure system fixed point is unstable to dilution. Since we believe quantum randomness and classical ran- domness are in different universality classes, we assume the percolation fixed point to be unstable in the g direction. Therefore there is a critical surface, whose projection in the g-p plane we show in Fig. 1, that surrounds the Ntel or- dered phase. The quantum random fixed point does not lie in the p-g plane because it requires a nontrivial fixed dis- tribution of coupling constants. This picture indicates that the critical concentration should in fact depend on l/S. However, our series were not long enough and well enough behaved to detect a nonzero value of Sp =pQ - pc. It might be possible to show that t3pfO by considering the high-dimensionality limit. Although we expect Ap to decrease with increasing dimension,i the high- dimensionality limit is often quite simple for classical mod- els, because there loopless, or tree, clusters dominate. In fact, for most classical models the two-point correlation function for high d is given in terms of chain diagrams. The case of lattice animals is a notable exception in which loop- less diagrams with all numbers of free ends are important12 in high d. In the present case we find that in the high- dimensionality limit our series have important contribu- tions from all loopless diagrams. In fact, most of our series alternate in sign, indicating the importance of an unphys- ical singularity at small negative p. Similar behavior was found previously3 for quantum percolation. For example, in the limit of high dimension the series for C(p) becomes a power series in the variable dp with,coefficients, starting with the constant term, which are 0.2500, 0.5000, - 0.3333, 1.7299, - 4.6647, 23.1059, - 115.3064, 664.3666, - 3935.6508, ii? 961.6815, - 147 319.3240, 912 274.4025, - 5 675 735.2061, and 35 459 841.9534. To summarize: Quantum fluctuations, like lattice animals, have interesting behavior in high dimension. Finally, we discuss the numerical values we found for the critical exponents. Since our analyses were not very sensitive to the exact choice ofpQ, we set pQ = pc In most cases the results given in Table I represent several indepen- dent methods of analysis and the error bars reflect the degree of consistency between different methods. The series for E(p) could not be analyzed, as is often the case for Wan, Harris, and Adler 5192 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 165.123.108.243 On: Thu, 09 Jul 2015 15:23:28 TABLE I. Critical exponents for quantum dilution. d=2 d=3 d=8 V 0.77*0.1 0.575 *0.05 . . . ZV 1.30*0.1 1.30*0.1 1.33 =ko.o3 4 1.6hO.3 ..I 1.5hO.05 4-d 0.25 f 0.2 0.25 kO.2 0.10~0.1 Y 0.70*0.1 0.35*0.1 0.19*0.01 The entries for 4 - EY are values obtained from the series A(p)/C(p). specific-heat series. For d=2 and 3 we used several of the above series including those for c2 and e to get estimates for the correlation length exponent v. The other exponents could then be determined from the series of Eq. ( 11) . We expected to find$=zv because zv is the exponent associ-
ated with the time axis and #J is associated with the equiv-
alent temperature axis. However, as can be seen from Ta-
ble I, we did not quite get this result. Perhaps this is an
indication of the reliability of our results. The unusually
small values of y are a consequence of Eq. (6)) which is a
unique reflection of quantum critical phenomena. Perhaps
the most important result in Table I is that v is very defi-
nitely much less than for percolation, where Y = $in d=2. The fact that v is different from its percolation value proves that the quantum antiferromagnet at percolation is not de- scribed at the percolation fixed point as the classical system is.112 Or in other words, at the percolation fixed point l/S is relevant. Although v is not too different from the non- linear sigma model result, (v-,0.7), our value of z that is greater than unity indicates that the nonlinear sigma model fixed point is indeed unstable to randomness. So although we have not shown that Sp is nonzero, our results are 5193 J. Appl. Phys., Vol. 69, No. 8,15 April 1991 otherwise consistent with Fig. 1. It is obvious that there is much to be understood, although our results are the first to indicate rough values for the critical exponents of this quantum dilution fixed point. This work was supported in part by the National Sci- ence Foundation (NSF) under Grant No. DMR-88-15469 and by a grant from the U.S.-Israel Binational Science Foundation. One of us (J. A.) acknowledges the support of the Israel Academy of Science. We thank B. I. Halperin and T. C. Lubensky for helpful discussions concerning scaling relations. We thank A. Aharony for helpful sugges- tions. We thank R. Fisch for some helpful comments on the location of po. The computations were done on a Star- dent Titan 3000 computer supported by the NSF MRL program under Grant No. DMR-88-19885. M. J. Stephen and G. S. Grest, Phys. Rev. Lett. 38, 567 (1977). A. B. Harris and T. C. Lubensky, Phys. Rev. B 35, 6964 (1987). ID. Boyanovsky and I. Cardy, Phys. Rev. 26, 154 (1982); D. Boy- anovsky and J. Cardy, Phys. Rev. 27, 6971 (1983). 4J. Hertz, Phys. Rev. B 14, 1165 (1976). J. H. Van Vleck, Theory of Electric and Magnetic Susceptibilities (Ox- ford University Press, Oxford, 1927). E. H. Lieb and D. C. Mattis, Ann. Phys. (NY) 16,407 (1961); E. H. Lieb and D. C. Mattis, J. Math. Phys. 3. 749 (1962). See M. E. Rose, Elementary Theory of Angular Momentum (Wiley, New York, 1957), p. 85. L. Klein, J. Adler, A. Aharony, A. B. Harris, and Y. Meir, Phys. Rev. B 40, 4824 (1989). D. L. Hunter and G. A. Baker, Jr., Phys. Rev: B 7, 3346 (1973); Y. Meir, J. Phys. A 20, L349 (1987). OS Chakravarty, B. I. Halperin, and D. R. Nelson, Phys. Rev. 39, 2344 (1989). R Raghavan and D. C. Mattis, Phys. Rev. B 23, 4791 (1981); Y. Shapir, A. Aharony, and A. B. Harris, Phys. Rev. Lett. 49,486 (1982). A. B. Harris, Phys. Rev. B 26, 337 (1982). I3 A. B. Harris, Phys. Rev. B 29, 2519 (1984), and Appendix A therein. Wan, Harris, and Adler 5193 [This article is copyrighted as indicated in the article. Reuse of AIP content is subject to the terms at: http://scitation.aip.org/termsconditions. Downloaded to ] IP: 165.123.108.243 On: Thu, 09 Jul 2015 15:23:28 ##### Citations More filters Journal ArticleDOI Owen Vajk, P. K. Mang1, Martin Greven1, Peter M. Gehring2 +1 moreInstitutions (2) 01 Mar 2002-Science TL;DR: Complementary neutron scattering and numerical experiments demonstrate that the spin-diluted Heisenberg antiferromagnet La2Cu1–z (Zn,Mg)zO4 is an excellent model material for square-lattice site percolation in the extreme quantum limit of spin one-half. Abstract: The study of randomness in low-dimensional quantum antiferromagnets is at the forefront of research in the field of strongly correlated electron systems, yet there have been relatively few experimental model systems. Complementary neutron scattering and numerical experiments demonstrate that the spin-diluted Heisenberg antiferromagnet La2Cu1– z (Zn,Mg) z O4is an excellent model material for square-lattice site percolation in the extreme quantum limit of spin one-half. Measurements of the ordered moment and spin correlations provide important quantitative information for tests of theories for this complex quantum-impurity problem. 65 citations Journal ArticleDOI Daniel Daboul1, Iksoo Chang2, Amnon Aharony1Institutions (2) Abstract: We calculate high-temperature graph expansions for the Ising spin glass model with 4 symmetric random distribution functions for its nearest neighbor interaction constants J ij . Series for the Edwards-Anderson susceptibility$\chi_{\mbox{}_\mathrm{EA}}$are obtained to order 13 in the expansion variable (J/(k B T))2 for the general d-dimensional hyper-cubic lattice, where the parameter J determines the width of the distributions. We explain in detail how the expansions are calculated. The analysis, using the Dlog-Pade approximation and the techniques known as M1 and M2, leads to estimates for the critical threshold (J/(k B T c ))2 and for the critical exponent$\gamma$in dimensions 4, 5, 7 and 8 for all the distribution functions. In each dimension the values for$\gamma$agree, within their uncertainty margins, with a common value for the different distributions, thus confirming universality. 21 citations Journal ArticleDOI Abstract: We generated/extended low concentration series for the second moment of distances between cluster points A 2 in site and bond percolation on the square lattice and used them to calculate series for the correlation length . We analysed these and other series for their critical amplitudes, and compared them with results from Monte Carlo simulations of triangular site, square site, and square bond percolation. The values obtained from the different models for the amplitude combination B 2 0 2 / 2 ( 0 , 2 , and B are the amplitudes of , the second moment of the cluster size distribution M 2 , and the strength of the infinite cluster P , respectively), are all within the range 2.23±0.10, confirming universality. 9 citations Journal ArticleDOI Chitoshi Yasuda1, Akihide Oguchi1Institutions (1) Abstract: The spin-wave theory and the coherent potential approximation are applied to a spin S Heisenberg antiferromagnet with nonmagnetic impurities on square lattice. The impurity effects are taken into account by substituting S (1- x ) for S and using the coherent potential approximation to the exchange interaction, where x is the impurity concentration. At T =0 for S =1/2 the critical impurity concentration x c of the Neel state is 0.303 and the percolation threshold x p is 0.500. The ground state in x c < x < x p is the disordered state with the spin gap. For S ≥1 the long range Neel order vanishes at x p =0.500. These results explain qualitatively the experimental results of La 2 Cu 1- x Mg x O 4 ( S =1/2) and K 2 Mn 1- x Mg x F 4 ( S =5/2). The difference of x c between these materials is caused by the decrease in the magnitude of the effective spin with impurity doping. The spin gap is expected to be observed for La 2 Cu 1- x Mg x O 4 in x c < x < x p at low temperatures. 5 citations Posted Content U. Kanbur1, Erol Vatansever1, Hamza Polat1Institutions (1) Abstract: The three-dimensional quenched random bond diluted$(J_1-J_2)$quantum Heisenberg antiferromagnet is studied on a simple-cubic lattice. Using extensive stochastic series expansion quantum Monte Carlo simulations, we perform very long runs for$L \times L \times L$lattice up to$L=48$. By employing standard finite-size scaling method, the numerical values of the Neel temperature are determined with high precision as a function of the coupling ratio$r=J_2/J_1$. Based on the estimated critical exponents, we find that the critical behavior of the considered model belongs to the pure classical$3DO(3)$Heisenberg universality class. 3 citations ##### References More filters Book 01 Dec 1957 3,531 citations Book 01 Jan 1932 2,053 citations Journal ArticleDOI J. A. Hertz1Institutions (1) Abstract: This paper proposes an approach to the study of critical phenomena in quantum-mechanical systems at zero or low temperatures, where classical free-energy functionals of the Landau-Ginzburg-Wilson sort are not valid The functional integral transformations first proposed by Stratonovich and Hubbard allow one to construct a quantum-mechanical generalization of the Landau-Ginzburg-Wilson functional in which the order-parameter field depends on (imaginary) time as well as space Since the time variable lies in the finite interval [$0,\ensuremath{-}i\ensuremath{\beta}$], where$\ensuremath{\beta}$is the inverse temperature, the resulting description of a$d$-dimensional system shares some features with that of a ($d+1$)-dimensional classical system which has finite extent in one dimension However, the analogy is not complete, in general, since time and space do not necessarily enter the generalized free-energy functional in the same way The Wilson renormalization group is used here to investigate the critical behavior of several systems for which these generalized functionals can be constructed simply Of these, the itinerant ferromagnet is studied in greater detail The principal results of this investigation are (i) at zero temperature, in situations where the ordering is brought about by changing a coupling constant, the dimensionality which separates classical from nonclassical critical-exponent behavior is not 4, as is usually the case in classical statistics, but$4\ensuremath{-}z$dimensions, where$z$depends on the way the frequency enters the generalized free-energy functional When it does so in the same way that the wave vector does, as happens in the case of interacting magnetic excitons, the effective dimensionality is simply increased by 1;$z=1$It need not appear in this fashion, however, and in the examples of itinerant antiferromagnetism and clean and dirty itinerant ferromagnetism, one finds$z=2, 3, \mathrm{and} 4$, respectively (ii) At finite temperatures, one finds that a classical statistical-mechanical description holds (and nonclassical exponents, for$dl4$) very close to the critical value of the coupling${U}_{c}$, when$\frac{(U\ensuremath{-}{U}_{c})}{{U}_{c}}\ensuremath{\ll}{(\frac{T}{{U}_{c}})}^{\frac{2}{z}}\frac{z}{2}$is therefore the quantum-to-classical crossover exponent 1,636 citations Journal ArticleDOI Abstract: It is argued that the long-wavelength, low-temperature behavior of a two-dimensional quantum Heisenberg antiferromagnet can be described by a quantum nonlinear$\ensuremath{\sigma}$model in two space plus one time dimension, at least in the range of parameters where the model has long-range order at zero temperature The properties of the quantum nonlinear$\ensuremath{\sigma}$model are analyzed approximately using the one-loop renormalization-group method When the model has long-range order at$T=0$, the long-wavelength behavior at finite temperatures can be described by a purely classical model, with parameters renormalized by the quantum fluctuations The low-temperature behavior of the correlation length and the static and dynamic staggered-spin-correlation functions for the quantum antiferromagnet can be predicted, in principle, with no adjustable parameters, from the results of simulations of the classical model on a lattice, combined with a two-loop renormalization-group analysis of the classical nonlinear$\ensuremath{\sigma}$model, a calculation of the zero-temperature spin-wave stiffness constant and uniform susceptibility of the quantum antiferromagnet, and a one-loop analysis of the conversion from a lattice cutoff to the wave-vector cutoff introduced by quantum mechanics when the spin-wave frequency exceeds$\frac{T}{\ensuremath{\hbar}}$Applying this approach to the spin-\textonehalf{} Heisenberg model on a square lattice, with nearest-neighbor interactions only, we obtain a result for the correlation length which is in good agreement with the data of Endoh et al on${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$, if the spin-wave velocity is assumed to be 067 eV$\frac{\AA{}}{\ensuremath{\hbar}}$We also argue that the data on${\mathrm{La}}_{2}$Cu${\mathrm{O}}_{4}$cannot be easily explained by any model in which an isolated Cu${\mathrm{O}}_{2}$layer would not have long-range antiferromagnetic order at$T=0$Our theory also predicts a quasielastic peak of a few meV width at 300 K when$k\ensuremath{\xi}\ensuremath{\ll}1$(where$k$is wave-vector transfer and$\ensuremath{\xi}\$ is the correlation length) The extent to which this dynamical prediction agrees with experiments remains to be seen In an appendix, we discuss the effect of introducing a frustrating second-nearest-neighbor coupling for the antiferromagnet on the square lattice

805 citations

Journal ArticleDOI
Elliott H. Lieb1, Daniel C. Mattis1Institutions (1)
Abstract: The total spin S is a good quantum number in problems of interacting spins. We have shown that for rather general antiferromagnetic or ferrimagnetic Hamiltonians, which need not exhibit translational invariance, the lowest energy eigenvalue for each value of S [denoted E(S)] is ordered in a natural way. In antiferromagnetism, E(S + 1) > E(S) for S > O. In ferrimagnetism, E(S + 1) > E(S) for S > S, and in addition the ground state belongs to S < S. S is defined as follows: Let the maximum spin of the A sublattice be S A and of the B sublattice S B; then S ≡ S A − S B. Antiferromagnetism is treated as the special case of S = O. We also briefly discuss the structure of the lowest eigen-functions in an external magnetic field.

599 citations

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