Two nonlinear Dicke models

Physica l&A (1977) 137-146 @ North-Holland Publishing Co.

TWO NONLINEAR DICKE MODELS*

R. GILMOREt

Institut de Physique Thtorique, Universiti de Louuain, E-1348 Louvain -la-Neuoe, Belgium

Received 12 July 1976

We consider two simple classes of nonlinear extensions of the Dicke model in order to

understand how sensitive the presence of the phase transition is to the structural form of the model

Hamiltonian. In both classes, second order phase transitions can occur for sufficiently large values

of the coupling constant A. Gap equations are derived for both classes of nonlinear extensions.

Second order phase transitions cannot occur in these classes of models unless the model

Hamiltonian contains a bilinear interaction term of the form originally proposed by Dicke.

1. Introduction

Following the proof that the Dicke model’) could support a phase transition for sufficiently large values of the coupling constant A2), a large number of other model Hamiltonians%‘) were studied for their critical properties in thermodynamic equilibrium. Each of the Hamiltonians studied in refs. 3-9 has the property that the interaction terms are bilinear in the operators of the two interacting subsystems.

In an effort to study how the nature of a phase transition depends on the functional form of the Hamiltonian, Gambardella’o) introduced a class of nonlinear extensions of the Dicke model of the form

H = ata +F(X),

x=&J;+h m (a taj + aa;)

j=l

(1.1)

(1.2)

and studied one member in this class, F(x) = x + ax*, (Y > 0. The function F(x)

* Work supported in part by the US Army Research Office, Durham, North Carolina, Grant DAHC04-72-0001.

t Permanent address: Physics Department, University of South Florida, Tampa, Florida, 33620, USA.

137

13x R. GILMORE

was assumed to be entire. Gambardella’s conclusions were:

1) The free energy exists if F(x) = x + F,(x), where F,(x) is an even function of x which is bounded below:

2) The free energy does not exist if F(-x) = -F(x), unless F(X) = x; 3) There is no phase transition for the particular mode1 F(_Y) = ,r + ax’; 4) Gambardella conjectured correctly that there is no phase transition for F(x) = x + F,(x), F,(x) f 0 and bounded below.

These results are all intimately related to the particular choice (1. I) made for the nonlinear extension of the Dicke model. The following heuristic argument illustrates this interconnection. If an ordered state exists, then in the ordered state the atomic and field operators behave to some extent like c-numbers: a, at -V/N. (r - I, and therefore X ‘- N. In the limit T + 0,

H - y,N + F(yzN), (1.3)

where y, - I in the ordered state and =0 in the disordered state. The energy per atom is an inhomogeneous function of N unless F(x) = x. It is this in- homogeneity which is directly responsible for conclusions 1-4 above.

From the foregoing heuristic homogeneity arguments, we would expect that nonlinear extensions of the Dicke model of the form

H = u tu + Nf(X/N) (1.4)

could exhibit critical behavior. Here f(x) is a real finite polynomial function of

the real variable x with finite coefficients and f(x) is bounded below. The particular choice f(x) = x + ax ‘, (Y > 0, leads under (I .4) to the Hamilto-

nian

H = uta +X+(a/N)X’ (1.5)

closest possible analog to the Hamiltonian studied in ref. 10, which is obtained from (I. 1) with F(x) = x + a.xz.

The critical behavior of the Hamiltonians (I .4) and the special case (1 S) is discussed in detail in section 2. The free energy F/N is computed to order In N/N. The result, expressed in a minimum principle over a compact two- dimensional space, is then used to discuss the critical properties of (1.5). A second order phase transition is possible for AZ > 4~’ when CYE 2 f, or for

(h’/.~)(l -a~)> 1 when GEE s;. In both cases the gap equation is given by

(2.21). The order parameters for the field and atomic subsystems are introduced in

section 3. The mean field system of coupled nonlinear equations which they obey is given for the general nonlinear extension (1.4). In the special case (1.5), the gap equation derived from the coupled nonlinear equations is identical to the gap equation derived for (I 3) in section 2.

Another class of nonlinear extensions of the Dicke model is considered in section 4. The free energy F/N is computed to B(ln N/N) and expressed in terms of a minimization in a compact two-dimensional space. The critical

TWO NONLINEAR DICKE MODELS 139

behavior is discussed and the gap equation for a second order phase transition is derived. A second order phase transition cannot occur unless the nonlinear extension contains a linear term (4.2) of the form originally introduced by Dicke’). In this case, the gap equation reduces to the Hepp-Lieb gap equation*).

2. The free energy

Lower and upper bounds on the free energy FIN can be computed using the method introduced by Hepp and Lieb4,“). First, the 2N dimensional atomic Hilbert space A = (C2)ON is decomposed into its 2J + 1 dimensional irreducible invariant subspaces, with J = N/2, N/2 - 1, . . . , l/2 or 0. The partition function is (p = l/k,T)

N/2

Z(p) = em fiN(F’N) = Tr, Tr, e- m = TrF 2 Y( N, J)Tr, e-OH. (2.1)

Here the multiplicityz3’*) Y(N, J) is the number of times the subspace J occurs in the reduction of (C2)oN :

Y(N, J) = N !(2J + 1)

(fN + J + l)!(iN -J)! = eNs(r)

’ (2.2)

where Ocr = J/N s$ and

s(r) = S(r)/N = -{(i+ r) In (i-t r) +(i- r) In (i-- r)}+ O(ln N/N). (2.3)

The asymptotic form s(r) is computed using Stirling’s formula13); s(r) has a natural interpretation as entropy14).

Within the subspace J, a lower bound on the free energy is provided by the inequality”)

23 + 1 Tr, Tr, e--OH < Tr,-

I e-BP](H) don.

4rr (2.4)

The integral is over the Bloch sphere, d0 = sin 8 do d4 is the invariant measure on the sphere surface, and 8 is measured from the north pole12.15). The operators P,(H) and Q,(H) are obtained from H by replacing each spherical tensor operator 9:(J) by its P- and Q- representatives’*), respectively’6):

9b(J) --L P,[~ii(J)I = (2J+ 1 +L)!

(25 + 1)!2L Yh(R 4)?

9;(J) --%= Q,[Sk,V)l= (2J)!

(25 - L)!2L Yi(R 4). (2.5)

The inequality (2.4) is reversed”) when P,(H) is replaced by Q,(H).

140 R. GILMORE

The polynomial operator (X)“ contains spherical tensors of order L s k. For finite L.

9ii(J)/NL = r'Yh + 0(1/N). (2.6)

Therefore, the Q- and P- representatives of the polynomial operator f(X/N) are equal to order l/N, and either may be used to compute the free energy. To

Wdeg f/N),

f WIN) Sf( +NsinB(aTe-im+aei+)). l r cos 19 + (2.7)

The partition function (2.1) can be computed by replacing the sum over J by an integral over r = J/N

:

(2.8)

0

The upper and lower bounds for F/N are given to B(ln N/N) for polynomial f

by the expression

e-BNWN) = Tr,? I I dr 47T

da exp -P[a”ia + Nf[Q(X/N)]] exp [Ns(r)].

0 (2.9)

The trace over Fock space Tr, can be estimated’) using the field coherent state representation”). For polynomial f, the estimate is accurate to 0(1/N) in the exponent, with the degree of rigor characteristic of ref. 3. For the special choice’@) f(x) = x + CYX*, a > 0, the trace can be carried out explicitly with the

aid of the useful formula

Trexp{$~(ata+aat)+R(at)S+La2+rut+la}

= (2 sinh idn 2 - 4RL))’ exp 1

R12 t Lr2 - qrl

q2-4RL I

valid for n < 0, 4RL - q 2 < 0. Then (2.9) reduces to the integral

(2.10)

0

@(r, 0,4; P) = h(r, &4)- s(r)//3 + O(ln N/N), (2.12)

h(r, &4) = (Y (er)* - r2(ae2 + A 2, sin2 0 + er cos 0

1 + 4ti (Ar)’ sin2 0

(2.11)

(2.13)


The asymptotic value of (2.11) is easily determined using Laplace’s method

FIN(P) = Min, @(r, 8,$; P). (2.14)

The function @(r, 8, C$ ; p) is independent of C$ because of the gauge invariance of the operator X (1.2). Therefore, the minimum has to be taken over the

compact two-dimensional domain r E [0, ;I, 8 E [O, nl. To investigate the possibility of critical behavior, we observe that for any

r E [O,!], 0 = 0 and 8 = P are always critical points of h(r, 0, t#a), with 8 = 0 a global maximum for any r and e = r a global minimum for r sufficiently small.

In the high temperature limit, r - -& cos 8, so that for high temperatures,

the free energy minimum occurs on the branch 0 = 7r. On this critical branch, r increases monotonically as a function of p E (0, a) to a maximum value

r I max = 29 (YE s 1,

1 r =- lmax 2ae’

(YESl, T-+0,8=7T. (2.15)

In order for a phase transition to occur at p,, the minimum of @(r, 644 ; P) must jump from the critical branch”) /3,(p) = 7~ to a critical branch 0,(B) # r. The phase transition is first order if lim8_bc 0,(p) f z-, otherwise it is second

order. For a second order phase transition to occur, the branch r!),(p) bifurcates”)

from the branch &,(p) = 7r at p,. Associated with this bifurcation is a

changeover in stability along the branch e,(p) = n. For T > T,, @(r, 0,+; /3) has a local and global minimum at 0 = r, for T < T,, @(r, 0, C$ ; p) has a local maximum at 8 = r. Thus, conditions under which a second order phase transition can occur are simply determined by investigating the coefficient of (0 - v)~ in the expansion of (2.12) about 8,, = n. This coefficient is

l r( 1 - 2aer)[t - r(h 2/~)( 1 - 2aer)l. (2.16)

In the range P E (0, m), r is positive and (1 - 2aer) is positive from (2.15). Therefore, the stability properties of (2.12) are governed entirely by the factor within [I. The minimum value of this coefficient is

(-2 , ( > for (YE 2 {, \

$1 -(A2/e)(1 -cue)], for (YE C$.

(2.17)

As a result, a phase transition will occur if

h21E > 4CXe, CYE z= +,

(A2/C)(1 -CE)> 1, LYE C;. (2.18)

When a second order phase transition can occur, the critical temperature is easily determined as follows. The value of r at which the stability changeover

142 R. GILMORE

on the branch e,,(p) = 7~ first occurs is

(2.19)

However, on this branch the value of r which minimizes @(r, 13,+; p) is determined from

1 i+r 0=2Lue’r-E+-In~

P f-r’ (2.20)

The critical temperature is given explicitly by

$+ r. pee (1 - 2a~r) = In *.

? - r. (2.2 I)

This is equivalent to the Hepp-Lieb gap equation*) in the limit (Y +O.

3. Order parameters

The order parameters19) describing the order-disorder phase transition are the expectation values of the shift operators (u), ((T;). These can be computed in a self-consistent mean field way from appropriate linearized Hamiltonians HF(w, V) for the field subsystem and HA(p, v) for the atomic subsystem. We define

where the subscripts indicate that the expectation values are to be taken with respect to the appropriate linearized Hamiltonian. These Hamiltonians are themselves expressed in terms of the order parameters CL, v’~):

H&L, u) = n 1-u + AV%f'(x){u tu + au*},

HA&. u) = 2 {&a; + A(~*cT; + pa;)}f’(x). (3.2)

]=I

Here f’(x) is the derivative of the polynomial f(x), evaluated at

x =x(/_& u) = $E(CC)A + h(l.L*v + WV*). (3.3)

The order parameters p., v are determined from the coupled nonlinear equations

p = -Avf’(x),

v = -(Ap/20) tanh @f’(x), (3.4)

t9* = &)2 + Ihk 12.


v(p) is related to the parameters r, 8, C$ of section 2 by

(3.5)

The order parameter

v = r sin 8 emi*,

where the values of For order parameters satisfying

r and 0 are chosen which minimize @(r, 8, c$; p) (2.12). (3.4) and (3.5),

lim ((a)-(a),)/*= 0, N-+=

lim ((a-) - ((T~)~) = 0. N-4

(3.6)

The gap equation for the critical temperature for a second order phase transition is obtained by linearizing (3.4) about the trivial solution CL = 0, v = 0:

1 = (A */e)f’(x,) tanh $,ef’(x,,),

x0 = x(w = 0, Y = 0) = & (a’), = -$e tanh $3,.$(x0). (3.7)

For the particular model Hamiltonian with f(x) = x + (YX’

f’(xJ = 1 + 2ax0 = 1 - (YE tanh $3,&(x0). (3.8)

The hyperbolic tangent may be eliminated between (3.7) and (3.8), resulting in a quadratic equation for f’(x,) with solution

f’(xo) = ; + +( 1 - 4crE2/A *$, (3.9a)

Cre(u*)r,= -t+#-4aE*/A*$. (3.9b)

The particular solution for f’(x,) is chosen which yields the correct value for (a,) in the limit a -0’).

It can easily be verified that the gap equation (3.7), (3.9a) is equivalent to (2.21).

4. Another nonlinear extension

We next consider nonlinear extensions of the Dicke model of the form”‘)

H = ata +i $xr~ + Ng(Y/N), j=l

(4.1)

N A Y=C-

k=, V% (a ta* + au:). (4.2)

The particular form of the nonlinear term in (4.1) is dictated by the homogeneity considerations of section 1. We assume that g(y) is a finite polynomial with finite coefficients which is bounded below for y real. This requirement is motivated by considerations leading to (4.3). We also assume, without loss of generality, that g(0) = 0.

144 R. GILMORE

Upper and lower bounds on the partition function for (4.1) can be determined using the methods outlined in section 2. The trace over Fock space can be carried out directly if g(y) is linear or quadratic, or can be estimated accurately using field coherent states, since g(y) is a finite polynomial. The free energy is

F/N = Mins{p*p +ET cos 6 in(y)-$s(r)}, (4.3)

y = Ar sin 8(~* em’* + p e’“). (4.4)

The minimum is taken in the 5 dimensional space F E C, (0, C#J) E S*, r E [0, $1.

The minimization (4.3) can be considerably simplified using

p = -hr sin 8 em’“g’(y), (4.5)

where g’ is the derivative of g(y). The gauge invariance of (4.2) reduces by a further dimension the parameter space over which the resulting minimization must be carried out:

F/N = Minz @(r, 0; p),

~(r,e;p)=h(r,8)-/3~‘s(r),

h(r, 0) = JArg’(y) sin 0/*+ Er cos 0 + g(y)

and y is now determined from the self-consistent equation

(4.6)

(4.7)

(4.8)

y = -21hr sin 012g’(y). (4.9)

The critical behavior analysis proceeds as in section 2. The function h(r, e) possesses critical points at 0 = 0, 8 = 7~ for all r E (0, i]. In the high temperature limit, r goes to zero asymptotically like -& cos 0. Therefore the high temperature minimum occurs at 0 = r. First and second order phase transitions occur as described in section 2. To investigate the occurrence of a second order phase transition we look for a stability changeover in the function @(r, 8; p) on the critical branch 0 = 7~. The coefficient of (0 - r)’ is

&r - [Arg’(O)(‘.

On the branch 0 = v the relation between r

r(P) = i tanh ~~~, 0 = %-.

As a result, a second order phase transition critical temperature is given by

, = 1hg’(o)12 tanh ‘p E. 2 c

E

and p is

(4.10)

(4.1 I)

can occur if IAg’(O)]‘/e > 1; the

(4.12)

Thus, no second order phase transition is possible unless the nonlinear extension (4.1) contains a linear term Y of the form originally proposed by Dicke’).

With order parameters p, v defined by (3.1), the linearized Hamiltonians


associated with (4.1) are

H,=ata+A~g’(y)[atv+av*],

HA = i {&CT; + Ag’(y)[+ + a;p*]}, j=l

y = A(/_L*V + pv*).

The coupled nonlinear order parameter equations

CL = -Avg’(y),

ha’(y) v = ------tanhp6,

28

e* = (‘2E)2+ IApg'(y)12.

(4.13)

are

(4.14)

The gap equation obtained by linearizing (4.14) about the thermal branch Jo = 0, v = 0 is (4.12).

5. Conclusions

Two classes of nonlinear extensions of the Dicke model have been consi-

dered. The particular structures of the models considered has been motivated by the heuristic discussion of the introduction. The free energy FIN, accurate to O(ln N/N), was computed for the first of these classes of models (1.4) in section 2. The free energy is expressed as the minimum value of a certain function @(r, 8,+ ; p) (2.12) over a compact two-dimensional domain r E 10, 11, 0 E [O, ~1. For the particular model Hamiltonian (1.5) in the class of models considered, a second order phase transition can occur for sufficiently large values of the coupling constant A (2.18). The critical temperature for a second order phase transition is given explicitly by eqs. (2.21) and (2.19).

Order parameters for the field and atomic subsystems were introduced in section 3. These obey a system of coupled nonlinear equations of the mean field type, which are derived from appropriate linearized Hamiltonians for the field and atomic subsystems. Linearization of these equations about the trivial solution provides an alternative expression (3.7), (3.9a) for the critical temperature which is equivalent to that determined in section 2.

A second class of nonlinear extensions of the Dicke model was considered in section 4. The free energy F/N was computed to 6(ln N/N) by the methods described in section 2, and the critical behavior was discussed. The coupled nonlinear order parameter equations were also derived from appropriate linearized Hamiltonians. In this class of models, a second order phase transition is possible only if the Hamiltonian (4.1) includes a bilinear coupling Y(4.2) of the form originally introduced by Dicke; the gap equation is given by (4.12).

146

References

1) R.H. Dicke, Phys. Rev. 93 (1954) 99.

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4) K. Hepp and E.H. Lieb, Phys. Rev. A8 (1973) 2517.

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1 I) E.H. Lieb, Commun. Math. Phys. 31 (1973) 327.

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13) M. Abramowitz and I.A. Stegun, eds., Handbook of Mathematical Functions (Dover, New

York, 1965), 6.1.37 p. 257.

14) See ref. 2, eq. (2.8).

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16) R. Gilmore, J. Phys. A9 (1976) 65.

17) R.J. Glauber, Phys. Rev. 138 (1963) 2529; 131 (1963) 2766.

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19) R. Gilmore and C.M. Bowden, Phys. Rev. Al3 (1976) 1898.

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Two nonlinear Dicke models