Generalized Cauchy determinant formula and its applications to Coulomb gas problems

Letters in Mathematical Physics 33: 195-206, 1995. 195 © 1995 Kluwer Academic Publishers. Printed in the Netherlands.

Generalized Cauchy Determinant Formula and its Applications to Coulomb Gas Problems

F L O R I N C O N S T A N T I N E S C U Fachbereich Mathematik, Johann Wolfgan 9 Goethe Universitiit Frankfurt, Rober-Mayer-Str. I0, 60054 Frankfurt am Main, Germany

(Received: 8 June 1994)

Abstract. Using only elementary methods, we generalize the classical Cauchy determinant formula and indicate applications to two-dimensional systems including the neutral asymmetric Coulomb gas and conformal quantum field theory.

Mathematics Subject Classifications (1991), 82B05, 82B21, 82D05, 81T40.

1. In troduct ion

There is a relatively simple determinant formula proved by Cauchy with applications

to several areas of mathematics and physics. It reads as follows: Let xi, Yi for

i = 1,. . . , N be complex variables. Denot ing

I 1 . ° .

xl - Yl X1 -- YN A = . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . (1 )

1 1

Xu - - Yl XU - - YN

the Cauchy determinant formula is

I ] (xi - x j ) (y~ - y j) 1 ~< i < j ~< N = ( -- 1) (N(S - 1 ) ) / 2 A . (2)

H ( x , - y j ) l <~i,j<N

In the case N = 2, it reduces to the simple relation

( x l - x 2 ) ( y 2 - y l )

( x l - y~)(x~ - y2)(x2 - y~)(x2 - y2)

1 1 1 1

- - ( X 1 - - Yl) (x2 -- Y2) ( x l - - Y2) (x2 -- Yl)"

196 FLORIN CONSTANTINESCU

The Cauchy determinant formula is related to other determinant formulas used in physics. For example, the connection between the Pfaffians and Haffnians

with zi E C, i = 1,..., 2N can be obtained by studying some limits of the Cauchy determinant formula (for definitions and proofs, see [10, 9]). The Cauchy determinant formula has applications in the complex function theory, representation theory of the symmetric group and symmetric functions, two-dimensional Coulomb gas, two-dimensional models in statistical mechanics, quantum field theory, etc. It can be looked at as being related to results stating an equivalence between bosons and fermions in two dimensions. For all that, the reader can consult [1, 3, 5-10].

In this Letter, we give some generalizations of the Cauchy determinant formula. Our leading idea goes back to the two-dimensional Coulomb gas but other models and interpretations like the ones mentioned above might be possible too. On the other hand, further generalizations related to Lie algebra symmetries are not un- likely.

In the second section, we discuss in some details the usual Cauchy determinant formula in the frame of the two-dimensional symmetric neutral Coulomb gas with only one species of particles preparing the third and the fourth sections. In the third section, we prove our generalizations of the Cauchy formula. In the fourth section, we apply these generalizations to the concrete example of the two-dimensional asymmetric neutral Coulomb gas. In a special case which turns out to be relatively simple, we are able to get some nice physical results. We close the Letter with a section containing a discussion and perspectives.

3. Cauchy Determinant Formula

We come back to the matrix A in (1). A proof of the Cauchy determinant formula (2) can be done by induction on n by using elementary operations on determinants (addition and subtractions of rows and columns). The inductive proof is covered in details in [-7]. Another proof can be given by resolution of singularities on the left and right sides of (2) followed by the determination of the remaining overall constant.

The interesting point of formula (2) is that coinciding points singularities on the left-hand side of (2) can be factorized in the expansion of the determinant on the right-hand side of the formula. In order to see how this works, let us consider the neutral symmetric two-dimensional gas with Coulomb point interaction. In contra- distinction to the three-dimensional classical gas, the two-dimensional classical Coulomb gas with the neutrality condition is stable [3, 5]. The interaction between the two charges is taken to be logarithmic and, as such, speaking about the two-dimensional Coulomb gas, we do not mean a three-dimensional system confined

GENERALIZED CAUCHY DETERMINANT FORMULA 197

to a plane. Nevertheless, the model is interesting in physics in the theory of infinitely long parallel charged wires or conducting wires, plasma physics and more recent theoretical investigations in two-dimensional physics. The thermodynamical proper- ties of two-dimensional gas are characterized by the partition function (qi = __- q)

Q . . . . exp qlqjln]rij] 1-I dr2, (4) 1 i = 1

where r~ is the two-dimensional position of the particle with charge q~, r~j is the relative distance ]ri - rj [, fl = ( k T ) - 1, T is the temperature, and k is the Boltzmann constant.

The volume V can be separated out by scaling giving [3]

Q = V2N+~'~J"/2)Pq'qJQ * (5)

with Q* (independent of V) given by

Q* = ~ ' " ( 1~ ]si-sj '~q'qJd2sl""dZsN (6) 3 Jo <~[si]<~l l < ~ i < j 4 2 N

where s~ is two-dimensional given by si = r iV -(1/2). The canonical pressure and the equation of state can be computed as in [3].

Certainly a condition for the existence of thermodynamics is

t "1 S 1 _flq2 ds < o(3 0

or

T > T~ = ½kq 2 (7)

obtained by looking at an N = 2 cluster (dipole) in (4). An important remark in the classical theory of two-dimensional Coulomb gas is

that Q* has no supplementary divergence at Tc arising from the collapse on N-particle clusters with N > 2 [3]. This can be neatly seen by remarking that the integrand in (6) is exactly the absolute value of some power of the left-hand side of (2) with + q-charges at x~ and -q-charges at yj (or the other way round). Using the Cauchy formula (2) and expanding the determinant, we get, by applying some elementary inequalities (including the Minkowski inequality for integrals), the following bound for Q*:

O * ~< (N !)m,x(pq2,1)(0.)N, (8)

where Q~' is a constant. Equation (8) proves the claim. Much better estimates on Q* concerning the factorial dependence can be given

showing that Q* goes with N as (N!) 1/2 and this, in turn, assures the existence of the thermodynamic limit (in fact, for all temperature) but this does not concern us here. The interested reader is referred to [3, 5].


In the next section, we raise the problem of an asymmetric classical neutral two- dimensional Coulomb gas as the starting point for the generalization of the Cauchy determinant formula contained in this Letter.

3. Generalized Cauchy Determinant Formula

As announced in the previous section, in order to present our generalization of the Cauchy determinant formula, we will consider an asymmetric two-dimensional neutral Coulomb gas consisting of N~ positive charges + n l q localized at the

positions x ~ , i = 1 . . . . , N 1 and N 2 negative charges - n 2 q at y j , j = 1 , . . . , N 2 .

We assume, without loss of generality, that nl > n2. Then the neutrality condition N i n t = N 2 n 2 = N gives N~ <N2 . If we evaluate, as in Section 2, the partition function of this ensemble, we are led to consider functions of the form

I-[ H (y i_y j )4 F = l<.i<j<.sl l<<.i<j<~N2 (9)

H - yj)"l" l <<.i<~Ni,l <~j<~N2

We show that F can be written as the determinant of a matrix very much resembling (1). In order to reduce the amount of writing, we restrict ourselves to the particular case nl = 3, n2 = 2. We will comment on the general case after completing the proofs in this particular case. The particular case retains most of the peculiarities of the problem.

The special logarithmic form of the two-dimensional Coulomb interaction reflec-

ted in the partition function, and therefore in F, enables us to apply a procedure which is best elucidated as 'point-splitting'. Indeed, the idea is to split each positive point charge + 3q located at x, into three point charges + q placed at x k, k = 1, 2, 3 and each negative point charge - 2 q located at yj into two point charges - q at y}, / = 1,2.

Let us consider the function G associated to the new neutral symmetric gas

G = ×

i ,k<k' j , l< l '

[I (xf -xf ' ) [I ( y l - y } ) x (i,k) < (j,k9 (i,z3 < (j,t') (10)

fI (x, - y}) i,k,j,l

where (i, k) < (j, l) means i < j and k < 1.


We remark that the first factor in G cancels part of the numerator of the second factor. In the limit x~ --+ xi, k = 1, 2, 3; y( j yj, l = l , 2 w e h a v e

lim G = f . (11) xki ~ X i

Y{--:'Yi

On the other hand, the second factor in G is a factor of type (1) which we denote again by A and which is characteristic of a two-dimensional symmetric gas with 2N charges +q. Using the Cauchy formula (2), it can be written as the following determinant (up to a sign). Altogether, we have

1 G = A, (12)

17I 04 - x, Fl (y} - y f ) i , k < k ' j , l < l '

where

A =

1 1 1 !

1 1 1 1

1 1 1 1

. . . . Y N 2

1 1 1 1

1 1 1 1

1 1 1 1

X~l--Yl X~l--YN2 X~l- 27 X~l--Y~2

A natural idea would be to take now x~ ~ xi and y}--+ yj in (12). This idea is obstructed by the fact that the first term in G strongly diverges in this limit. Fortu- nately, we will be able to discover these divergent factors in A and to cancel them without damaging the determinant structure of A. This goes as follows.

We subtract in A the first row of the determinant from the (N1 + 1)th row, the second row from the (N1 + 2)th row up to the Nl th row which we substract from the 2Nlth row. We continue by subtracting the first row from the (2N~ + 1)th row, the second from the (2N1 + 2)th up to 2N~th from the 3Nlth. We get

A

1-I ( x l - _ i


1 1 1 ~ I - y l x l - y L x I - y~

1 1 1

1 1 1 . . .

( ~ I - y l ) ( x ~ - y l ) ( X I 1 2 1 - - YN2)(X1 - - YN2) (x I -- y~)(x 2 -- y~)

1 1 1

( x ~ - y l ) (x~, l y l ) ( x L 1 2 - - YN2)(XN, -- Y}2) (XL -- Y~)(XL -- Y~)

1 1 1 ( x l - y { ) ( x ~ yl) (xl 1 3 _ _ Y N 2 ) ( X l _ y l ) (X 1 _ y 2 ) ( X ~ _ y 2 )

1 1 1

( X ~ 1 - - Y I ) ( X 3 1 - - Y l ) (.~N1 ~ 3 1 - y N ~ ) ( x ~ - y N J ( x L - y ~ ) ( x L - y~)

N o w substract the (N1 + 1)th row of the previous determinant from the (2Ni + 1)th row, etc, and get

A

[I (x~ - x ~ ' ) i,k <k'

1 1

x I - y l x l - y ~

1 1

(x{ - y l ) ( x ~ - y I ) ( x l - y ~ ) ( ~ - y~)

1 1 (xl - yl)(x 2 - y l ) ( x ~ - y l ) (xl - y 2 ) ( x ~ - y 2 ) ( x ~ - y 2 )

(14)

Taking into account relation (12), we In (14), we can perform the limit xi --, xi.

get

1 lira G - B, (15)

x>x, H (y~ - y~') j , l < l '

GENERALIZED CAUCHY DETERMINANT F O R M U L A 201

where

B =

1 1

x~ - y l x l - y2

1 1

( x l - y l ) 2 ( ~ - y~)~

1 1

(x~ - 21) 3 ( ~ - y ~ ?

(16)

We are left with the y-singularities in (15). In order to cancel them, we start with the determinant B and subtract the first column from the (N2 + 1)th column, the second column from the (N2 + 2)th column, etc. In this way, the factors y ) - y2 are segregated from B and cancel the corresponding divergent factors in (15). In the limit y} ~ y j , we get from (15) up to a sign

F = lim k l Xi ~ X i , Y j ~ y j

Finally, we get

1 1

X1 - - Yl (X1 - - y l ) 2

1 1

( x l - y l ) 2 ( x l - y l ) 3

2 3 (X 1 - - y l ) 3 (Xl - - y l ) 4

(17)

(x , - xj) 9 I] ( y i - yj)4 l ~ i < j ~ N a l <~i<j<~N2

where

N1,N2

I ] (xl - yj)6 i , j= 1

A

= A 2

A3

m 2

2A3

3A4

(18)

A i

1 1 ( x l - y l ) i ( x l - yu2) ~

1 1

(XN1 - - y l ) i ( X N 1 - - y N 2 ) i

, i = 1 , 2 , 3 , 4 .


In order to obtain the general formula for a gas with arbitrary charge clusters nlq, - n2q respecting the neutrality condition nlN1 = n2N2 = N, we perform first, as above, row subtractions in the usual Cauchy N x N determinant which emerges after the point splitting of the charged clusters. This subtraction produces the necessary factors to kill the singularities II(x~ - x~") for k < k'; k, k' = 1 .... , nl. The cancellation of the y-singularities has to be achieved by the corresponding column subtractions. Indeed a typical row in the determinant before starting column subtractions looks like

I f ( y i ) . . . f (y2) . . . f (y3) . . . I,

where f is a function of the type

1 f ( y t ) - ( . _ y l ) p, l 1 , . . . ,n2,p 1, . . . ,h i

with x-dependence suppressed. In the first step of the column subtraction, we subtract f ( y l ) from f(y2), f(y3),

etc., and use part of the y-singular factor to get the typical row

f ( 1, f ( y 2 ) _ f ( y i ) f ( y S ) _ f ( y i ) . y ~ - - ~ - T r ... y 3 _ y ~ . . .

Continuing the procedure, we get, in the next step,

f ( f ( y 2 ) _ f ( y i ) f ( y 3 ) _ f ( y l ) f ( y 2 ) _ f ( y l ) y l ) . . . ~ _ ~ T ... y 3 _ y ~ y 2 _ y l ""

and so on. In this way, all x- and y-singular factors have been cancelled. It remains to make sure that the above iterated differential quotients exist in the limit x~ -o xi and y j -o yj and to compute them. In order to solve this problem, we introduce the following simplified iterative notation:

y l ) = f ( y i ) ,

yl , y2) _ ( y 2 ) - (Y~) y2 __ yi ,

y~, y~, y~) = (yl, y~) _ ( # , y2) y3 _ yi

and, generally,

( y l , y2 . . . . . yn y , + l ) = ( y l , y 2 , . . . , 37, , y n + i ) _ (y~, y 2 , . . . , y,,) y n + l _ y,

where /~ means omission.


Now we use the following lemma which can be proved by induction using the Taylor formula and assuming the existence of enough derivatives of f :

In the limit of yl, y2 .... ,y" going to y, the generalized differential quotient ( yl, y2, . . . , y,) converges to (1/n!)f"(y) .

Putting all this information together and taking the trace of the sign, we get our general Cauchy determinant formula

1-I (x, x -- J H ( Y i - YJ) "~ l <~i<j~N1 l <<.i<j~N2

1-I (x, - yj). l .2 1 <~i<~Nt,1 <~j~N2

All ... Aln 2 = (__ 1)(N(N-1))/2 . . . . . . . . . . . . . . . . . . . . , (19)

rA.11 ... A . . . . I

Akl = ( k

where

+ 1 - 2 " ~ ( x l - )k+l-1 ( x l - - y 2) k+~-a

k - - 1 J

The expression in front of the determinant in the last equation stays for the binomial coefficient.

This formula is particularly simple if either nl or n2 is equal to unity. In this special case, the binomial coefficients are all equal to one. In the next section, we will give a simple physical application of this special case of the formula. Another more sophisticated application within the framework of the conformal perturbation theory for two-dimensional systems, is contained in [-2]. One can prove that off-critical perturbations of the two-dimensional rational eonformal field theory given by scalar primary operators with scaling dimensions smaller than (½, ½) produce a convergent perturbation expansion in finite volume.

In the general case, formula (19) shows charge symmetry as it should be. As a consequence, the binomial coefficients in the determinant are certainly symmetric under the permutation of k and 1.

Until now, we did not make restriction on nl and n 2. Let us remark that the new determinant formula is not economic if nl and n2 have a common divisor larger than one. In particular, one could imagine that nl = nz = n > 1. If the total number of particles is denoted by 2N, then the original Cauchy formula produces, for the nZ-power of the left hand side of (2), the nZ-power of the Cauchy N x N determinant, whereas by applying our generalized procedure, we get the first power of an nN x nN-determinant. Certainly, if the result is correct, these two determinants


should be equal. For the convenience of the reader, we give a hint to the proof of this statement in the very particular case in which N1 = N2 = 1. First, let us remark that the overall power of the difference x - y agrees because n 2 = 1 + 3 + ... + (2n + 1). As a consequence, the problem boils down to a determinant identity which is easy to prove by means of some elementary relations between binomial coefficients and rows (columns) manipulations. In the case n = 4, for instance, it reads

1 1 1 1

1 2 3 4 = 1 .

1 3 6 10

1 4 10 20

4. Application: The Asymmetric Two-Dimensional Coulomb Gas

We consider here the classical neutral asymmetric two-dimensional Coulomb gas of the charge-species nq, - q , n > 1. Following the analysis of the symmetric case in Section 2, we find, by applying the generalized Cauchy formula, that the critical temperature Tc is related to the formation of neutral (n + 1)-particle clusters and there is no divergence at Tc arising from the formation (collapse) of clusters with more than (n + D-particles. Furthermore, the critical temperature in the asymmetric case can be read off to be n-times the critical temperature in the symmetric case.

Indeed, after applying the scaling on the partition function as this was done in Section 2, we use our Cauchy formula to obtain in the scaled partition function the ]~q2-power of the determinant

A 1

n

where

1 ... 1 t (X 1 - - y l ) i (X1 - - yNz) i

A i = , i = 1,. . . ,n. l ' 1/ i i (xNi - y l ) (xN1 - yN2)

(20)

We expand the determinant (20) by applying the Laplace rule and obtain a product of Vandermonde determinants which can be computed explicitly.

The same procedure as the one described in Section 2 (inequalities for the power of the determinant absolute value), proves the claim.


In the general case of arbitrary nl and n2, the resolution of the singularities in the partition function is no longer elementary; some more involved analysis seems to be necessary. This is an interesting problem which deserves further study.

Before closing this section, let us remark that beside the applications mentioned in this Letter (classical asymmetric Coulomb gas and two-dimensional conformal quantum field theory), we believe that the generalized Cauchy determinant formula could also be used for further study of the fermionic-bosonic equivalence in two dimensions in its different disguises. Other possible applications will be mentioned in the next section.

5. Discussion

We have given a generalization of the Cauchy determinant formula keeping in mind the analogy with the partition function of the classical, neutral two-dimensional Coulomb gas. The present generalization can cope with a possible charge asymmetry of the gas components. Via Coulomb gas representation of the two-dimensional conformal quantum field theory [4], our formula is well-suited for the study of correlations in some particular cases of the conformal quantum field theory and its perturbations [2]. General minimal models and the Wess-Zumino-Witten model suggest searching or an even more general formula which could accommodate charges produced by more general representations of the corresponding infinite- dimensional Lie algebras of the Virasoro and Kac-Moody type. This is an open question.

Returning to the statistical mechanics of the classical, neutral (possible asymmetric) Coulomb gas in two dimensions, we have nothing to say here about more interesting physical phenomena like Debye screening and Kosterlitz-Thouless tran- sition (see, for instance, [9]). Nevertheless, the present results offer more possibilities for application than just those considered in Section 4. For instance, one could rigorously study the stability problem and thermodynamic limit of classical and quantum two-dimensional Yukawa and Coulomb systems with asymmetric charge distribution on the lines of [5] by using the equivalence of the Coulomb gas with the sine-Gordon (Toda) theory.

Acknowledgement

The author is thankful for discussions with R. Flume and W. Boenkost.

References

1. Coleman, S., Phys. Rev. D 11, 2088 (1975). 2. Constantinescu, F. and Flume, R., Phys. Lett. B 326, 101 (1994). 3. Deutsch, C. and Lavaud, M., Phys. Rev. A, 9, 2598 (1974). 4. Dotsenko, V1. S. and Fateev, V. A., Nuclear Phys. B 240, 312 (1984).


5. Fr6hlich, J., Comm. Math. Phys. 47, 233 (1976). 6. Fr6hlich, J. and Seiler, E., Helv. Phys. Acta 49, 889 (1976). 7. Hamermesh, M., Group Theory, Addison-Wesley, New York, 1962. 8. Hua, L. K., Harmonic Analysis of Functions of Several Complex Variables in the Classical Domains,

Amer. Math. Soc., Providence, 1963. 9. Itzykson, C. and Drouffe, J.-M., Statistical Field Theory, vols 1, 2, Cambridge Univ. press, 1989.

10. Itzykson, C. and Zuber, J.-B., Quantum Field Theory, McGraw-Hill, New York, 1980.

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Generalized Cauchy determinant formula and its applications to Coulomb gas problems