Geometric methods for the most general Ginzburg-Landau model with two order parameters

0.15 MB·English

by  I. P. Ivanov

Geometric methods for the most general Ginzburg-Landau model with two order
parameters
I. P. Ivanov∗
Interactions Fondamentales en Physique et en Astrophysique, Universit´e de Li`ege,
All´ee du 6 Aouˆt 17, bˆatiment B5a, B-4000 Li`ege, Belgium and
Sobolev Institute of Mathematics, Koptyug avenue 4, 630090, Novosibirsk, Russia
The Landaupotential in thegeneral Ginzburg-Landau theorywith two orderparameters andall
8 possible quadratic and quartic terms cannot be minimized with the straightforward algebra. Here,
0
a geometric approach is presented that circumventsthiscomputational difficulty and allows one to
0
get insight intomany properties of the model in the mean-field approximation.
2

n
a Introduction. The Ginzburg-Landau (GL) theory of- ing fact is that this question cannot be answered by a
J fers a remarkably economic description of phase tran- straightforward calculation. Differentiating the Landau

6 sitions associated with breaking of some symmetry [1]. potential in respect to ψ leads to a system of coupled
i
2
This breaking is described with an order parameter ψ: algebraic equations that cannot be solved explicitly.

the high symmetry phase corresponds to ψ = 0, while InthisLetterweshowthatdespitethiscomputational
]
n the low symmetry phase is described by ψ 6=0. In order difficulty,onecanstilllearnmuchaboutthemostgeneral
o to find when a given system is in high or low symmetry 2OP GL model. Namely, one can study the number and
c
phase, one constructs a Landau potential that depends the properties of the minima of the Landau potential,
-
r on the order parameter, and then find its minimum. Its classify possible symmetries and study when and how
p
classic form is they are broken. In short, one can describe the phase
u
s b diagram of the model, at least in the mean-field approx-
t. V(ψ)=−a|ψ|2+ 2|ψ|4+o(|ψ|4). (1) imation, without explicitly minimizing the potential.
a
There exists, in fact, an extensive literature dating
m Higherordertermso(|ψ|4)areusuallyassumedtobeneg-
back to 1970’s on minimization of group-invariant po-
- ligible. The values of the coefficients a and b and their
tentials with severalOPswith the aid ofstratificationof
d dependence on temperature, pressure, etc. can be either
n the orbit space, see e.g. [1, 13]. Here we show that in
calculated from a microscopic theory, if it is available,
o thecaseoftwoorderparametersrealizingthesamegroup
or considered as free parameters in a phenomenological
c representationtheanalysiscanbeextendedmuchfarther
[ approach. Thephasetransitionassociatedwiththesym-
than in the general case, with important physical conse-
metry breaking takes place when an initially negative a
1 quences. For a particular application of this formalism
v becomes positive, and the minimum of the potential (1) to the 2HDM, see [14, 15].
4 shifts from zero to |ψ|= a/b.
Theformalism. Letusfocusonthesimplestcasewhen
8 Many systems are knowpn, in which two competing or-
twoOPsψ andψ arejustcomplexnumbers. Themost
0 der parameters (OP) coexist. Among them are general 1 2
4 general quadratic plus quartic Landau potential is
O(m)⊕O(n)-symmetricmodels,[2],modelswithtwoin-
.
1 teracting N-vector OPs with O(N) symmetry, [3]; spin- V = −a |ψ |2−a |ψ |2−a (ψ∗ψ )−a∗(ψ∗ψ )
0 1 1 2 2 3 1 2 3 2 1
density-wavesincuprates,[4]; SO(5) theoryofantiferro- b b
08 magnetism and superconductivity, [5]; multicomponent, + 21|ψ1|4+ 22|ψ2|4+b3|ψ1|2|ψ2|2 (2)
: [6],spin-tripletp-wave,[7],andtwo-gap,[8,9],supercon- b
iv ductivity, with its application to magnetism in neutron + (cid:20) 24(ψ1∗ψ2)+b5|ψ1|2+b6|ψ2|2(cid:21)(ψ1∗ψ2)+c.c.
X stars, [10]; two-band superfluidity, [11]; various mech-
r anisms of spontaneous breaking of the electroweak sym- It contains 13free parameters: reala1, a2, b1, b2, b3 and
a
metrybeyondtheStandardModelsuchasthetwo-Higgs- complex a , b , b , b . For the illustration of the main
3 4 5 6
doublet model (2HDM), [12]. idea, we place no restriction on |ψ | from above. Note
i
To describe such a situation within the GL model, thatpotential(2)containsquartictermsthatmixψ and
1
one constructs a Landau potential similar to (1), which ψ . Such terms are usually absent in particular applica-
2
depends on two order parameters, ψ and ψ . Coeffi- tionsofthe2OPGLmodel(forarareexception,see[9]),
1 2
cients of this potential, a and b , can be considered in- but inthe approachpresentedhere it is essentialthatall
i i
dependent, although in each particular application they possible terms are included from the very beginning.
might obey specific relations. One thus arrives at the Once potential (2) is written, the physical nature of
mostgeneraltwo-order-parameter(2OP)GLmodelwith OPs becomes irrelevant. One can consider them as com-
quadratic and quartic terms. ponents of a single complex 2-vector Φ = (ψ ,ψ )T.
1 2
A natural question arises: what is the ground state The key observation is that the most general poten-
of the most general 2OP GL model? A rather surpris- tial (2) keeps its generic form under any regular linear
2
transformation between ψ and ψ : Φ → Φ′ = T ·Φ, Finding eigenvalues Bµν explicitly in terms of a , b
1 2 i i
T ∈ GL(2,C). It can be also accompanied with a suit- requires solution of a fourth-order characteristic equa-
able transformation of the coefficients a , b , so that one tion,whichconstitutesone ofthe computationaldifficul-
i i
arrives at exactly the same potential as before. Thus, ties of the straightforward algebra. We reiterate that in
the problem has some reparametrization freedom with ouranalysiswenever usethese explicit expressions. The
the reparametrization group GL(2,C). Among 13 free analysis relies only on the fact that the eigenvalues are
parameters,only 6 play crucialrole inshaping the phase real and satisfy (6).
diagram of the model, while the other 7 just reflect the Minima of the Landau potential. Let us first find how
way we look at it. many extrema the potential (4) can have in the orbit
Let us now introduce a four-vector rµ = (r , r ) = space. Since extrema lie on the surface of LC+, we use
0 i
(Φ†σµΦ) with components theLagrangemultipliermethodtoarriveatthefollowing
simultaneous equations:
r = (Φ†Φ)=|ψ |2+|ψ |2,
0 1 2
2Re (ψ1∗ψ2) Bµνhrνi−λhrµi=Aµ, hrµihrµi=0. (7)
ri = (Φ†σiΦ)=|2ψIm|2(−ψ1∗|ψψ2|)2  . (3) Here, hrµi labels the position of an extremum. This sys-
1 2
  tem cannotbe solvedexplicitly in the mostgeneralcase,
Here, index µ = 0,1,2,3 refers to components in the howeveronecanestablishhow manyextremaagivenpo-
internal space and has no relation with the space-time. tential has.
Multiplying ψi by a common phase factor does not To find it, we rewrite hrii=hr0ini, where ni is a unit
changerµ,soeachrµparametrizesaU(1)-orbitintheψi- 3D vector; then (7) becomes
space. Since the Landaupotentialis alsoU(1)-invariant,
it can be defined in this 1+3-dimensional orbit space. [A0−(B0−Bi)hr0i]ni =Ai. (8)
The SL(2,C)⊂GL(2,C) group of transformations of
Assume for simplicity that A > 0 and B < B <
ΦinducestheproperLorentzgroupSO(1,3)oftransfor- 0 1 2
mations of rµ. This group includes 3D rotations of the B3. The l.h.s. of (8) at fixed hr0i and all unit vec-
tors n parametrizes an ellipsoid with semiaxes A −
vector r as well as “boosts” that mix r and r , so the i 0
i 0 i
(B − B )hr i. As hr i increases from zero to infin-
orbit space gets naturally equipped with the Minkowski 0 i 0 0
space structure. Since r >0 and rµr ≡r2− r2 =0, ity, this ellipsoid first shrinks, then grows, collapsing at
the orbit space is given0by the “forwaµrdlig0htcoPne”iLC+ r0(i) ≡A0/(B0−Bi) to planar ellipses. One can see that
during these transformations for hr i = [0,∞) it sweeps
in the Minkowski space. 0
at least once the entire Minkowski space and at least
Allthisallowsustorewrite(2)inaverycompactform:
twice the interior of the sphere of radius A . In ad-
0
1
V =−A rµ+ B rµrν, (4) dition, there are two cusped regions, such as shown in
µ µν
2 Fig. 1, whose interior is swept twice more. So, by check-
with ing whether Ai lies inside these regions, one can get the
1 number of solutions of (8) without finding them explic-
Aµ = (a1+a2, −2Re a3, 2Im a3, −a1+a2) , itly.
2
b+12 +b −Re b+ Im b+ −b−12
2 3 56 56 2
Bµν = 1−Re b+56 Re b4 −Im b4 Re b−56 .(5)
2 Im b+ −Im b −Re b −Im b− 
 56 4 4 56 
 −b−12 Re b− −Im b− b+12 −b 
 2 56 56 2 3 
Here, b± ≡b ±b , b± ≡b ±b .
12 1 2 56 5 6
One usually requires that the quartic term of the po-
tentialincreasesin alldirections inthe OP-space. In the
orbit space, this was proved in [14] to be equivalent to FIG. 1: The envelope of ellipsoids for r(1) <r <r(2) in the
0 0 0
the statement that Bµν is diagonalizable by an SO(1,3) Ai-space.
transformation and after diagonalization it takes form
Bµν =diag(B0, −B1, −B2, −B3) with Inthe 1+3-dimensionalspaceofAµ, these 3Dregions
serve as bases of corresponding conical regions with dif-
B >B ,B ,B . (6)
0 1 2 3
ferent numbers of extrema of the potential. Namely, at
Sincerµr =0,thematricesBµν andB˜µν =Bµν−Cgµν least one non-trivial solution of (7) exists, if Aµ lies out-
µ
are equivalent. This degree of freedom in the definition side the past lightcone LC− (otherwise the global mini-
of Bµν shifts all the eigenvalues by a common constant. mumofthepotentialisattheorigin,hψ i=0). IfAµ lies
i
3
inside the future lightcone LC+, then there are at least also invariant under the same group of transformations,
two non-trivial solutions. If in addition Aµ lies inside we say that the symmetry is preserved; otherwise, we
one or both caustic cones, then two additional extrema saythattheexplicitsymmetryisspontaneouslyviolated.
per cone exist. In total, the potential can have up to six In most applications, the Landau potential does possess
non-trivialextremaintheorbitspace. Thisfactwasalso some explicit symmetry, so whether it is preservedor vi-
found independently in [16]. olated can have profound physical consequences.
The above construction does not distinguish a mini- An explicit symmetry corresponds to such a map of
mumfromasaddlepoint(withcondition(6),thereareno the Minkowski space that leaves invariant, separately,
non-trivialmaximaintheorbitspace). Astraightforward B rµrν andA rµ. Inageneralsituation,itmightbefar
µν µ
method for finding a minimum, which consists in check- from evident that the potential has any explicit symme-
ing that the hessian eigenvalues are all non-negative, is try. The following criterion helps recognize the presence
again of little use here. Instead, one can still use geome- of a hidden explicit symmetry and tells what symmetry
try to study the properties of the minima. it is:
As described above, physically realizable points of the
Proposition 2. Suppose that the potential is explicitly
orbit space lie on LC+. Nevertheless, let us consider ex-
invariant under some transformations of rµ. Let G be
pression(4) inthe entire Minkowskispace. Let us define
the maximal group of such transformations. Then:
an equipotential surface MV as a set of all vectors rµ
(a) G is non-trivial if and only if there exists an eigen-
with the same value of V. These equipotential surfaces
vector of B orthogonal to A ;
donotintersect,arenestedintoeachother,andhavevery µν µ
(b) G is one of the following groups: Z , (Z )2, (Z )3,
simple geometry: they are second-order 3-dimensional 2 2 2
O(2), O(2)×Z , O(3).
manifolds (3-quadrics). Since LC+ is also a specific 3- 2
quadric, finding points in the orbit space with the same Theproofwillbegivenin[17](seealso[18]forasimilar
value of V amounts to finding intersections of these two statementin 2HDM). In the case ofa discrete symmetry
3-quadrics. In particular, to find a local minimum of the the following Propositions can be easily proved:
potentialintheorbitspace,onehastofindanequipoten-
tialsurfacethattouchesLC+inanisolatedpoint(wesay Proposition 3. The maximal violation of any discrete
explicit symmetry consists in removing only one Z fac-
that two surfaces “touch” if they have parallel normals 2
tor: (Z )k →(Z )k−1.
at the intersection points). The global minimum cor- 2 2
responds to the unique equipotential surface that only Proposition 4. For any explicit discrete symmetry,
touches but never intersects LC+. The fact that the minimathatpreserveandspontaneouslyviolate thissym-
search for the global minimum is reformulated in terms metry cannot coexist.
ofcontactoftwo3-quadricsleadstoseveralPropositions
listed below. Both Propositions follow from Proposition 1 and the
Let us now find how many among the extrema are factthatthesetofallminimapreservestheexplicitsym-
minima. Let us fix Bµν and move Aµ continuously in metry group G.
the parameter space. We first note that the signature of If a discrete symmetry is spontaneously violated, then
the hessian can change only when the total number of there are two generate minima in the orbit space. One
extrema changes. A saddle point cannot simply become canalsoprovethe converse,i.e. the twodegeneratemin-
a minimum; it can only split into several extrema, one imacanariseonlyviaspontaneousviolationofadiscrete
of them being a minimum, or it can merge with other symmetry of the potential, [15]. Thus, the criterion for
extrema to produce a minimum. Therefore, the conical the spontaneous violation of a discrete symmetry is that
3-surfaces described above (LC−, LC+, caustic cones) Aµ lies inside a caustic cone associated with the largest
separate regions in the Aµ-space with a definite number eigenvalue.
of minima. One can then check a representative point For a concrete example, suppose that all eigenvalues
Aµ = (A0, 0, 0, 0) (in the basis where Bµν is diagonal) of Bµν are distinct and that A3 =0, while other compo-
of the innermost region in the Aµ space and find that nents Ai 6= 0. This potential has an explicit Z2 symme-
there are two distinct minima in this case. This proves try generatedby reflections of the third coordinate. The
the following Proposition: global minimum spontaneously violates this symmetry
(i.e. hr i=6 0), if B >B , B and
3 3 1 2
Proposition1. The most general quadratic plus quartic
potentialwith twoorderparameters can have at mosttwo A2 A2 A2
1 + 2 < 0 . (9)
distinct local minima in the orbit space. (B −B )2 (B −B )2 (B −B )2
3 1 3 2 0 3
Symmetries and their violation. The potential can The proofis basedonthe “shrinkingellipsoid”construc-
have an additional explicit symmetry, i.e. it can remain tion described above and will be given in detail in [17].
invariant under some transformations of ψ (or coeffi- Local order parameters. In this Letter we have illus-
i
cients) alone. If the position of the global minimum is trated the idea using the global OPs ψ . The approach
i
4
canbe easilyextendedtomodels,whereOPsaredefined ent symmetry properties (in 2HDM it corresponds to a
locally, ψ (x). In this case, one considers the free-energy charge-breakingvacuum). One can easilyformulate con-
i
functional F[ψ ] = d3x[K(ψ ) + V(ψ )] with kinetic ditions when it is the global minimum of the model, so
i i i
term K and potentiaRl V. In the general model the ki- thephasediagramremainsequallytractableinthiscase.
netic term must include all quadratic combinations of In conclusion, we considered the most general
the gradient terms: Ginzburg-Landau model with two order parameters, in-
cluding all possible quadratic and quartic terms in the
K =κ1|D~ψ1|2+κ2|D~ψ2|2+κ3(D~ψ1)∗(D~ψ2)+c.c., (10) Landau potential. Since the minimization of the poten-
tial cannot be done explicitly, we developed a geomet-
where D~ is either ∇~ or the covariant derivative. It can ric approach based on the reparametrization properties
berewritteninthereparametrizationinvariantformK = of the model and used it to study the ground state of
Kµρµ with ρµ ≡(D~Φ)†σµ(D~Φ) andKµ defined in terms the model. Future research should include dynamics of
of κi similarly to Aµ defined in terms of ai. the fluctuations above the ground state, corrections to
The presence of Kµ leads only to minor complications the potential beyond the quartic term, renormalization
oftheaboveanalysis. Alltheconclusionsaboutthenum- group flow, as well as modifications of the results at fi-
ber of extrema and minima remain unchanged, however nite temperature and in the presence of external fields.
one should now distinguish symmetries of the potential
and of the entire free energy functional, see [17] for de-
tails. I thank Ilya Ginzburgfor useful comments. This work
Two localOPsalso leadto the existence of quasitopo- wassupportedbyFNRS andpartlyby grantsRFBR05-
logical solitons. This possibility has been known for 02-16211and NSh-5362.2006.2.
some time; for example, in [19], a soliton in the one-
dimensional two-band superconductor with a simple in-
terband interaction term was described. Such a soliton
corresponds to the relative phase between the two con-
∗
densates that changes continuously from zero to 2π as Electronic address: [email protected]
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1 2 2 1 1 2
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