An interesting idea has been discussed in arXiv: 0801.4051 by a group of people from Graz, Austria and Regensburg, Germany (here). The idea is to construct operators using observables that are sensitive to both Chiral Symmetry breaking and Deconfinement transition.
Why is this necessary? Well, deconfinement and chiral symmetry breaking transitions are both non-perturbative, and it is important to study them non-perturbatively (via the lattice). Such systematic studies should use order parameters to probe the respective symmetries that are being broken/restored in such transitions, indeed if there are any. The common viewpoint is that in the deconfinement transition, the centre symmetry is spontaneously broken, while the chiral transition breaks the chiral symmetry spontaneously. While the Polyakov loop acts as an order parameter in the former transition, the chiral transition is the OP in the latter. The caveat here is the fact that these two transitions are defined in two opposite limits, the former for the pure gauge theory (or in the quenched theory, where the fermion masses are infinite) and the latter in the chiral limit of vanishing quark masses. At any intermediate quark masses, these quantities are not strictly order parameters.
The paper constructs an operator called the "Dual chiral condensate" by imposing (unphysical) arbitrary boundary conditions on the fermions and then taking the Fourier transform with respect to the boundary condition in such a way that they project on to the "string" operators that wind around the lattice only once. While the finite quark mass allows the quarks to jiggle about thereby creating operators which have spatial displacement as well, the fact that they wind around the temporal direction exactly once ensures that these have the same centre symmetry as the Polyakov loops. In fact, in the infinite mass limit these do become the Polyakov loops.
So far so good! They took the chiral condensate and obtained with it an operator sensitive to the center symmetry. Next, how does it behave? Does it help in understanding these phenomena any better? Within the framework of the SU(3) quenched lattice gauge theory, they show that even with quark masses about 100 MeV, this quantity vanishes below Tc, the deconfinement temperature, but is finite above it. Rewriting the chiral condensate in terms of the eigenvalues of the (staggered) Dirac operator (with the phase angle), they show that it is the IR modes which play any role. Larger eigenvalues contribute less since they appear in the denominator.
Carrying the investigations further (see here), the SU(2) lattice gauge theory with adjoint fermions is considered. In this theory, the confinement and the chiral symmetry transitions are widely separated with the latter (~870 MeV) occurring at about 4 times the former transition temperature. This is interesting since it gives a window where the system is no longer confined but chiral symmetry is broken, and therefore is an ideal testbed to check for the effects of these two transitions separately. In the SU(3) case (with quenched or 2+1 flavours) these transitions occur simultaneously. In their results, they see that while the chiral condensate is not sensitive to the deconfinement transition, it is sensitive to the boundary condition above the deconfinement transition (but not below the deconfinement transition), until all the way to the chiral transition when it has completely melted away. The spectral gap remains closed until after the chiral transition. Therefore, the dual chiral condensate vanishes below the deconfinement transition. Above it, and until the chiral transition, it keeps increasing, since the chiral condensate depends on the boundary condition. Beyond the chiral transition, it depends on the mass of the fermion (for heavy fermions it keeps increasing, but drops for lighter fermions). Thus, using light fermions,
the dual chiral condensate is actually able to distinguish the deconfinement and the chiral transitions.
A step further (see here) uses the configurations of the BMW group to look at this observable for full QCD. It is found that the low energy spectra of the Dirac operator is sensitive to the boundary conditions (above Tc), while the large energy spectra is not, which means that the dual condensate gets all the contributions from the IR modes. Again, while the quark condensate has no dependence on the boundary condition below Tc, it does so above Tc, which accounts for the vanishing of the dual condensate below Tc, and its non-zero value above it. It is (perhaps) interesting to note that the dual condensate is zero to the same level as the Polyakov loop is (at least looking from the figure 5 of the paper).
It seems natural now to expect several investigations using this operator and in this direction in the gauge theories!
Why is this necessary? Well, deconfinement and chiral symmetry breaking transitions are both non-perturbative, and it is important to study them non-perturbatively (via the lattice). Such systematic studies should use order parameters to probe the respective symmetries that are being broken/restored in such transitions, indeed if there are any. The common viewpoint is that in the deconfinement transition, the centre symmetry is spontaneously broken, while the chiral transition breaks the chiral symmetry spontaneously. While the Polyakov loop acts as an order parameter in the former transition, the chiral transition is the OP in the latter. The caveat here is the fact that these two transitions are defined in two opposite limits, the former for the pure gauge theory (or in the quenched theory, where the fermion masses are infinite) and the latter in the chiral limit of vanishing quark masses. At any intermediate quark masses, these quantities are not strictly order parameters.
The paper constructs an operator called the "Dual chiral condensate" by imposing (unphysical) arbitrary boundary conditions on the fermions and then taking the Fourier transform with respect to the boundary condition in such a way that they project on to the "string" operators that wind around the lattice only once. While the finite quark mass allows the quarks to jiggle about thereby creating operators which have spatial displacement as well, the fact that they wind around the temporal direction exactly once ensures that these have the same centre symmetry as the Polyakov loops. In fact, in the infinite mass limit these do become the Polyakov loops.
So far so good! They took the chiral condensate and obtained with it an operator sensitive to the center symmetry. Next, how does it behave? Does it help in understanding these phenomena any better? Within the framework of the SU(3) quenched lattice gauge theory, they show that even with quark masses about 100 MeV, this quantity vanishes below Tc, the deconfinement temperature, but is finite above it. Rewriting the chiral condensate in terms of the eigenvalues of the (staggered) Dirac operator (with the phase angle), they show that it is the IR modes which play any role. Larger eigenvalues contribute less since they appear in the denominator.
Carrying the investigations further (see here), the SU(2) lattice gauge theory with adjoint fermions is considered. In this theory, the confinement and the chiral symmetry transitions are widely separated with the latter (~870 MeV) occurring at about 4 times the former transition temperature. This is interesting since it gives a window where the system is no longer confined but chiral symmetry is broken, and therefore is an ideal testbed to check for the effects of these two transitions separately. In the SU(3) case (with quenched or 2+1 flavours) these transitions occur simultaneously. In their results, they see that while the chiral condensate is not sensitive to the deconfinement transition, it is sensitive to the boundary condition above the deconfinement transition (but not below the deconfinement transition), until all the way to the chiral transition when it has completely melted away. The spectral gap remains closed until after the chiral transition. Therefore, the dual chiral condensate vanishes below the deconfinement transition. Above it, and until the chiral transition, it keeps increasing, since the chiral condensate depends on the boundary condition. Beyond the chiral transition, it depends on the mass of the fermion (for heavy fermions it keeps increasing, but drops for lighter fermions). Thus, using light fermions,
the dual chiral condensate is actually able to distinguish the deconfinement and the chiral transitions.
A step further (see here) uses the configurations of the BMW group to look at this observable for full QCD. It is found that the low energy spectra of the Dirac operator is sensitive to the boundary conditions (above Tc), while the large energy spectra is not, which means that the dual condensate gets all the contributions from the IR modes. Again, while the quark condensate has no dependence on the boundary condition below Tc, it does so above Tc, which accounts for the vanishing of the dual condensate below Tc, and its non-zero value above it. It is (perhaps) interesting to note that the dual condensate is zero to the same level as the Polyakov loop is (at least looking from the figure 5 of the paper).
It seems natural now to expect several investigations using this operator and in this direction in the gauge theories!
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