- Email: [email protected]
Exclusion statistics and strongly correlated electron systems
I [IIrAB ELSEVIER
Physica B 212 (1995) 315 319
Exclusion statistics and strongly correlated electron systems M.V.N. Murthy*, R. Shankar The Institute of Mathematical Sciences, Madras 600113, India
Abstract
We generalize Haldane's definition of exclusion statistics to systems with infinite dimensional Hilbert spaces. We then show that the statistics parameter is completely determined by the high temperature limit of the second virial coefficient, when the virial expansion exists in this limit. Next we address the question as to when quasiparticles with nontrivial exclusion statistics can occur in interacting electron systems and argue that they would occur in systems where Fermi liquid theory breaks down. We then discuss some explicit examples in one dimensional systems where this is known to happen. From these we abstract general constraints on the occupation numbers for exclusion anyons. Finally we give examples of some physical systems, spin chains and impurity problems, where the phenomenon is predicted to occur.
A few years ago Haldane [1] proposed a definition of statistics of particles based on a generalized exclusion principle. He considered many particle systems with finite dimensional Hilbert spaces where the dimension of the single particle Hilbert space depends linearly on the total number of particles present. Then the exclusion statistics parameter g is defined by Ad = - gAN, where Ad is the change in the dimension of the single particle space and AN is the change in the number of particles. Thus g is a measure of the (partial) Pauli blocking in the system [2]. g = 0(1) corresponds to bosons (fermions). Haldane applied this definition to two cases: quasiparticles in fractional quantum Hall effect (FQHE) systems and spinons in quantum antiferromagnets. In both cases he argued that the exclusion statistics parameter, g, is equal to the exchange statistics parameter, ~. The generalized exclusion principle is thus expected to be realized in physical systems. Johnson and Canright [3] have tested this prediction numerically for F Q H E systems with ~ = 13 and find it to be true with the modification that, for quasiparticles, g = 2 - ½.
* Corresponding author.
In this paper we first briefly review our earlier work [5] where we had generalized Haldane's definition to the case where the Hilbert space of the particles is infinite dimensional and had shown that in general the exclusion statistics parameter g is proportional to the dimensionless second virial coefficient provided the system admits a virial expansion of the equation of state in the high temperature limit. We then qualitatively argue that in interacting fermion systems with nontrivial values of g, we should expect the breakdown of Fermi liquid theory. We then examine in detail the spectrum of a solvable model in one dimension, the Calogero-Sutherland model, which we have earlier shown [6] gives an explicit realization of the phenomenon. We abstract from this model the general constraints on the occupation numbers for exclusion anyons [4]. Finally, we briefly discuss some physical models (other than FQHE) where such quasiparticles are expected to occur. We now describe the generalization of Haldane's definition to systems with infinite dimensional Hilbert spaces. The motivation to do so is so that the definition can apply to continuum systems with no gaps in the spectrum where there is no natural way of defining a finite dimensional low energy space. The dimension of
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 0 4 8 - 8
M.V.N. Murthy, R. Shankar/Physica B 212 (1995) 315 319
316
the Hilbert space of N identical particles when y is the statistics parameter is given [1, 3] by (d DN(g)
+
(1
-
-
1 -
9)(N
1))!
-
=
(1) N!(d
g(N
-
1))!'
where d is the dimension of the single particle space. It is easy to see that g = 0(1) yields the dimension of the bosonic (fermionic) Hilbert space. It is now straightforward to extract the statistics parameter 9 from Eq. (1) in the limit when the single particle dimension d ~ ~ . We obtain lim
½--g=e~N(N-
d
1)
N!
dN
-- 1
"
(2)
A regulated definition of the dimension of the Hilbert space is given by the corresponding N-particle partition function since we have DN = l i m ~ o Z N = l i m ~ o Tr(e-~nN), where fl is the inverse temperature and HN denotes the N-particle Hamiltonian. Therefore, for dealing with infinite dimensional Hilbert spaces, we propose a generalization of Haldane's definition,
CZ1 l- ZN ½ - g = a -l oiNm( N-- 1- )/[ N _ ! ~Z?-
-]
lJ
'
(3)
where C = 2 a,, ds being the spatial dimension. The factor C occurs because the regulated definition cuts off the energy at 2fl when two particles occupy the same state; however for counting purposes we want to cut off all states at the same energy irrespective of their occupation. It is then possible to show [5] that if all the virial coefficients are finite in the f l - , 0 limit, then the righthand side of Eq. (3) is independent of N and is equal to CB2, B2 being the dimensionless second virial coefficient. Thus the Haldane exclusion statistics parameter V is completely determined by the high temperature limit of the second virial coefficient. We can now apply this result to some well known systems. We first consider anyon gas. Here the second virial coefficient is well known and is given [7] by
are finite, then the result given in Eq. (5) would be true for anyon gas in general. Next consider anyons confined in a magnetic field. Once again 9 is given by Eq. (5) when the partition function includes the trace over all Landau levels. This is easy to see since both the oscillator frequency o9 and the cyclotron frequency ogc act as regulators and at high temperatures one recovers the anyon gas result. If however we confine the trace to the lowest Landau level, then we obtain 9 = a for quasiholes and 9 = 2 - a for quasiparticles in conformity with the results obtained for F Q H E systems [3]. It is instructive to look at the second virial coefficient in this case as a function of temperature. As shown in Fig. 1, in the range flo9 ~ 1, flo9¢ >> 1, only the lowest Landau level contributes, Bz is independent of temperature and corresponds to g = 2 - ~. When flo9¢ is of the order of 1, the other Landau levels start contributing and finally in the range flogc .8 1, B2 is again temperature independent and corresponds to 9 = c~(2 - ~). We would like to stress here that this does not mean that 9 as defined by us is a temperature dependent property. Nor is it a high temperature property of the system. The high temperature limit of the partition function is taken in order to count all the states. All states, low and high energy states, contribute equally to the partition
1.5 a:O.2
1
-
0.5 "~.
0
2~ 2 -- 4c~ + 1 B2 -
4
(4)
where ~ = 0 is bosonic and c¢ = 1 is fermionic. Using the relation between B2 and g we have
-0.5
0.001
g
=
~(2 -
o¢)
(5)
which has the correct limit for ~ = 0, 1. This result would be true for t w o anyons. However, it is not conclusively proved that the higher virial coefficients are finite in the case of anyon gas. It has been proved that the third virial coefficient is finite [8, 9]. If indeed all virial coefficients
........
i o.ol
........
i
........
o.1 8 [ , 0 = o.1 , %
i 1 = z.o]
, , ,~
a=0.o , lo
1 loo
Fig. 1. Second virial coefficient B 2 of anyons in a magnetic field and confined in a harmonic oscillator potential as a function of inverse temperature/3. The values ofo9 (oscillator frequency) and ~o, (cyclotron frequency of the magnetic field) are explicitly shown.
317
M. KN. Murthy, R. Shankar/Physica B 212 (1995) 315 319
function in this limit. The Hamiltonian serves the purpose of ordering the states according to energy. To define a finite difference of two divergent series it is necessary to order them. Energy gives us a physical basis for doing so. Thus g does not depend on the details of the Hamiltonian but only on the ordering. However g can be scale dependent. This can be seen and understood in the ~ = 0 case (free bosons) of the above example (see Fig. 1). At high temperatures (/~oc ,~ 1), the magnetic field is irrelevant and the system behaves like a free Bose gas in two dimensions. In the range fl¢o ,~ 1, flo)c >> 1, the system effectively behaves like a one dimensional free Bose gas. Thus the second virial coefficient and hence g have the two corresponding plateaus. Exactly the same thing happens for the nonzero values of ~ also. Therefore g can be scale dependent. In fact this is how we expect nontrivial values of g in physical systems of interacting electrons. Although the full Hilbert space may obey the dimension formula corresponding to fermions (g = 1), the low energy sector could deviate from it and correspond to some nontrivial value of g. We now qualitatively address the question as to when can such quasiparticles occur in a system of interacting fermions. The analysis above shows that the first thing to do is to examine the high temperature limit of the second virial coefficient. Consider then a system of quasiparticles with a constant density of states (as would be the case for quasiparticle energies small compared to the Fermi energy). ZI is equal to y/[3, where ? is the density of states. We write the two particle energy as E2 = t:l + g2 + A(el,g2)"
(6)
el. 2 are the single particle energies and A the interaction energy. The second virial coefficient can then be calculated and we have 1--g=
~irno?
fl
d x , dx2e-¢X'+X~)A
.
(7)
Thus for g to have a value other than 1, the energy shift A should remain nonzero at all energies whatsoever high. If.it goes to a constant at high energies, i.e. if lim Z~(~ 1, ~2) --1" Z~, 1~1,152 ~
oD
then 1 -- g = 73.
(8)
What is essentially happening in this case is that the addition of a particle causes a shift in the energy of all the levels. Thus when we count the dimension of the single particle Hilbert space with any cutoff, it changes since some of the states have been pushed out of the cutoff. If the energy shift remains at all energies, then g is non-
trivial wherever the cutoff is. Now this is precisely the scenario associated with the breakdown of Fermi liquid theory [10]. Note that A is related to the Landau parameters, f, by A = (1/Las)f, where L is the system size and d~ the spatial dimension. So for A to survive in the thermodynamic limit, we must have singular Landau parameters. This is what is expected to cause the breakdown of Fermi liquid theory [11]. Thus these qualitative arguments lead us to expect quasiparticles with nontrivial values of exclusion statistics parameters in interacting fermion systems in non-Fermi liquid phases. In fact the concurrence of exclusion statistics and non-Fermi liquid phases has been earlier stated by Anderson [ 121 in the context of the normal state of the cuprate superconductors. The well-known and well-studied examples of such systems are interacting fermion systems in one dimension. Consider one such system, the Calogero-Sutherland model (CSM). This is a model of a system of interacting fermions. The hamiltonian is given by
i=1
-2~xZ, +
¢°2x~
+ 2 , < ~ = l ( x i - x j ) 2"
(9)
The particles are confined in a harmonic well and the thermodynamic limit is obtained by taking o) ~ 0. It has been shown earlier [6, 13, 14"1 that this system behaves like an ideal gas of particles with fractional exclusion statistics. We will now review that result and then clarify how the states obey the dimension formula in Eq. (1). We will then abstract a general rule for constraints on the occupation numbers that leads to the formula. The spectrum of CSM is exactly known. The states can be labelled by a set of fermionic occupation numbers {nk}, k = 1. . . . . oo, nk = 0, 1. The energy is given by E[{nk}] = k=l ~ eknk + o2 N(N2
1) '
(10)
where ek = ko), N = ~k% 1 nk. We note that this spectrum is identical to the spectrum of quasiparticle states in a Gaussian theory of compact bosons [5] with radius R = 1 / w / ~ and with the following identification while k in the above equation is a state label in CSM, in the Gaussian theory of compact bosons k refers to the box quantized momenta. The following discussion thus applies to this case also. Since the bosonic theory describes a Luttinger liquid, our considerations apply to the low energy physics of a large class of one dimensional interacting fermion systems. The simplicity of the spectrum makes it possible to explicitly compute the right-hand side of Eq. (3) explicitly, since we have Z ~ = e x p ( - f l o 2 ( N ( N - 1 ) / 2 ) ) Z ~ , where ZvN is the N-particle partition function of noninteracting fermions. The answer is indeed independent of
318
M. KN. Murthy, R. Shankar/Physica B 212 (1995) 315-319
N and we get g = 1 + 2. Thus the particles in the C S M are anyons with the exclusion statistics parameter, g, equal to 1 + 2. To bring out the ideal gas nature of the system, we define the shifted single particle energies as eA(ki,g ) = ek, -- O(1 -- g)(i -- 1).
(11)
Here k~ are the occupied states and are labelled such that k~+ ~ > k~. We identify this shifted energy with the anyon single particle energy since the total energy can now be written as N
E[{k,}] = ~ eA(k,,g).
(12)
i-I
We now break up the energy space into cells of size m and define the anyon occupancy n~ of the kth cell to be the number of anyons with energies in the range k o ~ eA < (k + 1)o). It has been shown in Refs. [6, 13] that the thermal average of these occupancies is given by
1 ~o(ek) + g"
fig - -
Here 8k equation
=
(13)
ko) and w(e) is given as the solution of the
wg(e,g)(1 + w(e,g))1-0 = et~(~-~),
(14)
where fl is the inverse temperature and/~ is the chemical potential of the system. Furthermore the grand partition function factorizes and can be written as lnZ A =
dxln(1 + w - a ( x ) ) ,
(15)
where x is the dimensionless energy defined as x = e/e). It follows from Eqs. (15) and (14) that n-A k =
~xk in zA"
(16)
This shows that the C S M has the basic properties of an ideal gas of particles with fractional exclusion statistics. The inverse square interaction can thus be looked upon as a purely exclusion statistical interaction in one dimension. To get some feel for the anyon number densities, let us consider the ground state of the N-particle system. The set {kl} is (1,2, 3 . . . . . N). The corresponding set of shifted energies is {~o~(1 + g)~o,(1 + 2g)~o. . . . . (1 + (N - 1)g)~o}. The number of particles in the interval e to e + co is then =l/g, nA(t~'O)
0,
e < ~F = gp, /: /> /~F,
(17)
where ~ - coN is the average density. This is the same as the zero temperature limit of the distribution function in
Eq. (13). It is also clear from the above example that the maximal occupancy of any cell, nmax A < 1/g + 1. We will now clarify the connection between the states of the C S M and the dimension formula in Eq. (1). Consider the single particle states with a cutoff on the unshifted single particle fermion energies, kmax = dE. The number of cells occupied by the shifted anyon energies is then d = dr - (1 - g)(N - 1) (a fractional value of d implies that the cell with the maximum energy is never maximally occupied). The total number of states is D = a~CN, since the anyon energies are in one to one correspondence with the unshifted single particle fermion energies. When dF is expressed in terms of d in the above expression, we recover Eq. (1). As we saw above the fermion occupancies, {nk} are in one to one correspondence with the anyon occupancies {hA}. It is also clear that-this results in constraints on the anyon occupancies. We have already come across one constraint, namely nmax A < 1/g + 1. We will now write down the complete set of constraints for the case g = P/q where p and q are coprime integers, p < q. To do so we note that A - - ~klA ~k~+l
= o)(ki+
1
-- ki + p/q).
(18)
The above equation says that neighbouring fermion levels, (ki+l - ki = 1), correspond to neighbouring anyon levels (ek,~, A -- e~) = O)~. Thus "close packed" fermion configurations go over to "closed packed" anyon configurations. Also we see that if ki÷ a and ki are not neighb o u r s t h e n e k~+l A and ekA have to lie in different cells. Thus if in the lth cell, the lowest anyon level is tn((l - 1) + re~q), then that cell can have an occupancy n A <~ ( q - m)/p + 1 = n m a x - m / p . An anyon can have an energy of ~o((l - l) + re~q) iffthe fractional part o f N i p / q is equal to re~q, i.e. if m = p N t m o d q, Nt being the number of anyons with energy less than o)(l - 1). The above two statements imply that n A satisfies the constraint p(nmax A -- np) >~ pN~ mod q. Also if the lowest anyon energy in the cells is ~o((l - 1) + re~q), then any allowed value of n~ is realized since there will always be a corresponding fermionic configuration with the appropriate number of fermions close packed above the one corresponding to the lowest anyon energy in the cell. Thus we have shown that if the sets of anyon occupancies {n A} satisfy the following two constraints: n A < q/p + 1,
(19)
A -- n A) >/ p N i mod q, p(nmax
(20)
then the number of states with N = Y.t n p is equal to the right-hand sideEq. (1). Therefore we can now remove the scaffolding of the model and define anyon occupancies by the above two constraints. These are then the general constraints on the anyon occupancies.
M.V.N. Murthy, R. Shankar/Physica B 212 (1995) 315-319
Note that with these constraints, the problem of "negative probabilities" pointed out in Ref. [16] does not arise. That problem comes if one tries to reproduce the distribution function in Eq. (13), assuming that there are no constraints on the anyon occupancies, e.g. it occurs when g N > d. This is explicitly prohibited by the constraints. Finally we mention some physical systems where such quasiparticles are predicted to occur. FQHE systems are of course an example. The normal state of the cuprate superconductors is also believed to be non-Fermi liquid state. However the theory of these two dimensional systems is as yet incomplete and inconclusive. The theory of interacting electron systems in one dimension is unambiguous. Fermi liquid theory does break down and such quasiparticles do exist. Thus they should occur in quantum wires, if the correlations play an important role in their physics. The isotropic quantum antiferromagnetic chain is another physically realizable model. This model can be mapped on to the Luttinger model by the Jordan-Wigner transformation. The low energy physics is then described by a Luttinger liquid with the radius parameter equal to 1/,~//2. Therefore the system would have quasiparticles with g = ½, the spinons. The overscreened Kondo impurity systems and another class of models where such quasiparticles are expected to exist. The recent solution of these models using conformal field theory [17] predicts nontrivial quantum numbers for quasiparticles. In particular, for the s = ½, two channel problem, which may have a physical realization [18], quasiparticles with g = ½ are predicted. In conclusion, Haldane's definition of exclusion statistics, when generalized to infinite dimensional Hilbert spaces, is determined by the high temperature of the second virial coefficient if all the viral coefficients are finite in this limit. It is then possible to argue qualitatively that the occurrence of quasiparticles with nontrivial values of g will concur with the breakdown of Fermi liquid theory. Indeed in one dimensional interacting fermion systems, both these phenomena do concur. Simple solvable models in one dimension have been analysed in detail to abstract the general rule for anyon occupancies. There are physically realizable models with quasipar-
319
ticles having a nontrivial value of g- It is clearly of interest and importance to pinpoint signatures of the nontriviat exclusion statistics in these models that can be experimentally seen.
References [1] F.D.M. Haldane, Phys. Rev. Lett. 67 (1991) 937. [2] See also A.G. Izergin and V. Korepin, Lett. Math. Phys. (1982) 283. [3] M.D. Johnson and G.S. Canright, Phys. Rev. B 49 (1994). [4] Note that, hereafter, we use the name anyon genericallyfor all particles with fraction statistics-exclusionof exchange. [5] M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. 72 (1994) 3629. [6] M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. 73 (1994) 3331. [7] D.P. Arovas, J.R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B 251 (1985) 117; J.S. Dowker, J. Phys. A 18 (1985) 3521. [8] J. Myrheim and K. Olaussen, Phys. Lett. 229 (1993) 267. [9] R.K. Bhaduri, R.S. Bhalerao, A Khare, J. Law and M.V.N. Murthy, Phys. Rev. Lett. 66 (1992) 523; Diptiman Sen, Phys. Rev. Lett. 68 (1992)2977; M. Sporre, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B 389 (1993) 645; A. Dasnieres de Veigy and S. Ouvry, Phys. Lett. B 291 (1992) 30; J. Law, A. Suzuki and R.K. Bhaduri, Phys. Rev. A 46 (1992) 4693. [10] P.W. Anderson, Phys. Rev. 164 (1967) 352. [11] P.W. Anderson, Phys. Rev. Lett. 64 (1990) 2306; 66 (1991) 3226; P.C.E. Stamp, Phys. Rev. Lett. 68 (1992) 2180. [12] P.W. Anderson, Progr. Theor. Phys. (Suppl.) 107 (1992)45. [13] S.B. Isakov, Phys. Rev. Lett. 73 (1994) 2150; Yong-ShiWu, Phys. Rev. Lett. 73 (1994) 922; D. Bernard and Yong-Shi Wu, University of Utah Preprint UU-HEP/94-03 {condmat/9404015). [14] Z.N.C. Ha, Phys. Rev. Lett. 73 (1994) 1574. [15] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197; B. Sutherland, J. Math. Phys. 12 (1971) 246, 251; Phys. Rev. A 4 (1971) 2019; 5 (1972) 1372. [16] C. Nayak and F. Wilczek,Phys. Rev. Lett. 73 (1994) 2740. [17] I. Altleckand A.W.W. Ludwig,Nucl. Phys. 360 (1991) 641. [18] D.C. Ralph, A.W.W. Ludwig, Jan Von Delft and R.A. Burham, Phys. Rev. Lett. 72 (1994) 1064.
Physica B 212 (1995) 315 319
Exclusion statistics and strongly correlated electron systems M.V.N. Murthy*, R. Shankar The Institute of Mathematical Sciences, Madras 600113, India
Abstract
We generalize Haldane's definition of exclusion statistics to systems with infinite dimensional Hilbert spaces. We then show that the statistics parameter is completely determined by the high temperature limit of the second virial coefficient, when the virial expansion exists in this limit. Next we address the question as to when quasiparticles with nontrivial exclusion statistics can occur in interacting electron systems and argue that they would occur in systems where Fermi liquid theory breaks down. We then discuss some explicit examples in one dimensional systems where this is known to happen. From these we abstract general constraints on the occupation numbers for exclusion anyons. Finally we give examples of some physical systems, spin chains and impurity problems, where the phenomenon is predicted to occur.
A few years ago Haldane [1] proposed a definition of statistics of particles based on a generalized exclusion principle. He considered many particle systems with finite dimensional Hilbert spaces where the dimension of the single particle Hilbert space depends linearly on the total number of particles present. Then the exclusion statistics parameter g is defined by Ad = - gAN, where Ad is the change in the dimension of the single particle space and AN is the change in the number of particles. Thus g is a measure of the (partial) Pauli blocking in the system [2]. g = 0(1) corresponds to bosons (fermions). Haldane applied this definition to two cases: quasiparticles in fractional quantum Hall effect (FQHE) systems and spinons in quantum antiferromagnets. In both cases he argued that the exclusion statistics parameter, g, is equal to the exchange statistics parameter, ~. The generalized exclusion principle is thus expected to be realized in physical systems. Johnson and Canright [3] have tested this prediction numerically for F Q H E systems with ~ = 13 and find it to be true with the modification that, for quasiparticles, g = 2 - ½.
* Corresponding author.
In this paper we first briefly review our earlier work [5] where we had generalized Haldane's definition to the case where the Hilbert space of the particles is infinite dimensional and had shown that in general the exclusion statistics parameter g is proportional to the dimensionless second virial coefficient provided the system admits a virial expansion of the equation of state in the high temperature limit. We then qualitatively argue that in interacting fermion systems with nontrivial values of g, we should expect the breakdown of Fermi liquid theory. We then examine in detail the spectrum of a solvable model in one dimension, the Calogero-Sutherland model, which we have earlier shown [6] gives an explicit realization of the phenomenon. We abstract from this model the general constraints on the occupation numbers for exclusion anyons [4]. Finally, we briefly discuss some physical models (other than FQHE) where such quasiparticles are expected to occur. We now describe the generalization of Haldane's definition to systems with infinite dimensional Hilbert spaces. The motivation to do so is so that the definition can apply to continuum systems with no gaps in the spectrum where there is no natural way of defining a finite dimensional low energy space. The dimension of
0921-4526/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSD1 0 9 2 1 - 4 5 2 6 ( 9 5 ) 0 0 0 4 8 - 8
M.V.N. Murthy, R. Shankar/Physica B 212 (1995) 315 319
316
the Hilbert space of N identical particles when y is the statistics parameter is given [1, 3] by (d DN(g)
+
(1
-
-
1 -
9)(N
1))!
-
=
(1) N!(d
g(N
-
1))!'
where d is the dimension of the single particle space. It is easy to see that g = 0(1) yields the dimension of the bosonic (fermionic) Hilbert space. It is now straightforward to extract the statistics parameter 9 from Eq. (1) in the limit when the single particle dimension d ~ ~ . We obtain lim
½--g=e~N(N-
d
1)
N!
dN
-- 1
"
(2)
A regulated definition of the dimension of the Hilbert space is given by the corresponding N-particle partition function since we have DN = l i m ~ o Z N = l i m ~ o Tr(e-~nN), where fl is the inverse temperature and HN denotes the N-particle Hamiltonian. Therefore, for dealing with infinite dimensional Hilbert spaces, we propose a generalization of Haldane's definition,
CZ1 l- ZN ½ - g = a -l oiNm( N-- 1- )/[ N _ ! ~Z?-
-]
lJ
'
(3)
where C = 2 a,, ds being the spatial dimension. The factor C occurs because the regulated definition cuts off the energy at 2fl when two particles occupy the same state; however for counting purposes we want to cut off all states at the same energy irrespective of their occupation. It is then possible to show [5] that if all the virial coefficients are finite in the f l - , 0 limit, then the righthand side of Eq. (3) is independent of N and is equal to CB2, B2 being the dimensionless second virial coefficient. Thus the Haldane exclusion statistics parameter V is completely determined by the high temperature limit of the second virial coefficient. We can now apply this result to some well known systems. We first consider anyon gas. Here the second virial coefficient is well known and is given [7] by
are finite, then the result given in Eq. (5) would be true for anyon gas in general. Next consider anyons confined in a magnetic field. Once again 9 is given by Eq. (5) when the partition function includes the trace over all Landau levels. This is easy to see since both the oscillator frequency o9 and the cyclotron frequency ogc act as regulators and at high temperatures one recovers the anyon gas result. If however we confine the trace to the lowest Landau level, then we obtain 9 = a for quasiholes and 9 = 2 - a for quasiparticles in conformity with the results obtained for F Q H E systems [3]. It is instructive to look at the second virial coefficient in this case as a function of temperature. As shown in Fig. 1, in the range flo9 ~ 1, flo9¢ >> 1, only the lowest Landau level contributes, Bz is independent of temperature and corresponds to g = 2 - ~. When flo9¢ is of the order of 1, the other Landau levels start contributing and finally in the range flogc .8 1, B2 is again temperature independent and corresponds to 9 = c~(2 - ~). We would like to stress here that this does not mean that 9 as defined by us is a temperature dependent property. Nor is it a high temperature property of the system. The high temperature limit of the partition function is taken in order to count all the states. All states, low and high energy states, contribute equally to the partition
1.5 a:O.2
1
-
0.5 "~.
0
2~ 2 -- 4c~ + 1 B2 -
4
(4)
where ~ = 0 is bosonic and c¢ = 1 is fermionic. Using the relation between B2 and g we have
-0.5
0.001
g
=
~(2 -
o¢)
(5)
which has the correct limit for ~ = 0, 1. This result would be true for t w o anyons. However, it is not conclusively proved that the higher virial coefficients are finite in the case of anyon gas. It has been proved that the third virial coefficient is finite [8, 9]. If indeed all virial coefficients
........
i o.ol
........
i
........
o.1 8 [ , 0 = o.1 , %
i 1 = z.o]
, , ,~
a=0.o , lo
1 loo
Fig. 1. Second virial coefficient B 2 of anyons in a magnetic field and confined in a harmonic oscillator potential as a function of inverse temperature/3. The values ofo9 (oscillator frequency) and ~o, (cyclotron frequency of the magnetic field) are explicitly shown.
317
M. KN. Murthy, R. Shankar/Physica B 212 (1995) 315 319
function in this limit. The Hamiltonian serves the purpose of ordering the states according to energy. To define a finite difference of two divergent series it is necessary to order them. Energy gives us a physical basis for doing so. Thus g does not depend on the details of the Hamiltonian but only on the ordering. However g can be scale dependent. This can be seen and understood in the ~ = 0 case (free bosons) of the above example (see Fig. 1). At high temperatures (/~oc ,~ 1), the magnetic field is irrelevant and the system behaves like a free Bose gas in two dimensions. In the range fl¢o ,~ 1, flo)c >> 1, the system effectively behaves like a one dimensional free Bose gas. Thus the second virial coefficient and hence g have the two corresponding plateaus. Exactly the same thing happens for the nonzero values of ~ also. Therefore g can be scale dependent. In fact this is how we expect nontrivial values of g in physical systems of interacting electrons. Although the full Hilbert space may obey the dimension formula corresponding to fermions (g = 1), the low energy sector could deviate from it and correspond to some nontrivial value of g. We now qualitatively address the question as to when can such quasiparticles occur in a system of interacting fermions. The analysis above shows that the first thing to do is to examine the high temperature limit of the second virial coefficient. Consider then a system of quasiparticles with a constant density of states (as would be the case for quasiparticle energies small compared to the Fermi energy). ZI is equal to y/[3, where ? is the density of states. We write the two particle energy as E2 = t:l + g2 + A(el,g2)"
(6)
el. 2 are the single particle energies and A the interaction energy. The second virial coefficient can then be calculated and we have 1--g=
~irno?
fl
d x , dx2e-¢X'+X~)A
.
(7)
Thus for g to have a value other than 1, the energy shift A should remain nonzero at all energies whatsoever high. If.it goes to a constant at high energies, i.e. if lim Z~(~ 1, ~2) --1" Z~, 1~1,152 ~
oD
then 1 -- g = 73.
(8)
What is essentially happening in this case is that the addition of a particle causes a shift in the energy of all the levels. Thus when we count the dimension of the single particle Hilbert space with any cutoff, it changes since some of the states have been pushed out of the cutoff. If the energy shift remains at all energies, then g is non-
trivial wherever the cutoff is. Now this is precisely the scenario associated with the breakdown of Fermi liquid theory [10]. Note that A is related to the Landau parameters, f, by A = (1/Las)f, where L is the system size and d~ the spatial dimension. So for A to survive in the thermodynamic limit, we must have singular Landau parameters. This is what is expected to cause the breakdown of Fermi liquid theory [11]. Thus these qualitative arguments lead us to expect quasiparticles with nontrivial values of exclusion statistics parameters in interacting fermion systems in non-Fermi liquid phases. In fact the concurrence of exclusion statistics and non-Fermi liquid phases has been earlier stated by Anderson [ 121 in the context of the normal state of the cuprate superconductors. The well-known and well-studied examples of such systems are interacting fermion systems in one dimension. Consider one such system, the Calogero-Sutherland model (CSM). This is a model of a system of interacting fermions. The hamiltonian is given by
i=1
-2~xZ, +
¢°2x~
+ 2 , < ~ = l ( x i - x j ) 2"
(9)
The particles are confined in a harmonic well and the thermodynamic limit is obtained by taking o) ~ 0. It has been shown earlier [6, 13, 14"1 that this system behaves like an ideal gas of particles with fractional exclusion statistics. We will now review that result and then clarify how the states obey the dimension formula in Eq. (1). We will then abstract a general rule for constraints on the occupation numbers that leads to the formula. The spectrum of CSM is exactly known. The states can be labelled by a set of fermionic occupation numbers {nk}, k = 1. . . . . oo, nk = 0, 1. The energy is given by E[{nk}] = k=l ~ eknk + o2 N(N2
1) '
(10)
where ek = ko), N = ~k% 1 nk. We note that this spectrum is identical to the spectrum of quasiparticle states in a Gaussian theory of compact bosons [5] with radius R = 1 / w / ~ and with the following identification while k in the above equation is a state label in CSM, in the Gaussian theory of compact bosons k refers to the box quantized momenta. The following discussion thus applies to this case also. Since the bosonic theory describes a Luttinger liquid, our considerations apply to the low energy physics of a large class of one dimensional interacting fermion systems. The simplicity of the spectrum makes it possible to explicitly compute the right-hand side of Eq. (3) explicitly, since we have Z ~ = e x p ( - f l o 2 ( N ( N - 1 ) / 2 ) ) Z ~ , where ZvN is the N-particle partition function of noninteracting fermions. The answer is indeed independent of
318
M. KN. Murthy, R. Shankar/Physica B 212 (1995) 315-319
N and we get g = 1 + 2. Thus the particles in the C S M are anyons with the exclusion statistics parameter, g, equal to 1 + 2. To bring out the ideal gas nature of the system, we define the shifted single particle energies as eA(ki,g ) = ek, -- O(1 -- g)(i -- 1).
(11)
Here k~ are the occupied states and are labelled such that k~+ ~ > k~. We identify this shifted energy with the anyon single particle energy since the total energy can now be written as N
E[{k,}] = ~ eA(k,,g).
(12)
i-I
We now break up the energy space into cells of size m and define the anyon occupancy n~ of the kth cell to be the number of anyons with energies in the range k o ~ eA < (k + 1)o). It has been shown in Refs. [6, 13] that the thermal average of these occupancies is given by
1 ~o(ek) + g"
fig - -
Here 8k equation
=
(13)
ko) and w(e) is given as the solution of the
wg(e,g)(1 + w(e,g))1-0 = et~(~-~),
(14)
where fl is the inverse temperature and/~ is the chemical potential of the system. Furthermore the grand partition function factorizes and can be written as lnZ A =
dxln(1 + w - a ( x ) ) ,
(15)
where x is the dimensionless energy defined as x = e/e). It follows from Eqs. (15) and (14) that n-A k =
~xk in zA"
(16)
This shows that the C S M has the basic properties of an ideal gas of particles with fractional exclusion statistics. The inverse square interaction can thus be looked upon as a purely exclusion statistical interaction in one dimension. To get some feel for the anyon number densities, let us consider the ground state of the N-particle system. The set {kl} is (1,2, 3 . . . . . N). The corresponding set of shifted energies is {~o~(1 + g)~o,(1 + 2g)~o. . . . . (1 + (N - 1)g)~o}. The number of particles in the interval e to e + co is then =l/g, nA(t~'O)
0,
e < ~F = gp, /: /> /~F,
(17)
where ~ - coN is the average density. This is the same as the zero temperature limit of the distribution function in
Eq. (13). It is also clear from the above example that the maximal occupancy of any cell, nmax A < 1/g + 1. We will now clarify the connection between the states of the C S M and the dimension formula in Eq. (1). Consider the single particle states with a cutoff on the unshifted single particle fermion energies, kmax = dE. The number of cells occupied by the shifted anyon energies is then d = dr - (1 - g)(N - 1) (a fractional value of d implies that the cell with the maximum energy is never maximally occupied). The total number of states is D = a~CN, since the anyon energies are in one to one correspondence with the unshifted single particle fermion energies. When dF is expressed in terms of d in the above expression, we recover Eq. (1). As we saw above the fermion occupancies, {nk} are in one to one correspondence with the anyon occupancies {hA}. It is also clear that-this results in constraints on the anyon occupancies. We have already come across one constraint, namely nmax A < 1/g + 1. We will now write down the complete set of constraints for the case g = P/q where p and q are coprime integers, p < q. To do so we note that A - - ~klA ~k~+l
= o)(ki+
1
-- ki + p/q).
(18)
The above equation says that neighbouring fermion levels, (ki+l - ki = 1), correspond to neighbouring anyon levels (ek,~, A -- e~) = O)~. Thus "close packed" fermion configurations go over to "closed packed" anyon configurations. Also we see that if ki÷ a and ki are not neighb o u r s t h e n e k~+l A and ekA have to lie in different cells. Thus if in the lth cell, the lowest anyon level is tn((l - 1) + re~q), then that cell can have an occupancy n A <~ ( q - m)/p + 1 = n m a x - m / p . An anyon can have an energy of ~o((l - l) + re~q) iffthe fractional part o f N i p / q is equal to re~q, i.e. if m = p N t m o d q, Nt being the number of anyons with energy less than o)(l - 1). The above two statements imply that n A satisfies the constraint p(nmax A -- np) >~ pN~ mod q. Also if the lowest anyon energy in the cells is ~o((l - 1) + re~q), then any allowed value of n~ is realized since there will always be a corresponding fermionic configuration with the appropriate number of fermions close packed above the one corresponding to the lowest anyon energy in the cell. Thus we have shown that if the sets of anyon occupancies {n A} satisfy the following two constraints: n A < q/p + 1,
(19)
A -- n A) >/ p N i mod q, p(nmax
(20)
then the number of states with N = Y.t n p is equal to the right-hand sideEq. (1). Therefore we can now remove the scaffolding of the model and define anyon occupancies by the above two constraints. These are then the general constraints on the anyon occupancies.
M.V.N. Murthy, R. Shankar/Physica B 212 (1995) 315-319
Note that with these constraints, the problem of "negative probabilities" pointed out in Ref. [16] does not arise. That problem comes if one tries to reproduce the distribution function in Eq. (13), assuming that there are no constraints on the anyon occupancies, e.g. it occurs when g N > d. This is explicitly prohibited by the constraints. Finally we mention some physical systems where such quasiparticles are predicted to occur. FQHE systems are of course an example. The normal state of the cuprate superconductors is also believed to be non-Fermi liquid state. However the theory of these two dimensional systems is as yet incomplete and inconclusive. The theory of interacting electron systems in one dimension is unambiguous. Fermi liquid theory does break down and such quasiparticles do exist. Thus they should occur in quantum wires, if the correlations play an important role in their physics. The isotropic quantum antiferromagnetic chain is another physically realizable model. This model can be mapped on to the Luttinger model by the Jordan-Wigner transformation. The low energy physics is then described by a Luttinger liquid with the radius parameter equal to 1/,~//2. Therefore the system would have quasiparticles with g = ½, the spinons. The overscreened Kondo impurity systems and another class of models where such quasiparticles are expected to exist. The recent solution of these models using conformal field theory [17] predicts nontrivial quantum numbers for quasiparticles. In particular, for the s = ½, two channel problem, which may have a physical realization [18], quasiparticles with g = ½ are predicted. In conclusion, Haldane's definition of exclusion statistics, when generalized to infinite dimensional Hilbert spaces, is determined by the high temperature of the second virial coefficient if all the viral coefficients are finite in this limit. It is then possible to argue qualitatively that the occurrence of quasiparticles with nontrivial values of g will concur with the breakdown of Fermi liquid theory. Indeed in one dimensional interacting fermion systems, both these phenomena do concur. Simple solvable models in one dimension have been analysed in detail to abstract the general rule for anyon occupancies. There are physically realizable models with quasipar-
319
ticles having a nontrivial value of g- It is clearly of interest and importance to pinpoint signatures of the nontriviat exclusion statistics in these models that can be experimentally seen.
References [1] F.D.M. Haldane, Phys. Rev. Lett. 67 (1991) 937. [2] See also A.G. Izergin and V. Korepin, Lett. Math. Phys. (1982) 283. [3] M.D. Johnson and G.S. Canright, Phys. Rev. B 49 (1994). [4] Note that, hereafter, we use the name anyon genericallyfor all particles with fraction statistics-exclusionof exchange. [5] M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. 72 (1994) 3629. [6] M.V.N. Murthy and R. Shankar, Phys. Rev. Lett. 73 (1994) 3331. [7] D.P. Arovas, J.R. Schrieffer, F. Wilczek and A. Zee, Nucl. Phys. B 251 (1985) 117; J.S. Dowker, J. Phys. A 18 (1985) 3521. [8] J. Myrheim and K. Olaussen, Phys. Lett. 229 (1993) 267. [9] R.K. Bhaduri, R.S. Bhalerao, A Khare, J. Law and M.V.N. Murthy, Phys. Rev. Lett. 66 (1992) 523; Diptiman Sen, Phys. Rev. Lett. 68 (1992)2977; M. Sporre, J.J.M. Verbaarschot and I. Zahed, Nucl. Phys. B 389 (1993) 645; A. Dasnieres de Veigy and S. Ouvry, Phys. Lett. B 291 (1992) 30; J. Law, A. Suzuki and R.K. Bhaduri, Phys. Rev. A 46 (1992) 4693. [10] P.W. Anderson, Phys. Rev. 164 (1967) 352. [11] P.W. Anderson, Phys. Rev. Lett. 64 (1990) 2306; 66 (1991) 3226; P.C.E. Stamp, Phys. Rev. Lett. 68 (1992) 2180. [12] P.W. Anderson, Progr. Theor. Phys. (Suppl.) 107 (1992)45. [13] S.B. Isakov, Phys. Rev. Lett. 73 (1994) 2150; Yong-ShiWu, Phys. Rev. Lett. 73 (1994) 922; D. Bernard and Yong-Shi Wu, University of Utah Preprint UU-HEP/94-03 {condmat/9404015). [14] Z.N.C. Ha, Phys. Rev. Lett. 73 (1994) 1574. [15] F. Calogero, J. Math. Phys. 10 (1969) 2191, 2197; B. Sutherland, J. Math. Phys. 12 (1971) 246, 251; Phys. Rev. A 4 (1971) 2019; 5 (1972) 1372. [16] C. Nayak and F. Wilczek,Phys. Rev. Lett. 73 (1994) 2740. [17] I. Altleckand A.W.W. Ludwig,Nucl. Phys. 360 (1991) 641. [18] D.C. Ralph, A.W.W. Ludwig, Jan Von Delft and R.A. Burham, Phys. Rev. Lett. 72 (1994) 1064.