THE QUASI-CRYSTALLINE QUANTUM SPIN-½ XY-CHAIN

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THE QUASI-CRYSTALLINE QUANTUM SPIN-½ XY-CHAIN

Th. M. Nieuwenhuizen

To cite this version:

Th. M. Nieuwenhuizen. THE QUASI-CRYSTALLINE QUANTUM SPIN-½ XY-CHAIN. Journal de Physique Colloques, 1986, 47 (C3), pp.C3-211-C3-216. �10.1051/jphyscol:1986322�. �jpa-00225733�

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JOURNAL D E PHYSIQUE

Colloque C3, supplement au no 7, Tome 47, juillet 1986

THE QUASI-CRYSTALLINE QUANTUM SPIN-% XY-CHAIN

Service de Physique Theorique, CEN-Saclay, F-91191 Gif-sur-Yvette Cedex, France

RLsum6 - On Ctudie une chaine de spins quantiques avec des couplages "quasi- cristallins dans le plan XY et un champ magndtique uniforme le long de l'axe des 2. L'airnantation P tempdrature nulle est une fonction de Cantor du champ -

magndtique. La chaleur spLcifique en champ nu1 se comporte comme 'IA 2 basse temperature ; l'exposant varie continhent entre z6ro et un.

Abstract - The quantum spin-k chain with "quasicrystalline" couplings in the XY-plane and a steady magnetic field in the 2-direction is studied. The zero-temperature magnetization is a Cantor-function. The zero-field specific -

heat behaves at T~ at low temperature with varying continuously between zero and one.

INTRODUCTION

Recently several one-dimensional quasicrystalline systems have been studied. Luck and Petritis [I] considered a quasicrystalline phonon model, which is presented in shorter form in this issue [Z]. Then Luck considered a quasicrystalline Ising model at zero temperature [3l. The present contribution summarizes a recent work done in collaboration with Luck [4],

We consider the Fibonacci tiling of the line, explained for instance in Ref.123 (this issue). To long bonds we attach a coupling strength Ji = P, and to short bonds a coupling .Ti = 1. Given this sequence {Ji), we consider the quantum spin-Y XY Hamiltonian

where h represents a magnetic field in the Z-direction. The Sy are half the Pauli matrices. They obey:

[ s ~ . s ! ] ⁼^is,,~ " 7 S: ; C ⁼

a

*present address : InstitUt fiir Theoretische Physik. RUTH Aachen. Templergraben 55, D-5100 Aachen, FRG

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1986322

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C3-212 JOURNAL DE PHYSIQUE

The reason why t h e model ( 1 ) can be solved i s t h a t i t can be reduced t o a f r e e fermion problem. Lieb e t a 1 [5] showed t h i s f o r t h e "pure" case ( J i = J) and Smith [6] noted t h a t i t can be done a s w e l l i n t h e case of a r b i t r a r y Ji. The r e s u l t i s t h a t t h e f r e e energy has t h e form

where H(E) i s t h e i n t e g r a t e d density of s t a t e s H ( E ) = l i m - 1 ( # E, < E)

N- N of t h e t i g h t - b i n d i n g problem

Since H(E) w i l l t u r n o u t t o be a Cantor function, one cannot introduce a density of s t a t e s p(E) by dH(E) = p(E)dE i n Eq.(3). I f E, i s an eigenvalue of Eq.(5), s o is (-E,), a s i s seen by s u s b s t i t u t i n g ui = ( - l ) i v i . Hence H(E) = 1-H(-E); H(0) = % and w e only have t o consider p o s i t i v e energies.

The spectrum of Eq.(5) can be s t u d i e d most conveniently by introducing

$, = J,J ,... ^Jnun. Its equation of motion can be w r i t t e n i n t h e t r a n s f e r matrix form

where Tn i s a product of (n+l) 2x2 matrices. See a l s o Ref.[2] f o r a c l o s e l y r e l a - ted s i t u a t i o n . For t h e Fibonacci chain obtained by p r o j e c t i o n and discussed i n [2]. t h e Tp s a t i s f y a s p e c i f i c recurrence r e l a t i o n . Their normalized t r a c e s

L

xL = f tr (Ti ) and determinants yL = d e t ( T F L ) s a t i s f y t h e recurrence r e l a t i o n s

L

derived by Kohmoto e t a 1 [7] and Luck [3]:

we f i n d yo = 1. yl = p2 and hence yL = p2FL+1

.

^where^FLa r e t h e Fibonacci numbers.

This l e a d s u s t o d e f i n e EL = P - ~ L * l x L , which s a t i s f i e s

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The boundary conditions a r e most e a s i l y expressed a s

The map (8) has a conserved q u a n t i t y [7]

1

I = E;-~ + + e: - ~E,_,C~-,E, = - ( p + ~ - l ) ~

4

Fjg.l: P l o t of t h e i n t e g r a t e d density of s t a t e s H(E) a g a i n s t energy E. f o r p=Z.

THE SPECTRUM

I n Fig.1 w e present a p l o t of the i n t e g r a t e d d e n s i t y of states H(E) f o r t h e value p=2. It i s seen t h a t H is a Cantor function. Its behavior near t h e ma- ximal energy Emax i s

where P i s a p e r i o d i c function with u n i t period. The exponent A and t h e s c a l e I x ~ I

follow from an argument given i n [I]: The map (8) has a six-cycle

ff -t -6 + -ff -. 13 + -a + -6. The l a r g e s t eigenvalue ( i n absolute value) is A$ with

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JOURNAL DE PHYSIQUE

A s c a l i n g argument then y i e l d s A = 22nr/ Zn lX2 I .

The very same behavior is present near gap edges Eg i n s i d e t h e spectrum.

I n p a r t i c u l a r A does not depend on E,; t h e periodic function P may do s o , however.

A t E=O one f i n d s t h a t t h e i n i t i a l conditions ( 9 ) already l i e on a six-cy- c l e

'

^-+⁰⁺⁰^-+^-

'

^-)⁰ 0 discussed by Kohmoto e t a 1 [7]. Hence one has a s i m i -

2 2

l a r r e s u l t

where t h e l a r g e s t eigenvalue of t h e l i n e a r i z e d map i s A$ with

-

and where A = 3Zn r/Zn X g . THERMODYNAMICS

From Eq.(3) one notes t h a t t h e e x t e r n a l f i e l d h plays t h e r o l e of the Fermi l e v e l . A t T=O a l l s t a t e s with E I h a r e f i l l e d up. One f i n d s , f o r instance, f o r t h e magnetization

Hence M is a Cantor function, with t h e s c a l i n g behavior (11) a t i t s gap edges E, # 0 and t h e behavior (14) a t h = 0. The z e r o temperature s u s c e p t i b i l i t y i s an i l l - d e f i n e d q u a n t i t y ( z e r o o r i n f i n i t e ) .

From ( 3 ) and ( 1 4 ) . we deduce f o r t h e s p e c i f i c h e a t

where Ro i s a periodic function with u n i t period, e x p r e s s i b l e i n t e r m s of Po.

- -

Since 0 < A < 1, Eq.(17) i n t e r p o l a t e s between t h e uniform case ( P = l : b1) and the t o t a l l y random c a s e ( C (Zn T ) - 2 * = 0 ' ) .

The s u s c e p t i b i l i t y behaves a s

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Plots of C(h=O;T) and x(h=O;T) are presented in Figs.2 and 3 for the case P=3. It is seen that C may exhibit an infinity of regions of non-monotonic behavior.

u: Log-Log plot of the zero-field specific heat against temperature, for p=3.

For h = E,, relations similar to Eqs.(17)-(18) hold, with A and [ A 2 ( re- -

plachg A and _{A 3 ,}respectively. For values of h inside one of the gaps, low-temperature excitations are damped exponentially: nothing interesting happens in a desert.

m: Same as Fig.2 for the zero-field susceptibility.

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ACKNOWLEDGEMENT

This work was supported by the Netherlands Organization for the Advance- ment of Pure Research (Z.W.O.).

REFERENCES

[I] J.M. Luck and D. Petritis. J. Stat. Phys. 42, 289 (1986).

[2] J.M. Luck (this issue)

131 J . M . Luck, J. Phys. A (to appear)

[4] J.M. Luck and Th. M Nieuwenhuizen, Europhysics Letters (to appear)

[5] E. Lieb, T. Schultz and D.C. Mattis. Ann. Phys. (N.Y. ) 16, 407 (1961).

[6] E.R. Smith, J. Phys. Q, 1419 (1970)

(71 ^M.Kohmoto and Y. Oono, ~hys. Letters m. 145 (1984)