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Infinite N phase transitions in one-plaquette (2+1)-dimensional models of lattice gauge theory with manton's action
Volume 129B, number 3,4
PHYSICS LETTERS
22 September 1983
INFINITE N PHASE TRANSITIONS IN ONE-PLAQUETTE (2 + I)-DIMENSIONAL MODELS OF LATTICE GAUGE THEORY WITH MANTON'S ACTION "~ Ole TRINHAMMER
Institute of TheoreticalPhysics,S-412 96 G6teborg,Sweden Received 9 May 1983
Pure gluon U(N) lattice gauge theory is solvedby the transfer matrix method in a semiclassicalapproximation for Manton's action showing a third order phase transition at he = g2N =it2/2x/~.
Introduction. With current Monte Carlo simulations for the hadronic spectrum, the lattice formulation of gauge theories [ 1,2] has regained interest. Even pure gluon models might be of relevance to the simulations [3]. Different actions have been proposed to put the continuous QCD theory on the lattice [ 4 - 6 ] , and in (1 + 1)-dimensional models they have shown different phase structure as to the presence of third order phase transitions with, e.g., the Wilson action showing a transition [7] and Manton's not [8]. For the I[N expansion [9] the existence of phase transitions in the infinite N limit is crucial for the possible interchange of the strong coupling and the large Nlimit [7,8]. It was conjectured in ref. [8] from (1 + 1)-dimensional calculations, that one might avoid a (thus artificial, scheme-dependent) phase transition by the use of Manton's action for higher dimensional systems also. My calculations, however, show that the semiclassical treatment [10] of Manton's action in 2 + 1 dimensions lends no hope. One encounters a third order phase transition, completely analogous to Wadia's Wilson treatment [ 11 ] and Azakov's [ 12] loop-space [13] treatment of Manton's action. Here I will follow a transfer matrix method [11] proven in ref. [14] to be valid for any gauge invariant action of the pla. quette variables. Work supported by Swedish Institute. 234
For the purposes here, it will become clear that Manton's action makes the mathematics of the semiclassical approximation much easier and its physics more transparent. This has been of great importance in the discussion of its relation to the exact quantum treatment in ref. [15].
Manton's action on a (2 + 1)-dimensional lattice. The partition function [ 1]
Z(N, /3)=f ~ dU~ exp( # ~p tr×2(p))
(1)
defines the model, where II~ dU~ is the Haar measure on the gauge group U(N) oI generalized pure gluon QCD. (1) delivers the dynamical matrix variables on the links £ and/3 tr ×2(p) is Manton's action defined via the shortest distance to 1N in group space (geodesic [16]): d2(1N, Up)= tr X2(p),
exp[ix(P)l= Up,
(2)
where Up is the usual Wilson plaquette variable Up = U(1)U(2)U+(3)U+(4), and ×(p) a hermitian matrix with all eigenvalues lying between -rr and n. (That this uniquely defines X to lie in the so-called first Brillouin zone of the magnetic weight lattice is best seen in ref. [16].)/3 = 1/2g2,go is the coupling constant, including the lattice constant, to ensure the correct continuum limit. The partition function (1) generates plaquettes on
0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 129B, number 3,4 Space
PHYSICS LETTERS
~=_:.- - r - - - - . - - -~=--_-- - - _ - _ - : S S S - S - - - - - - S ~ - - - ' ~
U (4)~' ~
[~___.~T,rne
',
. . i. . . . . . . . . . .
-L._:~-- -~-----s~ "a~-- _-~ . . . . . . . . . . . . .
Space
-'J'--_-.~'
U( I~"- . . . . .
Fig. 1.
the model lattice of fig. 1 and governs their dynamics. I impose periodic boundary conditions in time, period T. In the appendix of ref. [14] it is shown that Wadia's [11 ] generalized transfer matrix method [2] for oneplaquette dynamics works for Manton's action as well as for Wilson's (and in fact works for any action that is a class function of the plaquette variables). The result is z = Tr V T ,
(3)
where T is the time extension of the system and Tr indicates periodic time boundary and integration over initial configuration states. V is the (gauge invariant) transfer matrix which operates on the space generated by the N independent gauge invariant operators
Uk = N -1 t r U k ,
k =l,2,...,N,
U = U(1)U(2)U+(3)U+(4).
(4)
The definition of these operators is similar to the overcomplete set used in the loop-space representation of Jevicki and Sakita [13]. In the limit of continuous time, V becomes V = 1 - rn,
(5)
where ~---> 0 is a normalization constant proportional to the lattice constant in the time direction and where H is the Kogut-Susskind [ 17], one-plaquette hamiltonian: H = _2g2A + g - 2 tr X2(p),
(6)
with X(P) only spacelike plaquettes and with A = - E 2 being the Laplace operator on the group (where E l , the electric field of Kogut and Susskind, is generator of the U(N) rotations in the gauge invariant space of states [11]). Now this hamiltonian - explicitly time independent - is a generator of infinitesimal time translations and is thus the desired hamiltonian. By means of a transfer matrix seen as an evolution operator and by
22 September 1983
the fundamental identification of the hamiltonian as a generator of infinitesimal time translations, the lagranglan problem is recast in the hamiltonian formalism. At least for one-plaquette models the hamiltonian thus follows directly from the lagrangian frame of the partition function. Essentially only one assumption is set: The lattice dynamics is made extremely anisotropic (/3r-+ ~,/]x -+ 0) under the scaling conditions: /3r/3x = (l/g2) 2 ,
p r r = 1/g 2 ,
g fixed.
(7)
Thus the general expectation that different frames should show the same phenomenology in phase structure is confirmed when we find a third order phase transition for Manton's action soon. Recently Caldi [18] showed by use of the transfer matrix method in 1 + 1 dimensions that you get exactly the hamiltonian of (6) without the Manton (Wilson) term present. This is what you would expect since you cannot have spacelike plaquettes in 1 + 1 dimensions. Whether you would expect that from your general belief in equivalence between lagrangian and hamiltonian formulations from the explicit hamiltonian frame with no plaquettes available at all, a trivial phase structure, is a matter of taste. Caldi finds no transition. The absence of the phase transition in the calculations of tang et al. [8 ] using Manton's action in I + I dimensions and in the partition function frame, and the absence in Caldi's calculations using the Wilson action and the transfer matrix frame has been held paradoxical [12,18,19] to the presence of phase transitions in calculations based on Jevicki and Sakita's loop-space representation [13] using either Manton [12,19] or Wilson actions [12]. But since Jevicki and Sakita's formalism starts directly from the Kogut-Susskind hamiltonian (with the plaquette term present) one can understand the origin of the erroneous comparisons as being mistakes already at the level of dimensionality ,1. It is trivial to see that within the approximations of the loop-space formalism you get no phase transition if the plaquette term is dropped. Still, though, there is a paradox (of which I have not found a good explanation), namely between the original Gross-Witten phase transition [11 ] using the ,1 I thank Bo-Sture Skagerstam for this argument.
235
Volume 129B, number 3,4
PHYSICS LETTERS
Wilson action in 1 + 1 dimensions and the absence of it found by Lang et al. using the Manton action in 1 + 1 dimensions. But as similarity is reestablished, as we shall soon see in 2 + 1 dimensions, the solution of the 1 + 1 paradox looses some excitement.
22 September 1983 v(o)
:
Parametrization. Having proved the applicability of the transfer matrix it is straightforward to follow Wadia [11] through the transformation of the problem (6) in group space to a problem in parameter space [20]: N
N a2 lx-'x - - +~£.1 •_- a02 Xi=I t ]
N2
'
where X = g2N, -lr < 0 i < lr, 0 i being the N eigenangles of the U(N) plaquette variable, and E(O)/N 2 is the ground state energy/degree of freedom of this matrix, q~ = J ~ is a wavefunction, scaling the original, qJ, wavefunction in group space by the jacobian of the parametrization transformation. Since ~0 is symmetric (there is no preferred labelling of the N eigenangles) and J is antisymmetric, ep describes a problem of N"fermionic" entities. (8) easily separates to N equations: ( - 2 g 2 d2/dO 2 +g-20i)~o(Oi) = e,~(Oi),
(9)
where E (0) = ~'g l ei, -rt ~ 0 i < lr. In an exact treatment one would have to find the N lowest levels of (9) [ 15]. However, here I go on to analyse the problem semiclassically following Wadia [11] and Brezin et al. [10].
u(OI
Fig. 2. Introducing momentum.
the classical hamiltonian. See fig. 2. Each point in phase space (0, p) corresponds to an energy described by (12); and from the uniform distribution in phase space we can get back the eigenangle density [which replaces the average of the squares of the wavefunctions in (9)] : PF
u(O) = f u(O,p) dp, -PF
(13)
defining the "Fermi momentum" PF for a given angle as
h(O,PF) =EF'e'p F = (EF[2g 2 -- 02[294) 1/2 ,
(14)
and we get PF
u(O) = 2 f 0
dp u(O,p) = 2PF =~f2(N/X)(Xe F - 0 2 ) 1/2, (is)
where X = g2N and e F = E F IN.
Semiclassical calculation. We can treat the problem as a many-fermion system with the ground state uniformly populated by "fermions" up to the Fermi level [21]. Thus for large N one introduces a semiclassical distribution function in phase space given by: u(O,p) = 1
forh(O,p)<,EF ,
=0
for h(O, p) > E F ,
f dOdp u'O, t p)=N.
(10)
(11)
236
N =f
dO ~-~ u(O)
¢, N = (N/rrN/2X) {rr [XeF(3,) -- lr2 ] 1/2
E F is the Fermi level and h(0, p) = 2g2p 2 + g - 2 0 2 ,
Strong coupling. For e F ~> 7r2/X (i.e. X large for given eF) the allowed 0-interval certainly extends between - I t and rr and we can readily determine the Fermi level e F as a function of X by use of (15) and the normalization condition (11); for Manton's action there are no cumbersome elliptic integrals as for Wilson's:
(12)
+ XeF(X) arcsin (n/[XeF(X)] 1/2)).
(16)
V o l u m e 129B, n u m b e r 3,4
PHYSICS LETTERS
The ground state energy of the total system is calculated by weighting with our energy measure (12):
fdO - ~ do u(O,p)h(O,p)
E(O) =
22 September 1983
E(O)' N2
(17)
That is Es(0)
I
N2
N2
_
r
~r P F
r
J O
d 0 d p (2g2p 2 + g - 2 0 2 )
Fig. 3. Behavmur o f
2 lrvr2~. 2
f
dO [~(Xe F - 02)3/2 + 02(~ke F -- 02)1/2 ] •
o
(18)
After some calculations and the normalization result (16) as it stands, these two integrals readily yields
E(O)/N 2 = ~e F - X- 2 ~ 2-1/2(XeF - rr2) 3/2 .
(19)
Weak coupling. This corresponds to h(O, O) > E F for some 0 I define
E
]-,r, Ir [.
h(OF, O) = E F ¢==~OF = (XeF) 1/2 .
(20)
I still have the (0-dependent) Fermi m o m e n t u m (14) but now I have to normalize the density on [-O F, OF] OF
N=
E(O)/N2 as a f u n c t i o n
of h = g2N.
-rr -PF
f 2dO -~(0)
¢=~ e F = 2 V ~ ,
(21)
-O F
and with the similar calculations of ( 1 7 ) - ( 1 9 ) you get
E(0)/Ar2w ,'" = ~ev
= X/'~"
(22)
Introducing x2(•) = XeFfir2 , (16) and (19) might be written (x 2 - 1)1/2 + x 2 arcsin(1/Ixl) = ½1rX/Xc,
(23)
E(sO)/N 2 = rrZx2/2X - 7r3(x 2 - 1)3/2/3V~-X 2 ,
(24)
where Xc = rr2/2x/2. The strong coupling criterium e F i> rr2/X corresponds to x 2 ~> 1. Except for a factor 2 times the lattice constant in the energy expression, these are complete analogues of .Azakov's results [12] obtained by using the direct Kogut-Susskind hamiltonian solved by Jevicki-Sakita's loop-space method [13]. E(O)/N2(X) is shown qualitatively in fig. 3. Comparing (22) and (24) and using (23) for im-
plicit differentiation, it is not hard to show that E(sO)/ N 2 and E(~O)/N2 have common value at x = 1, common first and second derivative (=0) whereas the third derivative is discontinuous at x = 1, signalling a third order phase transition at the corresponding X-value X = Xc. That is, a phase transition takes place between strong and weak coupling domains. If one treats the potential in the full quantum mechanical problem (9) as a true (non periodic) harmonic oscillator potential, it is easily seen that the value Xc correspond to the situation where level N reaches the top of the potential [15].
Conclusion. It has been proven by a semiclassical argument, that there is no difference in phase structure for the Wilson [11 ] and Manton action when formulated in lagrangian language and then translated by the transfer matrix to the hamiltonian: The (2 + 1)-dimensional lattice model of pure gluon U(N) one-plaquette gauge theory exhibits a third order phase transition in semiclassical treatment o f the large N limit. Further it is seen that the method applied here yields also the same phase structure as the JevickiSakita [ 13] loop-space, collective field method applied by Azakov [12]. This confirms in (2 + 1)-dimensional one-plaquette models the validity of the general expectation, recently stressed by Caldi [18], that where the type of question is such that different schemes of formulation can give an answer, the answers should agree. The work was supported by a scholarship grant from the Swedish Institute, and I would like to thank Bo-Sture Skagerstam for helping to settle this investigation and for a lot of helpful discussions. I thank Per Salomonson for reading the manuscript, and the particle theory group at Chalmers in G6teborg for hospitality during my stay. 237
Volume 129B, number 3,4
PHYSICS LETTERS
References [1] G. Wilson, Phys. Rev. 10D (1974) 2445. [2] J.B. Kogut, Rev. Mod. Phys. 51 (1979) 659. [3] E.g.B. Berg and A. Bllloke, Glueball spectroscopy in 4-d SU(3) lattice gauge theory, DESY preprint 82-079 (December 1982); K. Ishikawa, A. Sato, G. Schierholz and M. Teper, Phys. Lett. 120B (1983) 387. [4] N.S. Manton, Phys. Lett. 96B (1980) 328. [5] E. Onofri, SU(N) gauge theory with Villain's action, CERN preprint (1981). [6] P. Menotti and E. Onofri, Nucl. Phys. B190 (1981) 288. [7] D.J. Gross and E. Witten, Phys. Rev. D21 (1980) 446. {8] C.B. Lang, P. Salomonson and B.S. Skagerstam, Nucl. Phys. B190 (FS3) (1981) 337. [9] G. 't Hooft. Nucl. Phys. B75 (1974) 461. [10] E. Br6zin, C. Itzykson, G. Parisi and J. Zuber, Commun. Math. Phys. 59 (1978) 35. [11] S.R. Wadm, Phys. Lett. 93B (1980) 403.
238
22 September 1983
[ 12 ] S.I. Azakov, The I[N expansion m the one-plaquette model with Manton's action. Laboratory of High Energy Physics preprint no. 46 (Baku, 1982). [13] A. Jevicki and B. Sakita, Phys. Rev. D22 (1980) 467. [14] O. Trinhammer, Gbteborg report (Chalmers, G6teborg, Sweden); Appendix (April 1983). [ 15] O. Trinhammer, Band structure approach to ground state energy m lattice gauge theory, G6teborg report (Chalmers, G6teborg, April 1983). [16] J. Minor, Ann. Math. Stud. 51 (1963). [17] J.B. Kogut and L. Susskind, Phys. Rev. D l l (1975) 395. [18] D.G. Caldi, Phys. Rev. D25 (1982) 3356. [19] J.P. Rodrigues, Variant actions and the presence of a Gross-Witten phase transition, preprints NSF-ITP-81122, BROWN-HET-462. [201 H. Weyl: The classicalgroups (Princeton U.P., Princeton, 1946) p. 194. [21 ] K. Osterwalder and E. Seller, Ann. Phys. (NY) 110 (1978) 440.
PHYSICS LETTERS
22 September 1983
INFINITE N PHASE TRANSITIONS IN ONE-PLAQUETTE (2 + I)-DIMENSIONAL MODELS OF LATTICE GAUGE THEORY WITH MANTON'S ACTION "~ Ole TRINHAMMER
Institute of TheoreticalPhysics,S-412 96 G6teborg,Sweden Received 9 May 1983
Pure gluon U(N) lattice gauge theory is solvedby the transfer matrix method in a semiclassicalapproximation for Manton's action showing a third order phase transition at he = g2N =it2/2x/~.
Introduction. With current Monte Carlo simulations for the hadronic spectrum, the lattice formulation of gauge theories [ 1,2] has regained interest. Even pure gluon models might be of relevance to the simulations [3]. Different actions have been proposed to put the continuous QCD theory on the lattice [ 4 - 6 ] , and in (1 + 1)-dimensional models they have shown different phase structure as to the presence of third order phase transitions with, e.g., the Wilson action showing a transition [7] and Manton's not [8]. For the I[N expansion [9] the existence of phase transitions in the infinite N limit is crucial for the possible interchange of the strong coupling and the large Nlimit [7,8]. It was conjectured in ref. [8] from (1 + 1)-dimensional calculations, that one might avoid a (thus artificial, scheme-dependent) phase transition by the use of Manton's action for higher dimensional systems also. My calculations, however, show that the semiclassical treatment [10] of Manton's action in 2 + 1 dimensions lends no hope. One encounters a third order phase transition, completely analogous to Wadia's Wilson treatment [ 11 ] and Azakov's [ 12] loop-space [13] treatment of Manton's action. Here I will follow a transfer matrix method [11] proven in ref. [14] to be valid for any gauge invariant action of the pla. quette variables. Work supported by Swedish Institute. 234
For the purposes here, it will become clear that Manton's action makes the mathematics of the semiclassical approximation much easier and its physics more transparent. This has been of great importance in the discussion of its relation to the exact quantum treatment in ref. [15].
Manton's action on a (2 + 1)-dimensional lattice. The partition function [ 1]
Z(N, /3)=f ~ dU~ exp( # ~p tr×2(p))
(1)
defines the model, where II~ dU~ is the Haar measure on the gauge group U(N) oI generalized pure gluon QCD. (1) delivers the dynamical matrix variables on the links £ and/3 tr ×2(p) is Manton's action defined via the shortest distance to 1N in group space (geodesic [16]): d2(1N, Up)= tr X2(p),
exp[ix(P)l= Up,
(2)
where Up is the usual Wilson plaquette variable Up = U(1)U(2)U+(3)U+(4), and ×(p) a hermitian matrix with all eigenvalues lying between -rr and n. (That this uniquely defines X to lie in the so-called first Brillouin zone of the magnetic weight lattice is best seen in ref. [16].)/3 = 1/2g2,go is the coupling constant, including the lattice constant, to ensure the correct continuum limit. The partition function (1) generates plaquettes on
0.031-9163/83/0000-0000/$ 03.00 © 1983 North-Holland
Volume 129B, number 3,4 Space
PHYSICS LETTERS
~=_:.- - r - - - - . - - -~=--_-- - - _ - _ - : S S S - S - - - - - - S ~ - - - ' ~
U (4)~' ~
[~___.~T,rne
',
. . i. . . . . . . . . . .
-L._:~-- -~-----s~ "a~-- _-~ . . . . . . . . . . . . .
Space
-'J'--_-.~'
U( I~"- . . . . .
Fig. 1.
the model lattice of fig. 1 and governs their dynamics. I impose periodic boundary conditions in time, period T. In the appendix of ref. [14] it is shown that Wadia's [11 ] generalized transfer matrix method [2] for oneplaquette dynamics works for Manton's action as well as for Wilson's (and in fact works for any action that is a class function of the plaquette variables). The result is z = Tr V T ,
(3)
where T is the time extension of the system and Tr indicates periodic time boundary and integration over initial configuration states. V is the (gauge invariant) transfer matrix which operates on the space generated by the N independent gauge invariant operators
Uk = N -1 t r U k ,
k =l,2,...,N,
U = U(1)U(2)U+(3)U+(4).
(4)
The definition of these operators is similar to the overcomplete set used in the loop-space representation of Jevicki and Sakita [13]. In the limit of continuous time, V becomes V = 1 - rn,
(5)
where ~---> 0 is a normalization constant proportional to the lattice constant in the time direction and where H is the Kogut-Susskind [ 17], one-plaquette hamiltonian: H = _2g2A + g - 2 tr X2(p),
(6)
with X(P) only spacelike plaquettes and with A = - E 2 being the Laplace operator on the group (where E l , the electric field of Kogut and Susskind, is generator of the U(N) rotations in the gauge invariant space of states [11]). Now this hamiltonian - explicitly time independent - is a generator of infinitesimal time translations and is thus the desired hamiltonian. By means of a transfer matrix seen as an evolution operator and by
22 September 1983
the fundamental identification of the hamiltonian as a generator of infinitesimal time translations, the lagranglan problem is recast in the hamiltonian formalism. At least for one-plaquette models the hamiltonian thus follows directly from the lagrangian frame of the partition function. Essentially only one assumption is set: The lattice dynamics is made extremely anisotropic (/3r-+ ~,/]x -+ 0) under the scaling conditions: /3r/3x = (l/g2) 2 ,
p r r = 1/g 2 ,
g fixed.
(7)
Thus the general expectation that different frames should show the same phenomenology in phase structure is confirmed when we find a third order phase transition for Manton's action soon. Recently Caldi [18] showed by use of the transfer matrix method in 1 + 1 dimensions that you get exactly the hamiltonian of (6) without the Manton (Wilson) term present. This is what you would expect since you cannot have spacelike plaquettes in 1 + 1 dimensions. Whether you would expect that from your general belief in equivalence between lagrangian and hamiltonian formulations from the explicit hamiltonian frame with no plaquettes available at all, a trivial phase structure, is a matter of taste. Caldi finds no transition. The absence of the phase transition in the calculations of tang et al. [8 ] using Manton's action in I + I dimensions and in the partition function frame, and the absence in Caldi's calculations using the Wilson action and the transfer matrix frame has been held paradoxical [12,18,19] to the presence of phase transitions in calculations based on Jevicki and Sakita's loop-space representation [13] using either Manton [12,19] or Wilson actions [12]. But since Jevicki and Sakita's formalism starts directly from the Kogut-Susskind hamiltonian (with the plaquette term present) one can understand the origin of the erroneous comparisons as being mistakes already at the level of dimensionality ,1. It is trivial to see that within the approximations of the loop-space formalism you get no phase transition if the plaquette term is dropped. Still, though, there is a paradox (of which I have not found a good explanation), namely between the original Gross-Witten phase transition [11 ] using the ,1 I thank Bo-Sture Skagerstam for this argument.
235
Volume 129B, number 3,4
PHYSICS LETTERS
Wilson action in 1 + 1 dimensions and the absence of it found by Lang et al. using the Manton action in 1 + 1 dimensions. But as similarity is reestablished, as we shall soon see in 2 + 1 dimensions, the solution of the 1 + 1 paradox looses some excitement.
22 September 1983 v(o)
:
Parametrization. Having proved the applicability of the transfer matrix it is straightforward to follow Wadia [11] through the transformation of the problem (6) in group space to a problem in parameter space [20]: N
N a2 lx-'x - - +~£.1 •_- a02 Xi=I t ]
N2
'
where X = g2N, -lr < 0 i < lr, 0 i being the N eigenangles of the U(N) plaquette variable, and E(O)/N 2 is the ground state energy/degree of freedom of this matrix, q~ = J ~ is a wavefunction, scaling the original, qJ, wavefunction in group space by the jacobian of the parametrization transformation. Since ~0 is symmetric (there is no preferred labelling of the N eigenangles) and J is antisymmetric, ep describes a problem of N"fermionic" entities. (8) easily separates to N equations: ( - 2 g 2 d2/dO 2 +g-20i)~o(Oi) = e,~(Oi),
(9)
where E (0) = ~'g l ei, -rt ~ 0 i < lr. In an exact treatment one would have to find the N lowest levels of (9) [ 15]. However, here I go on to analyse the problem semiclassically following Wadia [11] and Brezin et al. [10].
u(OI
Fig. 2. Introducing momentum.
the classical hamiltonian. See fig. 2. Each point in phase space (0, p) corresponds to an energy described by (12); and from the uniform distribution in phase space we can get back the eigenangle density [which replaces the average of the squares of the wavefunctions in (9)] : PF
u(O) = f u(O,p) dp, -PF
(13)
defining the "Fermi momentum" PF for a given angle as
h(O,PF) =EF'e'p F = (EF[2g 2 -- 02[294) 1/2 ,
(14)
and we get PF
u(O) = 2 f 0
dp u(O,p) = 2PF =~f2(N/X)(Xe F - 0 2 ) 1/2, (is)
where X = g2N and e F = E F IN.
Semiclassical calculation. We can treat the problem as a many-fermion system with the ground state uniformly populated by "fermions" up to the Fermi level [21]. Thus for large N one introduces a semiclassical distribution function in phase space given by: u(O,p) = 1
forh(O,p)<,EF ,
=0
for h(O, p) > E F ,
f dOdp u'O, t p)=N.
(10)
(11)
236
N =f
dO ~-~ u(O)
¢, N = (N/rrN/2X) {rr [XeF(3,) -- lr2 ] 1/2
E F is the Fermi level and h(0, p) = 2g2p 2 + g - 2 0 2 ,
Strong coupling. For e F ~> 7r2/X (i.e. X large for given eF) the allowed 0-interval certainly extends between - I t and rr and we can readily determine the Fermi level e F as a function of X by use of (15) and the normalization condition (11); for Manton's action there are no cumbersome elliptic integrals as for Wilson's:
(12)
+ XeF(X) arcsin (n/[XeF(X)] 1/2)).
(16)
V o l u m e 129B, n u m b e r 3,4
PHYSICS LETTERS
The ground state energy of the total system is calculated by weighting with our energy measure (12):
fdO - ~ do u(O,p)h(O,p)
E(O) =
22 September 1983
E(O)' N2
(17)
That is Es(0)
I
N2
N2
_
r
~r P F
r
J O
d 0 d p (2g2p 2 + g - 2 0 2 )
Fig. 3. Behavmur o f
2 lrvr2~. 2
f
dO [~(Xe F - 02)3/2 + 02(~ke F -- 02)1/2 ] •
o
(18)
After some calculations and the normalization result (16) as it stands, these two integrals readily yields
E(O)/N 2 = ~e F - X- 2 ~ 2-1/2(XeF - rr2) 3/2 .
(19)
Weak coupling. This corresponds to h(O, O) > E F for some 0 I define
E
]-,r, Ir [.
h(OF, O) = E F ¢==~OF = (XeF) 1/2 .
(20)
I still have the (0-dependent) Fermi m o m e n t u m (14) but now I have to normalize the density on [-O F, OF] OF
N=
E(O)/N2 as a f u n c t i o n
of h = g2N.
-rr -PF
f 2dO -~(0)
¢=~ e F = 2 V ~ ,
(21)
-O F
and with the similar calculations of ( 1 7 ) - ( 1 9 ) you get
E(0)/Ar2w ,'" = ~ev
= X/'~"
(22)
Introducing x2(•) = XeFfir2 , (16) and (19) might be written (x 2 - 1)1/2 + x 2 arcsin(1/Ixl) = ½1rX/Xc,
(23)
E(sO)/N 2 = rrZx2/2X - 7r3(x 2 - 1)3/2/3V~-X 2 ,
(24)
where Xc = rr2/2x/2. The strong coupling criterium e F i> rr2/X corresponds to x 2 ~> 1. Except for a factor 2 times the lattice constant in the energy expression, these are complete analogues of .Azakov's results [12] obtained by using the direct Kogut-Susskind hamiltonian solved by Jevicki-Sakita's loop-space method [13]. E(O)/N2(X) is shown qualitatively in fig. 3. Comparing (22) and (24) and using (23) for im-
plicit differentiation, it is not hard to show that E(sO)/ N 2 and E(~O)/N2 have common value at x = 1, common first and second derivative (=0) whereas the third derivative is discontinuous at x = 1, signalling a third order phase transition at the corresponding X-value X = Xc. That is, a phase transition takes place between strong and weak coupling domains. If one treats the potential in the full quantum mechanical problem (9) as a true (non periodic) harmonic oscillator potential, it is easily seen that the value Xc correspond to the situation where level N reaches the top of the potential [15].
Conclusion. It has been proven by a semiclassical argument, that there is no difference in phase structure for the Wilson [11 ] and Manton action when formulated in lagrangian language and then translated by the transfer matrix to the hamiltonian: The (2 + 1)-dimensional lattice model of pure gluon U(N) one-plaquette gauge theory exhibits a third order phase transition in semiclassical treatment o f the large N limit. Further it is seen that the method applied here yields also the same phase structure as the JevickiSakita [ 13] loop-space, collective field method applied by Azakov [12]. This confirms in (2 + 1)-dimensional one-plaquette models the validity of the general expectation, recently stressed by Caldi [18], that where the type of question is such that different schemes of formulation can give an answer, the answers should agree. The work was supported by a scholarship grant from the Swedish Institute, and I would like to thank Bo-Sture Skagerstam for helping to settle this investigation and for a lot of helpful discussions. I thank Per Salomonson for reading the manuscript, and the particle theory group at Chalmers in G6teborg for hospitality during my stay. 237
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References [1] G. Wilson, Phys. Rev. 10D (1974) 2445. [2] J.B. Kogut, Rev. Mod. Phys. 51 (1979) 659. [3] E.g.B. Berg and A. Bllloke, Glueball spectroscopy in 4-d SU(3) lattice gauge theory, DESY preprint 82-079 (December 1982); K. Ishikawa, A. Sato, G. Schierholz and M. Teper, Phys. Lett. 120B (1983) 387. [4] N.S. Manton, Phys. Lett. 96B (1980) 328. [5] E. Onofri, SU(N) gauge theory with Villain's action, CERN preprint (1981). [6] P. Menotti and E. Onofri, Nucl. Phys. B190 (1981) 288. [7] D.J. Gross and E. Witten, Phys. Rev. D21 (1980) 446. {8] C.B. Lang, P. Salomonson and B.S. Skagerstam, Nucl. Phys. B190 (FS3) (1981) 337. [9] G. 't Hooft. Nucl. Phys. B75 (1974) 461. [10] E. Br6zin, C. Itzykson, G. Parisi and J. Zuber, Commun. Math. Phys. 59 (1978) 35. [11] S.R. Wadm, Phys. Lett. 93B (1980) 403.
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[ 12 ] S.I. Azakov, The I[N expansion m the one-plaquette model with Manton's action. Laboratory of High Energy Physics preprint no. 46 (Baku, 1982). [13] A. Jevicki and B. Sakita, Phys. Rev. D22 (1980) 467. [14] O. Trinhammer, Gbteborg report (Chalmers, G6teborg, Sweden); Appendix (April 1983). [ 15] O. Trinhammer, Band structure approach to ground state energy m lattice gauge theory, G6teborg report (Chalmers, G6teborg, April 1983). [16] J. Minor, Ann. Math. Stud. 51 (1963). [17] J.B. Kogut and L. Susskind, Phys. Rev. D l l (1975) 395. [18] D.G. Caldi, Phys. Rev. D25 (1982) 3356. [19] J.P. Rodrigues, Variant actions and the presence of a Gross-Witten phase transition, preprints NSF-ITP-81122, BROWN-HET-462. [201 H. Weyl: The classicalgroups (Princeton U.P., Princeton, 1946) p. 194. [21 ] K. Osterwalder and E. Seller, Ann. Phys. (NY) 110 (1978) 440.