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Phase structure of strongly coupled lattice Yukawa theories
LKcit Nuclear Physics B (Proc. Suppl.) 26 (1992) 519-521 North-Holland
[P
q1M
r_
C8 [_~ ~~.: L) Uh~i
L I=IL L-1 L uj I-L ~
PHASE STRUCTURE OF STRONGLY COUPLED LATTICE YUKAWA THEORIES Toru Ebiharat* and Kei-Ichi Kondot t Graduate School of Science and Technology, Chiba University, 1-58 Yayoi-cho, Chiba 260, JAPAN t Department of Physics, Faculty of Science, Chiba University, 1-95 Yayoi-cho, Chiba 260, JAPAN
Using weak and strong Yukawa coupling expansions combined with the mean field theory, we determine the p structure of Yukawa, models with Z(2), U(1), O(4) scalars. We have taken into account all the terms in the relevant expansions up to eighth order. In the leading order, all the Yukawa models have four phases and do not possess the ferrimagnetic (FI) phase. This feature persists for Z(2) and U(1) Yukawa models to all orders. In the O(4) Y a model, on the other hand, the FI phase appears in the next to leading order and all higher orders, and two symmetric phases are completely separated by the FI phase. This result suggests that the existence of the FI phase is not a lattice artifact .
We shall study the lattice SU(2)L x SU(2)R Yukawa theory with Nf species staggered fermions X, Xx coupled to the Higgs field 4oz via a 1ocal Yukawa coupling g. , which is defined by the partition function on a d-dimensional hypercubic lattice Za: Z =
X'f
dobx dx .d2ie -S ; S = SB + SF, (1)
SB = -X E SF -
2
N112
Nj x~
p
f=1
Xs ix,p(X- a - Xz- i )
(-1)"+"'+"' with unit length, -tz t = 1 where o=0,1,2,3 denotes O(4) index (;,4,,+A _ Ea=0 r +,). For the staggered fermion X, R we treat an even number of fermion species Nf, which corresponds to 2dl2 Nf species of continuum Dirac fermions due to fermion species doubling . We can easily carry out the fermion integration in the partition function Z as a result of the bilinear structure of the action (1) in the fermionic fields . The Yukawa model is thus converted into a purely bosonic model containing the contribution of the fermion determinant:
NI
2: 1: gY Xx4> .Xx + 1, + f=1 s f=N
Li, k."
I 12+1 s
--Xi'
where rlx,a = 1 (p = 1), rhj, = (-1)"+* ..+s'-i (p = 2, 3, . . . , d). The Higgs field has a global O(4) symmetry and both SU(2) and O(4) notations are used for the radially froze. -- scalar field: + z(.T'Oz (s = 1, 2, 3),
-`e -Satf , Se~ = SB-Nf ln DetM .(4)
* Speaker at the Conference
Although In DetM cannot be calculated analytically in closed form, we may easily perform weak and strong Yukawa coupling expacsions.[1,4] In the strong coupling region (91 , » 1), In DetM can be expanded in powers of the inverse Yukawa
0920-5632/'92/$05.00 0 1992- Elsevier Science Publishers B.V
All rights reserved.
T. Kb°
a, K.-1. Kondo /P/urw structure ofstrongly coupled lattice Mikawa theories
A combinatorial exercise in the Fig. 1 shows
cou ling as fo ows: In
t(K~y + gY
et
-_ g2N Y
+
C2 = (-
0 (_1)n+1 x ngn
4 ) 2d,
C4 = (-
j)2
2á(4d 2á(4d-2),
Cs = (- )3 2á(12d2 r 12á - 7),
Y
]' (5) where K--,, = 2 F 0 and "tr" is taken over SU(2) indices. Here we have
C8 = (- j) 2á(32d3 + 72d2 -117d + 41) , where the factor (-!.)n = (-f- i )n(- z )n comes from the product of K$y matrix elements . In the graphical rewesentation, one can calculate the factor Den easily for each graph . As we would like to take the second order phase transition line, we have only to compute the factor Den to the order H2 . Thus we obtain
*)2 = l and used the fact that s fi = that Det ( ) = et ( O) for , E SU(2). Now we apply the mean field theory[8] to our model. Equation (5) has a nice graphical interpretation, as shown in Fig. 1: Each term can be rep_ E2 1)n-1 +O(H 3). Den 8 (8) resented by a graph consisting of a closed chain with vertices on sites X1, z2i . . . , z2n and links In the strong coupling region we have percorresponding to the fermion matrix K.,, . formed the 1/g, expansion exactly up to order
Fig. 1: Various contributions to C6 at strong Yukawa coupling : (a)the two-link straight path graph (b)1he overlaid link graph (c)the plaquefe graph .
1/988 . To classify the phase we have introduced two order parameters : (A~) and (-tsi.) # 1 whose nonzero values signal the breaking of the symmetry of the model. Here we summarize the critical lines obtained for our O(4) Yukawa model with Nf species staggered fermions in d dimension, which is exact up to order 1/g" in the strong coupling region, i. e., gY / d/2 > 1:
Each graph is associated with a factor [1 2d-1 + 6d2 +6d-7 K = +? T Nfd 1 :E (trot, s, . .384g 2.)H - Den, and a combinato2 6 d (4 32g4 gY gY gY rial factor C2n which is equal to the number of 32d3 + 72d2 - 117d + 41 (9) possible ways to put the graph of this type on 8192g8 )~' the lattice. Using these factors, we can rewrite Y equation (5) as follows : where the plus sign refers to the FM(S)-PM(S) transition line and the minus sign to the AFM(S)(In DetMay )N PM(S) transition . (_1)2n+1 n=1
. . - C2n Den N, 2 0 2n Y
where N is the number of lattice sites.
i (Oat .) denotes the staggered magnetization defined by (tae) . N with N being the number of lattice sites.
T. Ebiharq K-I. Kondo/Phase structure of strongly coupled lattice Hikawa theories
521
Similarly, the critical line in the weak coupling region, i.e., g, / d/2 < 1, is obtained as follows.
up only the double chain graph (leading order of the 1/d expansion) appearing in every order of 1/g,, expansion in the framework of the F 2 _ _2 _1 2 2d- 1 4 6d +6d-7 6 proximation. This approach, even if it is perhaps d [1 Nf d3 gy -1g (dgy useful qualitatively, cannot be expected to give 6d5 quantitatively correct results. 3243 + 72d2 _ 117d + 41 8 (10) 9Y) . 32d7 Actually our result shows that the existence of FI phase is independent of the fermion species where the plus sign refers to the FM(W)-PM(W) N1 > NÎ = 2 .4 (d = 4), while FI phase ex transition line and the minus sign to the AFM(W)ists only for Nf > 4 according to Stephanov and PM(W) transition . Tsypin .[1] By using the above equations for the four critIt is very interesting to compare the phase ical lines, we get the new phase, a ferrimagnetstructure of the O(4) Yukawa model with that ically Ordered (FI) phase «0) îé 0, (0,t) 0) of the corresponding Z(2)[2,6,7] and U(1)[4,5] in the intermediate region of Yukawa coupling models . For Z(2) and U(1) Yukawa models, FI (see solid line in Fig. 2), in sharp contrast with phase does not appear to all order in the weak the leading order result where the FI phase is and the strong Yukawa coupling expansions (up not prodtieed (see broken line in Fig. 2). The to eighth order) and there are four phases : FM, presence of this FI phase implies that P phase PMW, PMS and AFM. with unbroken symmetry is not a single phase, Our calculations show that the most interbut consist of two parts; PM phase with weak esting phenomena in the lattice Higgs-fermion Yukawa coupling and with strong one, both of theory may take place in the intermediate rethem are completely separated by FI phase. gion,i. e., gY = Z.
e
d
2
r;
References [1] M.A . Stephanov and M.M . Tsypin, Phys. Lett.
Fig. 2: Phase Diagram for the O(4) Yukawa model (Nf = 8, d = 4) . Our results should be compared with those of Stephanov and Tsypin.[1] They combined the 1/d expansion with 1/g ß, expansion and summed
242B (1990) 432; 236B (1990) 344; 261B (1991) 109. [2] T. Ebihara and K,1. Kondo, Preprint CH1UA-EP55 (1991) . [3] W. Boric, A.K . De, K. Jansen, J. Jeisdk, T. Neuhaus and J. Smit, Nucl. Phys . B344 (1990) 207. [4] A. Hasenfratz, W. Liu and T. Neuhaus, Phys . Lett . 236B (1990) 339. [5] L. Lin, I. Montvay, H. Wittig and G. Mùnster, DESY preprint 90-142 . [6] I-H . Lee, J. Shigemitsu and R.E.Shrock, Nucl. Phys . 334 (19901 265: [7] T. Ebihara, Master Thesis, Chiba University (1991) . [8] J.M . Drouffe and J.B . Zuber, Phys . Rep. 102 (1983) 1.
[P
q1M
r_
C8 [_~ ~~.: L) Uh~i
L I=IL L-1 L uj I-L ~
PHASE STRUCTURE OF STRONGLY COUPLED LATTICE YUKAWA THEORIES Toru Ebiharat* and Kei-Ichi Kondot t Graduate School of Science and Technology, Chiba University, 1-58 Yayoi-cho, Chiba 260, JAPAN t Department of Physics, Faculty of Science, Chiba University, 1-95 Yayoi-cho, Chiba 260, JAPAN
Using weak and strong Yukawa coupling expansions combined with the mean field theory, we determine the p structure of Yukawa, models with Z(2), U(1), O(4) scalars. We have taken into account all the terms in the relevant expansions up to eighth order. In the leading order, all the Yukawa models have four phases and do not possess the ferrimagnetic (FI) phase. This feature persists for Z(2) and U(1) Yukawa models to all orders. In the O(4) Y a model, on the other hand, the FI phase appears in the next to leading order and all higher orders, and two symmetric phases are completely separated by the FI phase. This result suggests that the existence of the FI phase is not a lattice artifact .
We shall study the lattice SU(2)L x SU(2)R Yukawa theory with Nf species staggered fermions X, Xx coupled to the Higgs field 4oz via a 1ocal Yukawa coupling g. , which is defined by the partition function on a d-dimensional hypercubic lattice Za: Z =
X'f
dobx dx .d2ie -S ; S = SB + SF, (1)
SB = -X E SF -
2
N112
Nj x~
p
f=1
Xs ix,p(X- a - Xz- i )
(-1)"+"'+"' with unit length, -tz t = 1 where o=0,1,2,3 denotes O(4) index (;,4,,+A _ Ea=0 r +,). For the staggered fermion X, R we treat an even number of fermion species Nf, which corresponds to 2dl2 Nf species of continuum Dirac fermions due to fermion species doubling . We can easily carry out the fermion integration in the partition function Z as a result of the bilinear structure of the action (1) in the fermionic fields . The Yukawa model is thus converted into a purely bosonic model containing the contribution of the fermion determinant:
NI
2: 1: gY Xx4> .Xx + 1, + f=1 s f=N
Li, k."
I 12+1 s
--Xi'
where rlx,a = 1 (p = 1), rhj, = (-1)"+* ..+s'-i (p = 2, 3, . . . , d). The Higgs field has a global O(4) symmetry and both SU(2) and O(4) notations are used for the radially froze. -- scalar field: + z(.T'Oz (s = 1, 2, 3),
-`e -Satf , Se~ = SB-Nf ln DetM .(4)
* Speaker at the Conference
Although In DetM cannot be calculated analytically in closed form, we may easily perform weak and strong Yukawa coupling expacsions.[1,4] In the strong coupling region (91 , » 1), In DetM can be expanded in powers of the inverse Yukawa
0920-5632/'92/$05.00 0 1992- Elsevier Science Publishers B.V
All rights reserved.
T. Kb°
a, K.-1. Kondo /P/urw structure ofstrongly coupled lattice Mikawa theories
A combinatorial exercise in the Fig. 1 shows
cou ling as fo ows: In
t(K~y + gY
et
-_ g2N Y
+
C2 = (-
0 (_1)n+1 x ngn
4 ) 2d,
C4 = (-
j)2
2á(4d 2á(4d-2),
Cs = (- )3 2á(12d2 r 12á - 7),
Y
]' (5) where K--,, = 2 F 0 and "tr" is taken over SU(2) indices. Here we have
C8 = (- j) 2á(32d3 + 72d2 -117d + 41) , where the factor (-!.)n = (-f- i )n(- z )n comes from the product of K$y matrix elements . In the graphical rewesentation, one can calculate the factor Den easily for each graph . As we would like to take the second order phase transition line, we have only to compute the factor Den to the order H2 . Thus we obtain
*)2 = l and used the fact that s fi = that Det ( ) = et ( O) for , E SU(2). Now we apply the mean field theory[8] to our model. Equation (5) has a nice graphical interpretation, as shown in Fig. 1: Each term can be rep_ E2 1)n-1 +O(H 3). Den 8 (8) resented by a graph consisting of a closed chain with vertices on sites X1, z2i . . . , z2n and links In the strong coupling region we have percorresponding to the fermion matrix K.,, . formed the 1/g, expansion exactly up to order
Fig. 1: Various contributions to C6 at strong Yukawa coupling : (a)the two-link straight path graph (b)1he overlaid link graph (c)the plaquefe graph .
1/988 . To classify the phase we have introduced two order parameters : (A~) and (-tsi.) # 1 whose nonzero values signal the breaking of the symmetry of the model. Here we summarize the critical lines obtained for our O(4) Yukawa model with Nf species staggered fermions in d dimension, which is exact up to order 1/g" in the strong coupling region, i. e., gY / d/2 > 1:
Each graph is associated with a factor [1 2d-1 + 6d2 +6d-7 K = +? T Nfd 1 :E (trot, s, . .384g 2.)H - Den, and a combinato2 6 d (4 32g4 gY gY gY rial factor C2n which is equal to the number of 32d3 + 72d2 - 117d + 41 (9) possible ways to put the graph of this type on 8192g8 )~' the lattice. Using these factors, we can rewrite Y equation (5) as follows : where the plus sign refers to the FM(S)-PM(S) transition line and the minus sign to the AFM(S)(In DetMay )N PM(S) transition . (_1)2n+1 n=1
. . - C2n Den N, 2 0 2n Y
where N is the number of lattice sites.
i (Oat .) denotes the staggered magnetization defined by (tae) . N with N being the number of lattice sites.
T. Ebiharq K-I. Kondo/Phase structure of strongly coupled lattice Hikawa theories
521
Similarly, the critical line in the weak coupling region, i.e., g, / d/2 < 1, is obtained as follows.
up only the double chain graph (leading order of the 1/d expansion) appearing in every order of 1/g,, expansion in the framework of the F 2 _ _2 _1 2 2d- 1 4 6d +6d-7 6 proximation. This approach, even if it is perhaps d [1 Nf d3 gy -1g (dgy useful qualitatively, cannot be expected to give 6d5 quantitatively correct results. 3243 + 72d2 _ 117d + 41 8 (10) 9Y) . 32d7 Actually our result shows that the existence of FI phase is independent of the fermion species where the plus sign refers to the FM(W)-PM(W) N1 > NÎ = 2 .4 (d = 4), while FI phase ex transition line and the minus sign to the AFM(W)ists only for Nf > 4 according to Stephanov and PM(W) transition . Tsypin .[1] By using the above equations for the four critIt is very interesting to compare the phase ical lines, we get the new phase, a ferrimagnetstructure of the O(4) Yukawa model with that ically Ordered (FI) phase «0) îé 0, (0,t) 0) of the corresponding Z(2)[2,6,7] and U(1)[4,5] in the intermediate region of Yukawa coupling models . For Z(2) and U(1) Yukawa models, FI (see solid line in Fig. 2), in sharp contrast with phase does not appear to all order in the weak the leading order result where the FI phase is and the strong Yukawa coupling expansions (up not prodtieed (see broken line in Fig. 2). The to eighth order) and there are four phases : FM, presence of this FI phase implies that P phase PMW, PMS and AFM. with unbroken symmetry is not a single phase, Our calculations show that the most interbut consist of two parts; PM phase with weak esting phenomena in the lattice Higgs-fermion Yukawa coupling and with strong one, both of theory may take place in the intermediate rethem are completely separated by FI phase. gion,i. e., gY = Z.
e
d
2
r;
References [1] M.A . Stephanov and M.M . Tsypin, Phys. Lett.
Fig. 2: Phase Diagram for the O(4) Yukawa model (Nf = 8, d = 4) . Our results should be compared with those of Stephanov and Tsypin.[1] They combined the 1/d expansion with 1/g ß, expansion and summed
242B (1990) 432; 236B (1990) 344; 261B (1991) 109. [2] T. Ebihara and K,1. Kondo, Preprint CH1UA-EP55 (1991) . [3] W. Boric, A.K . De, K. Jansen, J. Jeisdk, T. Neuhaus and J. Smit, Nucl. Phys . B344 (1990) 207. [4] A. Hasenfratz, W. Liu and T. Neuhaus, Phys . Lett . 236B (1990) 339. [5] L. Lin, I. Montvay, H. Wittig and G. Mùnster, DESY preprint 90-142 . [6] I-H . Lee, J. Shigemitsu and R.E.Shrock, Nucl. Phys . 334 (19901 265: [7] T. Ebihara, Master Thesis, Chiba University (1991) . [8] J.M . Drouffe and J.B . Zuber, Phys . Rep. 102 (1983) 1.