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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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