- Email: [email protected]
The phase diagram of higher-charge Abelian Higgs models and external fields
Nuclear Physics B (Proc. Suppl.) 9 (1989) 49-52 North-Holland, Amsterdam
49
THE PHASE DIAGRAM OF HIGHER-CHARGE ABELIAN HIGGS MODELS AND EXTERNAL FIELDS* Poul H. Damgaard The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Urs M. Hellerf:l: Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, U.S.A. We establish the phase diagram of multiply-charged U(1) Higgs models when the scalar field can fluctuate ,n magnitude. The response of these models to external electromagnetic fields is found to be very similar to what has previously been observed in the fundamental U(1) Higgs model, and in qualitative agreement with expectations based on the analogy to superconducting systems.
A charge-Q U(1) Higgs model on the lattice can be defined by the following action: s
=
~ Z : [1 - cosO.~(.)]
x,U
+A ~-~(q~ - 1) z 4- ~ 022 ~c
(1)
x
where/3 = 1/e ~, and a and A are the usual lattice coupling constants. In addition, we see that the scalar field ¢~ is given a charge Qe, which in effect plays the role of one new coupling constant in this model. In the continuum the introduction of this higher charge is just a trivial rescaling of the usual Q = 1 theory. In the model (1), where the gauge potential 8~(x) is an angular variable on the compact interval [-Tr, 7r], on the other hand, the presence of a Q =~ i clearly modifies the model drastically. So far the phase diagram of this model has only been established in the fixed-length limit 1, in our notation corresponding to to the special case A = oo. Actually, from experience with the fundamental ( Q = I ) U(1) Higgs model 2, we would expect that the main features of the A = c~ phase diagram continue to be valid for values of A down to A ~ I. We have confirmed these expectations by numerical simulations. 3 But for A << I it is clear that
the phase diagram will look fundamentally different. We shall concentrate in this talk on the small A region of the phase diagram. The easiest way to see the difference between small and large A behavior is to focus on one boundary line (plane) of the phase diagram, the one with /3 = 0. Perform the change of variables ¢=,¢* --* p(x),w= in the path integral of the action (1) by ~= = p(x)d ~t. After going to unitary gauge, wx = 0, the integral over ~= is trivial. At /3 = 0 the integral over the U(1) gauge links then reduces to a simple product over one-link integrals. These can be done exactly, and we are left with an effective action Vef/(p) which in detail reads:
VeII(P) = A(P2 - 1) 2 + P: - ln(p) - 4 ln[I0(2~p2)]. (2) Here Io(x) is the modified Bessel function, the ln(p)-term is an "entropy term" from the measure, and the only approximation we have made in (2) is to use translational invariance of VefI(P) to replace p(x)p(x + I~) -~ p2 (the standard tree-level prescription for the effective potential; higher loop corrections can clearly be computed systematically, if needed). The effective potential Vcff(p) displays clear 1st order phase transitions for 0 < A < 0.13, for details see ref.3. Note that V~ff(p) (and hence the location of phase transitions for/~ ~ 0) is completely independent of Q!
"Talk presented by P. H. Damgaard. tWork supported by the National Science Foundation under grant No. PHY82-17853, supplemented by funds from NASA. ~Current address: Supercomputer Computations Research Institute, The Florida State University, Tallahassee, FL 32305, U.S.A.
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
P.H. Damgaard, U.M. Heller/ The phase diagram
50
*¢ Q =6 O=2, ,75
k=OOI,
.50
n~-o
J~ • O.OI ~4
flex • O
I@tllcll
64 I | t t ~
~ B . ~ltn(l*,'i ) .25
.ZS
veff
trem
%" O
o~
,'~
z~o
,'~
1.5
I 2.0
Figure 1: P h a s e d i a g r a m of t h e Q = 2 theory.
Figure 2: P h a s e d i a g r a m of t h e Q = 6 theory.
Thus, for A in the above range all charge-Q models have phase transitions which reach (for fixed )~) the same point on the /3 = 0 axis. Phase transitions on other limiting axes are equally easy to establish. At /3 = oo the links freeze into pure gauge configurations, so there we expect to find the 4-D phase transition of the global 0(2) Higgs model. Note that also in this limit the phase transition occurs at a point which is independent of Q. Similarly, on the m = 0 line (plane) the Higgs fields are decoupled and we thus have again Q-independence. The only phase transition we can encounter then is that of the 4-D pure U(1) gauge theory (at/3c ~ 1.0). However, in the other extreme limit of the phase diagram, at ~ = ~ , things are clearly strongly dependent on the charge Q. Now the gauge potentials freeze into
are above the Coulomb-Higgs phase transition, i.e., the large-g limit is, because of the ¢4 potential, effectively reached quite early. Similarly fig.2 shows the corresponding phase diagram for the Q = 6 theory. All phase transitions here fall on almost perfectly straight lines, agreeing on the borders nicely with the expectations outlined above. In the Z(6) gauge theory the two phase transitions occur at 4/3 ~ 1.0 and /3 ~ 1.6, and as can be seen from the figure, these transitions continue straight down in the phase diagram until they hit the vertical line of phase transitions from/3 = 0 to/3 = o~. Next we would like to discuss the changes found in the phase diagrams once we couple the systems to external electromagnetic sources. The primary motivation for this study is the analogy to superconductivity, mainly as it is described in the phenomenological Ginzburg-Landau theory 5. We have already explored this analogy in detail for the fundamental U(1) Higgs model S, but the case Q > 2 in this lattice model has not been investigated before. To implement a coupling to an external field, we simply make the change
e,(x) = 2rrnu(x)/Q, n,(x) = 0 , 1 , 2 , . . . , Q - 1.
(3)
For Q = 1 this leaves us with a trivial model without any phase transitions. For Q >_ 2 we find an effective Z(Q) gauge theory, which for 2 <_ Q _< 4 is known to have just one phase transition, while for Q >_ ,5 two phase transitions are observed 4, separating a confining phase at low/3 and an ordered large-/3 phase with a Coulomb phase in between. All these expectations turn out to be confirmed by detailed Monte Carlo studies, To illustrate the two different types of phase diagrams for 2 < Q _< 4 and Q >_ ,5. we have concentrated on Q = 2 and Q = 6. Shown in fig.1 is the observed phase diagram for A = 0.01 and Q = 2. We find that the approach to the ~ = oo transition point of the Z(2) gauge theory (at /3 = ~ ln(1 + v~), known from duality transformations) is reached surprisingly early. This can perhaps be understood from eq. (I) by ¢*¢ effectively becoming large at these small x-values since they
I1 - cos~,,~(x)] ~ x,p>v
~
[1 - cos(8,j~(x) - ~ ( x ) ) ]
x,#>v
(4) in the action (1). It is easy to show that in the naive continuum limit eq.(3) indeed reproduces the usual current coupling j~XA#(x) to the vector potential. For simplicity we restrict ourselves to constant external fields, and we take these to be non-vanishing in all 1-4 planes. We can clearly choose any ~ with the prescription (3), but we shall further demand that there exist lattice field configurations { ~ ( x ) } such that ~ ( x ) = 0~ (mod 27r). Only external fields which can satisfy this requirement can be considered to be physical on our lattice. Working with
P.H. Damgaard, U.M. Heller/ The phase diagram
51
periodic boundary conditions on a lattice of size L 4, one finds that this restricts us to
Ore, = 2r:ne~/L2, ne~ ex
E Z.
(5)
Thus the total flux through the lattice will equal 27rne~. and it will jump only in steps of 2zr. This is simply the lattice version of the theorem that on a 2-D torus N ---- e ¢ / 2 ~ r ---- (e/41r)
/ d2xeu.Fm,(x)
(6)
is a topological invariant (here ¢ is the magnetic flux). As in the fundamental U(1) Higgs model 3, we have found also in these Q > 1 models many of the celebrated phenomena known from the theory (and experiments) of superconductivity. For example, deep in the Higgs phase we find a complete expulsion of magnetic flux, the Higgs model analogue of the Meissner effect 5. In contrast, the low-~ Coulomb phase shows the expected "trivial" behavior under the influence of an external field: magnetic flux simply penetrates completely. (This is only moderately changed as we move very close to the strong coupling phase where magnetic monopole fluctuations cause a certain amount of magnetic screening). Perhaps one of the most interesting effects in the higher-charge models is the Q-multiplicity of magnetic vortices 6 for given fixed magnetic flux. Let us now see in detail how this comes about. By defining a reduced plaquette variable
O~(x) = O,~(x) + 2~n~(x)
(7)
such that Ou~(x) C (-Tr, Tr] (by choosing n ~ appropriately), we can continuously measure the total flux
e¢ :
~ ~.~(x) = 2~ E ~,~(x) x,#,~
(8)
x,l~,v
through all lattice planes. What kind of vortex structure should we now expect from the Q ~ 1 theories? In the continuum, vortices are known to carry a quantized magnetic flux which (in the appropriate units) is simply given by the expression for the integer-valued topological charge above (for U(1) gauge field configurations on a torus). One convenient lattice version of this expression is7.8:
Q
x ,p.,~,
which, as can be seen from above, preserves the relation to magnetic flux. Thus, for a total flux of 21rn~, we see that the topological charge equals N
=
Q.n¢~.
(10)
F i g u r e 3: T w o - v o r t e x solution for n ~ = 1 a n d Q = 2.
Phrased differently, even if we introduce a minimalamount of constant external flux (corresponding to n~= = I), we will obtain a configuration with N = Q, i.e., presumably, Q isolated vortices. This can also be seen directly at the action level if we perform standard duality transformations in the Villain limit of the original action 9. It is intriguing to see this Q-fold degeneracy of vortices more explicitly. To do this, we have performed a series of lattice coolings in the region of the Higgs phase of these models where magnetic flux is not expelled (i.e., close to the Coulomb phase, but still in the Higgs phase). For example, consider the flux density plot in fig.3, obtained after cooling of an equilibrium configuration in the Q = 2 theory and with minimal amount of constant magnetic flux, n~x = I. Two clear peaks of precisely the form expected of lattice vortices have emerged. Similarly, the Higgs field magnitude is found to have pronounced holes at the points where magnetic flux has peaks, in agreement with classical expectations of such vortices. The above considerations can be carried through for all higher-Q models, with the appropriate caveats required by the more complex phase straucture of Q _> 5 theories. Details of such studies will be presented elsewhere I°. To summarize our findings, we have mapped out the
52
P.B. Damgaard, U.M. Belier/The phase diagram
phase diagram of two of the higher-charge U(1) Higgs models (those with Q = 2 and Q = 6), which clearly display the different behavior expected (and found) for 1 < Q _< 4 and 5 < Q. In the large-A region (which actually extends down to A ~, 1) we have found no qualitative differences from the limiting case A = oo which previously had been studied in the literature 1, and we have therefore not presented the phase diagram for this case here. In the smalI-A region this is no longer the case, and we find in fact marked differences, as we have illustrated in figs.1 and 2. Particularly interesting is the fact that in the highdomain, where < ~ = > takes large values (and in this sense represents a "broken" Higgs phase), we find a nontrivial set of phase boundaries, which, for fixed A, isolate distinct regions in this part of the phase diagram. As we have explained, these phase boundaries are easily interpreted (and understood) in terms of the residual Z(Q) gauge degrees of freedom that are left upon taking the "freezing" limit ~ -~ oc. Finally we have investigated the behavior of this class of models as we subject the systems to external electromagnetic fields. As already observed in the fundamental U(1) Higgs model S, the "classical" expectations based on the tree-level analogy to the phenomenological GinzburgLandau theory of superconductivity turn out to be fulfilled to a large extent in this fully quantized set of gauge theories as well. A striking phenomenon such as the complete magnetic flux expulsion (a relativistic analogue of the Meissner effect) deep in the Higgs phase was observed directly. Similarly, we have identified regions of the Higgs phase where magnetic flux is allowed to penetrate (at least partially). For consistency with the classical theory of superconductivity this should only be possible for a type-II superconductor, and then the magnetic flux should be restricted to narrow flux tubes, the vortices. Although we have made no attempt to isolate the full "quantum vortices" (any classical picture would indeed break down here anyway), we have checked that the closest classical solutions of any equilibrium configuration prepared in this part of the phase diagram really does correspond precisely to a set of classical lattice vortex solutions (and these solutions are only found there). This is particularly interesting in the case of Q # 1 models, because now even the minimal amount of constant magnetic flux on our periodic lattice corresponds to topological charge _IV = Q. In perfect agreement with this, we have found Q isolated vortices for the minimal magnetic flux. The only remaining piece of work needed to complete this whole picture appears to be a working definition of the quantized Higgs model equivalent of type-I and type-II phases. This seems to be a challenging problem.
References [i] K.C. Bowler, G.S. Pawley, B.J. Pendleton, D.J. Wallace and G.W. Thomas, Phys. Lett. 104B (1981) 481; D.J.E. Callaway and L.J. Carson, Phys. Rev. D25 (1982) 531; J. Ranft, J. Kripfganz and G. Ranft, Phys. Rev D28 (1983) 360; J.M.F. Labastida, E. Sanchez-Velasco, R.E. Shrock and P. Wills, Phys. Rev. D34 (1986) 3156. [2] Y. Munehisa, Phys. Rev. D30 (1984) 1310; Phys. Rev. D31 (1985) 1522; K. Jansen, J. Jersi~k, C.B. Lang, T. Neuhaus and G. Vones, Phys. Lett. 155B (1985) 268; Nucl. Phys. B265[FS15] (1986) 129; D. Espriu and ].F. Wheater, Nucl. Phys. B258 (1985) 101; V.P. Gerdt, A.S. Ilchev and V.K. Mitryushkin, Sov. J. Nucl. Phys. 43 (1986) 468. [3] P.H. Damgaard and U.M. Heller, Phys. Rev. Lett. 60 (1988) 1246; preprint NBI-HE-88-19 and NSF-ITP88-27i, to appear in Nucl. Phys. B. [4] M. Creutz, L. Jacobs and C. Rebbi, Phys. Rep. 95 (1983) 201, and references therein. [5] See e.g., 'Superconductivity' (R.D. Parks, Ed.), M. Dekker inc., New York, 1969; D. Saint-James, E.J. Thomas and G. Sarma: 'Type II Superconductivity', Pergamon Press, New York, 1965. [6] H.B. Nielsen and P. Olesen, Nucl. Phys. B61 (1973) 45. [7] M. Lilscher, Comm. Math. Phys. 85 (1982) 39. [8] C. Panagiotakopoulos, Nucl. Phys. ]3251 (1985) 61. [9] P.H. Damgaard, preprint NBI-HE-88-51, to appear in Phys. Lett. B. [10] P.H. Damgaard and U.M. Heller, in preparation.
49
THE PHASE DIAGRAM OF HIGHER-CHARGE ABELIAN HIGGS MODELS AND EXTERNAL FIELDS* Poul H. Damgaard The Niels Bohr Institute, University of Copenhagen, Blegdamsvej 17, DK-2100 Copenhagen, Denmark Urs M. Hellerf:l: Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, U.S.A. We establish the phase diagram of multiply-charged U(1) Higgs models when the scalar field can fluctuate ,n magnitude. The response of these models to external electromagnetic fields is found to be very similar to what has previously been observed in the fundamental U(1) Higgs model, and in qualitative agreement with expectations based on the analogy to superconducting systems.
A charge-Q U(1) Higgs model on the lattice can be defined by the following action: s
=
~ Z : [1 - cosO.~(.)]
x,U
+A ~-~(q~ - 1) z 4- ~ 022 ~c
(1)
x
where/3 = 1/e ~, and a and A are the usual lattice coupling constants. In addition, we see that the scalar field ¢~ is given a charge Qe, which in effect plays the role of one new coupling constant in this model. In the continuum the introduction of this higher charge is just a trivial rescaling of the usual Q = 1 theory. In the model (1), where the gauge potential 8~(x) is an angular variable on the compact interval [-Tr, 7r], on the other hand, the presence of a Q =~ i clearly modifies the model drastically. So far the phase diagram of this model has only been established in the fixed-length limit 1, in our notation corresponding to to the special case A = oo. Actually, from experience with the fundamental ( Q = I ) U(1) Higgs model 2, we would expect that the main features of the A = c~ phase diagram continue to be valid for values of A down to A ~ I. We have confirmed these expectations by numerical simulations. 3 But for A << I it is clear that
the phase diagram will look fundamentally different. We shall concentrate in this talk on the small A region of the phase diagram. The easiest way to see the difference between small and large A behavior is to focus on one boundary line (plane) of the phase diagram, the one with /3 = 0. Perform the change of variables ¢=,¢* --* p(x),w= in the path integral of the action (1) by ~= = p(x)d ~t. After going to unitary gauge, wx = 0, the integral over ~= is trivial. At /3 = 0 the integral over the U(1) gauge links then reduces to a simple product over one-link integrals. These can be done exactly, and we are left with an effective action Vef/(p) which in detail reads:
VeII(P) = A(P2 - 1) 2 + P: - ln(p) - 4 ln[I0(2~p2)]. (2) Here Io(x) is the modified Bessel function, the ln(p)-term is an "entropy term" from the measure, and the only approximation we have made in (2) is to use translational invariance of VefI(P) to replace p(x)p(x + I~) -~ p2 (the standard tree-level prescription for the effective potential; higher loop corrections can clearly be computed systematically, if needed). The effective potential Vcff(p) displays clear 1st order phase transitions for 0 < A < 0.13, for details see ref.3. Note that V~ff(p) (and hence the location of phase transitions for/~ ~ 0) is completely independent of Q!
"Talk presented by P. H. Damgaard. tWork supported by the National Science Foundation under grant No. PHY82-17853, supplemented by funds from NASA. ~Current address: Supercomputer Computations Research Institute, The Florida State University, Tallahassee, FL 32305, U.S.A.
0920-5632/89/$03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
P.H. Damgaard, U.M. Heller/ The phase diagram
50
*¢ Q =6 O=2, ,75
k=OOI,
.50
n~-o
J~ • O.OI ~4
flex • O
I@tllcll
64 I | t t ~
~ B . ~ltn(l*,'i ) .25
.ZS
veff
trem
%" O
o~
,'~
z~o
,'~
1.5
I 2.0
Figure 1: P h a s e d i a g r a m of t h e Q = 2 theory.
Figure 2: P h a s e d i a g r a m of t h e Q = 6 theory.
Thus, for A in the above range all charge-Q models have phase transitions which reach (for fixed )~) the same point on the /3 = 0 axis. Phase transitions on other limiting axes are equally easy to establish. At /3 = oo the links freeze into pure gauge configurations, so there we expect to find the 4-D phase transition of the global 0(2) Higgs model. Note that also in this limit the phase transition occurs at a point which is independent of Q. Similarly, on the m = 0 line (plane) the Higgs fields are decoupled and we thus have again Q-independence. The only phase transition we can encounter then is that of the 4-D pure U(1) gauge theory (at/3c ~ 1.0). However, in the other extreme limit of the phase diagram, at ~ = ~ , things are clearly strongly dependent on the charge Q. Now the gauge potentials freeze into
are above the Coulomb-Higgs phase transition, i.e., the large-g limit is, because of the ¢4 potential, effectively reached quite early. Similarly fig.2 shows the corresponding phase diagram for the Q = 6 theory. All phase transitions here fall on almost perfectly straight lines, agreeing on the borders nicely with the expectations outlined above. In the Z(6) gauge theory the two phase transitions occur at 4/3 ~ 1.0 and /3 ~ 1.6, and as can be seen from the figure, these transitions continue straight down in the phase diagram until they hit the vertical line of phase transitions from/3 = 0 to/3 = o~. Next we would like to discuss the changes found in the phase diagrams once we couple the systems to external electromagnetic sources. The primary motivation for this study is the analogy to superconductivity, mainly as it is described in the phenomenological Ginzburg-Landau theory 5. We have already explored this analogy in detail for the fundamental U(1) Higgs model S, but the case Q > 2 in this lattice model has not been investigated before. To implement a coupling to an external field, we simply make the change
e,(x) = 2rrnu(x)/Q, n,(x) = 0 , 1 , 2 , . . . , Q - 1.
(3)
For Q = 1 this leaves us with a trivial model without any phase transitions. For Q >_ 2 we find an effective Z(Q) gauge theory, which for 2 <_ Q _< 4 is known to have just one phase transition, while for Q >_ ,5 two phase transitions are observed 4, separating a confining phase at low/3 and an ordered large-/3 phase with a Coulomb phase in between. All these expectations turn out to be confirmed by detailed Monte Carlo studies, To illustrate the two different types of phase diagrams for 2 < Q _< 4 and Q >_ ,5. we have concentrated on Q = 2 and Q = 6. Shown in fig.1 is the observed phase diagram for A = 0.01 and Q = 2. We find that the approach to the ~ = oo transition point of the Z(2) gauge theory (at /3 = ~ ln(1 + v~), known from duality transformations) is reached surprisingly early. This can perhaps be understood from eq. (I) by ¢*¢ effectively becoming large at these small x-values since they
I1 - cos~,,~(x)] ~ x,p>v
~
[1 - cos(8,j~(x) - ~ ( x ) ) ]
x,#>v
(4) in the action (1). It is easy to show that in the naive continuum limit eq.(3) indeed reproduces the usual current coupling j~XA#(x) to the vector potential. For simplicity we restrict ourselves to constant external fields, and we take these to be non-vanishing in all 1-4 planes. We can clearly choose any ~ with the prescription (3), but we shall further demand that there exist lattice field configurations { ~ ( x ) } such that ~ ( x ) = 0~ (mod 27r). Only external fields which can satisfy this requirement can be considered to be physical on our lattice. Working with
P.H. Damgaard, U.M. Heller/ The phase diagram
51
periodic boundary conditions on a lattice of size L 4, one finds that this restricts us to
Ore, = 2r:ne~/L2, ne~ ex
E Z.
(5)
Thus the total flux through the lattice will equal 27rne~. and it will jump only in steps of 2zr. This is simply the lattice version of the theorem that on a 2-D torus N ---- e ¢ / 2 ~ r ---- (e/41r)
/ d2xeu.Fm,(x)
(6)
is a topological invariant (here ¢ is the magnetic flux). As in the fundamental U(1) Higgs model 3, we have found also in these Q > 1 models many of the celebrated phenomena known from the theory (and experiments) of superconductivity. For example, deep in the Higgs phase we find a complete expulsion of magnetic flux, the Higgs model analogue of the Meissner effect 5. In contrast, the low-~ Coulomb phase shows the expected "trivial" behavior under the influence of an external field: magnetic flux simply penetrates completely. (This is only moderately changed as we move very close to the strong coupling phase where magnetic monopole fluctuations cause a certain amount of magnetic screening). Perhaps one of the most interesting effects in the higher-charge models is the Q-multiplicity of magnetic vortices 6 for given fixed magnetic flux. Let us now see in detail how this comes about. By defining a reduced plaquette variable
O~(x) = O,~(x) + 2~n~(x)
(7)
such that Ou~(x) C (-Tr, Tr] (by choosing n ~ appropriately), we can continuously measure the total flux
e¢ :
~ ~.~(x) = 2~ E ~,~(x) x,#,~
(8)
x,l~,v
through all lattice planes. What kind of vortex structure should we now expect from the Q ~ 1 theories? In the continuum, vortices are known to carry a quantized magnetic flux which (in the appropriate units) is simply given by the expression for the integer-valued topological charge above (for U(1) gauge field configurations on a torus). One convenient lattice version of this expression is7.8:
Q
x ,p.,~,
which, as can be seen from above, preserves the relation to magnetic flux. Thus, for a total flux of 21rn~, we see that the topological charge equals N
=
Q.n¢~.
(10)
F i g u r e 3: T w o - v o r t e x solution for n ~ = 1 a n d Q = 2.
Phrased differently, even if we introduce a minimalamount of constant external flux (corresponding to n~= = I), we will obtain a configuration with N = Q, i.e., presumably, Q isolated vortices. This can also be seen directly at the action level if we perform standard duality transformations in the Villain limit of the original action 9. It is intriguing to see this Q-fold degeneracy of vortices more explicitly. To do this, we have performed a series of lattice coolings in the region of the Higgs phase of these models where magnetic flux is not expelled (i.e., close to the Coulomb phase, but still in the Higgs phase). For example, consider the flux density plot in fig.3, obtained after cooling of an equilibrium configuration in the Q = 2 theory and with minimal amount of constant magnetic flux, n~x = I. Two clear peaks of precisely the form expected of lattice vortices have emerged. Similarly, the Higgs field magnitude is found to have pronounced holes at the points where magnetic flux has peaks, in agreement with classical expectations of such vortices. The above considerations can be carried through for all higher-Q models, with the appropriate caveats required by the more complex phase straucture of Q _> 5 theories. Details of such studies will be presented elsewhere I°. To summarize our findings, we have mapped out the
52
P.B. Damgaard, U.M. Belier/The phase diagram
phase diagram of two of the higher-charge U(1) Higgs models (those with Q = 2 and Q = 6), which clearly display the different behavior expected (and found) for 1 < Q _< 4 and 5 < Q. In the large-A region (which actually extends down to A ~, 1) we have found no qualitative differences from the limiting case A = oo which previously had been studied in the literature 1, and we have therefore not presented the phase diagram for this case here. In the smalI-A region this is no longer the case, and we find in fact marked differences, as we have illustrated in figs.1 and 2. Particularly interesting is the fact that in the highdomain, where < ~ = > takes large values (and in this sense represents a "broken" Higgs phase), we find a nontrivial set of phase boundaries, which, for fixed A, isolate distinct regions in this part of the phase diagram. As we have explained, these phase boundaries are easily interpreted (and understood) in terms of the residual Z(Q) gauge degrees of freedom that are left upon taking the "freezing" limit ~ -~ oc. Finally we have investigated the behavior of this class of models as we subject the systems to external electromagnetic fields. As already observed in the fundamental U(1) Higgs model S, the "classical" expectations based on the tree-level analogy to the phenomenological GinzburgLandau theory of superconductivity turn out to be fulfilled to a large extent in this fully quantized set of gauge theories as well. A striking phenomenon such as the complete magnetic flux expulsion (a relativistic analogue of the Meissner effect) deep in the Higgs phase was observed directly. Similarly, we have identified regions of the Higgs phase where magnetic flux is allowed to penetrate (at least partially). For consistency with the classical theory of superconductivity this should only be possible for a type-II superconductor, and then the magnetic flux should be restricted to narrow flux tubes, the vortices. Although we have made no attempt to isolate the full "quantum vortices" (any classical picture would indeed break down here anyway), we have checked that the closest classical solutions of any equilibrium configuration prepared in this part of the phase diagram really does correspond precisely to a set of classical lattice vortex solutions (and these solutions are only found there). This is particularly interesting in the case of Q # 1 models, because now even the minimal amount of constant magnetic flux on our periodic lattice corresponds to topological charge _IV = Q. In perfect agreement with this, we have found Q isolated vortices for the minimal magnetic flux. The only remaining piece of work needed to complete this whole picture appears to be a working definition of the quantized Higgs model equivalent of type-I and type-II phases. This seems to be a challenging problem.
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