Canonical partition functions in lattice QCD
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1 Canonical partition functions in lattice QCD (an appetizer) Julia Danzer, Christof Gattringer, Ludovit Liptak, Marina Marinkovic
2 Canonical partition functions Grand canonical partition function: Z(µ) = D[U] e S g[u] det[d(µ)] det[d(µ)] is complex and cannot be used as a positive weight factor. Fugacity expansion: Z(µ) = Q e µ N t Q Z Q, N t = β/a Canonical partition functions: Z Q = D[U] e S g[u] det[d] Q Fermion determinant with fixed quark number Q: det[d] Q = 1 dϕ e i ϕ N t Q det[d(µ = iϕ)] 2π
3 Fermion determinant on the lattice The Dirac operator on the lattice approximates the derivative terms by nearest neighbors: D(µ) xy = 3 j=1 γ j [ δ x+ĵ,y U j(x) δ x ĵ,y U j(x ĵ) ] + γ 4 [ e µ δ x+ˆ4,y U 4(x) e µ δ x ˆ4,y U 4(x ˆ4) ] The gauge fields are coupled as group valued link variables: U ν (x) = exp(ia ν (x)) The chemical potential µ weights forward propagation in time with e µ and backward propagation in time with e µ.
4 Fermion determinant as a collection of loops The fermion determinant is a gauge invariant quantity and as such given as a sum of closed loops dressed with the gauge links. To introduce the chemical potential on the lattice, temporal gauge links are multiplied with e ±µ. Thus the chemical potential enters according to the winding number Q of the loops around the compact time direction as a factor e µ N t Q, e i ϕ N t Q (for µ = iϕ) Fourier transformation projects to a fixed winding number Q. det[d] Q = 1 dϕ e i ϕ N t Q det[d(µ = iϕ)] 2π A numerical evaluation of the Fourier transform, in particular for large Q, needs many intermediate values for ϕ and is numerically expensive.
5 Graphical illustration 4^ x Trivial loops Non trivial loop exp() = 1 exp( i ϕn ) t
6 Dimensional reduction formula for the fermion determinant J. Danzer, C. Gattringer, PRD 8 It is possible to reduce the part of the fermion determinant which depends on the chemical potential to a 3-dimensional determinant: det[d(µ)] = A det[ 1 H e µ N t H +1 e µ N t H 1 ] The matrices H, H ±1 live on a 3-dimensional time slice and are made from hopping terms and propagators on four sub-domains of the lattice. The numerical evaluation of the dimensionally reduced matrix is by a factor of Nt 3 O(1) cheaper. Calculating canonical determinants is accelerated by this factor: det[d] Q = A dϕ e i ϕ N t Q det[ 1 H e i ϕ N t H +1 e i ϕ N t H 1 ] 2π
7 Graphical representation of the terms det[d(µ)] = A det[ 1 H e µ N t H +1 e µ N t H 1 ] H +1 Λ (1) Λ (4) H Λ (3) Λ (2)
8 Separation of scales det[d(µ)] = A det[ 1 H e µ N t H +1 e µ N t H 1 ] W = det[ 1 H e µ N t H +1 e µ N t H 1 ] We analyze the different factors using quenched SU(3) gauge configurations on lattices. m A det[d] µ= W µ= det[d] µ=.1 W µ=.1 det[d] µ=.2 W µ= A is a µ-independent bulk factor which carries most of the quark mass dependence. W is a µ-dependent factor of order 1, which shows only a weak variance with the quark mass.
9 Center symmetry The gauge action is invariant under center transformations: U 4 ( x, x 4) z U 4 ( x, x 4) where x 4 is a fixed time coordinate and z is in the center Z 3 of SU(3). In pure gauge theory the center symmetry and its breaking are essential for understanding the finite temperature deconfinement transition. An order parameter for center symmetry is the Polyakov loop P: P = x 4 U 4 ( x, x 4 ) which transforms non-trivially P z P
10 Center symmetry In the deconfined phase the Polyakov loop assumes a finite value and spontaneously chooses one of the center angles..5 T < T c T > T c
11 Center symmetry and canonical determinants The canonical determinant det[d] Q consists of loops that wind Q-times and thus under center transformations behaves as: det[d] Q z Q det[d] Q With our new techniques for canonical determinants we can study the role of center symmetry for the canonical approach systematically. We find that the canonical determinants det[d] Q indeed show a very strong center pattern. Center symmetry is important also for full QCD!
12 Center symmetry and canonical determinants -2e+44 2e+44-1e+44 1e+44-2e+43 2e+43 2e+44 Q = Q = 1 Q = 2 2e+43-2e+43-2e+44 T < T c 1e+57 Q = Q = 1 Q = 2 5e+56-1e+57 T > T c -5e+56-1e+57 1e+57-1e+57 1e+57-5e+56 5e+56
13 An application: Grand canonical determinant and center symmetry Can we understand the symmetry breaking pattern of the full theory from the center properties of the canonical determinants? Fugacity expansion at µ = : det[d] = Q det[d] Q After a center transformation: det[d] = Q det[d] Q z Q Above T c for configurations in sectors with complex Polyakov loop give rise to z = exp(± i 2π/3), while for the real sector one has z = 1. Phase averaging leads to a much smaller fermion determinant for the complex sectors.
14 Distribution of the grand canonical determinants 2 T > T c θ P = θ P = 2π/3 θ P = - 2π/ Center symmetry properties of the canonical determinants explain why the real Polyakov loop sector is selected in full QCD. The center symmetry arguments may be extended to understand the behavior of observables, e.g., the non-vanishing value of the Polyakov loop below T c and the mechanism for its increase at T c.
15 Summary Outlook An exact dimensional reduction formula gives rise to a very efficient evaluation of canonical fermion determinants on the lattice. This allows for a detailed study of their properties. Center symmetry and its breaking are important also for full QCD. Currently we use the dimensional reduction formula also for a systematic study of the sign problem. Plan: Explore further the fugacity expansion and the role of the center for observables. Plan: Go beyond the quenched analysis. Plan: Reweighting strategies with canonical determinants. Plan: Different dimensional reduction formula for very high densities.
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