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Chiral symmetry breaking in strongly coupled quenched QED
Nuclear Physics B (Proc . Suppl .) 17 (1990) 675-678 North-Holland
675
CHIRAL SYMMETRY BREAKING IN STRONGLY COUPLED QUENCHED QED Simon HANDS Physics Department, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA . Results for the chiral condensate as a function of electron - photon coupling e in quenched non-compact lattice QED, obtained using both conjugate-gradient and Lanczos algorithms, are presented. There is a small but significant departure from mean field scaling, although the evidence for a fixed point of the Miransky type is inconclusive. It is shown that the symmetry-breaking mechanism may be associated with effective monopole loops generated by the non-compact dynamics . In this talk I will present results obtained in collaboration with John Kogut, 1 Elbio Dagotto, 1 and Roy Wensley.2 We have investigated the phenomenon of chi ral symmetry breaking in quenched non-compact lattice QED, with the aim of determining the critical scaling behaviour of the condensate < ~0 >, and understanding the underlying physics. The action for the quenched theory is S8 = - 1 EWjn),
M 40
o aa
0. Cu
0 ô
N
where 0, , is the oriented sum of real link variables ®,,(n) around elementary plaquettes of the lattice. S9 is invariant under local gauge transformations defined by the additive group of real numbers R. Notice that as written Ss has no coupling constant dependence, and simply describes free modes which resemble photons in the small momentum limit. This makes it possible to generate statistically independent configurations {0} at each saaeep by generating the (gauge-fixed) photon modes in momentum space and using a fourier transformation . 3 Critical slowing down is not a problem for this system . The chiral condensate < ~0 > is given by < ~0(m) >= Ntr(4D[U] + m) -1 ,
(2)
where N is the number of lattice sites, and P is the usual staggered fermion kinetic operator. The dependence on the coupling constant ,Q - 1/e2 is smuggled in via the definition of the connection U,,(n) = exp(ie8,,(n)), required for gauge invariance of the fermi action . Notice that the fermions are only sensitive to a compact for0920-5632/90/$3 .50 © Elsevier Science Publishers B .V . North-Holland
0 0 ô
0 .00
0 .01
I
0 .02
I
0 .03
-.
0.04
0.05
Figure 1: p(A)/N vs. A at Q = 0.25. mulation of the gauge variables; we shall return to this point. There are two numerical strategies to evaluate
< 00 > in the chiral limit; either a direct inversion of (P + m) using the conjugate-gradient algorithm, which
requires an extrapolation to m = 0, or an estimation of the density p of eigenvalues A of the hermitian operator i49 using the Lanczos algorithm, which yields the condensate via the relation < ~«m = 0) >= rP(a = 0) .
(3)
The advantage of the second method is that the m -e 0 extrapolation is performed analytically, although its applicability beyond the quenched approximation is unclear.
S. Hands/Chiral symmetry breaking
676
0 0
o,
ô-
o-
C
0 C~
i -,Zb.
#V4T U
4A
ô
+K+
® _
++ttt ~®m~ I
CD ~ QO 11110
o
®
® 0® 0 m it - Go IN
mm
t
'+mv
O O ô
0 0
o 0.10
0.15 Figure 2:
0.20
0.25
0.30
< 4 > vs. ~6.
I will concentrate on results obtained using the Lanczos algorithm, but will also show conjugate-gradient data for comparison .1 Figure 1 shows p(A) obtained from the 25 smallest eigenvalues calculated on ensembles of configurations at Q = 0.25 on lattice sizes 104 (dots), 12 4 (squares) and 164 (triangles) . The intercept on the vertical axis is clearly non-zero, signalling broken chiral symmetry. The most striking feature of this plot is the complete absence of finite volume effects: going to larger lattices merely increases the resolution of p(A) in the vicinity of A = 0. The lack of finite volume effects is also seen in conjugate-gradient calculations; it may either be an artifact of the quenched approximation (since the underlying dynamics is free), or be a sign that chiral symmetry breaking is driven by short-ranged effects. Figure 2 shows < ~ lr > vs. ß as calculated by the Lanczos (filled circles) and the conjugate-gradient algorithms (open circles), on lattices ranging in size up to 154. The two methods are in good agreement, except in the immediate neighbourhood of the transition : here we may expect the difficulties associated with the m -+ 0 extrapolation to lead to an overestimate in the latter case . The critical coupling & ti 0.26. Also shown is a mean field fit to the data at strong coupling of the form ~1S_MR - 106 "
(4)
The resulting PMF = 0.236 : there is a significant departure from mean field scaling near the transition. To home in on this region, we performed Lanczos runs on a 164 lattice at 6 values separated by only 0.0025. The
ô
0.000
0.005
A
0.010
0.015
Figure 3: p(A)/N vs . A for P ranging from 0.2350 (diamonds) in steps of 0.0025 to 0.26 (filled squares). 0 = 0.24 is omitted. results for p(A) are shown in Figure 3. For P > 0.25 the curvature of p(A) makes it difficult to extract p(0) using only the simplest assumptions (a straight line fit) . In the absence of a theoretical understanding, we tried two fitting schemes which use different assumptions, and quote the spread in values as a systematic error.l Our results for < ~0 > in the vicinity of the transition are shown in figure 4. The scaling seems to be pretty well fitted by a straight line, although a critical scaling behaviour of the form < ~O >« exp(-b/ P~ -,0), (5) inspired by the self-consistent analytic methods of Miransky et al,4 and with Q,, ^_- 0.27, cannot be excluded . We feel that the uncertainty would be best resolved not by going to larger lattices, but by using the existing data with a better theoretical understanding of the shape of
P(A) . At this stage, since the numerical evidence for Miransky-type scaling is not strong, let us stop to ask if there might not be another mechanism responsible for the transition in the lattice model.2 Recall that fermions enter via an action invariant under compact gauge transformations, ie . the gauge group is U(1)~_-R/Z, which is smaller than the non-compact gauge group R. Whenever a gauge symmetry has a larger global covering symmetry, we can expect topological excitations such as monopoles to be important. In the lattice model, this arises because the fermions are unable to distinguish between flux 0
S. Hands/Chiral symmetry breaking 0
677
ô
ô O
-a . . . . . .a . . . . . . O
o~ i
N w Vo
0 .23
O O 0 .24
0.25
0 .26
0 .27
O>
and 27r through any particular plaquette: in the latter case we say there is a "Dirac string" present. Magnetic monopoles may be identified by the following procedure. 5 First, define the integer variable a,,Y(n), which turns out to be dual to the Dirac sheet, by
where Ô,,Y E current m,,(nl, givedi by
0 .19
0.23
0.27
0 .31
ß
Figure 4: < ~ vs . P near the transition . The two sets of points arise from different fitting procedures .
e0, ,(n) = e0, ,(n) -I- 2reum(n),
0 .15
(6)
Then the magnetic monopole defined on links of the dual lattice, is m,(n) ` E/IVKA~YBKA(n)1
with the kinematic constraint ~,,m,,(n) = 0, implying that the integer-valued current forms closed loops. Monopole currents defined in this way depend on the coupling A: figure 5 shows resultsfrom a 12 4 lattice. The filled circles show the fraction of dual links < I > /NI occupied by monopole current elements, and the triangles show the Dirac sheet area spanning the loops as a fraction of the perimeter < a > / < I > (the position of the sheet is unambiguously defined in the non-compact theory). We see that monopoles are present in measurable concentrations in the coupling range of interest, and that as the coupling gets stronger the ratio area :perimeter rises from - 1/4, which we might expect if the loops are just isolated plaquettes, to - 1/2, which might indicate some sort of coalescing to form larger loops. However, the variation with i0 is smooth: there is no sign of a discontinuity which might signal a connection with the chiral
Figure 5: Monopole activity as a function of Q. 0
®
°
/0
vm
o ô m
°
O
aoo o ?t ô
°
~go V
o 0
°
0 CD
CD
0 0.15
0 .19
0 ô 0
0 .23
0 .27
0 .31
Figure 6: < nmar/ntot > (squares) and X (filled circles) vs. i0 on a 12 4 lattice. transition. INe can more clearly expose discontinuous behaviour by neglecting the vector nature of m,, and simply looking for clusters of dual sites connected together by current links. Figure 6 shows the size of the largest such cluster as a fraction of all occupied sites < nmar/ntoe > (squares), and also a "susceptibility" X=
` z z ( L.n gnn - nm ar ~ ntot
where gn is the number of clusters per configuration of size n. There is spectacular evidence for a percolation threshold at A = 0 .24, very close to the position of the chiral transition .
678
S. Hands/Chiral symmetry breaking
1 end with some speculative remarks . Whilst all the evidence is circumstantial, it is difficult to escape the conclusion that the two transitions are related, and that large tangled monopole loops somehow drive the chiral transition . In any case, it is clear that the self-consistent treatments 4 are not appropriate for describing the transition in the lattice model, since they take no account of the compact nature of the fermi action . It will be interesting to extend these measurements to the unquenched case, where we might expect the two transitions to occur at the same f, If a continuum limit does exist, it may well be very difficult to describe in terms of simple electrons and photons. Perhaps the scaling behaviour predicted by Miransky lies in a different universality class, which could be probed by some linearused form of electron-photon interaction . To implement this in a fashion which has any meaningful remnant of gauge symmetry appears technically challenging. Finally, a word of warning about simulating at 1Vf = 2 . The arguments that the flavor symmetry of staggered fermions is restored in the continuum limit may not hold in the strongcoupling phase if the fermions are confined and do not appear as asymptotic states in the spectrum. One should be even more circumspect than usual in taking the square root of the determinant.
ACKNOWLEDGEMENTS This work was funded by NSF grant PHY 87-01775 . Numerical work was performed using the resources of the National Center for Supercomputing Applications, the Pittsburgh Supercomputing Center, and the Magnetic Fusion Energy Research Center. I've greatly enjoyed and benefitted from discussions with the other participants in this session . REFERENCES 1. S.J. Hands, J .B . Kogut and E. Dagotto, Illinois preprint ILL-(TH)-89-#44 (1989) 2. S.J. Hands and R.J. Wensley, Illinois preprint ILL(TH)-89-#38 (1989) 3. E. Dagotto, A. Kocic and J.B. Kogut, Illinois preprint ILL-(TH)-89-#34 (1989) 4. P.I. Fomin, V.P. Gusynin, V.A. Miransky and Yu.A. Sitenko, Riv. Nuovo Cimento 6 (1983) 1; V.A. Miransky, II Nuovo Cimento 90A (1985) 149 5. T.A. DeGrand and D. Toussaint, Phys. Rev . D22 (1980) 2478
675
CHIRAL SYMMETRY BREAKING IN STRONGLY COUPLED QUENCHED QED Simon HANDS Physics Department, University of Illinois at Urbana-Champaign, 1110 West Green Street, Urbana, IL 61801, USA . Results for the chiral condensate as a function of electron - photon coupling e in quenched non-compact lattice QED, obtained using both conjugate-gradient and Lanczos algorithms, are presented. There is a small but significant departure from mean field scaling, although the evidence for a fixed point of the Miransky type is inconclusive. It is shown that the symmetry-breaking mechanism may be associated with effective monopole loops generated by the non-compact dynamics . In this talk I will present results obtained in collaboration with John Kogut, 1 Elbio Dagotto, 1 and Roy Wensley.2 We have investigated the phenomenon of chi ral symmetry breaking in quenched non-compact lattice QED, with the aim of determining the critical scaling behaviour of the condensate < ~0 >, and understanding the underlying physics. The action for the quenched theory is S8 = - 1 EWjn),
M 40
o aa
0. Cu
0 ô
N
where 0, , is the oriented sum of real link variables ®,,(n) around elementary plaquettes of the lattice. S9 is invariant under local gauge transformations defined by the additive group of real numbers R. Notice that as written Ss has no coupling constant dependence, and simply describes free modes which resemble photons in the small momentum limit. This makes it possible to generate statistically independent configurations {0} at each saaeep by generating the (gauge-fixed) photon modes in momentum space and using a fourier transformation . 3 Critical slowing down is not a problem for this system . The chiral condensate < ~0 > is given by < ~0(m) >= Ntr(4D[U] + m) -1 ,
(2)
where N is the number of lattice sites, and P is the usual staggered fermion kinetic operator. The dependence on the coupling constant ,Q - 1/e2 is smuggled in via the definition of the connection U,,(n) = exp(ie8,,(n)), required for gauge invariance of the fermi action . Notice that the fermions are only sensitive to a compact for0920-5632/90/$3 .50 © Elsevier Science Publishers B .V . North-Holland
0 0 ô
0 .00
0 .01
I
0 .02
I
0 .03
-.
0.04
0.05
Figure 1: p(A)/N vs. A at Q = 0.25. mulation of the gauge variables; we shall return to this point. There are two numerical strategies to evaluate
< 00 > in the chiral limit; either a direct inversion of (P + m) using the conjugate-gradient algorithm, which
requires an extrapolation to m = 0, or an estimation of the density p of eigenvalues A of the hermitian operator i49 using the Lanczos algorithm, which yields the condensate via the relation < ~«m = 0) >= rP(a = 0) .
(3)
The advantage of the second method is that the m -e 0 extrapolation is performed analytically, although its applicability beyond the quenched approximation is unclear.
S. Hands/Chiral symmetry breaking
676
0 0
o,
ô-
o-
C
0 C~
i -,Zb.
#V4T U
4A
ô
+K+
® _
++ttt ~®m~ I
CD ~ QO 11110
o
®
® 0® 0 m it - Go IN
mm
t
'+mv
O O ô
0 0
o 0.10
0.15 Figure 2:
0.20
0.25
0.30
< 4 > vs. ~6.
I will concentrate on results obtained using the Lanczos algorithm, but will also show conjugate-gradient data for comparison .1 Figure 1 shows p(A) obtained from the 25 smallest eigenvalues calculated on ensembles of configurations at Q = 0.25 on lattice sizes 104 (dots), 12 4 (squares) and 164 (triangles) . The intercept on the vertical axis is clearly non-zero, signalling broken chiral symmetry. The most striking feature of this plot is the complete absence of finite volume effects: going to larger lattices merely increases the resolution of p(A) in the vicinity of A = 0. The lack of finite volume effects is also seen in conjugate-gradient calculations; it may either be an artifact of the quenched approximation (since the underlying dynamics is free), or be a sign that chiral symmetry breaking is driven by short-ranged effects. Figure 2 shows < ~ lr > vs. ß as calculated by the Lanczos (filled circles) and the conjugate-gradient algorithms (open circles), on lattices ranging in size up to 154. The two methods are in good agreement, except in the immediate neighbourhood of the transition : here we may expect the difficulties associated with the m -+ 0 extrapolation to lead to an overestimate in the latter case . The critical coupling & ti 0.26. Also shown is a mean field fit to the data at strong coupling of the form ~1S_MR - 106 "
(4)
The resulting PMF = 0.236 : there is a significant departure from mean field scaling near the transition. To home in on this region, we performed Lanczos runs on a 164 lattice at 6 values separated by only 0.0025. The
ô
0.000
0.005
A
0.010
0.015
Figure 3: p(A)/N vs . A for P ranging from 0.2350 (diamonds) in steps of 0.0025 to 0.26 (filled squares). 0 = 0.24 is omitted. results for p(A) are shown in Figure 3. For P > 0.25 the curvature of p(A) makes it difficult to extract p(0) using only the simplest assumptions (a straight line fit) . In the absence of a theoretical understanding, we tried two fitting schemes which use different assumptions, and quote the spread in values as a systematic error.l Our results for < ~0 > in the vicinity of the transition are shown in figure 4. The scaling seems to be pretty well fitted by a straight line, although a critical scaling behaviour of the form < ~O >« exp(-b/ P~ -,0), (5) inspired by the self-consistent analytic methods of Miransky et al,4 and with Q,, ^_- 0.27, cannot be excluded . We feel that the uncertainty would be best resolved not by going to larger lattices, but by using the existing data with a better theoretical understanding of the shape of
P(A) . At this stage, since the numerical evidence for Miransky-type scaling is not strong, let us stop to ask if there might not be another mechanism responsible for the transition in the lattice model.2 Recall that fermions enter via an action invariant under compact gauge transformations, ie . the gauge group is U(1)~_-R/Z, which is smaller than the non-compact gauge group R. Whenever a gauge symmetry has a larger global covering symmetry, we can expect topological excitations such as monopoles to be important. In the lattice model, this arises because the fermions are unable to distinguish between flux 0
S. Hands/Chiral symmetry breaking 0
677
ô
ô O
-a . . . . . .a . . . . . . O
o~ i
N w Vo
0 .23
O O 0 .24
0.25
0 .26
0 .27
O>
and 27r through any particular plaquette: in the latter case we say there is a "Dirac string" present. Magnetic monopoles may be identified by the following procedure. 5 First, define the integer variable a,,Y(n), which turns out to be dual to the Dirac sheet, by
where Ô,,Y E current m,,(nl, givedi by
0 .19
0.23
0.27
0 .31
ß
Figure 4: < ~ vs . P near the transition . The two sets of points arise from different fitting procedures .
e0, ,(n) = e0, ,(n) -I- 2reum(n),
0 .15
(6)
Then the magnetic monopole defined on links of the dual lattice, is m,(n) ` E/IVKA~YBKA(n)1
with the kinematic constraint ~,,m,,(n) = 0, implying that the integer-valued current forms closed loops. Monopole currents defined in this way depend on the coupling A: figure 5 shows resultsfrom a 12 4 lattice. The filled circles show the fraction of dual links < I > /NI occupied by monopole current elements, and the triangles show the Dirac sheet area spanning the loops as a fraction of the perimeter < a > / < I > (the position of the sheet is unambiguously defined in the non-compact theory). We see that monopoles are present in measurable concentrations in the coupling range of interest, and that as the coupling gets stronger the ratio area :perimeter rises from - 1/4, which we might expect if the loops are just isolated plaquettes, to - 1/2, which might indicate some sort of coalescing to form larger loops. However, the variation with i0 is smooth: there is no sign of a discontinuity which might signal a connection with the chiral
Figure 5: Monopole activity as a function of Q. 0
®
°
/0
vm
o ô m
°
O
aoo o ?t ô
°
~go V
o 0
°
0 CD
CD
0 0.15
0 .19
0 ô 0
0 .23
0 .27
0 .31
Figure 6: < nmar/ntot > (squares) and X (filled circles) vs. i0 on a 12 4 lattice. transition. INe can more clearly expose discontinuous behaviour by neglecting the vector nature of m,, and simply looking for clusters of dual sites connected together by current links. Figure 6 shows the size of the largest such cluster as a fraction of all occupied sites < nmar/ntoe > (squares), and also a "susceptibility" X=
` z z ( L.n gnn - nm ar ~ ntot
where gn is the number of clusters per configuration of size n. There is spectacular evidence for a percolation threshold at A = 0 .24, very close to the position of the chiral transition .
678
S. Hands/Chiral symmetry breaking
1 end with some speculative remarks . Whilst all the evidence is circumstantial, it is difficult to escape the conclusion that the two transitions are related, and that large tangled monopole loops somehow drive the chiral transition . In any case, it is clear that the self-consistent treatments 4 are not appropriate for describing the transition in the lattice model, since they take no account of the compact nature of the fermi action . It will be interesting to extend these measurements to the unquenched case, where we might expect the two transitions to occur at the same f, If a continuum limit does exist, it may well be very difficult to describe in terms of simple electrons and photons. Perhaps the scaling behaviour predicted by Miransky lies in a different universality class, which could be probed by some linearused form of electron-photon interaction . To implement this in a fashion which has any meaningful remnant of gauge symmetry appears technically challenging. Finally, a word of warning about simulating at 1Vf = 2 . The arguments that the flavor symmetry of staggered fermions is restored in the continuum limit may not hold in the strongcoupling phase if the fermions are confined and do not appear as asymptotic states in the spectrum. One should be even more circumspect than usual in taking the square root of the determinant.
ACKNOWLEDGEMENTS This work was funded by NSF grant PHY 87-01775 . Numerical work was performed using the resources of the National Center for Supercomputing Applications, the Pittsburgh Supercomputing Center, and the Magnetic Fusion Energy Research Center. I've greatly enjoyed and benefitted from discussions with the other participants in this session . REFERENCES 1. S.J. Hands, J .B . Kogut and E. Dagotto, Illinois preprint ILL-(TH)-89-#44 (1989) 2. S.J. Hands and R.J. Wensley, Illinois preprint ILL(TH)-89-#38 (1989) 3. E. Dagotto, A. Kocic and J.B. Kogut, Illinois preprint ILL-(TH)-89-#34 (1989) 4. P.I. Fomin, V.P. Gusynin, V.A. Miransky and Yu.A. Sitenko, Riv. Nuovo Cimento 6 (1983) 1; V.A. Miransky, II Nuovo Cimento 90A (1985) 149 5. T.A. DeGrand and D. Toussaint, Phys. Rev . D22 (1980) 2478