On the finite-size scaling in quantum critical phenomena

Physica A 171 (1991) 374-383 North-Holland

ON THE FINITE-SIZE SCALING IN QUANTUM CRITICAL

PHENOMENA N.S. TONCHEV Nadjakov institute o f Solid State Physics, Bulgarian Academy of Sciences, 1784 Sofia, Bulgaria

Received 8 August 1990

An exactly solvable d-dimensional model of a structural phase transition, confined to a geometry Ld-d'x ~0d and subjected to periodic boundary conditions, is considered in the low-temperature region. It is shown that in this region quantum effects are essential and that borderline dimensionalities between which finite-size scaling holds do not coincide with the lower and upper critical dimensionalities for the bulk system. A generalized version of the dimensional cross-over rule is established, i.e., the finite-size scaling behaviour of a ddimensional system finite in d*= d - d' directions in the quantum limit is equivalent to the finite-size scaling behaviour of the classical (d + 1)-dimensional system, also finite in d* directions.

I. Introduction

It is well known from the theory of critical phenomena that for T > 0 quantum effects are unimportant near a critical point [1]. However, quantum systems may have critical points at temperature T = 0. The dimensional crossover rule asserts that the critical singularities of such a system with dimensionality tl are equivalent to those of a classical system with dimensionality d and T > 0. The relation between tt and d may be different for various types of systems (see, e.g. refs. [1-3] and references therein). It is important to recall that all the singularities in the thermodynamic functions associated with the critical points occur after taking the thermodynamic limit, N--> :~, V---~0% N / V = const., where N is the numbcr of l.,a~"'~"~:"lu,..~.,.~.oHu,4 V = '-'It x -.- × L d IS" the volume of the system. If some or all sizes L / of the system remain finite, the critical point singularities will be modified: rounded and shifted [4]. In such a situation the following question is reasonable: is there any correspondence between the finite-size effects related to the classical and quantum systems, respectively? An application of finite-size scallng ideas to low-temperature (quantum) 0378-4371/91/$03.50 © 1991- Elsevier Science Publishers B.V. (North-Holland)

N.S. Tonchev I Finite-size scaling in quantum critical phenomena

375

systems was announced by Carry (see introduction of ch. 1 in ref. [5l). In ref. [2] some concrete steps in the area of quantum finite-size critical phenomena on the basis of an exactly soluble model were taken. A d-dimensional model was considered in fully finite (block) geometry and under periodic boundary conditions. Its finite-size critical behaviour can be studied exactly both in classical and quantum limits. In the present study, we consider the same model in the most general geometry L d - d ' × ~d' with periodic boundary conditions imposed along the d * = d - d' finite-size directions (with some length scale L), d and d' being arbitrary real parameters (0 ~ d ' < d). The general geometry allowed us to reformulate the dimensional crossover rule for finite-size systems. We also consider the role of the quantum fluctuations in the low-temperature region. The model Hamiltonian reads (see refs. [5, 6])

1 ~(P_p~

UN = ~

- A

0.~)

1

~

~ ,

B (~(2~)

+ -~ ~, alp(l, l ' ) ( Q t - Qr)" + " ~

•

(1) Here, 0 1 E ~ and /5I = - i V o (h = 1), are operators of the displacement and momentum of a particle with mass m placed at the lth site of a finite d-dimensional lattice 7/d = L 1 x . . . x L a of N=IIj__~ a Lj sites, subject to periodic boundary conditions. The parameter A =---yore > 0 determines the frequency of a mode unstable in the harmonic approximation, and the parameter B > 0 "describes" an anharmonic interaction, which is inversely proportional to the number of particles N. The harmonic force constant ~b(/, l') determines a short-range interaction of particles in the lattice. The inverse susceptibility per particle a = x - ~(q ~ o, T) (see the notations and eqs. (4) and (5) in ref. [6]) obeys the following self-consistency equation: (2)

1 + za - A w ( N , d, A) = O,

where w(N, da)=

- -1 ~, (A +

2N q

Iq12) -''2 coth ~ (A + Iql- )

)

(3)

Here the dimensionless temperature t = T / 4 E o and the quantum parameter A = vo/4E o ( E o = A2/4B is the barrier height of the double-well potential in eq. (1)) are introduced, and

q=

2"rrpl 2aXPd } LI . . . . ' Ld '

p~ = 0 , ---1, ---2,... (mod L k ) ,

(4) k=l,...

,d.

376

N.S. Tonchev I Finite-size scaling in quantum critical phenomena

From the previous investigation [6, 7] it is known that the critical properties of the model in the thermodynamic limit are determined by the asymptotic behaviour of the integral 'IT

•

d"q(a+lql'-)-'"-coth

2(2,rr)d

(a+lql2) a'2 ,

(5)

--T~

when (for a different value of the ratio A/t) A-->O +.

2. Large-L asymptotic form of the self-consistency equation In the finite-size scaling theory it is essential to evaluate the asymptotic (large LI) behaviour of the function (3) for finite values of the parameter y

=

AI'2L/2,tr,

L=

L/]

.

The approach suggested in refs. [8, 9] may be develeped for the study of quantum finite-size effects. First, using the identity z - 1 / 2 coth z 1/2

03(ilrx 112) e -:~ d x ,

=

(6)

o

where 03(iarx)=

exp(-'rt2kZx 2)

~ k~

(7)

-og

is the reduced Jacobi 03 function, we obtain from eq. (3)

e -ax ,

(8)

o

where 1

Qt. i x ) = ~

~

1

(L/.,

-

I

)/2

e x p [ - ( 2 w n / L ~ )2x]. Z expt-qZx) =Z qx tk n--(Lk-I)/2

Then setting L t = . . . .

(9)

L d. = L and taking the limit L~-* ~ for j = d* + 1,

N.S. Tonchev / Finite-size scahng in quantum critical phenomena

377

. . . , d, we present eq. (8) in the form

w(Ld', d, A) = ~

dx e-aX03 i~ ~

[Q(x)l

[~b(~rx )l (4"trx)

o where ~b('rrx1/2) =

", (10)

(X)"2

dz e -xz2 = (4"n'x)~:2 Jim L.-..~ uc

QL(X)

(11)

--T¢

is the error function. Now, we note that the following asymptotic expression holds (see refs. [8, 9]), when L --> 0o:

iEzd*

Here the prime in the summation denotes that the term with l = 0 is omitted. Further, in all terms in eq. (10) of the form

[~0rx,,2)l~'exp( Illzt2) 4X

'

Ill~0,

one may replace ~b(~rx~:2) by unity, since the contribution from small values of x is exponentially small. Thus from eqs. (10)-(12) we obtain

w(U', d, a) = w(~, d, ~) + ~w(L ~', d, a ) ,

(13)

where

tf

w(o% d,/t) = ~

(

dx e-a*03 i~r 2t

X1/2)(4,1Tx)_d/2[~(,tTXI/2)]d

0

and

~w(Ld*,d, A) = ~

dx (4rrx) -d/2 e-a~03 0

The application of the Jacobi identity

1(1)

03(i~x ) = ~ 0 3

ix

-~-

(14)

378

N.S. Tonchev I Finite-size scaling in quantum critical phenomena

to the function between brackets in eq. (15) yields after some algebra the result [

"lTd't2 L4-d f

~w(La"d'A)= ~t (2-~-L-)2

dxx

-d'/2

0

x exp~[-AL2X~ ~ } o3(i tL xl/2 ) "iT L4_ d + At (2xrL) 2

x_d,12

dx

exp

--AL2x] ~

l

0

x 03(i "-~--tLxl'2)[[O3(ix'/2)]d* --xX/('rr]d*/2 -- 1] "

(16)

We see that the lattice sum in eq. (3) can be divided into two parts: a bulk term w(oo, d, A) (eqso (5) and (14)) anti a finite-size term ~w(L d', d, A) (eq. (16)). Eqs. (2), (14) and (16) are the basis for studying finite-size effects in the model under consideration.

3. Finite-size scaling and quantum fluctuations

From the previous study [6, 7] it is known that the model system (1) exhibits a phase transition at the bulk critical temperature t c = tc(A ). In the t-A-plane, the phase diagram is sketched in fig. 1. According to the finite-size hypothesis, the thermodynamic functions close to the critical point are controlled by the scaled variable X = L / ~ , where ~:~ is the bulk correlation length. Let us see how this hypothesis will be modified in the presence of quantum fluctuations. The bulk term w(00, d, A) has an asymptotic form for small A with the leading low-HA behaviour given by (see ref. [10])

w(oo, d, A) ~ a + bA

+ c A(d-1)/2 +

qb(AA2/t 2) ,

where a, b and c are finite constants,

a,(z) = i (z + y2)-,~yd-,[exp(z

_ y:),J2 _ lldY

0

and l < d < 3 . Now the finite-size self-consistency eq. (2) ca~ be written in the form

(17)

N.S. Tonchev I Finite-size scaling in quantum critical phenomena

379

te

0

?~c

X

Fig. 1. Dimensionless critical temperature t c plotted against quantum parameter A (schematically), for d > 2 .

1 + A - A(a +

bA + c A ( d - i ) / 2 )

AL1-afa,(Lt/A,

-

A(t/A)a-'fI~(AA2]t 2)

-

(18)

A L 2) = 0 ,

where

r (Lt/Z,

A L 2) =

ld'

"ri"d'12 47r 2

tL f dz Z -d'/2 exp ( A

4rr 2 /

~-A

o

×[ [03(izt'2)]a'-(~-)a''2-1]\z/ __

1

e,

zrd*

- Ll-a°wl't'

,d,A).

Denoting by AL the solution of eq. (2) and by zl~ the solution of the corresponding bulk equation, which follows from eq. (2) in the limit I.---) 0%we define the correlation lengths ~t. = A~-1~2 and ~,: = A~:1/2 for the finite and infinite systems, respectively. In terms of the correlation lengths eq. (18) may be rewritten in the form,

(t/A)~ 1, L ] d-I

+fa,

(},

L d-' ]

L

+ff

A2

1

+ff ~

=0

From eq. (19) it follows that ~:L will scale like

A2,

l
(19)

380

N.S. Tonchev I Finite-size scaling in quantum critical phenomena ~L = L X

,

(20)

,

where X ( x , y) is a universal scaling function. Expanding this form for (t/A) ~ 1 yields the correction to quantum (t = 0) finite-size scaling [3], due to classical fluctuations.

4. F'mite-size correction and dimensional crossover

In the classical limit, the model system (1) exhibits a phase-transition at the bulk critical temperature tc = ( d - 2)(2ar)alSdxao-2, where S a = 2~rdt2/F(dl2) is the area of the d-dimensional unit sphere and x o = 2rr(d/Sd) TM is the radius of the effective sphere replacing the Brillouin zone. In the quantum limit t = 0 the phase transition is driven by the parameter A; it occurs at the bulk critical value Ac = 2 ( d - 1)(2~r)alSdxdo-1 . Let us now consider the equation 1 + A - A[w(m, d, A) + ~ w ( L a', d, A)] = 0

(21)

in the vicinity of the critical points tc(A = O) and Ac(t = O) ",cspec_,""ely. 4.1. Classical case: (t/A)--->oo

In this limit we have 03(i'tr2 ~t ZII2 ) ~ 1 ) and from eqs. (14) and (16) A--,01im[;twC'(~, a,

a)l = t

f

d z e-aZ(4~rz)-a'z[4~Orz~/2)]a

(22)

0

and

lim [A~wCt(Ld', d, A)] = t A-*O

?t

Z 4-d

(

dz

Z -d'/2

exp

o

7t t 4_ d + t (2,trL)2

dz

z_d./2

t

AL2z) 47r 2 ALZz)

exp

4,tr 2

0

X [[Os(izli2)]d'+(z)e'12--rr

1] .

(23)

N.S. Tonchev / Finite.size scaling in quantum critical phenomena

381

It is possible to express the integral in the r.h.s, of eq. (22) in terms of a hypergeometric function, i.e.

Sa

a--,o lim [Aw¢~(oo, d, A)] = t d(2.tr)d Up to terms of order I+A=

_

¢7(A/x2),

xdo [ d Xn2 ) F 1 , 1 " 1 + ; ~2 +A k ' 2 xo+A

xna

near t--- t~, we obtain (0 ~< d' < 2 < d < 4) L -d*'fftA(d"-2)t2

t + F(1 - ½d) t Atd_2~/2 +

tc +

2d,.ttd/2

(4ar)a"2F(½d') sin(½d'~)

tL4-a'trd''2 (2~L)2

fdu o

(24)

u_a,/Zexp[(-AL2] ] 4~ /ul

x[od*(iul'Z)_(~r)d*'2_l]

(25)

\ -H !

4.2. Quantum case: t/A-->O In this limit we have . (i'n'2 ~t z t / 2 ) ~ A/2~ i/2tzl ~2 O~ From eqs. (14) and (16) we thus obtain wq(oo, d,

A) = f dz (4arz)-¢d+l)'2[~b(~rz1,2)]d exp(-Az)

(26)

o

and

~a'÷1''2 L3_ef dzz_,d,+,,12exp( ~wq(L a*, d, A) = ~r (2rrL) 2 0

,ff(d'+l)/2 L 3-d

I,z:'a-L )

i

J

o

AL~) 4at-

dz Z - ( d ' + l ) f a e x p

4"a'-

The integral in the r.h.s, of eq. (26) also may be expressed in terms of the hypergeometric function, i.e.,

382

N.S. Tonchev I Finite-size scaling in quantum critical phenomena

wq(~, d , / . 1 ) -

Sd xd ( 1 d(2rr) d (x n + z ) , , z F 1 , ~ ; 1 +

d x~ ) 2'x n+A '

(27)

and up to terms of order 6(A/Xo), near A ~ Ac, we obtain (0<~ d' < 1 < d < 3) A /"(½(1 - tl)) X A(a_,):z 1 + A = ~ + 2a+~rr(a+~)/2 L - d * , f f A A ( c t ' - I )/2

+

(4~r)~a'+l)/ZF(~(d'+ 1)) s i n I l ( d ' + 1)~rl A L 3-'t"n'('~' + I)/2 f

+

du u -t~''+i):"

(2~L)Z o

×[od3"(iu"Z)--(a'r) \ ' ~ / a'/z

-- 1] .

(28)

In eq. (28) d is used for convenience. The tilde denotes that the dimensionality of the system in the quantum limit may be different from the classical one. Fro m e q s . (25) and (28) immediately follow some relations between d, d', and d, d' which ensure the finite-size dimensional crossover. They are d = d + 1,

d ' = el' + 1

(d* = d * ) .

(29)

5. Discussion We have analyzed the low-temperature behaviour of the correlation length near the critical point Ac. It is interesting to note that finite-size scaling persists in the presence of quantum duc!uations but now it takes place between d f s.s fs = 1 and d u = 3. These borderline dimensionalities differ from the lower and upper critical dimensionalities which for t > 0 are d~ = 2 and d u = 3 respectively. In the pure quantum limit, t c = 0, it has been proved [6] that a dimensional crossover occurs for the bulk system according to which the critical behaviour of the quantum d-dimensiona~ system is equivalent to the behaviour of the (d + 1)-dimensional system at t c > 0. From the relations in eq. (29) it follows that if some of the dimensions of the system remain finite the finite-size scaling behaviour of the d-dimensional system (finite in d* = d - d' directions) in the quantum limit ( a --- A~, tc = 0) is equivalent to the finite-size behaviour of the classical (A = 0, t ~ to) (d + 1)-

N.S. Tonchev / Finite-size scaling in quantum critical phenomena

383

dimensional system (also finite in d* directions). Certainly, finite-size dimensional crossover includes as a particular case the bulk dimensional crossover when d* = 0. Since model (1) is relatively simple the verification of the above formulated finite-size dimensional crossover for more complicated models is of great interest. For example the model (1) may be obtained from the Hamiltonian = A^2 1 B ~ t~(l,l')(Qt - Qr)2+ HN 2l ~t \(mP-~- Qt ) + 4 (t,r) 4 ,~tQ: '

(30) which is frequently used in the theory of structural phase transitions, by the rule (see ref. [11]) 1 I

This model (30) should be mentioned as a candidate for the future application of finite-size scaling theory in the low-temperature (T >I 0) region.

References [1] A.D. Bruce and R.A. Cowley, Structural Phase Transitions (Taylor and Francis, London, 1981). [2] G- Busiello, L.De Cesare and I. Rabuffo, Physica A 117 (1983) 445. [3] N.S. Tonchev, Physica A 148 (1988) 356. [4] M.N. Barber, in: Phase Transitions and Critical Phenomena, C. Domb and J. Lebowitz, eds., vol. 8 (Academic Press, New York, 1984). [5] J.L. Cardy, ed., Current Physics Sources and Comments, vol. 2, Finite-Size Scaling (NorthHolland, Amsterdam, 1988). [6] N.M. Plakida and N.S. Tonchev, Physica A 136 (1986) 176 [7] N.M. Plakida and N.S. Tonchev, Teor. Mat. Flz. (USSR) 72 (1987) 269. [8] J.G. Brankov and N.S. Tonchev, J. Stat. Phys. 52 (1988) 143. [9] J.G. Brankov and N.S. Tonchev, J. Stat. Phys. 59 (1990) 1431. [10] T. Schneider, H Beck and E. Sto!!, Phys. Rev, B 13 (!976) !!23. [11] S Stamenkovi6, N.S. Tonchev and V.A. Zagrebnov, Physica A 145 (1987) 262.