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Received 20 September 2006; accepted 23 March 2007. We present a mathematically simple procedure to explain spontaneous symmetry breaking in quantum ...
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Frontier of Statistical Physics and Information Processing 2013. Spontaneous Symmetry Breaking at the Fluctuating. Level in Classical and Quantum Systems.
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Dynamical spontaneous symmetry breaking (DSSB) has been extremely ... physics, chiral symmetry breaking occurs when the self-consistent Schwinger-Dyson.
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Jun 28, 1999 - (Received 30 December 1998). Classes of spontaneous symmetry breaking at zero and low magnetic fields in single quantum dots (QD's) and ...
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SPONTANEOUS SYMMETRY BREAKING PHENOMENA WITH NON-EQUIVALENT VACUA N. C. Bobillo-Ares Departamento de Matemticas. Facultad de Ciencias. Universidad de Oviedo. Calvo Sotelo s.n. 33007 Oviedo, SPAIN. and J. Casahorrn Departamento de Fsica. Facultad de Ciencias. Universidad de Oviedo. Calvo Sotelo s.n. 33007 Oviedo, SPAIN. FFUOV-97/04
Abstract We study the existence of bidimensional bosonic models for which the spontaneous symmetry breaking phenomenon yields, at classical level, non-equivalent vacua. Once we introduce the concept of vacuum manifold V , de ned in terms of the vacuum eld con gurations, the behavior of the models at issue can be analyzed by means of the so-called classical moduli space Mc(V ). The aforementioned structure allows us to classify the kink-like excitations into loops and links. To be precise, the loops interpolate smoothly between equivalent minima of the classical potential while the links connect vacua located at dierent points in Mc (V ). Although exact results are very hard to come by, we resort to models nice enough to provide us with the solitary waves in closed form.
Since the mid-seventies the search for classical solutions of the corresponding non-linear equations has been one of the most successful tools in both classical and quantum eld theories. Unless otherwise noted, we restrict ourselves to non-dissipative con gurations with nite energy because such objects are candidates to describe new sectors of the models at issue beyond the standard perturbative analysis. These classical solutions have received the names of solitary waves, energy lumps, kinks or solitons 1 . Moreover, they appear in any number of dimensions: the 4 kink for d = 1+1, the Nielsen-Olesen vortex in d = 2+1 or the celebrated 't Hooft-Polyakov monopole solution if considering d = 3 + 1 spaces. On the other hand, a semiclassical treatment of the tunneling phenomena (ranging from periodic potentials in quantum mechanics to nonAbelian Yang-Mills in eld theories) can be carried out by means of the so-called instantons, i.e. localized nite-action solutions of the classical euclidean eld equations. Though more massive than the elementary excitations, the kink-like con gurations become stable since an in nite energy barrier separates them from the ordinary sector. The stability is reinforced by the existence of a topological conserved charge which does not arise by Noether's theorem from a well-behaved symmetry, but characterizes the large distance behavior of the elds involved. Accordingly, it comes by no surprise that these classical solutions have been considered in the literature under the generic name of topological con gurations. (The brave reader can take advantage of the comprehensive description of the subject contained in the book of Rajaraman (1982)). From a general point of view, these topological con gurations emerge in connection with the spontaneous symmetry breaking phenomenon so that the exact symmetries of the Lagrangian density do not leave the vacuum invariant. In more physical terms, a spontaneous broken symmetry requires a degeneracy of vacuum states. When the symmetry at issue is a continuous one, the most remarkable consequence of this phenomenon is the appearance of one spinless particle of zero mass for each broken symmetry, as rst stated by Nambu (1960) and Goldstone (1961). These excitations correspond to operators which would rotate the vacuum by an in nitesimal amount to a degenerate vacuum because such a transformation does not cost any energy. Nevertheless, it is the case that the dimensionality of the space-time has an important role to play. In the context of statistical mechanics, the long-established theorem of Mermin&Wagner (1966) states that a continuous symmetry can only be spontaneously broken in a dimension larger than two. This result was reformulated by Coleman (1973) in the framework of eld theories: otherwise one should encounter the Goldstone boson associated with the spontaneous breaking of a continuous symmetry. However, a massless scalar eld theory is unde ned if d = 1 + 1 due to an infrared behavior which leads to meaningless divergent integrals. In summary, only discrete symmetries can appear spontaneously broken for a two-dimensional space-time world. On the other hand, it may be interesting at this point to recall the recent advances in four-dimensional quantum eld theories for non-Abelian Yang-Mills models with or without matter. Although one of the most important tools incorporated is the degree of supersymmetry present, it can happen that some of these new mechanisms work when going down to non-supersymmetric models. These theories exhibit a space of degenerate vacua so that in principle they mimic the behavior of the models just based on the standard spontaneous symmetry breaking phenomenon. The underlying structure that one nds behind the rst picture is however both surprising and subtle: the vacua at issue are inequivalent according to the schemes introduced by Seiberg&Witten (1994). Physically it means that for instance dierent vacua yield a well dierent spectra of excitations. In more mathematical terms, we can say that these four-dimensional quantum eld theories have a continuous manifold of non-equivalent exact ground states, i.e. a moduli space of vacua. The purpose of this article is to describe the way in which a general family of bidimensional bosonic models exhibits at classical level the aforementioned behavior. We resort to potentials V () which are a polynomial in 2 so that the discrete symmetry involved corresponds to the re ection ! ?, where stands for a real scalar eld. To elucidate the main properties of these models we take advantage of mathematical concepts like the vacuum manifold V and the classical moduli space Mc(V ). In doing so, we go beyond the long-established sine-Gordon or 4 theories where the distinction between inequivalent vacua The term soliton must be applied only to solutions which maintain their form after scattering processes. The sine-Gordon theory for instance exhibits a complete family of solitons in this rigorous sense. 1
is out of place. Particular attention is also devoted to classify the kink-like excitations into loops and links. While the loops interpolate between equivalent vacua, the links serve to connect vacua located at dierent points in Mc(V ). The arrangement of the article is as follows. In the second section we review the main features concerning the existence of solitary waves in bidimensional models which exhibit at classical level the spontaneous symmetry breaking phenomenon. Section 3 is devoted to the analysis of the mathematical tools we are in need to discuss the spontaneous symmetry breaking with non-equivalent vacua. Among other things, the ne distinction between loops and links is considered in detail. As the arguments in such section are somewhat abstract, it seems plausible to give a tangible example in terms of sextic and octic models. We conclude with several remarks which should shed light on further developments of the subject.
2. Solitary waves in d = 1+1.
To begin with, let us give some words about the notation. Space-time coordinates are represented by
x ( = 0; 1; x0 = t; x1 = x). As regards the metric tensor we have g = g , where g00 = ?g11 = 1 and g = 0 if 6= . Moreover @ refers to space-time derivatives @[email protected] . We consider models with a single real scalar eld (x; t) whose dynamics is governed by the Lagrangian density
L = 21 (@ )(@ ) ? V ()
2 1 @ 2 ? 1 U 2 () L = 21 @ ? @t 2 @x 2
where V () represents a well-behaved function of . As the potential is essentially semi-de nite positive, we write V () = U 2 ()=2. So the absolute minima of V (), which we identify with vacua of the subsequent quantum theory, are located at the zeroes of U () for some value or values of . According to the metric tensor g chosen we have As regards the dynamical equation derived from (2.2) we can write
@ 2 ? @ 2 = ?V 0 () = ?U () U 0 () @t2 @x2
(2:3) where the prime denotes as usual the derivative with respect to the eld . In terms of the energy density E (x; t) de ned as 2 1 @ 2 + 1 U 2 () + E (x; t) = 21 @ @t 2 @x 2
the energy E [(x; t)] associated with a eld con guration (x; t) is given by the integral
E [(x; t)] =
E (x; t) dx
Before going to the solitary waves themselves, it may be interesting to recall the way in which the degeneracy vacuum creates the right scenario for the topological con gurations to exist. Let the zeroes of U () occur at P points (P > 1), i.e.
U () = 0 if = cj (j = 1; :::; P ) (2:6) so that the con gurations constant in space-time = cj lead to E [cj ] = 0. Next, we are interested in static solutions for which (2.3) reduces to
d2 = U () U 0 () dx2
As now (2.5) reads
1 @ E [(x)] = ?1 2 @x Z
+ 12 U 2 () dx
it is the case that the dynamical equation for static solutions (x) derives from the minimization of the energy itself. Looking for solitary waves (x) with nite energy, the eld must approach (as x ! 1) one of the values cj written in (2.6). Thinking of the spatial variable x as the time and the eld as the coordinate of a unit-mass particle, (2.7) represents the dynamical equation for the motion in a potential ?V (). According to the boundary conditions required (@[email protected] ! 0 and V () ! 0 as x ! 1), the energy of such a motion is zero so that it is the mechanical action that provides us with the energy of the eld con guration (x). Except for the trivial solutions (x) = cj (j = 1; :::; P ), once the particle takes o from one of the zeroes of U (), the condition of zero energy imposes that it must end up at a neighboring minima of V (). In other words, the eld tends to dierent minima of the potential V () at in nity (hence the topological character of the con guration (x)). Almost from the very beginning of the subject, the static solitary waves in d = 1+1 have been obtained by solving the rst-order dierential equation associated with the Bogomol'nyi bound (1976). First of all, the energy of the solitary waves has been shown to be bounded from below by the magnitude of a topological charge (more on this later). On the other hand, eld con gurations which saturate the Bogomol'nyi bound (a rst-order equation) necessarily satisfy the ordinary second-order equation of motion. Although a description in detail of the Bogomol'nyi arguments is out the scope of this article, our objective is to keep the exposition self-contained. Accordingly, we can sketch a simple derivation of the Bogomol'nyi bound in terms of the energy functional written in (2.8). Looking for static con gurations (x), the energy density reads 2 U ( ) d E (x) = 21 d dx dx U ()
In this way we recover the Bogomol'nyi inequality
1 d U ( ) dx ?1 dx
E  with a saturated bound whenever
d = U () dx In addition, the solution of (2.11) would have energy E  given by Z 1 d U () dx E  = ?1 dx
(2:10) (2:11) (2:12)
a magnitude which, to some extent, can be interpreted in terms of an identically conserved topological current. In this context it is customarily assumed the existence of the standard topological current j given by j = @ . To make clear the connection between the Bogomol'nyi bound and the topological character of the con gurations we are dealing with, it proves convenient to introduce the so-called improved topological current ji , i.e.
ji = U () @
which is in fact automatically conserved because of the antisymmetry of . The topological charge results from the integration of the ji0 component along the real line so that
1 d U () dx ?1 dx
We remark that this topological conservation law is not merely a consequence of a well-behaved symmetry, but rather takes into account the large distance behavior of the eld con guration. This sketch concerning the Bogomol'nyi results closes with the relation between the energy E  and the topological charge Q, namely E  = jQj so that
E  = jT ( = +1) ? T ( = ?1)j 4
for T 0() = U (). The general solution of (2.11) will contain just an integration constant, namely Z x ? xo = Ud ()
where xo indicates the point at which we center the topological excitation at issue. On the other hand, generic solitary p waves [x; t] can be obtained from the static ones via a Lorentz-boost so that [x; t] = [(x ? vt)= 1 ? v2 ], where 1 > v > ?1 is the velocity.
3. Potentials with non-equivalent vacua.
The purpose of this section is to describe the properties of a family of bidimensional bosonic theories which exhibit, at classical level, a non-standard spontaneous symmetry breaking phenomenon. When a general choice for the U () function is made, we obtain in a systematic way models where a broken discrete symmetry appears associated with a set of physically inequivalent vacua. In doing so, we go beyond the long-established sine-Gordon and 4 models. The best way to analize the behavior of these theories takes advantage of the so-called vacuum manifold V , de ned as the set of all constant vacuum eld con gurations. According to the conditions imposed over U () in section 2, it is the case that the vacuum manifold V pays attention to the elds for which the potential vanishes. In this scheme two points P; P in V lead to equivalent physical situations whenever we can reach P from P via the action of the discrete group of symmetry G. Next, we introduce the classical moduli space Mc(V ) as the space of equivalence classes of vacua. To make clear the inequivalent character of the set of minima of V (), we resort to the standard Taylor expansion of the potential to get the mass of the corresponding excitations. In addition, the aforementioned structure of V provides the right scenario for the static solitary waves to exist. We classify the classical solutions with nite energy into loops and links. While the loops themselves connect minima represented by a single point in Mc(V ), the links move along the classical moduli space. Our description takes over functions U () which are polynomial in 2 . Accordingly, we are involved with a discrete Z2 symmetry ! ? in the spirit of the 4 theory. To be more speci c, we consider the two families of models given by
U1() = U2 () =
(2 ? c2k )
(2 ? c2k )
k=1 n Y
where we have c2m > c2n if m > n. Notice that for n = 1 the rst member of (3:1a) precisely corresponds to 4 . On the other hand, U2 () when n = 1 represents a well-behaved version of the sextic model (more on this later). When adding more vacua for higher n, the structure of the vacuum manifold V is enriched in a systematic way. Classically, we nd a spontaneous breaking phenomenon since not all the eld con gurations for which the potential is zero respect the symmetry of the corresponding Lagrangian density. As a matter of fact, the vacuum manifolds read
V1 = f = ck ; k = 1; :::; ng V2 = f = 0; = ck ; k = 1; :::; ng
(3:2b) With regards to V1 , it is the case that eld con gurations like = ck and = ?ck will describe equivalent physical situations since they are just connected by the action of the symmetry at issue. However, our choice in (3:1b) allows a more sophisticated scheme: the structure associated with V2 includes the existence of the vacuum = 0 at which the symmetry is enhanced because is unbroken. In more physical terms, at classical level the system can share the same life with both observed and broken symmetries depending on the vacuum the model chooses between. Next, we go to the classical moduli space Mc (V ) de ned as the space of equivalence classes of vacua, i.e. Mc (V ) = V =Z2 . The above arguments allow us to write that 5
Mc1 (V ) = fk ; k = 1; :::; ng
(3:3) where as expected the generic equivalence class k contains the two con gurations = ck . A similar structure appears for V2 , the dierence being now the existence in Mc2 (V ) of the class 0 associated with = 0. So we have
Mc2(V ) = f0 ; k ; k = 1; :::; ng
(3:4) To reinforce the idea of inequivalent vacua, it seems plausible to consider the low-lying excitations built around the absolute minima of the potential V (). For such a purpose, it suces a Taylor expansion which should represent as usual the starting point for a perturbative treatment. In doing so, we get masses given by m0 = jU 0 ( = 0)j (3:5a)
mk = jU 0 ( = ck )j; k = 1; :::; n
(3:5b) so that as anticipated in the introduction one encounters situations in which the spontaneous symmetry breaking of a discrete symmetry yields dierent spectra of excitations according to the vacuum the system chooses to live. For models exhibiting n discrete degenerate vacua, the conventional wisdom asserts that one can nd 2(n ? 1) static solitary waves. We recall that these kink-like excitations serve to connect any two neighboring minima of the potential as the spatial coordinate x varies from ?1 to 1. Once the set of vacua has been put along the straight line, as corresponds to models just based on a single real scalar eld , we immediately notice the existence of two well dierent topological con gurations. For the function U1 () written in (3.1a) one nds a rst solution which smoothly interpolates between = ?c1 and = c1 , two points of the vacuum manifold V1 which share identical equivalence class in Mc1(V ). On the other hand, the rest of solitary waves connect vacua located at dierent points in the classical moduli space. The above discussion allows us to establish a classi cation into loops and links. In summary, the loops interpolate between equivalent vacua while the links serve to connect minima belonging to dierent points in the classical moduli space. To complete this section it would be interesting to discuss in the framework of V and Mc (V ) the long-established sine-Gordon and 4 models. For instance, if we start from the function U () given by
U () = 2 cos 2
the Lagrangian density enjoys the global symmetry ! + 2p (p Z ). As regards the vacuum manifold V , the sine-Gordon model possesses an in nite number of vacua so that
V = fv = (2v + 1); v Z g (3:7) Accordingly, two generic vacua (q ; r ) in V can always be related via the Z symmetry at issue. Therefore the classical moduli space Mc(V ), de ned as usual in terms of the space of equivalence classes of vacua, i.e. Mc (V ) = V =Z , reduces itself to a single point. Although generous information on the multisoliton sector of sine-Gordon is available, we point out the well representative static solution (x) given by
(x) = 2 arcsin (tanh x) (3:8) which interpolates between the adjacent vacua ?1 = ? and 1 = . Of course, it represents a loop solution as corresponds to a trivial classical moduli space Mc(V ). When considering the 4 theory, our scheme requires to start from a U () of the form
U () = (2 ? 1) 6
so that, as anticipated, we recover the rst member of the series written in (3:1a). Proceeding along the same lines than before, we get again trivial classical moduli space Mc (V ) because the Z2 symmetry ! ? just relates the two vacua of V , i.e. = 1. The well-known kink (x) of this model given by
(x) = tanh x
comes to represent the solitary wave (loop) which connects the minima of the quartic potential.
4. Two tangible examples: sextic and octic models.
As the arguments exposed in the last section have been somewhat abstract, it seems plausible to give some tangible examples where the mathematical structures become more understandable. Although exact results are very hard to come by, we take advantage of the well-behaved versions of sextic and octic models. In doing so, the kink-like excitations can be presented in closed form. i) The sextic model. Let us start by considering the rst member of the general series written in (3:1b), i.e.
U () = (2 ? 1)
where we have put c = 1 for the sake of simplicity. As regards the vacuum manifold V we nd
V = f = 0; = 1g (4:2) with two vacua = 1 just related by ! ?, together with the eld con guration = 0 at which the aforementioned symmetry is observed. To close this formal exposition we write the classical moduli space
Mc (V ), i.e.
Mc f0; 1 g
(4:3) where 0 contains the vacuum = 0 while the couple = 1 belongs to 1 . In more physical terms, we would like to remark the inequivalent character of the vacua we are dealing with. For such a purpose, it suces to write the mass of the low-lying excitations around them, i.e. m0 = 1; m1 = 2. The analysis of the topological con gurations makes use of (2.11) and (4.1) to get the kink pro le
(x) = [(tanh x + 1)=2]1=2 (4:4) once as usual we center the solution at xo = 0. This solitary wave connects the adjacent vacua = 0 and = 1. In addition, we obtain other solutions by putting x ! ?x, ! ? to complete the four possibilities anticipated by a survey of (4.1). For this speci c model the topological sector reduces itself to links since no well-behaved solutions exist inside 0 or 1 . ii) The octic model. To discuss in brief this version of 8 we begin with
U () = (2 ? 1)(2 ? 2) (4:5) which corresponds to the second member of (3:1a) with c21 = 1 and c22 = 2. For the vacuum manifold V we have
p V = f = 1; = 2g
so that the classical moduli space Mc (V ) reads
Mc(V ) = f1 ; 2 g (4:7) p where 1 and 2 contain the vacua = 1 and = 2 respectively. As regards the excitations over
constant eld con gurations we nd that m1 = 2 and m2 = 12. 7
When going to the topological sector it is the case that the Bogomol'nyi equation leads us to + )2 (2 ? ) e12x = (1 (2 + )(1 ? )2
so that the kinks are obtained by solving the cubic equation 12x
3 ? 3 + 2((ee12x +?1)1) = 0
The solitary waves which derive from (4.9) are of the following form. The rst root reads 12x ? 1 1 (x) = ?2 cos 31 arccos ee12x + 1
where the solution with the sign minus in the exponential stands for the kink interpolating between = ?2 and = ?1. For the second root we have 12x ? 1 2 (x) = 2 cos 3 + 13 arccos ee12x + (4:11) 1 though now we need the sign plus to make the transition from = ?1 to = 1. Finally, we write for the
12x ? 1 3 (x) = 2 cos 3 ? 13 arccos ee12x + (4:12) 1 where again we take the sign plus for the increasing kink between = 1 and = 2. The so-called antikinks come out to be the result of the parity transformation x ! ?x. For reference, we simply point out that both loops (2 (x)) and links (1 (x) and 3 (x)) emerge in this octic model. For computing the classical energy
associated with the solitary waves, we can take advantage of (2.15). To end up this section, we simply recall that these topological con gurations are the basis for constructing extended, non-perturbative, quantum particle states.
5. Concluding remarks.
We have considered in detail a family of bidimensional bosonic models which exhibit, at classical level, the non-standard spontaneous breaking of a discrete symmetry of the form ! ?. As compared with the cases usually analyzed in the literature, we encounter a novel feature: the corresponding minima of the function potential (vacua in the subsequent quantum theory) are non-equivalent. Physically it means that for instance dierent vacua yield a well dierent spectra of excitations. To present a formal treatment of this situation we resort to mathematical concepts like the vacuum manifold V and the classical moduli space Mc (V ). In doing so, we go beyond the models where the classical moduli space reduces itself to a single point. We also devote attention to the kink-like excitations once the ne distinction between loops and links has been established. Up to now we have described theories just exhibiting the spontaneous symmetry breaking phenomenon at classical level. However, it is often the case that in several models the symmetry breaking does not occur at classical level but only comes into play when considering quantum corrections. In such a situation one refers to a radiative spontaneous symmetry breaking phenomenon, thus following the ideas nicely exposed by Coleman&Weinberg (1973). For instance, we can start from a classical potential V () for which the discrete symmetry at issue is exact. When incorporating quantum corrections up to leading order in the standard loop expansion for the eective potential, the true ground state of the model can be degenerate so that the aforementioned symmetry appears spontaneously broken. To sum up, the quantum contributions can dramatically change the way in which a symmetry of the Lagrangian density is realized as compared with the scheme the classical approximation gives. Therefore one needs the introduction of the so-called quantum moduli space Mq (V ) which contains information about the vacuum degeneracy pattern once the quantum corrections have been taken into account. In such a case one can speak of quantum kink-like objects as the 8
nite-energy eld con gurations which minimize the eective action. This is the way in which the standard method of section 2 can also be extended to study the models associated with radiative breakings.
Acknowledgment.- The authors thank the Comisin Interministerial de Ciencia y Tecnologa for nan-
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