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Quantum magnets: a brief overview

Indrani Bose

February 1, 2008

Department of Physis

Bose Institute

93/1, A. P. C. Road

Calutta-700009

Abstrat

Quantum magnetism is one of the most ative areas of researh in on-

densed matter physis. There is signiant researh interest speially in

low-dimensional quantum spin systems. Suh systems have a large num-

ber of experimental realizations and exhibit a variety of phenomena the

origin of whih an be attributed to quantum eets and low dimensions.

In this review, an overview of some aspets of quantum magnetism in

low dimensions is given. The emphasis is on key onepts, theorems and

rigorous results as well as models of spin hains, ladders and frustated

magneti systems.

1 Introdution

Quantum magnets are spin systems in whih the spins interat via the well-

known exhange interation. The interation is purely quantum mehanial in

nature and the form of the interation was derived simultaneously by Heisenberg

and Dira in 1926 [1. The most well-known model of interating spins in an

insulating solid is the Heisenberg model with the Hamiltonian

H =ij

JijSi.Sj (1)

Si is the spin operator loated at the lattie site i and Jij denotes the strength

of the exhange interation. The spin

Si an have a magnitude 1/2, 1, 3/2, 2,...et. The lattie, at the sites of whih the spins are loated, is d-dimensional.

Examples are a linear hain (d = 1), the square lattie (d = 2 ) and the ubi

lattie (d =3 ). Ladders have strutures interpolating between the hain and the

square lattie. An n-hain ladder onsists of n hains (n = 2, 3, 4,...et.) oupled

by rungs. Real magneti solids are three-dimensional (3d) but an be eetively

onsidered as low-dimensional systems if the exhange interations have dierent

1

strengths in dierent diretions. To give an example, a magneti solid may

onsist of hains of spins. The solid may be onsidered as a linear hain (d=1)

ompound if the intra-hain exhange interations are muh stronger than the

inter-hain ones. In a planar (d=2) magneti system, the dominant exhange

interations are intra-planar. Several examples of low-dimensional magneti

systems are given in [2.

The strength of the exhange interation Jij in Eq.(1) falls down rapidly

as the distane between interating spins inreases. For many solids, the sites

i and j are nearest-neighbours (n.n.s) on the lattie and Jij 's have the same

magnitude J for all the n.n. interations. The Hamiltonian in (1) then beomes

H = Jij

Si.Sj (2)

There are, however, examples of magneti systems in whih the strengths of the

exhange interations between su

essive pairs of spins are not the same. Also,

the interation Hamiltonian (1) may inlude n.n. as well as further-neighbour

interations. The well-known Majumdar-Ghosh hain [3 is desribed by the

Hamiltonian

HMG = JNi=1

Si.S i+1 +

J

2

Ni=1

Si.S i+2 (3)

and inludes n.n. as well as next-nearest-neighbour (n.n.n.) interations.The

Haldane-Shastry model [4 has a Hamiltonian of the form

H = Jij

1

|i j|2Si.Sj (4)

and inludes long-ranged interations. Real materials are haraterised by var-

ious types of anisotropy. The fully anisotropi n.n. Heisenberg Hamiltonian in

1d is given by

HXY Z =Ni=1

[JxSxi S

xi+1 + JyS

yi S

yi+1 + JzS

zi S

zi+1] (5)

The speial ases of this Hamiltonian are: the Ising (Jx = Jy = 0) , the XY(Jz = 0), the XXX or isotropi Heisenberg (Jx = Jy = Jz ) and the XXZ oranisotropi Heisenberg (Jx = Jy 6= Jz ) models. There is a huge literatureon these models some of whih are summarised in Refs. [1, 2, 5, 6, 7. Other

anisotropy terms may be present in the full spin Hamiltonian besides the basi

exhange interation terms.

Consider the isotropi Heisenberg Hamiltonian in (2) where ij denotesa n.n. pair of spins. The sign of the exhange interation determines the

favourable alignment of the n.n. spins. J > 0(J < 0) orresponds to anti-ferromagneti (ferromagneti) exhange interation. To see how exhange in-

teration leads to magneti order, treat the spins as lassial vetors. Eah n.n.

2

spin pair has an interation energy JS2cos where is the angle between n.n.

spin orientations. When J is < 0, the lowest energy is ahieved when = 0,i.e., the interating spins are parallel. The ferromagneti (FM) ground state

has all the spins parallel and the ground state energy Eg = J NzS22 where zis the oordination number of the lattie. When J is > 0, the lowest energy is

ahieved for = , i.e., the n.n. spins are antiparallel. The antiferromagneti(AFM) ground state is the Nel state in whih n.n. spins are antiparallel to

eah other. The ground state energy Eg = JNzS22 .Magnetism , however, is a purely quantum phenomenon and the Hamiltonian

(2) is to be treated quantum mehanially rather than lassially. For simpli-

ity, onsider a hain of spins of magnitude

12 . Periodi boundary ondition is

assumed, i.e.,

S N+1 =

S1 . The Hamiltonian (2) an be written as

H = J

Ni=1

[Szi Szi+1 +

1

2(S+i S

i+1 + S

i S

+i+1)] (6)

where

Si = Sxi iSyi (7)

are the raising and lowering operators. It is easy to hek that in the ase of a

FM, the lassial ground state is still the quantum mehanial ground state with

the same ground state energy. However, the lassial AFM ground state (the

Nel state) is not the quantum mehanial ground state. The determination of

the exat AFM ground state is a tough many body problem and the solution

an be obtained with the help of the Bethe Ansatz tehnique (Setion 2).

For a spin-1/2 system, the number of eigenstates of the Hamiltonian is 2N

where N is the number of spins. In a real solid N is 1023 and exat determi-nation of all the eigenvalues and the eigenfuntions of the system is impossible.

There are some lasses of AFM spin models for whih the ground state and in

a few ases the low-lying exitation spetrum are known exatly (Setion 2). In

the majority of ases, however, the ground state and the low-lying exited states

are determined in an approximate manner. Knowledge of the low-lying exita-

tion spetrum enables one to determine the low-temperature thermodynamis

and the response to weak external elds. The usual thermodynami quantities

of a magneti system are magnetisation, spei heat and suseptibility. Ex-

hange interation an give rise to magneti order below a ritial temperature.

However, for some spin systems, the ground state is disordered, i.e., there is no

magneti order even at T = 0. Long range order (LRO) of the Nel-type exists

in the magneti system if

limRS (0).

S (R )6= 0 (8)

where

R denotes the spatial loation of the spin. At T = 0, the expeta-

tion value is in the ground state and at T 6= 0 , the expetation value is theusual thermodynami average. The dynamial properties of a magneti system

3

are governed by the time-dependent pair orrelation funtions or their Fourier

transforms. Quantities of experimental interest inlude the dynamial orre-

lation funtions in neutron sattering experiments, the NMR spin-lattie re-

laxation rate, various relaxation funtions and assoiated lineshapes as well as

the dynamial response of the magneti system to various spetrosopi probes

[8. Knowledge of the ground and low-lying states and the orresponding energy

eigenvalues is essential to determine the thermodynami and dynami properties

of a magneti system.

The disovery of high-temperature uprate superondutors in 1987 has

given a tremendous boost to researh ativity in magnetism. The dominant

eletroni and magneti properties of the uprate systems are assoiated with

the opper-oxide (CuO2) planes. The Cu2+

ions arry spin-

12 and the spins

interat via the Heisenberg AFM exhange interation Hamiltonian. This fat

has given rise to a large number of studies on 2d antiferromagnets. The uprates

exhibit a variety of novel phenomena in their insulating, metalli and superon-

duting phases some of whih at least have links to quantum magnetism. The

subjet of magnetism has, as a result, expanded signiantly in sope and on-

tent. A rih interplay between theory and experiments has led to the disovery

of materials exhibiting hitherto unknown phenomena, formulation of new theo-

retial ideas, solution of old puzzles and opening up of new researh possibilities.

In this review, a brief overview of some of the important developments in quan-

tum magnetism will be given. The fous is on quantum antiferromagnets and

insulating solids.

2 Theorems and rigorous results

(i) Theorems :

A. Lieb-Mattis theorem [9

For general spin and for all dimensions and also for a bipartite lattie, the

entire eigenvalue spetrum satises the inequality

E0(S) E0(S + 1) (9)where E0(S) is the minimum energy orresponding to total spin S. The weakinequality beomes a strit inequality for a FM exhange oupling between spins

of the same sublattie. The theorem is valid for any range of exhange oupling

and the proof does not require PBC. The ground state of the S = 12 HeisenbergAFM with an even number N of spins is a singlet a

ording to the Lieb-Mattis

theorem.

B. Marshall's sign rule [10

The rule speies the struture of the ground state of a n.n. S = 12 Heisen-berg Hamiltonian dened on a bipartite lattie. The rule an be generalised to

spin S, n.n.n. FM interation but not to n.n.n. AFM interation. A bipartite

lattie is a lattie whih an be divided into two sublatties A and B suh that

the n.n. spins of a spin belonging to the A sublattie are loated in the B sub-

lattie and vie versa. Examples of suh latties are the linear hain, the square

4

and the ubi latties. A

ording to the sign rule, the ground state has the

form

| =

C | (10)

where | is an Ising basis state. The oeient C has the form

C = ()pa (11)with a real and 0 and p is the number of up-spins in the A sublattie.C. Lieb, Shultz and Mattis (LSM) theorem [11:

A half-integer S spin hain desribed by a reasonably loal Hamiltonian re-

speting translational and rotational symmetry either has gapless exitation

spetrum or has degenerate ground states, orresponding to spontaneously bro-

ken translational symmetry.

In the ase of a gapless exitation spetrum, there is at least one momentum

wave vetor for whih the exitation energy is zero. For a spetrum with gap,

the lowest exitation is separated from the ground state by an energy gap .The temperature dependene of thermodynami quantities is determined by

the nature of the exitation spetrum (with or without gap). The LSM theorem

does not hold true for integer spin hains. For suh hains, Haldane made a

onjeture that the spin exitation spetrum is gapped [12. This onjeture

has been veried both theoretially and experimentally [13.

D. Oshikawa, Yamanaka and Aek theorem [14

This theorem extends the LSM theorem to the ase of an applied magneti

eld. The ontent of the theorem is : translationally invariant spin hains in an

applied eld an have a gapped exitation spetrum, without breaking transla-

tional symmetry, only when the magnetization per site m (m = 1N

Ni=1 S

zi , N

is the total number of spins in the system ) obeys the relation

S m = integer (12)where S is the magnitude of the spin. The proof is an easy extension of that of

the LSM theorem. The gapped phases orrespond to magnetization plateaux in

the m vs. H urve at the quantized values of m whih satisfy (12). Whenever

there is a gap in the spin exitation spetrum, it is obvious that the magnetiza-

tion annot hange in hanging external eld. Frational quantization an also

o

ur, if a

ompanied by (expliit or spontaneous) breaking of the translational

symmetry. In this ase, the plateau ondition is given by

n(S m) = integer (13)where n is the period of the ground state. Hida [15 has onsidered a S = 12HAFM hain with period 3 exhange oupling. A plateau in the magnetization

urve o

urs at m = 16 (13 of full magnetization ). In this ase, n =3, S =

12 and

m = 16 and the quantization ondition in (13) is obeyed. Ref. [16 gives a reviewof magnetization plateaux in interating spin systems. Magnetization plateaux

5

have been observed in the magneti ompound NH4CuCl3 at m =14 and

34

[17. Possible extensions of the LSM theorem to higher dimensions have been

suggested [18. The ompound SrCu2(BO3)2 is the rst AFM ompound in 2din whih magnetization plateaux have been observed experimentally [19. Like

the quantum Hall eet, the phenomenon of magnetization plateaux is another

striking example of the quantization of a physially measurable quantity as a

funtion of the magneti eld.

E. Mermin-Wagner's theorem [20

There annot be any AFM LRO at nite T in dimensions d =1 and 2. The

LRO an, however, exist in the ground state of spin models in d =2. LRO exists

in the ground state of the 3d HAFM model for spin S 12 [21. At nite T, theLRO persists upto a ritial temperature Tc . For square [22 and hexagonal

[23 latties, LRO exists in the ground state for S 1 . The above results arebased on rigorous proofs. No suh proof exists as yet for S = 12 , d =2 (this

ase is of interest beause the CuO2 plane of the high-Tc uprate systems is a

S = 12 2d AFM).(ii) Exat Results :

A. the Bethe Ansatz [24

The Bethe Ansatz (BA) was formulated by Bethe in 1931 and desribes a

wave funtion with a partiular kind of struture. Bethe onsidered the spin 12Heisenberg linear hain in whih only n.n. spins interat. In the ase of the

FM hain, the exat ground state is simple with all spins aligned in the same

diretion, say, pointing up. An exitation is reated in the system by deviating

a spin from its ground state arrangement, i.e., replaing an up-spin by a down-

spin. Due to the exhange interation, the deviated spin does not stay loalised

at a partiular site but travels along the hain of spins. This exitation is the

so-alled spin wave or magnon. For the isotropi FM Heisenberg Hamiltonian,

the exat one-magnon eigenstate is given by

=

Nm=1

eikmSm | ....... (14)

where m denotes the site at whih the down-spin is loated and the summation

over m runs from the rst to the last site in the hain. The k's are the momenta

whih from periodi boundary onditions have N allowed values

k =2

N, = 0, 1, 2, ..., N 1 (15)

The exitation energy k, measured w.r.t. the ground state energy and in units

of J, is

k = (1 cosk) (16)In the ase of r spin deviations (magnons), the eigenfuntion an be written as

(r) =

m1

The amplitudes are given by the BA

a(m1,m2, ...,mr) =P

ei

jkPjmj+

12i

1,r

j

as in the FM ase but the sign of the exhange integral hanges from J to J(J > 0 ). The total spin of the AFM ground state is S = 0 a

ording to theLieb-Mattis theorem. In the ground state,

N2 spins are up and

N2 spins down

(r = N2 ). The ground stae is non-degenerate and there is a unique hoie of thei 's as

1 = 1, 2 = 3, 3 = 5, ...., N2= N 1 (23)

The ground state is also spin disordered, i.e., has no AFM LRO. The exat

ground state energy Eg is

Eg =NJ

4 JNln2 (24)

The low-lying exitation spetrum has been alulated by des Cloizeaux and

Pearson (dCP) [27 by making appropriate hanges in the distribution of i 's

in the ground state. The spetrum is given by

=

2|sink| , k (25)

for spin 1 states. The wave vetor k is measured w.r.t. that of the ground

state. A more rigorous alulation of the low-lying exitation spetrum has

been given by Faddeev and Takhtajan [28. There are S = 1 as well as S = 0states. We give a qualitative desription of the exitation spetrum, for details

Ref. [28 should be onsulted. The energy of the low-lying exited states an be

written as E(k1, k2) = (k1) + (k2) with (ki) =pi2 sinki and total momentum

k = k1 + k2. At a xed total momentum k, one gets a ontinuum of satteringstates. The lower boundary of the ontinuum is given by the dCP spetrum

(one of the kis = 0 ). The upper boundary is obtained for k1 = k2 =k2 and

Uk =

sink2 (26)

The energy-momentum relations suggest that the low-lying spetrum is atually

a ombination of two elementary exitations known as spinons. The energies

and the momenta of the spinons just add up, showing that they do not interat.

A spinon is a S = 12 objet, so on ombination they give rise to both S = 1 andS = 0 states. In the Heisenberg model, the spinons are only noninterating inthe thermodynami limit N . For an even number N of sites, the total spinis always an integer, so that the spins are always exited in pairs. The spinons

an be visualised as kinks in the AFM order parameter. Due to the exhange

interation, the individual spinons get deloalised into plane wave states. In-

elasti neutron sattering study of the linear hain S = 12 HAFM ompoundKCuF3 has onrmed the existene of unbound spinon pair exitations [29.

The Haldane-Shastry model [4 is another spin 12 model in 1d for whih theground state and low-lying exitation spetrum are known exatly. The ground

state has the same funtional form as the frational quantum Hall ground state

and is spin-disordered. The elementary exitations are spinons whih are non-

interating even away from the thermodynami limit, i.e., in nite systems.

8

The individual spinons behave as semions, i.e., have statistial properties inter-

mediate between fermions and bosons. In the ase of integer spin hains, the

spinons are bound and the exitation spetrum onsists of spin-wave-like modes

exhibiting the Haldane gap. The BA tehnique desribed in this Setion is the

one originally proposed by Bethe. There is an algebrai version of the BA whih

is in wide use and whih gives the same nal results as the earlier tehnique. For

an introdution to the algebrai BA method, see the Refs. [30, 31. A tutorial

review of the BA is given in Ref. [32. The BA was originally proposed for

the Heisenberg model in magnetism. Later, the method was applied to other

interating many body systems in 1d suh as the Fermi and Bose gas models

in whih partiles on a line interat through delta funtion potentials [33, the

Hubbard model in 1d [34, 1d plasma whih rystallizes as a Wigner solid [35,

the Lai-Sutherland model [36 whih inludes the Hubbard model and a dilute

magneti model as speial ases, the Kondo model in 1d [37, the single impurity

Anderson model in 1d [38, the supersymmetri t-J model (J = 2t) [39 et. In

the ase of quantum models, the BA method is appliable only to 1d models.

The BA method has also been applied to derive exat results for lassial lattie

statistial models in 2d.

B. The Majumdar-Ghosh hain [3, 7

The Hamiltonian is given in Eq. (3). The exat ground state of HMG is doubly

degenerate and the states are

1 [12][34]...[N 1N ], 2 [23][45]...[N1] (27)where [lm] denotes a singlet spin onguration for spins loated at the sitesl and m. Also, PBC is assumed. One nds that translational symmetry is

broken in the ground state. The proof that 1 and 2 are the exat ground

states an be obtained by the method of `divide and onquer'. One an verify

that 1 and 2 are exat eigenstates of HMG by applying the spin identityS n.(

S l +

S m)[lm] = 0 . Let E1 be the energy of 1 and 2 . Let Eg be the

exat ground state energy. Then Eg E1 . One divides the Hamiltonian H intosub-Hamiltonians , Hi 's, suh that H =

iHi . Hi an be exatly diagonalised

and let Ei0 be the ground state energy. Let g be the exat ground state wave

funtion. By variational theory,

Eg = g |H |g =i

g |Hi|g i

Ei0 (28)

One thus gets, i

Ei0 Eg E1 (29)

If one an show that

i Ei0 = E1 , then E1 is the exat ground state energy.

For the MG-hain, the sub-Hamiltonian Hi is

Hi =J

2(S i.

S i+1 +

S i+1.

S i+2 +

S i+2.

S i) (30)

9

There are N suh sub-Hamiltonians. One an easily verify that Ei0 = 3J8 andE1 = 3J4 N2 ( - 3J4 is the energy of a singlet and there are N2 VBs in 1 and2). From (29), one nds that the lower and upper bounds of Eg are equal and

hene 1 and 2 are the exat ground states with energy E1 = 3JN8 . Thereis no LRO in the two-spin orrelation funtion in the ground stae:

K2(i, j) =Szi S

zj

=

1

4ij 1

8|ij|,1 (31)

The four-spin orrelation funtion has o-diagonal LRO.

K4(ij, lm) =Sxi S

xj S

yl S

ym

= K2(ij)K2(lm)

+1

64|ij|,1|lm|,1exp(i(

i+ j

2 l +m

2)) (32)

Let T be the translation operator for unit displaement. Then

T1 = 2, T2 = 1 (33)

The states

+ =12(1 + 2),

=12(1 2) (34)

orrespond to momentum wave vetors k = 0 and k = . The exitationspetrum is not exatly known. Shastry and Sutherland [40 have derived the

exitation spetrum in the basis of `defet' states. A defet state has the wave

funtion

(p,m) = ...[2p3, 2p2]2p1[2p, 2p+1]...[2m2, 2m1]2m[2m+1, 2m+2]...(35)

where the defets (2p1 and 2m) separate two ground states. The two defetsare up-spins and the total spin of the state is 1. Similarly, the defet spins an

be in a singlet spin onguration so that the total spin of the state is 0. Beause

of PBC, the defets o

ur in pairs. A variational state an be onstruted

by taking a linear ombination of the defet states. The exitation spetrum

onsists of a ontinuum with a lower edge at J(52 2 |cosk|). A bound stateof the two defets an o

ur in a restrited region of momentum wave vetors.

The MG hain has been studied for general values J of the n.n.n. interation

[41. The ground state is known exatly only at the MG point = 12 . Theexitation spetrum is gapless for 0 < < cr( 0.2411). Generalizations ofthe MG model to two dimensions exist [42, 43. The Shastry-Sutherland model

[42 is dened on a square lattie and inludes diagonal interations as shown

in Figure 1. The n.n. and diagonal exhange interations are of strength J1and J2 respetively. For

J1J2

below a ritial value 0.7 , the exat ground state

onsists of singlets along the diagonals. At the ritial point, the ground state

10

hanges from the gapful disordered state to the AFM ordered gapless state. The

ompound SrCu2(BO3)2 is well-desribed by the Shastry-Sutherland model[19. Bose and Mitra [43 have onstruted a J1 J2 J3 J4 J5 spin- 12model on the square lattie. J1, J2, J3, J4 and J5 are the strengths of the n.n.,

diagonal, n.n.n., knight's-move-distane-away and further-neighbour-diagonal

exhange interations (Figure 2). The four olumnar dimer states (Figure 3)

have been found to be the exat eigenstates of the spin Hamiltonian for the

ratio of interation strengths

J1 : J2 : J3 : J4 : J5 = 1 : 1 :1

2:1

2:1

4(36)

It has not been possible as yet to prove that the four olumnar dimer states are

also the ground states. Using the method of `divide and onquer', one an only

prove that a single dimer state is the exat ground state with the dimer bonds

of strength 7J . The strengths of the other exhange interations are as speiedin (36). For a 4 4 lattie with PBC, one an trivially show that the four CDstates are the exat ground states.

C. The Aek-Kennedy-Lieb-Tasaki (AKLT) model [44

We have already disussed the LSM theorem, the proof of whih fails for

integer spin hains. Haldane [12, 13 in 1983 made the onjeture, based on

a mapping of the HAFM Hamiltonian, in the long wavelength limit, onto the

nonlinear model, that integer-spin HAFM hains have a gap in the exitation

spetrum. The onjeture has now been veried both theoretially and exper-

imentally [45. In 1987, AKLT onstruted a spin-1 model in 1d for whih the

ground state ould be determined exatly [44. Consider a 1d lattie, eah site

of whih is o

upied by a spin-1. Eah suh spin an be onsidered to be a

symmetri ombination of two spin-

12 's. Thus, one an write down

++ = |++ , Sz = +1 = | , Sz = 1+ =

12(|++ |+ , Sz = 0

+ = + (37)

where `+' (`') denotes an up (down) spin.AKLT onstruted a valene bond solid (VBS) state in the following man-

ner. In this state, eah spin-

12 omponent of a spin-1 forms a singlet (valene

bond) with a spin-

12 at a neighbouring site. Let

(, = + or ) be the

antisymmetri tensor:

++ = = 0, + = + = 1 (38)A singlet spin onguration an be expressed as

12 | , summation over

repeated indies being implied. The VBS wave funtion (with PBC) an be

written as

11

|V BS = 2N2 11122223 .....iiii+1NN N1 (39)

|V BS is a linear superposition of all ongurations in whih eah Sz = +1 isfollowed by a Sz = 1 with an arbitrary number of Sz = 0 spins in betweenand vie versa. If one leaves out the zero's, one gets a Nel-type of order. One

an dene a non-loal string operator

ij = Si exp(ij1l=i+1

Sl )Sj , ( = x, y, z) (40)

and the order parameter

Ostring = lim|ij|ij

(41)

The VBS state has no onventional LRO but is haraterised by a non-zero

value

49 of O

string. After onstruting the VBS state, AKLT determined the

Hamiltonian for whih the VBS state is the exat ground state. The Hamiltonian

is

HAKLT =i

P2(S i +

S i+1) (42)

where P2 is the projetion operator onto spin 2 for a pair of n.n. spins. The

presene of a VB between eah neighbouring pair implies that the total spin

of eah pair annot be 2 (after two of the S = 12 variables form a singlet, theremaining S = 12 's ould form either a triplet or a singlet). Thus, HAKLTating on |V BS gives zero. Sine HAKLT is a sum over projetion operators,the lowest possible eigenvalue is zero. Hene, |V BS is the ground state ofHAKLT with eigenvalue zero. The AKLT ground state (the VBS state) is spin-

disordered and the two-spin orrelation funtion has an exponential deay. The

total spin of two spin-1's is 2, 1, 0. The projetion operator onto spin j for a

pair of n.n. spins has the general form

Pj(S i +

S i+1) =

l 6=j

[l(l + 1)S 2

][l(l + 1) j(j + 1)] (43)

where

S =

S i +

S i+1 . For the AKLT model, j = 2 and l = 1, 0 . From (42)

and (43),

HAKLT =i

[1

2(S i.

S i+1) +

1

6(S i.

S i+1)

2 +1

3

](44)

The method of onstrution of the AKLT Hamiltonian an be extended to higher

spins and to dimensions d > 1. The MG Hamiltonian (apart from a numerial

prefator and a onstant term) an be written as

12

H =i

P 32(S i +

S i+1 +

S i+2) (45)

The S = 1 HAFM and the AKLT hains are in the same Haldane phase, har-aterised by a gap in the exitation spetrum. The physial piture provided

by the VBS ground state of the AKLT Hamiltonian holds true for real systems

[46. The exitation spetrum of HAKLT annot be determined exatly. Arovas

et al. [47 have proposed a trial wave funtion

|k = N 12Nj=1

eikjSj |V BS , = z,+, (46)

and obtained

(k) =k |HV BS | kk | k =

25 + 15cos(k)

27(47)

The gap in the exitation spetrum = 1027 at k = . Another equivalent wayof reating exitations is to replae a singlet by a triplet spin onguration [48.

3 Spin Ladders

A. Undoped ladders

In the last Setion, we disussed some exat results for interating spin sys-

tems. The powerful tehnique of BA was desribed. The BA annot provide

knowledge of orrelation funtions. There is another powerful tehnique for 1d

many body systems known as bosonization [49 whih enables one to alu-

late various orrelation funtions for 1d systems. After the disovery of high-

Tc uprate systems, the study of 2d AFMs aquired onsiderable importane.

There are, however, not many rigorous results available for 2d spin systems.

Ladder systems interpolate between a single hain (1d) and the square lattie

(2d) and are ideally suited for the study of the rossover from 1d to 2d. Consider

a two-hain spin ladder (Figure 4) desribed by the AFM Heisenberg exhange

interation Hamiltonian

HJJR =ij

JijS i.

S j (48)

The n.n. intra-hain and the rung exhange interations are of strength J and

JR respetively. When JR = 0, one obtains two deoupled AFM spin hains for

whih the exitation spetrum is known to be gapless. Dagotto et al. [50 derived

the interesting result that the lowest exitation spetrum is separated by an

energy gap from the ground state. The result is easy to understand in the simple

limit in whih the exhange oupling JR along the rungs is muh stronger than

the exhange oupling J along the hains. The intra-hain oupling may thus be

treated as perturbation. When J = 0 , the exat ground state onsists of singlets

13

along the rungs, eah singlet having the spin onguration

12[| |]. The

ground state energy is 3JRN4 , where N is the number of rungs in the ladder.In rst order perturbation theory, the orretion to the ground state energy is

zero. The ground state has total spin S = 0. A S = 1 exitation may be reatedby promoting one of the rung singlets to a S = 1 triplet. A triplet has the spin

onguration | (Sz = +1), 1

2| + (Sz = 0 ) and | (Sz = 1). A

triplet osts an exhange energy equal to JR . The weak oupling along the

hains gives rise to a band of propagating S = 1 magnons with the dispersionrelation

(k) = JR + Jcosk (49)

in rst order perturbation theory (k is the wave vetor). The spin gap, dened

as the minimum exitation energy is given by

SG = () (JR J) (50)The two-spin orrelations deay exponentially along the hains showing that the

ground state is a quantum spin liquid (QSL). As the rung exhange oupling JRdereases, one expets that the spin gap will also derease and ultimately be-

ome zero at a ritial value of JR. Barnes et al. [51, however, put forward the

onjeture that SG > 0 for allJRJ

> 0, inluding the isotropi limit JR = J . Avariety of numerial tehniques like exat diagonalization of nite-sized ladders

[50, Quantum Monte Carlo (QMC) simulations [52 and density-matrix renor-

malization group (DMRG) [53 have veried the onjeture. We now onsider

the ase of an n-hain ladder. A surprising fat emerging out of several theoret-

ial studies [54, 55 is: the exitation spetrum has spin gap (is gapless) when n

is even (odd). In the rst (seond) ase, the two-spin orrelation funtion has

an exponential (power-law) deay. For odd n, the ladder has properties similar

to those of a single hain. The strong oupling limit (JR J) again provides aphysial piture as to why this is true. When n is even, the S =

12 spins along

a rung ontinue to form a singlet ground state. Hene the reation of a S =1

exitation requires a nite amount of energy as in the ase of the two-hain

ladder. The gap should derease as n inreases so that the gapless square lat-

tie limit is reahed for large n. When n is odd, eah rung onsists of an odd

number of spins, eah of magnitude

12 . The inter-rung (intra-hain) oupling J

generates an eetive interation between the S = 12 rung states, whih beauseof rotational invariane, should be of the Heisenberg form with an eetive ou-

pling Jeff setting the energy sale. The equivalene of an odd-hain ladder to

the single Heisenberg hain leads to a gapless exitation spetrum. Rojo [56

has given a rigorous proof of the gaplessness of the exitation spetrum when

n is odd. Khveshhenko [57 has shown that for odd-hain ladders, a topolog-

ial term governing the dynamis at long wavelengths appears in the eetive

ation, whereas, it exatly anels for even-hain ladders. The topologial term

has similarity to the one that auses the dierene between integer and half-

odd integer spin hains. In the rst ase, the spin exitation spetrum has the

14

well-known Haldane gap. In the latter ase, the LSM theorem shows that the

exitation spetrum is gapless. Ghosh and Bose [58 have onstruted an n-hain

spin ladder model for whih the exat ground state an be determined for all

values of n. For n even (odd), the exitation spetrum has a gap (is gapless).

This is true even for large n, thus the square lattie limit annot be reahed in

the model. Thermodynami properties of the S = 12 two-hain ladder have beenrst studied by Troyer et al. [59. Using a quantum transfer matrix method,

they obtained reliable results down to temperature T 0.2J . The AFM or-relation length AFM has been found to be 3-4 lattie spaings. The magneti

suseptibility (T ) shows a rossover from a Curie-weiss form, (T ) = CT+at

high temperature to an exponential fall-o, (T ) eSGTT

as T 0. Thefall-o is a signature of a nite spin gap SG. Frishmuth et al. [60, using apowerful loop algorithm, have alulated the magneti suseptibility and found

evidene for the gapped (gapless) exitation spetrum in the ase of an even

(odd)-hain ladder.

A major interest in the study of ladder systems arises from the fat that

there is a large number of experimental realizations of ladder systems. A om-

prehensive review of major experimental systems is that by Dagotto [61. We

disuss here only a few interesting ladder systems. Hiroi et al. [62 were the

rst to synthesize the family of layer ompounds Srn1Cun+1O2n. Rie et al.[54 subsequently reognized that these ompounds ontained weakly-oupled

ladders of

n+12 hains. For n =3 and 5, respetively, one gets the two-hain and

three-hain ladder ompounds. Azuma et al. [63 have determined the tem-

perature dependene of the magneti suseptibility in these ladder ompounds

experimentally. A spin gap is indiated by the sharp fall of (T )for T < 300Kin the two-hain ladder ompound SrCu2O3 . The magnitude of the spin gap is

SG 420K. This is approximately in agreement with the theoretial result ofSG J2 , if an exhange oupling J 1200K is assumed. For the three-hainladder ompound Sr2Cu3O5 , Azuma et al. found that (T ) approahes a on-stant as T 0, , as expeted for the 1d Heisenberg AFM hain. Muon spinrelaxation measurements by Kojima et al. [64 shows the existene of a long

range ordered state with Nel temperature TN = 52 K, brought about by the

interlayer oupling. No sign of long range ordering was found in the two-hain

ladder ompound, onrming the dierene between odd and even hain ladders.

The ompound LaCuO2.5 is formed by an array of weakly interating two-hain

ladders [65. The evidene of spin-liquid formation at intermediate tempera-

tures (onrmed by the existene of a spin gap) and an ordered Nel state at

low temperatures, shows that the spin singlet state is in lose ompetition with

a Nel state. Spin ladders, belonging to the organi family of materials, have

also been synthesized. A reent example is the ompound (C5H12N)2CuBr4[66. This ompound is a good example of a strongly oupled (

JRJ 3.5 ) ladder

system. The phase diagram of the AFM spin ladder in the presene of an ex-

ternal magneti eld is partiularly interesting. In the absene of the magneti

eld and at T = 0, the ground state is a quantum spin liquid with a gap in theexitation spetrum. At a eld Hc1 , there is a transition to a gapless Luttinger

15

liquid phase (gBHc1 = SG , the spin gap, B is the Bohr magneton and g theLand splitting fator). There is another transition at an upper ritial eld Hc2to a fully polarised FM state. Both Hc1 and Hc2 are quantum ritial points.

The phase transitions that o

ur at these points are quantum phase transitions

as they o

ur at T = 0. At a quantum ritial point, the system swithes fromone ground state to another. The transition is brought about by hanging a

parameter (magneti eld in the present example) other than temperature. At

small temperatures, the behaviour of the system is determined by the rossover

between two types of ritial behaviour: quantum ritial behaviour at T = 0and lassial ritial behaviour at T 6= 0. Quantum eets are persistent inthe rossover region at small nite temperature and suh eets an be probed

experimentally. Refs. [67, 68, 69 give extensive reviews of quantum ritial

phenomena. In the ase of the ladder system (C5H12N)2CuBr4, the magneti-zation data, obtained experimentally, exhibit universal saling behaviour in the

viinity of the ritial elds, Hc1 and Hc2 . We remember that in the viinity of

ritial points, physial quantities of a system exhibit saling behaviour. Quan-

tum spin systems provide several examples of quantum phase transitions and

organi spin ladders are systems whih provide experimental testing grounds of

theories of suh transitions. For inorgani spin ladder systems, the value of Hc1is too high to be experimentally a

essible.

B. Frustrated spin ladders

Bose and Gayen [70 have studied a two-hain spin ladder model with frus-

trated diagonal ouplings (Figure 5, frustrated spin systems are dened in Se-

tion 4). The intra-hain and diagonal spin-spin interations are of equal strength

J . The exhange interations along the rungs are of strength JR . It is easy

to show that for JR 2J , the exat ground state onsists of singlets along therungs with the energy Eg = 3JRN4 where N is the number of rungs. Xian [71pointed out that the Hamiltonian of the frustrated ladder model an be written

as

H = JRi

S 1i.

S 2i + J

i

P i.

P i+1 (51)

where,

Pi =

S 1i +

S 2i , represents a omposite operator at the i-th rung

and `1' and `2' refer to the lower and upper hains respetively. Due to the

ommutativity of the rung interation part of the Hamiltonian with the seond

part, the eigenstates of the Hamiltonian an be desribed in terms of the total

spins of individual rungs. The energy eigenvalue for the state with singlets on all

the rungs is Esg = 3JRN4 . The seond term in the Hamiltonian (Eq.(51)) doesnot ontribute in this ase. If the two rung spins form a triplet, the seond term

is equivalent to the Hamiltonian of a spin-1 Heisenberg hain with a one-to-one

orrespondene between a rung of the ladder and a site of the S = 1 hain.Beause the two parts of H ommute, the eigenvalue, when the rung spins form

a triplet is

ETg = (Je0 +JR

4)N (52)

16

where e0 = 1.40148403897(4) is the ground state energy/site of the spin 1Heisenberg hain. Comparing the energy ETg with the energy E

sg of the rung

singlet state, one nds that as long as

JRJ> (JR

J)c = e0 , the latter state is the

exat ground state. At the ritial value (JRJ)c , there is a rst order transition

from the rung singlet state to the Haldane phase of the S = 1 hain. Thelowest spin exitation in the rung singlet state an be reated by replaing a

rung singlet (S = 0) by a triplet (S = 1). The triplet exitation spetrumhas no dynamis. In a more general parameter regime, i.e., when the intra-

hain exhange interation is not equal in strength to the diagonal exhange

interation, the ground and the exited states an no longer be determined

exatly. In this ase, one takes reourse to approximate analytial and numerial

methods. Kolezhuk and Mikeska [72 have onstruted a lass of generalised

S = 12 two-hain ladder models for whih the ground state an be determinedexatly. The Hamiltonian H is a sum over plaquette Hamiltonians and eah

plaquette Hamiltonian ontains various two-spin as well as four-spin interation

terms. They have further introdued a toy model, the Generalised Bose-Gayen

(GBG) model whih has a rih phase diagram in whih the phase boundaries

an be determined exatly. Reently, some integrable spin ladder models with

tunable interation parameters have been introdued [73, 74, 75. The integrable

models, in general, ontain multi-spin interation terms besides two-spin terms.

C. Doped spin ladders

A major reason for the strong researh interest in ladders is that doped

ladder models are toy models of strongly orrelated systems. The most well-

known examples of the latter are the high-Tc uprate systems. As already

mentioned in the Introdution, these systems exhibit a rih phase diagram as

a funtion of the dopant onentration. Doping eetively replaes the spin-

12

's assoiated with the Cu2+ ions in the CuO2 planes by holes. The holes are

mobile in a bakground of antiferromagnetially interating Cu spins. Also,

due to strong Coulomb orrelations, the double o

upany of a site by two

eletrons, one with spin up and the other with spin down, is prohibited. This

is a non-trivial many body problem beause it involves a ompetition between

two proesses: hole deloalization and exhange energy minimization. A hole

moving in an antiferromagnetially ordered spin bakground, say, the Nel state,

gives rise to parallel spin pairs whih raise the exhange interation energy of the

system. The questions of interest are: whether a oherent motion of the holes

is possible, whether two holes an form a bound state (in the superonduting

(SC) phase of the doped uprates, harge transport o

urs through the motion

of bound pairs of holes), the development of SC orrelations, the possibility

of phase separation of holes et. For the uprates, a full understanding of

many of these issues is as yet laking (see [76 for a reent review of high-Tcsuperondutivity). The doped ladders are simple model systems in whih the

onsequenes of strong orrelation an be studied with greater rigour than in

the ase of the struturally more omplex uprate systems. Reent experimental

evidene [61 suggests that some phenomena are ommon to ladder and uprate

systems. The study of ladder systems is expeted to provide insight on the

17

ommon origin of these phenomena. Some ladder ompounds an be doped

with holes. Muh exitement was reated in 1996 when the ladder ompound

Sr14xCaxCu24O41 was found to beome SC under pressure at x = 13.6. Thetransition temperature Tc 12K at a pressure of 3 GPa. As in the ase ofSC uprate systems, holes form bound pairs in the SC phase of ladder systems.

The possibility of binding of hole pairs in a two-hain ladder system was rst

pointed out by Dagotto et al. [50. The strongly orrelated doped ladder system

is desribed by the t-J Hamiltonian

HtJ = ij,

tij(C+iCj +H.C.) +

ij

Jij(S i.

S j 1

4ninj) (53)

The C+iand Ci are the eletron reation and annihilation operators whih at

in the redued Hilbert spae (no double o

upany of sites).

C+i = C+i(1 ni)

Ci = Ci(1 ni) (54)

is the spin index and ni , nj are the o

upation numbers of the i-th and j-th

sites respetively. The term proportional to ninj is often dropped. The rst

term in Eq.(53) desribes the motion of holes with hopping integrals tR and t for

motion along the rung and hain respetively. In the onventional t J laddermodel, i and j are n.n. sites. The seond term (minus the 14ninj term) isthe usual AFM Heisenberg exhange interation Hamiltonian. The t J modelthus desribes the motion of holes in a bakground of antiferromagnetially

interating spins. In the undoped limit, eah site of the ladder is o

upied by a

spin-

12 and the tJ Hamiltonian redues to the AFM Heisenberg Hamiltonian.

Removal of a spin reates an empty site, i.e., a hole. A large number of studies

have been arried out on t J ladder models. These are reviewed in Refs.[61, 77, 78. We desribe briey some of the major results. A hole-doped

single AFM hain is an example of a Luttinger Liquid (LL) whih is dierent

from a Fermi liquid. The latter desribes interating eletron systems in higher

dimensions and at low temperatures. A novel harateristi of a LL is spin-

harge separation due to whih the harge and spin parts of an eletron (or

hole) move with dierent veloities and thus beome separated in spae. The

undoped two-hain ladder has a spin gap. This gap remains nite but hanges

disontinuously on doping. This is beause there are now two distint triplet

exitations (remember that the spin gap is the dierene in energies of the lowest

triplet exitation and the ground state). One triplet exitation is obtained

by exiting a rung singlet to a rung triplet as in the undoped ase. A new

type of triplet exitation is obtained in the presene of two holes. A lear

physial piture is obtained in the limit JR J . In this ase, the ground statepredominantly onsists of singlets along the rungs. On the introdution of a

hole, a singlet spin pair is broken and the hole exists with a free spin-

12 . In the

presene of two holes on two separate rungs, the two free spin-

12 's ombine to

18

give rise to an exited triplet state. The ground state is a singlet and onsists

of a bound pair of holes. The binding of holes an be understood in a simple

manner. Two holes loated on two dierent rungs break two rung singlets and

the exhange interation energy assoiated with two rungs is lost in the proess.

If the holes are loated on the same rung, the exhange interation energy of only

one rung is lost. If JR is muh greater than the other parameters of the system,

the holes preferentially o

upy rungs in pairs. As JR dereases in strength, the

hole bound pair has a greater spatial extent. The lightly doped ladder system

is not in the LL phase, i.e., no spin-harge separation o

urs. The system is in

the so-alled Luther-Emery phase with gapless harge exitations and gapped

spin exitations. A variety of numerial studies show that the hole pairs and the

spin gap are present even in the isotropi limit JR = J . Also, the relative stateof hole pairs has approximate d-wave symmetry with the pairing amplitude

having opposite signs along the rungs and the hains. The d-wave symmetry

is a feature of strong orrelation and is onsidered to be the symmetry of the

pairing state in the ase of uprate systems.

Bose and Gayen [70, 79, 80 have onstruted a two-hain tJ ladder modelwith frustrated diagonal ouplings. The intra-hain n.n. and the diagonal hop-

ping integrals have the same strength t. The other parameters have been dened

earlier. The speial struture of the model enables one to determine the exat

ground and exited states in the ases of one and two holes. The most signif-

iant result is an exat, analyti solution of the eigenvalue problem assoiated

with two holes in the innite t J ladder. The binding of holes has been ex-pliitly demonstrated and the existene of the Luther-Emery phase established.

For onventional t J ladders (the diagonal bonds are missing), the only ex-at results that have been obtained are through numerial diagonalization of

nite-sized ladders. Derivation of exat, analytial results in this ase has not

been possible so far. The reason for this is that as a hole moves in the anti-

ferromagnetially interating spin bakground, spin exitations in the form of

parallel spin pairs are generated. Proliferation of states with spin exitations

makes the solution of the eigenvalue problem extremely diult. In the ase of

the frustrated t J ladder model, there is an exat anellation of the terms

ontaining parallel spin pairs [80. Thus the hole has a perfet oherent motion

through the spin bakground. Frahm and Kundu [81 has onstruted an inte-

grable tJ ladder model and obtained the phase diagram. The model ontainsterms desribing orrelated hole hopping in hains whih may not be realizable

in real systems. Several studies have been arried out on the two-hain Hubbard

ladder as well as on multi-hain Hubbard and tJ ladders. Referenes of someof the studies may be obtained from [61.

4 Frustrated spin models in 2d

In Setions 2 and 3 we have disussed quasi-1d interating spin systems, namely,

spin hains and ladders. As already mentioned in the Introdution, the CuO2plane of the undoped uprate systems is a 2d AFM. The undoped uprates

19

exhibit AFM LRO below a Nel temperature TN . On the introdution of a

few perent of holes, the AFM LRO is rapidly destroyed leaving behind spin-

disordered states in the CuO2 planes. This fat has triggered lots of interest in

the study of spin systems with spin disordered states as ground states. Frus-

trated spin models are ideal andidates for suh systems. To understand the

origin of frustration, onsider the AFM Ising model on the triangular lattie.

An elementary plaquette of the lattie is a triangle. The Ising spin variables

have two possible values, 1, orresponding to up and down spin orientations.An antiparallel spin pair has the lowest interation energy J . A parallel spinpair has the energy +J . In an elementary triangular plaquette, there are threeinterating spin pairs. Due to the topology of the plaquette, all the three pairs

annot be simultaneously antiparallel. There is bound to be at least one paral-

lel spin pair. The parallel spin pair may be loated along any one of the three

bonds in the plaquette and so the ground state is triply degenerate. The Ising

model on the full triangular lattie has a highly degenerate ground state suh

that the entropy/spin is a nite quantity. As a result, the system never orders

inluding at T = 0 . Frustration o

urs in the system sine all the spin pairinteration energies annot be simultaneously minimised. On the other hand,

onsider the AFM Ising model on the square lattie. All the four spin pairs

in an elementary square plaquette an be made antiparallel and so there is no

frustration. The system exhibits magneti order below a ritial temperature.

If one of the spin pair interations in eah elementary square plaquette is FM

and the rest AFM, frustration o

urs in the square lattie spin system. A spin

system with mixed FM and AFM interations is frustrated if the sign of the

produt of exhange interations around an elementary plaquette is negative.

In the ase of a purely AFM model, frustration o

urs if the number of bonds

in an elementary plaquette of the lattie is odd. Examples of suh latties in 2d

are the triangular and kagom latties. In 3d, the pyrohlore lattie, the elemen-

tary plaquette of whih is a tetrahedron provides an example. A spin system

is also frustrated due to the presene of both n.n. as well as further-neighbour

interations. Consider AFM n.n. as well as n.n.n. interations between a row

of three Ising spins. Again, all the three spin pairs annot simultaneously be

made antiparallel.

Let us now treat the spins as lassial vetors (S ). For AFM spin-spininteration, the lowest energy is ahieved for an antiparallel spin onguration.

In the lassial limit, the spins on a bipartite lattie are ordered in the AFM

Nel state. On a non-bipartite lattie, suh as the triangular lattie, the lassial

ground state represents a ompromise between ompeting requirements. In the

ground state, the spins form an ordered three-sublattie struture with 1200

between n.n. spins on dierent sublatties. The ground state of the lassial

Heisenberg model on the kagom lattie is, however, highly degenerate and

disordered. We now onsider the full quantum mehanial spin Hamiltonian and

ask the question how the lassial ground states are modied when quantum

utuations are taken into a

ount. In the ase of the triangular lattie, it is now

believed that the quantum mehanial ground state of the S = 12 HAFM modelhas AFM LRO of the Nel-type, i.e., quantum utuations do not destroy the

20

three-sublattie order of the lassial ground state. In the seond senario, when

the lassial ground state is highly degenerate and disordered, thermal/quantum

utuations selet a subset of states whih tend to inorporate some degree of

long range order. This is the phenomenon of `order from disorder' whih is

ounterintuitive sine order is brought about by utuations whih normally

have disordering eets. The lassial kagom lattie HAFM ground states

inlude both oplanar as well as nonoplanar spin arrangements and utuations

lead to the seletion of oplanar order. This kind of ordering is partiularly true

for large values of the spin S. As the magnitude of the spin is dereased towards

S = 12 , the quantum utuations inrease in strength. These utuations oftendestroy the ordered struture obtained for large S. The quantum mehanial

ground states of the S = 12 HAFM on the kagom and pyrohlore latties havebeen found to be spin disordered. Some reent referenes of frustrated magneti

systems are [82, 83. The triangular lattie S = 12 HAFM is the rst exampleof a spin model in whih frustration o

urs due to lattie topology [84. The

S = 12 HAFM model has also been studied on a partially frustrated pentagonallattie [? and a parameter region identied in whih the ground state has AFM

LRO of the Nel-type.

Two well-known examples of spin-disordered states are the quantum spin

liquid (QSL) and dimer or valene bond (VB) states. A QSL state is a spin

singlet with total spin S = 0 and has both spin rotational and translationalsymmetry. In a VB state, spin rotational symmetry is present but tanslational

symmetry is broken. In suh states pairs of spins form singlets whih are alled

VBs or dimers with the VBs being frozen in spae. A well-known example of a

QSL state is the resonating-valene-bond (RVB) state [84 whih is a oherent

linear superposition of VB states (Figure 6). The RVB state is the starting

point for the well-known RVB theory of high-Tc SC. Spin-disordered (no AFM

LRO as dened in Eq. (8)) states with novel order parameters are:

(a) Chiral states

In these states, the spins are arranged in ongurations haraterised by the

order parameter

i =S i.(

Si+x

Si+y

(55)

with the three spins belonging to one plaquette of the square lattie and x, y

denoting unit vetors in the x and y diretions respetively. The hiral state

breaks time reversal symmetry or a reetion about an axis (parity).

(b) Dimer states

These are the VB states in whih the VBs are frozen in spae. A well-known

example of suh states is the olumnar dimer (CD) states. In suh states,

the VBs are arranged in olumns. On the square lattie, four suh states are

possible. The order parameter of CD states is

Dl =(l)

S l.(

Sl+x

+ iSl+y

Slx i

Sly)

(56)

where the l-sites are even and (l) = +1(1) if both lx and ly are even (odd).The order parameter takes the values 1, i,1,i for the four CD states shown

21

in Figure 3.

() Twisted states:

At the lassial level (S ), the spins in the twisted state are arrangedin inommensurate strutures. These ongurations an be visualised as spins

lying in a plane and with a twist angle in some diretion. It is possible that

suh states survive the inlusion of quantum utuations. The order parameter

is vetorial in nature and is given by

Tl =S l (S l+x +

Sl+y

)

(57)

These states are alled spin nematis and are dierent from helimagnets in whih

both Tl and the spin-spin orrelation funtions show LRO.

(d) Strip or Collinear States

In a lassial piture, the spins are ferromagnetially ordered in the x diretion

and antiferromagnetially ordered in the y diretion. The onguration obtained

by rotating the previous one by

pi2 is also possible. The order parameter is given

by

Cl =S l.(

Sl+x

Sl+y

+Slx

Sly)

(58)

Cl takes the values 1,1 in the two dierent strip ongurations.The spin-disordered states deribed above are quantum-oherent states and

are haraterised by novel order parameters. The term `quantum paramagnet'

is often used to desribe suh states.

Examples of real frustrated systems are many [82, 83. The best studied ex-

perimental kagom system is the magnetoplumbite, SrCr8xGa4+xO19 . Thesystem onsists of dense kagom layers of S = 32 Cr ions, separated by dilute tri-angular layers of Cr. In a mean-eld theory of the HAFM, the high temperature

suseptibility is given by

=C

T + cw, T TN (59)

Here, the Curie onstant C =2Bp

2

3kB, where B is the Bohr magneton and,

p = g[s(s + 1)]12, g being the Land splitting fator governing the splitting of

the spin multiplet in a magneti eld. Also, cw is the Curie-Weiss temperature.

The Nel ordering temperature TN is dened experimentally using bulk probes.

One looks for singularities in either the spei heat C(T) or the temperature

derivative of the suseptibility (T ) . In the ase of non-frustrated systems,TN cw and in the seond ase, TN

Frustration orresponds to f > 1 . For the kagom AFM SrCr8Ga4O19 , f is ashigh as 150.

SrCrGaO displays unonventional low-T behaviour [86. One of these is

the insensitivity of the spei heat C(T ) to applied magneti elds H as largeas twie the temperature. Suseptibility measurements show the existene of a

gap SG in the triplet spin exitation spetrum. For T < SG, eSGkBT

.

The spei heat, however, does not derease exponentially for T < SG, i.e.,does not have a thermally ativated behaviour. It has a T 2 dependene. This

fat along with the experimental observation of insensitivity of C(T ) to externalmagneti eld have been explained by suggesting that a large number of singlet

exitations fall within the triplet gap [86 . Numerial evidene of suh exita-

tions has been obtained in the ase of the S = 12 HAFM on the kagom lattie[87. The number of suh exitations has been found to be (1.15)N whereN is the number of spins in the lattie. Mambrini and Mila [88 have reently

established that a subset of short-range RVB states aptures the spei low

energy physis of the kagom lattie HAFM and the number of singlet states

in the singlet-triplet gap is (1.15)N in agreement with the numerial results.The appearane of singlet states in the singlet-triplet gap ould be a generi

feature of strongly frustrated magnets. Other examples of suh systems are:

the S = 12 frustrated HAFM on the15 -depleted square lattie desribing the 2d

AFM ompound CaV4O9 [89, 90, the HAFM on the 3d pyrohlore lattie and

a 1d system of oupled tetrahedra [91. Bose and Ghosh [90 have onstruted a

frustrated S = 12 AFM model on the15depleted square lattie and have shown

that in dierent parameter regimes the plaquette RVB (PRVB) and the dimer

states are the exat ground states. In the PRVB state, the four-spin plaquettes

(Figure 7) are in a RVB spin onguration whih is a linear superposition of

two VB states. In one suh state, the VBs (spin singlets) are horizontal and

in the other state the VBs are vertial. In the dimer state, VBs or dimers

form along the bonds joining the four-spin plaquettes. Both the PRVB and

the dimer states are spin disordered states. The state intermediate between the

PRVB and the dimer states has AFM LRO. Both the PRVB and dimer phases

are haraterised by spin gaps in the exitation spetrum. For the unfrustrated

HAFM model on the

15 -depleted square lattie, Troyer et al. [92 have arried

out a detailed study of the quantum phase transition from an ordered to a dis-

ordered phase. In the ordered phase, the exitation spetrum is gapless. The

spin gap SG ontinuously goes to zero in a power-law fashion at the quan-tum ritial point separating a gapped disordered phase from a gapless ordered

phase. Chung et al. [93 have arried out an extensive study on the possible

paramagneti phases of the Shastry-Sutherland model [19. In addition to the

usual dimer phase, they nd the existene of a phase with plaquette order and

also a topologially ordered phase with deonned S = 12 spinons and helialspin orrelations. Takushima et al. [94 have studied the ground state phase

diagram of a frustrated S = 12 quantum spin model on the square lattie. Thismodel inludes both the Shastry-Sutherland model as well as the spin model on

the

15depleted square lattie as speial ases. The nature of quantum phase

23

transitions among the various spin gap phases and the magnetially ordered

phases has been laried.

Lieb and Shupp [95 have derived some exat results for the fully frustrated

HAFM on a pyrohlore hekerboard lattie in 2d. This lattie is a 2d version

of the 3d pyrohlore lattie. The ground states have been rigorously proved to

be singlets. The Lieb-Mattis theorem is appliable only for bipartite latties.

Hene, the proof for the pyrohlore hekerboard lattie is a new result. Lieb and

Shupp have further proved that the magnetization in zero external eld vanishes

separately for eah frustrated tetrahedral unit. Also, the upper bound on the

suseptibility is

18 in natural units for both T = 0 and T 6= 0 . Frustration an

also o

ur from a ompetition between exhange anisotropy and the transverse

eld terms as in the ase of the transverse Ising model. Moessner et al. [96

have shown that the transverse Ising model on the triangular (kagom) lattie

has an ordered (disordered) ground state.

Frustrated AFMs with short-range dimer or RVB states as ground states

have a gap in the spin exitation spetrum and the two-spin orrelation fun-

tion has an exponential deay as a funtion of the distane separating the spins.

A single branh desribes the S = 1 triplet exitation spetrum. In Setion 2we pointed out that in the ase of the S = 12 HAFM hain, a pair of spinons(eah spinon has spin S = 12 ) are the fundamental exitations. The lowestexitation spetrum is thus not a single branh of S = 1 magnon exitationsbut a ontinuum of sattering states with well-dened lower and upper bound-

aries. AFM ompounds in 2d, in general, have exitation spetra desribed

by S = 1 magnons. Anderson [84 suggested that a RVB state may supportpairs of spinons as exitations whih are deonned via a rearrangement of the

VBs. In this ase, an extended and highly dispersive ontinuum of exitations

is expeted. Reently, Coldea et al. [97 have investigated the ground state or-

dering and dynamis of the 2d S = 12 frustrated AFM Cs2CuCl4 using neutronsattering in high magneti elds. The dynami orrelations exhibit a highly

dispersive ontinuum of exited states whih are harateristi of the RVB state

and arise from pairs of S = 12 spinons. A reent paper `RVB Revisited' byP.W.Anderson [98 fouses on the relevane of the RVB state to desribe the

normal state of the CuO2 planes in the high-Tc uprate superondutors.

5 Conluding remarks

In Setions 1-4, a brief overview of low-dimensional quantum magnets, speially,

antiferromagnets has been given. The subjet of quantum magnetism has wit-

nessed an unpreedented growth in researh ativity in the last deade. This is

one of the few researh areas in whih rigorous theories an be worked out and

experimental realizations are not diult to nd. Coordination hemists and

material sientists have prepared novel materials and onstruted new applia-

tion devies. Experimentalists have employed experimental probes of all kinds

to rene old data and unover new phenomena. Theorists have taken reourse

to a variety of analytial and numerial tehniques to explain the experimen-

24

tally observed properties as well as to make new preditions. Powerful theorems

have been proposed and exat results obtained. This trend is still ontinuing

and will lead to further breakthroughs in materials, phenomena and tehniques

in the oming years. The study of doped magneti materials whih inlude

uprates, ladders and spin hains has aquired onsiderable importane in re-

ent times. The dopants an be magneti and nonmagneti impurities as well as

holes. To give one example, it has been possible to dope the spin-1 Haldane-gap

AFM ompound Y2BaNiO5 with holes. Neutron sattering experiments reveal

the existene of midgap states and an inommensurate double-peaked struture

fator [99. Several new experimental results on the Haldane-gap AFMs have

been obtained in the last few years [100 whih add to the rihness of phenom-

ena observed in magneti syatems. In this overview, only a few of the aspets

of quantum magnetism have been highlighted. Conventional 2d and 3d mag-

nets have not been disussed at all as a good understanding of these materials

already exists. We have not disussed reent advanes in material appliations

whih inlude materials exhibiting olossal magnetoresistane, moleular mag-

nets, nanorystalline magneti materials, magnetoeletroni devies (the study

of whih onstitutes the new subjet of spintronis) et. A report on some of

these developments as well as some reent issues in quantum magnetism may

be obtained from Ref. [101.

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29

Figure Captions

Figure 1. The Shastry-Sutherland model

Figure 2. Five types of interation in the J1 J2 J3 J4 J5 modelFigure 3. Four olumnar dimer states

Figure 4. A two-hain spin ladder

Figure 5. The two-hain frustrated spin ladder model

Figure 6. An example of a RVB state

Figure 7. The

15depleted square lattie.

30

ab

.......

+