Lattice QCD at finite density

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Dec 22, 1988 - Volume 215, number 3. PHYSICS LETTERS B. 22 December 1988. LATTICE QCD AT FINITE DENSITY. I.M. BARBOUR, C.T.H. DAVIES and Z.
Volume 215, number 3

PHYSICS LETTERS B

22 December 1988

L A T T I C E Q C D AT F I N I T E D E N S I T Y I.M. B A R B O U R , C.T.H. DAVIES and Z. SABEUR Department of Physics and Astronomy, University of Glasgow, GlasgowGI 2 8QQ, UK

Received 18 August 1988

We describe a new method of studying the behaviour of QCD at finite density. It involves an expansion of the grand canonical partition function in terms of partition functions at fixed particle number. Results compatible with mean field theory are obtained on small lattices at strong coupling. We discuss how to extend the method to large lattices and weaker couplings.

1. Introduction A study of the phase structure of Q C D at non-zero baryon density is important for our understanding of the conditions prevailing in nuclear matter at high pressure. Lattice simulations of Q C D which include a chemical potential for quarks, /~, have, however, proved extremely difficult. Our physical intuition suggests that there should be a phase transition to a chirally symmetric phase at a value of ¢t related to the lowest lying baryon mass. The simplest numerical approach, that of a lattice simulation of an SU (3) gauge theory in the quenched approximation, gives results in conflict with this expectation [1 ]. It appears that chiral symmetry is restored for all # > 0 in the limit that the quark mass goes to zero. From a study of different size lattices, however, it is clear that no phase transition actually occurs but that the behaviour of the chiral condensate is an artifact of the quenched approximation. This is also seen from the fact that a U (3) gauge theory, which has no baryons, gives the same behaviour as an SU (3) gauge theory in the quenched approximation. We shall show below that it is the fermion determinant (ignored in the quenched theory) which provides the underlying mechanism for a phase transition at # > 0 . Simulations in the full theory with quark loops have been plagued by the fact that, for non-zero /z, the determinant is complex. Gibbs [ 2 ] has argued that it is precisely the fact that the determinant has a phase that causes quenched simulations to behave incorrectly. Monte Carlo simulations with importance 0 3 7 0 - 2 6 9 3 / 8 8 / $ 03.50 © Elsevier Science Publishers B.V. ( North-Holland Physics Publishing Division )

sampling based on the modulus of the determinant have found that the phase o f the determinant fluctuates wildly [ 3 ]. Indeed, for values of/~ close to the expected phase transition, the real part of the determinant can be positive or negative with almost equal probability. Determination of the position o f the phase transition using such methods then becomes very difficult. Numerical work has concentrated on the behaviour of the theory at gauge coupling, fl, equal to zero. There mean field theory can be used to derive a critical value for/za, with which we can compare numerical results [ 1 ]. Recently Gocksch [4] has reported a measurement based on importance sampling using the modulus o f the determinant but collecting data in bins according to its phase. Although problems similar to those reported above are seen, he obtains a result in agreement with mean field theory after many measurements on a small 24 lattice. To extend this method to larger lattices will be very time-consuming. Similar agreement is also obtained by a different approach [ 5 ] in which the dimer method is extended to SU (3) and applied to a 44 lattice at small quark mass. This method is only valid in strong coupling. Although both these results show that the full theory does behave differently from the quenched theory we still need to find the position o f the phase transition at more realistic values of the gauge coupling and lattice size. In this paper we describe a new method of searching for a phase transition at finite density. By studying the partition function more directly, and in a way 567

Volume 215, number 3

PHYSICS LETTERSB

which removes much of the noise from the signal, we obtain very clear results at fl= 0, again in agreement with mean field theory. We believe that this method can be extended to obtain physically meaningful resuits at larger values of ft. The method works by expanding the grand canonical partition function, ~, in terms of the canonical partition functions, Z~, for the system with a fixed number of fermions, n. Measurements of the important Z, allow ~ to be reconstructed as a function of /ta and phase transitions in the chiral condensate and particle number are then apparent. In section 2 the method is described in more detail and in section 3 some results are presented. The partition functions Z, have been measured for all n on a 24 lattice at fl= 0 and various quark masses, 0.1 ~