Thermodynamics of a mixture of fermions and bosons in one dimension with arepulsive δfunction potentialC. K. Lai Citation: Journal of Mathematical Physics 15, 954 (1974); doi: 10.1063/1.1666778 View online: http://dx.doi.org/10.1063/1.1666778 View Table of Contents: http://scitation.aip.org/content/aip/journal/jmp/15/7?ver=pdfcov Published by the AIP Publishing Articles you may be interested in Extension of Bethe ansatz to multiple occupancies for one-dimensional SU(4) fermions with δ-functioninteraction J. Math. Phys. 43, 5977 (2002); 10.1063/1.1515380 Completeness of the energy eigenfunctions for the one-dimensional δ-function potential Am. J. Phys. 68, 712 (2000); 10.1119/1.19532 Existence of solutions of integral equations in the thermodynamics of onedimensional fermions with repulsivedelta function potential J. Math. Phys. 24, 133 (1983); 10.1063/1.525583 TwoBody Orbitals for OneDimensional Fermion Gas with Application to Repulsive δFunction Interactions J. Math. Phys. 8, 1915 (1967); 10.1063/1.1705437 Interacting Fermions in One Dimension. I. Repulsive Potential J. Math. Phys. 6, 432 (1965); 10.1063/1.1704291

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Thermodynamics of a mixture of fermions and bosons in one dimension with a repulsive a-function potential

c. K. Lai

Department of Physics, University of Utah, Salt Lake City, Utah 84112 (Received 19 October 1973)

The thermOdynamics of a mixture of fermions and bosons is derived on the basis of two ansiitze about the roots of a set of algebraic equations.

I. INTRODUCTION

In a previous paper 1 (to be called I), the ground state energy of a mixture of fermions and bosons in one dimension with repulsive o-function potential was ob­tained. In this paper, we would like to show that the thermodynamics could alternatively be derived on the basis of two ansatze about the roots of the set of alge­braic equations.

The Hamiltonian of the system to be considered is

N 0 2 H = -'}] -- + 2e '}] O(Xi -x),

1 ox2i i<j c>O (1)

for Ml fermions of species 1, M2 fermions of species 2, and Mb bosons. The energy levels of the system are de­termined by the algebraic equations (1. 11a, 1. llb, 1. 11g)

II. THERMODYNAMICS

We substitute Eq. (5) into Eqs. (2)-(4) to obtain

eipL= n (-P +~' -in7]\ C(l.',n) -P +~' + in7])'

(-P' + ~ - imT/\

J) - P' + ~ + im T/ J = (_ 1)m n (~ - A' + imTl)

A' ~ -A' - im7]

x n exp ftamnl e(~ - e)], nl [ n -1

n (A - ~' - inTI) = 1, C(t',n) A - ~' + inTI

where

for 1 = ± m, 1 '" n,

(6)

(7)

(8)

eiPL - n(iP - iA' - e') - A' iP - iN + e' '

number of P = N; (2) for 1 = - (m - 2), - (m - 4)"", (m - 2),

n ( iA - iP' + e') __ n (iA - iN + e) n (iA - iA' - e') p' iA - iP' - c' A' iA - iN - e A' iA - iA' + e '

n (iA - iA' + e') - 1 A' iA - iA' - e' - ,

number of A = M;

number of A = M b ;

whereN = Ml + M2 + M b , M = M? + M b , and e' = ie. Concerning the solutions of (2)-(4), we propose the following ansatze:

(3)

(4)

Ansatz 1: When L is very large, the A's in the complex plane are located in strings, which are fermion­like: That is, a string C(~, m) is of the form

Ansatz 2: The A's in Eqs. (3) and (4) are real numbers.

( 5)

Ansatz 1 is precisely the one we used to obtain the thermodynamics of the fermion problem2 (we will refer to this work as II). As for ansatz 2, it is not obvious from the structure of Eqs. (3) and (4) that the A's should be real. But if one considers that these variables might serve as a kind of pseudomomenta of the bosons, then ansatz 2 is a plausible one.

In the following section, we will obtain the integral equations on the basis of the above ansatze. In Sec. In, the integral equations are solved exactly in special cases [c --> 0 and e --> 00]. In Sec . IV, the second vi rial coeffi­cient in the fugacity expanSion is computed. Both the special cases and the second vi rial coefficient give cor­rect results, and thus help to confirm the ansatze pro­posed.

954 Journal of Mathematical Physics, Vol. 15, No.7, July 1974

otherwise;

e(x) = 2 tan-1 (X/T/) , 1/ = e/2.

Taking the logarithm, one has

PL = 2rrlp + '}] e -- , ( ~' -P) C(t'.n) n

'E e (~ - P ') = 2rrJf + '}] 'E P' m C(f',m) I

Xa e (t - f) - 'E e(t -mA'), mnl n-1 A'

'E e ---- = 2rr KA , (A e)

C(f ',n) n

where Ip ' J" KA are integers or half-integers coming from the multiples of 2rr in logarithm. As L --> 00, the above become integral equations:

(9)

(10)

(11)

(12)

r(A) dA,

(14)

where un' unh, etc. are the "particle" density and the "hole" density of the strings C(~, m), etc. Taking the Fourier transform of Eqs. (13)-(15), one obtains

Copyright © 1974 by the American Institute of Physics

(15)

954

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c. K. Lai: Thermodynamics of a mixture of fermions and bosons

Equation (16) can be converted into

(17)

(18)

(am + (7mh) 2 cosh1]w = am+ 1 •h + am- 10k , (19)

where aolo ==P + T. Then (17) becomes

5(w)/21T = P + Pk - i cosh 1]w (e-~Iwl aOh - alh)· (20)

Now it is straightforward to obtain the thermodynamics following the method of Yang and Yang. 3 By the use of Eqs. (18)-(20) one writes

Pk/P = exp[€(p)/T], ank/ak = exp[CPn(p)/T],

Tk/T = exp[X(P)T];

and minimizes the free energy F = E - TS with the constraints:

N - = const, (M1 - M2)/L = const, MiL = const. L

One then obtains integral equations for the €(P) etc., as in (II. 23):

a =p2 - € - iT to G1 1n(1 + e-€/T)dk -00

- iT Joo G1 In(1 + e-I/I/T) dk -00

- iT 1: Go In [1 + exp(CP; )] dk, (21a)

C = -1J; + a - p2 + €, (21b)

CP1 = iT 1: Go~ln[1 + exp(~)] -In(1 + e- dT )

- In(1 + e-I/I/T)~ dk, (21c)

CP" = iT 1: Go~ In [1 + eXPe~l )] + In [1 + exp(CPn;l)] \ dk, n'?: 2, (21d)

with the asymptotic condition

(22)

The G's are kernels defined as in (II.ll). Once the € and CP's are obtained, it can easily be shown that from Eqs. (13)-(15), p, etc. are given by the following as in (n. 24):

P = - (21T)-1(1 + e€/T)-I~, T = - (21T)-1 (1 + el/l/T)-1 o1J; , oa oa

and

an •k = - (21T)-1 [1 + exp(CPn/T)]-l :~; (23)

E Joo N_ = (M1 + M2 + M b) __ Joo pdrp, - = P2p(P)dP, L -00 L L -00

Mb 100

-= TdP, L -00

J. Math. Phys., Vol. 15. No.7. July 1974

955

Finally, the free energy is given by

F N M b T 100 (Ml - M2) -=a-+C--- In(l+e-dT )dk-B . L L L 21T -00 L

(24)

III. SPECIAL CASES

Equation (21) can be solved exactly in the cases c -> 0 and c -> CXJ.

A. c"" 0

The solution in this case should give the results of a mixture of free bosons and free fermions. As c -> 0 , the kernels become delta functions and Eq. (21d) yields

exp(2CPv) = [1 + exp(CPv+l)][1 + exp(cpv-l)]'

Equation (25) has solutions

1 + exp(cp) = sinh2(n.\. + 1-I)/sinh2.\.,

(25)

(26)

with .\.T == B, and I-! is to be determined from Eqs. (21a)­(21c). This leads to

[1 + exp(- €)] [1 + exp(-1J;)] = sinh2Ajsinh21-!, (27)

exp(- E + p2 - a) = exp(- 1/.1 - C) = sinh(.\. + I-!)/sinh.

After some algebraic manipulation, one readily obtains

th cosh.\. + expC + 2(cosh.\. + coshC) co Il = -sinhA exp(p2 - a - C) - 1

and

21TT = [exp(p2 - a - C) - 1]-1,

21TP = 21TT + [exp(p 2 - a - B) + 1]-1

(28)

(29)

+ [exp(p 2 - a + B) + 1]-1, (30)

N - 2M = 1... ~ exp(p 2 _ a _ B) + 1]-1 L 21T

- [exp(p 2 - a - B) + 1]-1}. (31)

Equation (29) is just the distribution function for free bosons and Eqs. (30)-(31) are the distribution function for a mixture of free bosons and fermions in a magne­tic field B.

B. c -+ ex>

In this case, one would expect that all particles be­have like identical free fermions. This is because as the interaction strength c --7 CXJ, the exchange force due to the symmetry of the wavefunction becomes unimportant. As c -> CXJ, the integrals JG1 In(1 + e-€/T) do not contri­bute, allowing the 1J; and the CP' s to be constants:

1 + eXPCPn = sinh2(n.\. + 1l)/sinh2,\,

1 + exp(-1J;) = sinh2,\/sinh21l.

I-! is to be determined from Eqs. (21a)-(21b):

expC = sinh(.\. - Il)/ sinhI-!.

Finally Eq. (21a) gives

p2 - E - T In(expC + 2coshA) = a,

21TP = (1 + e€/T)-l.

(32)

(33)

(34)

(35)

Equation (35) with Eq. (34) gives simply the distribution

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966 C. K. Lai: Thermodynamics of a mixture of fermions and bosons

function for free fermions with each energy level being occupied by only one particle.

IV. THE FUGACITY EXPANSION

One can obtain the fugacity expansion as follows: Let 00

Substituting the above into Eq. (21), to zeroth order in z, one obtains

p2 + ln~ = HGl lnwl + Go lnbl ],

C = In(wl - 1) - HGl Inwl + Go lnbl ],

In(bv - 1) = HGo InbV+ l + Go lnbv_d;

bo == 1/wl , II ~ 1.

To the first order of z, one obtains

~ = !. [G IL + G (W2) + G (Cl)] n... 2 l-~ 1 W 0 b ' -.l 1 1

Co "'2 -==-a,.--, bo wl

(37a)

(37b)

(37c)

(38a)

(38b)

II ~ 1. (38c)

Equation (38c) has solutions

bn =ln2 = sinh2(n>. + /L)/sinh2>., (39)

where /L can be expressed in terms of C from Eqs. (38b) and (38c), Equation (38c) has solutions

Cv = 1/21T jcv(w) exp(iwk)dw,

cv(w) = A(w) {Ivlv-l exp[- I (II + 2)7]w I] (40)

-Ivlv+l exp(- IIIW7]I)},

J. Math. Phys., Vol. 15, No.7, July 1974

956

where A(w) can be expressed in terms of the a's and W 's from the initial condition of colbo in Eq. (38c). After some algebraic manipulation, one can readily obtain

expC = sinh(>. - /L)/sinh/L,

~ = (expC + 2coshA) exp(- P2), (41)

~ = [(expC + coshA)2 - sinh2>.] exp(-p2/T) j1<2

. exp(-p2/T)dP.

For simplicity, let B = 0 (>. = 0), then the pressure is given by

p =.I... j dk[~z + (a 2 -l2:t 2/2)z2 + ... ] 21T

= (1TT)1I2 ~ (2 + expC)z + [(1 + expC)2 jK exp(- 2JJ2)dfJ 211' I -./2 1 T

_ (2 + eXPC)2]Z2 + ... t • 22/3 \

This agrees with the result obtained by standard methods. For higher orders in z, the procedure is more tedious and will not be presented here.

V. CONCLUSION

We have obtained the thermodynamics of a mixture of fermions and bosons on the basis of two ansatze. Ansatz 1 has been used successfully in the pure fermion prob­lem. Ansatz 2 is new. These ansatze are likely to be the correct ones as demonstrated by the correct solu­tions given in the special cases of the integral equations. The E(P), IPn(P)'s, and 1/I(P) in the equations can be inter­preted as excitation energies in the excitation spectrum at finite temperature. The excitation .spectrum has been computed in the pure fermion problem,4 and thus will not be repeated here.

'c. K. Lai and C. N. Yang, Phys. Rev. A 3, 393 (1971). 'c. K. Lai, Phys. Rev. Letters 26,1472 (1971). 3c. N. Yang and C. P. Yang, J. Math. Phys. 10, 1115 (1969). 'c. K. Lai (to be published).

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