A comparison between compact and noncompact formulation of the three dimensional lattice QED

Transcription

1 A comparison between compact and noncompact formulation of the three dimensional lattice QED Pietro Giudice niversità della Calabria & INN - Cosenza Swansea, 0/06/200 p.1/26

2 IN COLLABORATION WITH: Roberto iore, Domenico Giuliano, Donatella Marmottini, Alessandro Papa, Pasquale Sodano niversità della Calabria & INN - Cosenza niversità di Perugia & INN - Perugia Swansea, 0/06/200 p.2/26

3 Plan of the talk Introduction QED as an effective theory for underdoped high- superconductors QED as a laboratory for the theoretical investigation of mechanisms of confinement The model and its properties Lattice formulation Compact versus noncompact formulation [H.R. iebig and R.M. Woloshyn, Phys. Rev. D 42, 20-2 (1990)] Numerical results Summary and conclusions Swansea, 0/06/200 p./26

4 underdoped high- QED as an Eff. Theory for superconductors Doped cuprates are superconductors which manifest a superconducting behaviour at high temperature and exhibit a pseudogap phase T (a) CSB dsc T* (b) CSB dsc A/ SDW (CSB) QED T c dsc A/SDW -SC x [M. ranz et al., Phys. Rev. B 66, 04 (2002)] or a gap opens What does happen at? antiferromagnet / spin density wave -wave superconductor is the SDW order parameter [I.. Herbut, Phys. Rev. B 66 (2002) 09404] Swansea, 0/06/200 p.4/26

5 QED as an effective theory for underdoped high- superconductors An effective theory of NDERDOPED cuprates is provided by QED 1. ermion fields spin excitations of the superconducting states (SPINONS) minimally coupled to a massless gauge field where 2. Gauge field field derived from the fluctuating topological defects (vortex) in the superconducting phase This Eff. Theory does NOT describe the condensation comes into play phase as soon as vortex The quantum phase transition in cuprates is an example of spontaneous breaking of continuous global symmetry in QED, the chiral symmetry Chiral symmetry breaking (CSB) in QED is strongly related to. Physical case. There are several studies and numerical attempts to state this, but the question is still open [S.J. Hands et al., Nucl. Phys. B64, 21-6 (2002)] Swansea, 0/06/200 p./26

6 QED and confinement A particular formulation of QED in Euclidean dimensions is proven to possess the property of permanent confinement [A.M. Polyakov, Nucl. Phys. B120 (1977) ] Any pair of test electric charge and anti-charge is confined by a linear potential This has been obtained by considering instanton solutions of the theory, which in dimensions are magnetic monopoles Confinement of electrically charged particles is caused by a plasma of monopoles Monopoles are topological defects which appear when we consider a compact gauge group (cqed ) Moreover, this is true for cqed pure gauge, but it is not clear what happens when the gauge field is coupled to matter or example Herbut et al. argue that cqed with massless fermions is always in the confined phase [I.. Herbut and B.H. Seradjeh, Phys. Rev. Lett. 91, (200)] Swansea, 0/06/200 p.6/26

7 Compact and noncompact QED In order to study the chiral condensate and the confinement in QED, which are nonperturbative issues, we need to discretize the theory on a lattice. There are two formulations Compact QED Noncompact QED The d.o.f. are the link variables: [ The d.o.f. are the phases of the link variables: [ ] ] " Noncompact QED is widely used as an effective theory for underdoped high-temperature superconductors [S.J. Hands, J.B. Kogut and C.G. Strouthos, Nucl. Phys. B64 (2002) 21-6] [I.. Herbut, Phys. Rev. B 66 (2002) 09404] [N.E. Mavromatos, J. Papavassiliou, cond-mat/011421] [T. Appelquist, L.C.R. Wijewardhana, hep-ph/04020] [J. Alexandre, K. arakos, S.J. Hands, G. Koutsoumbas and S.E. Morrison, Phys. Rev. D 64 (2001) 0402] Swansea, 0/06/200 p.7/26

8 Compact and noncompact QED The compact formulation of the theory is suitable for the study of the mechanism of confinement [A.M. Polyakov, Nucl. Phys. B120 (1977) ] [I.. Herbut and B.H. Seradjeh, Phys. Rev. Lett. 91 (200) ] [P.D. Coddington, A.J.G. Hey, A.A. Middleton, J.S. Townsend, Phys. Lett. B 17 (1986) 64] Swansea, 0/06/200 p.8/26

9 $ ' & / 9 : / :< / : < - ; + ; + ; + : / :< / : :( The model The continuum Lagrangian density describing QED is given in Euclidean metric by [T.W. Appelquist et al., Phys. Rev. D (1986) 704] ( (,- ' ( + * )( where *. 2 /10 and $ & is the field strength The fermions ( ( 6 4 ) are 4-component spinors Note that QED is a super-renormalizable theory, dim does not display any energy dependence ' 8 7 0, so the coupling A convenient representation for ( + ) matrices is: where are Pauli matrices Swansea, 0/06/200 p.9/26

10 - + + = / / ; + ; :, :,, - -, - The model : < + matrices that anticommute with There are two So the massless theory is invariant under the chiral transformations: B ( (?> < the mass term becomes: Writing a 4-component spinor as D < D < and E :< E < E < E E < E and under parity transformation is parity conserving Swansea, 0/06/200 p.10/26 so E < E - E < :< E E < E

11 H, ( M ' G M R & S MK L P8, D 7 & O & R Q & MK L & M L L Lattice formulation The lattice action using staggered fermion field is given by N K1L ( <K1L (JI P8 < I & N K1L This action allows to simulate flavors of staggered fermions corresponding to flavors of the 4-component fermions considered in T 4 the previous slide [C. Burden and A.N. Burkitt, Europhys. Lett. (1987) 4] Swansea, 0/06/200 p.11/26

12 G G 8 " 7 G " & " & \ Lattice formulation depends on the type of formulation: Compact: D & ' & 8 7 K V& L < XTYW is the plaquette variable and & where Noncompact: $ & $ & K V& L " Z1[ ' " Z 9 ' $ & and is ^ ] K (?> 0 where is the phase of the link variable, related to gauge field by " 2 0 " Swansea, 0/06/200 p.12/26

13 _, Lattice formulation We used Hybrid Algorithm with staggered fermions: T (4-component fermions) Observables: monopole density [T.A. DeGrand and D. Toussaint, Phys. Rev. D 22 (1980) 2478] chiral condensate Lattice: and ; =0.01, 0.02, 0.02, 0.0, 0.04, 0.0 The continuum limit of the theory: QED is a super-renormalizable theory, dim not display any a dependence ' 8 7 0, so the coupling does The continuum limit is reached for < W X`Y Swansea, 0/06/200 p.1/26

14 b f ' f f ' f N k N M j N f Lattice monopoles We find the monopoles using the Gauss s law By measuring the total magnetic flux emanating from a closed surface in the lattice we can determine whether or not the surface enclose a monopole The flux is defined by $ & W Y ed c 1a The Bianchi identity tells us that summed over an closed surface always give zero (we measure: the monopole flux + the Dirac string flux) $ & We decompose into physical fluctuations which lie in the range Dirac strings which carry units of flux: $ & to and $ & g& h & (where g& is the number strings through the plaquette) We can now measure the monopole number inside a surface: i, W Y ed c f $ & (monopole density) Swansea, 0/06/200 p.14/26

15 β 2 <χχ> lattice 8 N f =2 stat. 1.8x10 compact form. is a dimensionless variable so in the continuum limit becomes a constant β We see two regimes: or or the theory is in the continuum limit (plateau) the theory describes a lattice system with finite spacing Swansea, 0/06/200 p.1/26

16 l M j _ M j M j Compact versus noncompact QED [H.R. iebig and R.M. Woloshyn, Phys.Rev. D 42, 20-2 (1990)] The chiral condensate and the monopole density are calculated for lattice QED (, ) for both compact and noncompact formulation of the theory at finite lattice spacing 6 6 compact noncompact lattice Plotting versus the monopole density, data points for both theories fall on the same curve to a very good approximation The physics of the chiral symmetry breaking is the same in the two theories or when vanishes retains a small value, indicating that the quark loop effects screen the forces that produce CSB Our program: The study of the relation between both in compact and noncompact QED continuum limit and in the Swansea, 0/06/200 p.16/26

17 0, m 0 o n m Continuum limit In the continuum limit all physical quantities are expressible in terms of the scale set by the coupling It is natural to work in terms of dimensionless variables such as or that depend on As the continuum limit is approached data taken at different should overlap on a single curve when plotted in dimensionless units [S.J. Hands et al., Nucl.Phys. B64, 21-6 (2002)], β 2 <χχ> lattice 8 N f =2 stat. 2.0x x10 compact form. β=0.9 β=1.1 β=1. β=1.7 β=1.8 β=1.9 β=2.0 β=2.2 β=2.4 β=2.6 β= βm The effects of finite volume are significant: this is related to the presence of a massless particle, the photon Thermodynamic limit ( ) Swansea, 0/06/200 p.17/26

18 p m B S r r _ B S _ o Numerical results Linear fit with /d.o.f. Linear fit with d.o.f. - Mq - Mq o β 2 <χχ> lattice 12 N f =2 stat. 1.0x10-1.x10 compact form. β=1.8 β=1.9 β=2.0 β=2.1 β= βm Swansea, 0/06/200 p.18/26

19 n r _ n B r _ B n p p r _ B n B p S = r S Numerical results 0.08 p m lattice 12 N f =2 stat. 7.4x x10 noncompact form r n Linear fit with Mq 0.0 S 0.04 β=0.6 β=0.7 β=0.7 β=0.8 β=0.9 β 2 <χχ> n d.o.f Linear fit with Mq S 0.00 βm d.o.f. Linear fit with p , e n p d.o.f. - Mq S in to be compared with [J. Alexandre et al., Phys. Rev. D 64, 0402 (2001)] - Mq in [S.J. Hands et al., Nucl.Phys. B64, 21-6 (2002)] Swansea, 0/06/200 p.19/26

20 Numerical results <χχ>.0e-0 2.e-0 2.0e-0 1.e-0 compact noncompact lattice 12 N f =2 1.0e-0.0e e ρ m It is not evident that the two formulations are equivalent in the continuum limit (at least with these preliminary results) Swansea, 0/06/200 p.20/26

21 j Numerical results <χχ>.0e-02 2.e e-02 1.e e-02 noncompact compact lattice 2 N f =2 stat.1000 linear fit in <χχ> <χχ> 1e-02 e-0 noncompact compact lattice 2 N f =2 stat quadratic fit in <χχ>.0e-0 0.0e+00 0e+00 1e-0 2e-0 e-0 4e-0 e-0 6e-0 ρ m 0e ρ m iebig and Woloshin plot using lattice Very difficult to extrapolate the chiral condensate to zero mass M does not change from to Swansea, 0/06/200 p.21/26

22 M j Numerical results Data not fall on a universal curve The continuum limit is not reached? β ρ m 1e-02 1e-02 1e-02 8e-0 6e-0 lattice 12 N f =2 stat. 1.0x10-1.x10 compact form. β=1.8 β=1.9 β=2.0 β=2.1 β=2.2 is independent from the 4e-0 fermion mass 2e-0 0e βm This supports the arguments by Herbut about the confinement in the presence of massless fermion Swansea, 0/06/200 p.22/26

23 Numerical results β ρ m 1.4e-0 1.2e-0 1.0e-0 8.0e-04 lattice 12 N f =2 stat. 7.4x x10 noncompact form. β=0.6 β=0.7 β=0.7 β=0.8 β= e e βm Noncompact formulation Swansea, 0/06/200 p.2/26

24 S n n r Numerical results β ρ m 7.0e-0 6.0e-0.0e-0 4.0e-0.0e-0 2.0e-0 1.0e-0 lattice 12 N f =2 stat. 10 am=0.0 compact form. 0.0e β β ρ m 8e-06 7e-06 6e-06 e-06 4e-06 e-06 2e-06 1e-06 lattice 12 N f =2 stat. 10 am=0.0 noncompact form β 1 couple of monopole Constant physical volume means ts In this case the physical volume decrease =const Swansea, 0/06/200 p.24/26

25 u Summary We discussed about QED as an effective theory for underdoped highv superconductors a laboratory for the theoretical investigation of mechanisms of confinement We showed the properties of the continuum Lagrangian and its lattice formulation We presented H.R. iebig and R. M. Woloshyn results (at finite lattice spacing) We discussed about our preliminary numerical results in the continuum limit Swansea, 0/06/200 p.2/26

26 w u u x Conclusions Our numerical results for are compatible with those by other research groups, although it is still questionable if the continuum limit has been reached and if the chiral limit is stable Chiral condensate extrapolation to zero mass is not naive The comparison between compact/noncompact lattice formulations is not ultimated x does not change varying the volume We have verified that there is a weak dependence of the monopole density by the fermion mass, this supporting the scenario of confinement even in presence of massless fermion (Herbut s arguments) Continuum limit of is not reached Swansea, 0/06/200 p.26/26