Critical phenomena and renormalization-group flow of multi-parameter Φ^4 field theories
aa r X i v : . [ h e p - l a t ] S e p Critical phenomena and renormalization-group flowof multi-parameter F field theories Ettore Vicari ∗ † Dipartimento di Fisica, Universitá di Pisa and INFNE-mail: [email protected]
In the framework of the renormalization-group (RG) approach, critical phenomena can be in-vestigated by studying the RG flow of multi-parameter F field theories with an N -componentfundamental field, containing up to 4th-order polynomials of the field. Some physically interest-ing systems require F field theories with several quadratic and quartic parameters, dependingessentially on their symmetry and symmetry-breaking pattern at the transition. Results for theirRG flow apply to disorder and/or frustrated systems, anisotropic magnetic systems, density wavemodels, competing orderings giving rise to multicritical behaviors.The general properties of the RG flow in multi-parameter F field theories are discussed. Anoverview of field-theoretical results for some physically interesting cases is presented, and com-pared with other theoretical approaches and experiments. Finally, this RG approach is applied toinvestigate the nature of the finite-temperature transition of QCD with N f light quarks. The XXV International Symposium on Lattice Field TheoryJuly 30-4 August 2007Regensburg, Germany ∗ Speaker. † I acknoledge the important contributions of my collaborators, in particular Andrea Pelissetto, on the topics dis-cussed in this talk. c (cid:13) Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-ShareAlikeLicence. http://pos.sissa.it/
G flow of multi-parameter F field theories Ettore Vicari
1. Introduction
In the framework of the renormalization-group (RG) approach to critical phenomena, a quanti-tative description of many continuous phase transitions can be obtained by considering an effectiveLandau-Ginzburg-Wilson (LGW) F field theory, containing up to fourth-order powers of the fieldcomponents. The simplest example is the O( N )-symmetric F theory, defined by the Lagrangiandensity L O ( N ) = (cid:229) i ( ¶ m F i ) + r (cid:229) i F i + u ( (cid:229) i F i ) (1.1)where F is an N -component real field. These F theories describe phase transitions characterizedby the symmetry breaking O( N ) → O( N − N = XY universality class for N = He, transitions in magnets with easy-plane anisotropy, and in superconductors), theHeisenberg universality class for N = N → N ) models, there are also other physically interestingtransitions described by more general Landau-Ginzburg-Wilson (LGW) F field theories, char-acterized by more complex symmetries and symmetry breaking patterns. The general LGW F theory for an N -component field F i can be written as L = (cid:229) i ( ¶ m F i ) + (cid:229) i r i F i + (cid:229) i jkl u i jkl F i F j F k F l (1.2)where the number of independent parameters r i and u i jkl depends on the symmetry group of thetheory. Here, we are only assuming a parity symmetry which forbids third-order terms. In the field-theoretical (FT) approach the RG flow is determined by a set of RG equations for the correlationfunctions of the order parameter. This approach has been applied to investigate the critical behaviorof disorder and/or frustrated systems, magnets with anisotropy, spin and density wave models,competing orderings giving rise to multicritical behaviors, and also the finite-temperature transitionin hadronic matter.The main issue discussed in this talk is the RG flow of general multi-parameter F field theo-ries, and its applications to the study of critical phenomena in statistical systems. In particular, weconsider FT perturbative approaches based on expansions in powers of renormalized quartic cou-plings. An overview of FT results for physically interesting cases is presented and compared withexperiments and other approaches such as lattice techniques. Finally, this RG approach is appliedto the study of the nature of the finite-temperature transition in QCD with N f light flavours.
2. Renormalization-group theory of critical phenomena
Critical phenomena are observed in many physical systems when they undergo second-order2
G flow of multi-parameter F field theories Ettore Vicari
H TT C PositiveMagnetizationNegativeMagnetization P TP T
C CCriticalPointTriplePoint
Solid GasLiquid
Figure 1:
Typical phase diagrams of ferromagnetic materials (left) and liquids (right) (continuous) transitions characterized by a nonanalytic behavior due to a diverging length scale. Classical examples are transitions in ferromagnetic materials and liquids, whose phase diagramsare sketched in Fig. 1. The first general framework to understand critical phenomena was proposedby Landau [3]; it was essentially a mean-field approximation. A satisfactory understanding waslater achieved by the Wilson RG theory [4].The main ideas to describe the critical behavior at a continuous transition are (i) the existenceof an order parameter which effectively describes the critical modes; (ii) the scaling hypothesis:singularities arise from the long-range correlations of the order parameter, which develop a diverg-ing length scale; (iii) universality: the critical behavior is essentially determined by a few globalproperties, such as the space dimensionality, the nature and the symmetry of the order parameter,the symmetry breaking, the range of the effective interactions. The RG theory of critical phenom-ena [4, 5, 6] provides a general framework where these features naturally arise. It considers a RGflow in a Hamiltonian space. The critical behavior is associated with a fixed point (FP) of the RGflow where only a few perturbations are relevant. The corresponding positive eigenvalues of thelinearized theory around the FP are related to the critical exponents n , h , etc...According to the RG theory, the singular part of the Gibbs free energy obeys a scaling law F sing ( u , u , . . . , u k , . . . ) = b − d F sing ( b y u , b y u , . . . , b y k u k , . . . ) (2.1)where u i are the nonlinear scaling fields, which are analytic functions of the model parameters. Ina standard continuous transition there are two relevant scaling fields with y i > u t and u h , and aninfinite set of irrelevant fields w i with y i <
0. The relevant fields may be identified with the reducedtemperature t ≡ T / T c − H , i.e. u t ∼ t and u h ∼ H for t , H → b y t | u t | =
1, we can write F sing = | u t | d / y t F sing ( u h | u t | − y h / y t , w i | u t | − y i / y t ) (2.2) Continuous transitions are generally characterized power-law behaviors. For example, defining the reduced tem-perature as t ≡ T / T c −
1, in the disordered phase the magnetic susceptibility c and the correlation length x diverge as c ∼ t − g and x ∼ t − n respectively, the specific heat behaves as C H ∼ | t | − a , the two-point function W ( ) ( x ) decays as1 / x d − + h at T = T c , the magnetization vanishes as M ∼ ( − t ) b in the ordered phase, etc... In standard continuous transi-tions only two critical exponents are independent, because there are several scaling relations: g = ( − h ) n , a = − d n , b = n ( d − + h ) / G flow of multi-parameter F field theories Ettore Vicari
Since w i | u t | − y i / y → t →
0, we can expand with respect to the arguments containing theirrelevant scaling fields, obtaining F sing ≈ | t | d / y t f ( | H || t | − y h / y t ) + | t | d / y t + D i f ( , i ) ( | H || t | − y h / y t ) + ... (2.3)where y t = / n , y h = ( d + − h ) / D i = − y i / y t >
0. The first term of the r.h.s. of Eq. (2.3) isthe universal asymptotic critical behavior, while the other terms give rise to nonuniversal scalingcorrections. The above scaling relations can be easily extended to allow for finite-size systems.In the presence of other relevant perturbations beside t and H , one observes more complicatedmulticritical behaviors.
3. Field-theoretical perturbative approach
The RG theory provides the basis for the FT approaches to the study of critical phenomena.The critical behavior can be determined by the RG flow of a corresponding Euclidean quantumfield theory (QFT).
Let us consider the Ising model defined on a d -dimensional lattice: H = − J (cid:229) h xy i s x s y , s x = ± , Z = (cid:229) { s x } exp ( − H / T ) , (3.1)where the sum in the Hamiltonian is over the nearest-neighbor sites of the lattice. The criticalbehavior is due to the long-range modes, with l ≫
1. In order to describe these critical modes,one may perform blocking averages over sizes a ≪ l , and derive an effective Hamiltonian for theblock average variables, which can be considered as real variables j x attached to the blocks. Thesimplest effective Hamiltonian of the block variables j x which preserves the Z symmetry may bewritten as H eff = a d − (cid:229) x , m ( j x + m − j x ) + ua d (cid:229) x ( j x − v ) (3.2)This blocking procedure does not affect the long range modes at the scale l , and therefore it doesnot change the universality class of the transition. Then, we may consider the limit a → H eff : H j = Z d d x H ( j ) , H ( j ) = ( ¶ m j ) + r j + u j (3.3)where r − r c (cid:181) T − T c . Again, this limit does not change the universality class. The correspondingpartition function Z = Z [ d j ] exp [ − Z d d x H ( j )] (3.4)is the path integral of an Euclidean QFT with L ( j ) = H ( j ) . Therefore, the critical behavior ofthe original Ising model is related to the behavior of the correlation functions of the F QFT fora real one-component field in the massless limit. Analogous euristic arguments can be applied toother statistical systems and corresponding QFTs.Note that the way back provides a nonperturbative formulation of an Euclidean QFT, from thecritical behavior of a statistical model. An important example is the Wilson lattice formulation ofQCD [7] defined in the critical (continuum) limit of a 4D statistical model.4
G flow of multi-parameter F field theories Ettore Vicari
A successful approach to the study of critical phenomena exploits the relation with the RGflow of a corresponding QFT, and therefore the typical techniques used in QFTs. We are interestedin the critical behavior of the “bare” correlation functions G n ( p ; r , u , L ) , which correspond to the“physical” correlation functions of the statistical system, unlike high-energy physics where thephysical correlation functions are the renormalized ones.In this section we consider the simplest case of the O( N ) F field theory (1.1). For d <
4, i.e. d = ,
2, the theory is super-renormalizable since the number of primitively divergent diagrams isfinite. Such divergences are related to the necessity of performing an infinite renormalization of theparameter r appearing in the bare Lagrangian, see, e.g., the discussion in Ref. [8]. This problemcan be avoided by replacing the quadratic parameter r of the Lagrangian with the mass m (inversesecond-moment correlation length) defined by m − = G ( ) ( ) ¶ G ( ) ( p ) ¶ p (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p = , (3.5)where G ( ) ab ( p ) = G ( ) ( p ) d ab is the one-particle irreducible two-point function. Perturbation the-ory in terms of m and the bare quartic coupling u is finite in d <
4. In the massive zero-momentum(MZM) scheme, [9] one considers a set of zero-momentum renormalization conditions in the mas-sive (disordered) phase, G ( ) ( p ) = Z − f (cid:2) m + p + O ( p ) (cid:3) , G ( ) ( ) = Z − f m − d g (3.6)where G ( n ) are one-particle irreducible correlation functions, and g is the renormalized MZM cou-pling. Moreover, one defines Z − t = G ( , ) ( ) , where G ( , ) ( p ) is the one-particle irreducible two-point function with an insertion of f . The critical limit m → G ( n ) r = Z n / j G ( n ) , (cid:20) m ¶¶ m + b ( g ) ¶¶ g − n h j ( g ) (cid:21) G ( n ) r ( p ) = [ − h j ( g )] m G ( , n ) r ( p ; 0 ) , (3.7)see, for example, Ref. [2]. The RG functions b ( g ) = m ¶ g ¶ m , h f , t ( g ) = ¶ ln Z f , t ¶ ln m (3.8)can be computed as power series of g . For the 3D O( N ) F models they have been computed to sixand seven loops respectively [10]. When m → g is driven toward an infrared-stablefixed point (FP), i.e. a zero g ∗ of the b -function, b ( g ) = − w ( g ∗ − g ) + O [( g ∗ − g ) x ] with x > b -function b ( g ) is nonanalytic at g = g ∗ [11]. Using the RG equations, onecan also identify h = h f ( g ∗ ) and 1 / n = − h f ( g ∗ ) + h t ( g ∗ ) .One may also consider an alternative perturbative scheme: the MS renormalization scheme [13],defined at T = T c , i.e. in the massless theory. This is based on the dimensional regularization andthe subtraction of the 1 / e poles ( e ≡ − d ) to obtain the renormalized correlation functions. Onesets F = [ Z f ( g )] / F r , u = A d m e Z g ( g ) , where A d is a d -dependent constant, and g is the MS5 G flow of multi-parameter F field theories Ettore Vicari
3D Ising exponents n a h b
EXPT liquid-vapour 0.6297(4) 0.111(1) 0.042(6) 0.324(2)fluid mixtures 0.6297(7) 0.111(2) 0.038(3) 0.327(3)uniaxial magnets 0.6300(17) 0.110(5) 0.325(2)PFT 6,7- l MZM [16] 0.6304(13) 0.109(4) 0.034(3) 0.326(1) O ( e ) exp [16] 0.6290(25) 0.113(7) 0.036(5) 0.326(3)Lattice HT exp [17] 0.63012(16) 0.1096(5) 0.0364(2) 0.3265(1)MC [18] 0.63020(12) 0.1094(4) 0.0368(2) 0.3267(1) Table 1:
Estimates of the critical exponents of the 3D Ising universality class, from experiments (takenfrom the review [1]), resummation of the FT 6,7-loop calculations within the MZM scheme and of O ( e ) expansions, and from lattice techniques: 25th order high-temperature (HT) expansion and Monte Carlo (MC)simulations. renormalized quartic coupling. Moreover, one defines a mass renormalization constant Z t by re-quiring Z t G ( , ) to be finite when expressed in terms of g . Then one derives the RG functions b ( g ) = m¶ g / ¶m and h f , t ( g ) = ¶ ln Z f , t / ¶ ln m . Analogously to the MZM scheme, the nontrivialzero g ∗ of the b -function is the stable FP, and the critical exponents are obtained by evaluating theRG functions h f , t ( g ) at g = g ∗ . Within this scheme, one can perform the so-called e ≡ − d expan-sion [14], i.e. and expansion about d =
4. Alternatively, one can fix d after renormalization [15],for example d =
3, obtaining a fixed-dimension expansion in the MS coupling g .FT perturbative expansions are divergent. If we consider a quantity S ( g ) that has a per-turbative expansion S ( g ) ≈ (cid:229) s k g k , the large-order behavior of the coefficients is given by s k ∼ k ! ( − a ) k k b (cid:2) + O ( k − ) (cid:3) , with a > d <
4. Thus, in order to obtain accurate results, an appro-priate resummation is required before evaluating the RG functions at the fixed-point value g ∗ . Thiscan be done by exploiting their Borel summability, which has been proved for the fixed-dimensionexpansion in d <
4, see e.g. Refs. [2, 1] and references therein, and has been conjectured forthe e expansion. Moreover, one can also exploit knowledge of the high-order behavior of the ex-pansion, which is computed by semiclassical instanton calculations, see, e.g., Refs. [12, 2]. Notethat the results of this resummation are essentially nonpertubative, because it uses nonperturbativeinformation. XY universality classes The effectiveness of the RG approach based on F QFTs can be appreciated by looking at theresults for the 3D O( N ) models, and in particular for the Ising and XY universality classes.The Ising universality class corresponds to a F theory with a real one-component field. Itdescribes transitions in several physical systems, such as liquid-vapor systems, fluid mixtures,uniaxial magnets. Table 1 shows selected results from experiments, field-theoretical and latticecomputations. A more complete review of results can be found in Ref. [1]. FT results are quiteprecise and in good agreement with experiments and lattice computations, which provide the mostprecise theoretical estimates.The 3D XY universality class is characterized by a two-component order parameter and thesymmetry U ( ) . It corresponds to the O( N ) model (1.1) with N =
2. An interesting represen-6
G flow of multi-parameter F field theories Ettore Vicari XY exponents a n h EXPT He [19] − ∗ PFT 6,7- l MZM [16] − O ( e ) exp [16] − − ∗ − ∗ Table 2:
Results for the critical exponents of the 3D XY universality class, from experiments on He,resummation of the FT 6,7-loop calculations within the MZM scheme and of O ( e ) expansion, and fromlattice techniques: a sinergy of HT expansions and MC simulations (MC+HT) and MC simulations (MC).Results marked by an asterisk are obtained by using the hyperscaling relation a = − d n . tative of this universality class is the superfluid transition of He along the l -line T l ( P ) , wherethe quantum amplitude of helium atoms is the order parameter. The superfluid transition of Heprovides an exceptional opportunity for a very accurate experimental test of the RG predictions.Moreover, experiments in a microgravity environment, for instance on the Space Shuttle [19], canachieve a significant reduction of the gravity-induced broadening of the transition. Exploiting thesefavorable conditions, the specific heat of liquid helium was measured to within a few nK from the l -transition [19]. In Table 2 we show selected results from experiments, field-theoretical and lat-tice computations. We note that there is a significant difference between the experimental resultand the best theoretical estimates using lattice techniques. FT results are in good agreement withexperiments and lattice computations, but they are not sufficiently precise to distinguish them. Thisdiscrepancy should not be necessarily considered as a failure of the RG theory, however it calls forfurther investigations. A proposal of a new space experiment has been presented in Ref. [22].
4. The RG flow in multi-parameter F field theories and the h conjecture Beside transitions belonging to the O( N ) universality classes, many other physically inter-esting transitions are described by more general multi-parameter F field theories, cf. Eq. (1.2),characterized by more complex symmetries [23, 24, 1]. The number of independent parameters r i and u i jkl depends on the symmetry group of the theory. An interesting class of models are those inwhich (cid:229) i F i is the unique quadratic polynomial invariant under the symmetry group of the theory,corresponding to the case all field components become critical simultaneously. This requires thatall r i are equal, r i = r , and u i jkl must be such not to generate other quadratic invariant terms underRG transformations, for example, it must satisfy the trace condition [25] (cid:229) i u iikl (cid:181) d kl . In thesemodels, criticality is driven by tuning the single parameter r , which physically may correspond tothe reduced temperature. More general LGW F theories, which allow for the presence of indepen-dent quadratic parameters r i , must be considered to describe multicritical behaviors where there areindependent correlation lengths that diverge simultaneously, which may arise from the competitionof distinct types of ordering. Note that, like the simplest O( N ) models (1.1), all multi-parameter F field theories are expected to be trivial in four dimensions.7 G flow of multi-parameter F field theories Ettore Vicari (cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)
O(n ) O(n ) flop line g T orderedordered disord. phase 1 (cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:0)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1)(cid:1) O(n ) O(n ) mixed g orderedordered T disord. phase Figure 2:
Phase diagrams with multicritical points where different transition lines meet. F field theories Some physically interesting examples of multi-parameter F field theories are: • The O( M ) ⊗ O( N ) models, which is the most general F theory with a N × M matrix funda-mental field F ai ( a = , ..., M , i = , ..., N ), and the symmetry O( M ) ⊗ O( N ), L = (cid:229) ai [( ¶ m F ai ) + r F ai ] + u ( (cid:229) ai F ai ) + v [ (cid:229) a , b ( (cid:229) i F ai F bi ) − ( (cid:229) ai F ai ) ] (4.1)They are relevant for transitions in noncollinear frustrated magnets and in He. See Sec. 5. • The MN models with a real M × N matrix field F ai : L = (cid:229) i , a (cid:2) ( ¶ m F ai ) + r F ai (cid:3) + (cid:229) i j , ab ( u + v d i j ) F ai F b j (4.2)They describe transitions in randomly diluted M -component spins for N →
0, see below inSec. 6, and magnets with cubic anisotropy for M = N = ,
3, see e.g. Refs. [23, 26]. • The spin-density wave model has two complex N -component order-parameter fields: L = | ¶ m F | + | ¶ m F | + r ( | F | + | F | ) + u , ( | F | + | F | )+ u , ( | F | + | F | ) + w , | F | | F | + w , | F · F | + w , | F ∗ · F | It is relevant for transitions where spin-density waves play an important role, as, for example,in high- T c superconductors (cuprates), see e.g. Refs. [27, 28]. • Physically interesting examples of multicritical behavior arise from the competition of or-derings with symmetries O( n ) and O ( n ) , at the multicritical point where the transitionlines meet, as shown Fig. 2. The corresponding LGW F theory must have the symmetryO( n ) ⊕ O( n ) and two O( n ) and O( n ) vector fields f and f , L = ( ¶ m ~ f ) + ( ¶ m ~ f ) + r ~ f + r ~ f + u ( ~ f ) + u ( ~ f ) + w ~ f ~ f (4.3)The multicritical behavior is observed by tuning two relevant scaling fields, related to r and r [29]. Multicritical behaviors of this type occur in several physical contexts: in anisotropicantiferromagnets, high- T c superconductors, multicomponent polymer solutions, etc..., seeRefs. [29, 30, 31, 32]. 8 G flow of multi-parameter F field theories Ettore Vicari
AG H C v u
Figure 3:
Example of RG flow in the coupling space of a two-parameter F field theory. In particular, thefigure sketches the RG flow in the renormalized coupling u - v plane of the O( M ) ⊗ O( N ) F theory in thelarge- N limit, which has four FPs: the Gaussian (G), O( M × N ) (H), chiral (C) and antichiral (A) FPs; thestable FP is the chiral one. The RG flow, and other interesting quantities as the critical exponents, can be determined byusing perturbative approaches, extending the methods employed in the case of the O( N ) models,see Sec. 3.2. In the massive (disordered-phase) MZM scheme, one expands in powers of the MZMquartic couplings g i jkl , defined by G ( ) i j ( p ) = d i j Z − f (cid:2) m + p + O ( p ) (cid:3) , G ( ) i jkl ( ) = m Z − f g i jkl (4.4)The massless (critical) MS scheme is based on a minimal subtraction procedure within the dimen-sional regularization, and can give rise to an e ≡ − d expansion, and also 3D expansions in therenormalized MS couplings g i jkl by setting e = F theories, see e.g. Refs. [33,26, 34, 35, 36, 30, 37, 39, 38, 40, 41, 28], to five or six loops, which requires the calculation ofapproximately 1000 Feynman diagrams. Again, the resummation of the series is essential. It canbe done by exploiting Borel summability and calculation of the large-order behavior [26]. This isachieved by extending the techniques employed for O( N ) models. The comparison of the resultsobtained from the analyses of the MZM and MS expansions provides nontrivial checks.The RG flow is determined by the FPs, which are common zeroes g ∗ i jkl of the b -functions, b i jkl ( g abcd ) ≡ m ¶ g i jkl ¶ m ( MZM ) , b i jkl ( g abcd ) ≡ m ¶ g i jkl ¶m ( MS ) , (4.5)in the MZM and MS schemes respectively. A FP is stable if all eigenvalues of its stability matrix S i j = ¶b i / ¶ g j | g = g ∗ have positive real part. Fig. 3 shows an example of RG flow in the quartic-coupling plane of a two-quartic-parameter F field theory.The existence of a stable FP implies that (i) physical systems with the given global proper-ties can undergo a continuous transition, (ii) the asymptotic behavior in continuous transitions is9 G flow of multi-parameter F field theories Ettore Vicari controlled by the stable FP (apart from cases requiring further tunings). The absence of a stableFP predicts first-order transitions between the disordered and ordered phases of all systems. Notethat, even in the presence of a stable FP, first-order transitions can be observed in systems that areoutside the attraction domain of the stable FP. h conjecture Multi-parameter F theories have usually several FPs. An interesting question is whether aphysical quantity exists such that the comparison of its values at the FPs identifies the most stableFP. In 2D unitary QFT, such a quantity is the central charge c , which is related to the correlationfunction of the stress tensor h T ( z ) T ( z ) i = c z , (4.6)and to the behavior of the free-energy of a slab, which is f ( L ) | L × ¥ = f ¥ L − c p / ( L ) at T = T c .The c -theorem [42] implies that the stable FP in a 2D unitary QFT is the one with the least valueof c . But, despite several attempts and some progress, see, for example, Refs. [43], no conclusiveresults on the extension of this theorem to higher dimensions has been obtained yet.Within general F theories, the following conjecture has been put forward [44]: In generalunitary F theories the infrared stable FP is the one that corresponds to the fastest decay of corre-lations. This corresponds to the FP with the largest value of the critical exponent h which charac-terizes the power-law decay of the two-point correlation function W ( ) ( x ) at criticality, W ( ) ( x ) (cid:181) x d − + h . (4.7)The exponent h is related to the RG dimension of the field, d F = ( d − + h ) /
2. Since in d < F theories may have more than one stable FP with separate attraction domains, the h conjecture should be then refined by comparing FP that are connected by RG trajectories startingfrom the Gaussian FP: among them the stable FP is the one with the largest value of h .The h conjecture holds in the case of the O( N )-symmetric F theory. For d <
4, the GaussianFP, for which h =
0, is unstable against the non-trivial Wilson–Fisher FP for which h > h in unitary theories follows rigorously from the spectral representation of the two-pointfunction [2]). It has been proven within the e ≡ − d expansion, i.e. close to d =
4. It remainsa conjecture at fixed dimension d <
4, but its validity has been confirmed by several analytic andnumerical results in lower dimensional cases. Several checks are reported in Ref. [44]. No coun-terexample has been found in all F theories studied so far. This conjecture has also been extendedto F theories describing multicritical behaviors, such as the O( n ) ⊕ O( n ) F theory (4.3). Inthis situation, the exponent h is replaced by a matrix and the conjecture applies to the trace of thematrix, i.e. Tr h .
5. The O( M ) ⊗ O( N ) F field theory In this section we discuss the RG flow of the O( M ) ⊗ O( N ) F theory (4.1). For M = v >
0, the symmetry breaking is O(2) ⊗ O( N ) → O(2) ⊗ O( N − N ) → O( N −
2) (although the O(2) groups in the symmetry-breaking pattern are not the same).10
G flow of multi-parameter F field theories Ettore Vicari lefthanded righthanded
Figure 4:
The chiral 120 o structure of the ground state of the antiferromagnetic XY model in a triangularlattice. These cases describe transitions in frustrated spin systems with noncollinear order [45], wherefrustration may arise either because of the special geometry of the lattice, or from the competitionof different kinds of interactions. Typical examples of systems of the first type are stacked triangularantiferromagnets (STA’s), for example CsMnBr , CsVBr , where magnetic ions are located at eachsite of a three-dimensional stacked triangular lattice. In STA’s frustration gives rise to a furtherdegeneracy of the ground state, which presents a chiral structure. This can be easily seen in thetwo-component XY antiferromagnetic model H = (cid:229) h xy i ~ s x · ~ s y defined on a triangular lattice. Theminimal energy configurations have the chiral 120 o structure shown in Fig. 4. Another interestingcase is given by M = , N = v <
0, which is relevant for the superfluid transition of He, seee.g. Ref. [48].The nature of the transition in frustrated systems with noncollinear order has been longlydebated. Experiments on several physical systems show continuous transitions, with exponents n = . ( ) , . ( ) for N = n = . ( ) for N = ⊗ O( N ) F theory. This issue was investigated byusing the e ≡ − d expansion. Calculations within the e = − d expansion do not find any stableFP close to d =
4, see e.g. Refs. [45, 1, 46] and references therein. The extension of this resultto the physical dimensions d = d = d . d =
3. Fixed-dimension d = b -functions. Theright Fig. 5 shows the RG trajectories in the u , v plane from the unstable Gaussian to the stablechiral FP (which can be obtained by solving the RG equation − l dg i / d l = b i ( g j ) , with l ∈ [ , ¥ ) and g j ( ) = dg i / d l | l = = u i ). Critical exponents n = . ( ) for N = n = . ( ) for N = M ) ⊗ O( N ) F theory can be also studied analytically in the large- N limitkeeping fixed M , for any 2 < d <
4. In the large- N limit one finds four FPs as shown in Fig. 3,the stable FP is the chiral one denoted by the letter C . The values of the critical exponent h at11 G flow of multi-parameter F field theories Ettore Vicari b u Zeros of b v v O(4) v O(6)
N=2 u N=3 v s=3/2s=1s=2/3s->0 + u v O(4)O(6)N=3N=2
Figure 5:
Some results for the RG flow of 3D O(2) ⊗ O( N ) models obtained by the analysis of the five-loop MS series [47]. The left figure shows the zeroes of the MS b functions. The right figure shows RGtrajectories in the u , v plane from the unstable Gaussian and O(2 N ) FPs to the stable chiral FP for variousvalues of the ratio s ≡ v / u of the bare quartic parameters. the FPs provide a further confirm of the h conjecture [44], indeed setting h = h e d / N + O ( / N ) where e d is a constant depending on d , one finds [51] h = h = / M , h = ( M + ) /
2, and h = ( M − )( M + ) / ( M ) respectively for the Gaussian, O( M × N ), chiral ad antichiral FPs. Allnumerical results at fixed M , N satisfy the h conjecture.
6. Ferromagnetic transitions in disordered spin systems
The FT approach allows us to also successfully describe ferromagnetic transitions in disor-dered spin systems, which are of considerable theoretical and experimental interest. Such transi-tions are observed in spin systems with impurities, such as mixing of antiferromagnetic materialswith non magnetic ones, for example Fe u Zn − u F , Mn u Zn − u F (uniaxial), Fe x Er z , Fe x Mn y Zr z (isotropic), and also He in porous materials. See e.g. Refs. [54, 1, 55, 56] for experimental andtheoretical reviews. These systems can be modeled by the lattice Hamiltonian H r = − J (cid:229) h xy i r x r y ~ s x · ~ s y (6.1)where the sum is over nearest-neighbor sites, ~ s x are M -component spin variables, and r x = , p and 1 − p , respectively. The disorder is quenched: it mimicks the physical situationin which the relaxation time of the diffusion of impurities is much larger than other typical scales.This implies that the free energy F ( r ) (cid:181) ln Z ( r ) must be averaged over the disorder. Accordingly,the expectation value of an observable must be computed by h O i ( b , { r } ) = (cid:229) { s } O e − b H ( s ; r ) (cid:229) { s } e − b H ( s ; r ) , h O i = Z [ d r ] P ( r ) h O i ( b , { r } ) In the FT approach randomly-dilute spin models can be described by a F field theory for an M -component field F i with and external random field y ( x ) coupled to the energy-density operator: L y = (cid:229) i ( ¶ m F i ( x )) + ( r + y ( x )) (cid:229) i F i ( x ) + g ( (cid:229) i F i ( x ) ) (6.2)12 G flow of multi-parameter F field theories Ettore Vicari v -30-25-20-15-10-50 u s=-0.01s=-0.1s=-0.25s=-0.5s=-0.9Ising-to-RDIs Ising RDIsG
Figure 6:
The RG flow of the MN model for M = N → s = u / v of the bare quartic parameters. Results obtained by the analysis of the six-loop MZM series [53]. where y ( x ) is a spatially uncorrelated random field, with probability P ( y ) ∼ exp ( − y / w ) . Then,using the replica trick, ln Z = lim n → ( Z n − ) / n , one can formally integrate over the disorder vari-ables, arriving at a translation invariant F theory, which is the MN model (4.2) with u = − w < MN model in thenonunitary limit N →
0. Therefore, it can be determined by analyzing the high-order MZM andMS series for N =
0, which have been computed respectively to six loops [26, 35, 34] and to fiveloops [33, 40].An interesting physical issue is whether the presence of impurities, and in general of quencheddisorder coupled to the energy density, can change the critical behavior. General RG arguments [52,23] show that the asymptotic critical behavior remains unchanged if the specific-heat exponent a of the pure spin system is negative, which is the case of multicomponent O( M )-symmetric spinsystems, i.e. M >
1. On the other hand, a different critical behavior is expected in the case ofIsing-like systems ( M = a Ising = . ( ) . This is confirmed by the resultsof the analyses of the high-order FT perturbative series, which show that the RG flow of the MN model in the limit N → u < M =
1, see Fig. 6, implying theexistence of a 3D randomly-dilute Ising (RDIs) universality class.Experiments, see, e.g., Refs. [54, 1] and references therein, confirm this scenario. The asymp-totic critical behavior remains unchanged for multicomponent systems. Table 3 reports results forIsing-like systems: from experiments [54], the analysis of the six-loop FT expansion in the MZMscheme [35], and recent Monte Carlo simulations [57]. The global agreement is very good.It is worth mentioning that the RDIs universality class also describes ferromagnetic transitionsin the presence of frustration, when frustration is not too large. This is for exmaple found in the 3D ± J Ising model [58], defined on a simple cubic lattice by the Hamiltonian H ± J = − (cid:229) h xy i J xy s x s y , s x = ± , (6.3)where J xy = ± J with probability P ( J xy ) = p d ( J xy − J ) + ( − p ) d ( J xy + J ) . This is a simplified13 G flow of multi-parameter F field theories Ettore Vicari
RDIs exponents n b
EXPT [54] 0.69(1) 0.359(9)PFT 6- l MZM [35] 0.678(10) 0.349(5)MC [57] 0.683(2) 0.354(1)
Table 3:
Results for the critical exponents of the 3D RDIs universality class. model [59] for disordered and frustrated spin systems showing glass behavior in some region oftheir phase diagram. Unlike model (6.1), the ± J Ising model is frustrated for any p . Neverthless,the paramagnetic-ferromagnetic transition line extending for 1 > p > p N = . ( ) belongsto the RDIs universality class, i.e., frustration turns out to be irrelevant at this transition [58]. For p < p N the low-temperature phase is glassy with vanishing magnetization, thus the critical behaviorat the transition belongs to a different Ising-glass universality class.
7. The finite-temperature transition in hadronic matter
The thermodynamics of Quantum Chromodynamics (QCD) is characterized by a transitionat T ≃
200 Mev from a low- T hadronic phase, in which chiral symmetry is broken, to a high- T phase with deconfined quarks and gluons (quark-gluon plasma), in which chiral symmetry isrestored [60]. Our understanding of the finite- T phase transition is essentially based on the relevantsymmetry and symmetry-breaking pattern. In the presence of N f light quarks the relevant symmetryis the chiral symmetry SU ( N f ) L ⊗ SU ( N f ) R . At T = N f ) V with a nonzero quark condensate h ¯ yy i . The finite- T transition is related to the restoringof the chiral symmetry. It is therefore characterized by the simmetry breakingSU ( N f ) L ⊗ SU ( N f ) R → SU ( N f ) V . (7.1)If the axial U(1) A symmetry is effectively restored at T c , the expected symmetry breaking becomesU ( N f ) L ⊗ U ( N f ) R → U ( N f ) V . (7.2)A suppression of the anomaly effects at T c is however unlikely in QCD. Semiclassical calculationsin the high-temperature phase [61] show that instantons are exponentially suppressed for T ≫ T c , implying a suppression of the anomaly effects in the high-temperature limit. Some latticestudies [62] suggest a significant reduction of the effective U(1) A symmetry breaking around T c ,but not a complete suppression. Since the anomaly, ¶ m J m (cid:181) N c Q , gets suppressed in the large- N c limit, the symmetry-breaking pattern (7.2) may be relevant in the large- N c limit.Other interesting QCD-like theories are SU ( N c ) gauge theories with N f Dirac fermions in theadjoint representation (aQCD). They are asymptotically free only for N f < /
4, thus only the cases N f = , Z N c transformationsrelated to the center of the gauge group SU( N c ), as in pure SU ( N c ) gauge theories. There are twowell-defined order parameters in the light-quark regime, related to the confining and chiral modes,i.e. the Polyakov loop and the quark condensate. Therefore, one generally expects two transitions:a deconfinement transition at T d associated with the breaking of the Z N c symmetry, and a chiral14 G flow of multi-parameter F field theories Ettore Vicari
U(1) A anomaly suppressed anomaly at T c QCD SU ( N f ) L ⊗ SU ( N f ) R → SU ( N f ) V U ( N f ) L ⊗ U ( N f ) R → U ( N f ) V N f = N f = L ⊗ U(2) R /U(2) V or first order N f ≥ ( N f ) → SO ( N f ) U ( N f ) → O ( N f ) N f = N f = Table 4:
Summary of the RG predictions. We report the possible types of transition for each case, indicatingthe universality class when the transition can also be continuous. transition at T c in which chiral symmetry is restored. In aQCD with N f massless flavors the chiral-symmetry group extends to [63] SU ( N f ) . At T = ( N f ) , due to quark condensation. Therefore the symmetry breaking at the finite- T chiral transition is SU ( N f ) → SO ( N f ) (7.3)with a symmetric 2 N f × N f complex matrix as order parameter related to the bilinear quarkcondensate. If the axial U(1) A symmetry is restored at T c , the symmetry-breaking pattern isU ( N f ) → O ( N f ) . MC simulations for N c = N f = T d is first order, while the chiral transition appears continuous. The ratio betweenthe two critical temperatures turns out to be quite large: T c / T d ≈ T chiral transition in QCD and aQCD, one can exploituniversality and renormalization-group (RG) arguments. [66, 37, 41, 67](i) Let us first assume that the phase transition at T c is continuous for vanishing quark masses.In this case the length scale of the critical modes diverges approaching T c , becoming even-tually much larger than 1 / T c , which is the size of the euclidean “temporal” dimension at T c .Therefore, the asymptotic critical behavior must be associated with a 3D universality classwith the same symmetry breaking. The order parameter must be an N f × N f complex-matrixfield F i j , related to the bilinear quark operators ¯ y Li y R j .(ii) The existence of such a 3D universality class can be investigated by considering the mostgeneral LGW F theory compatible with the given symmetry breaking, which describes thecritical modes at T c . Neglecting the U(1) A anomaly, it is given by L U ( N ) = Tr ( ¶ m F † )( ¶ m F ) + r Tr F † F + u (cid:0) Tr F † F (cid:1) + v (cid:0) F † F (cid:1) . (7.4)If F i j is a generic N × N complex matrix, the symmetry is U ( N ) L ⊗ U ( N ) R , which breaks toU ( N ) V if v >
0, thus providing the LGW theory relevant for QCD with N f = N . If F i j is alsosymmetric, the global symmetry is U ( N ) , which breaks to O( N ) if v >
0, which is the caserelevant for aQCD with N f = N /
2. The reduction of the symmetry to SU ( N f ) L ⊗ SU ( N f ) R G flow of multi-parameter F field theories Ettore Vicari v ustable FP
Gaussian O(n)unstable b zeroes of zeroes of b vu Figure 7:
Zeroes of the b -functions associated with the quartic couplings of the Lagrangian (7.4) for N = for QCD [SU ( N f ) for aQCD], due to the axial anomaly, is achieved by adding determinantterms, such as L SU ( N ) = L U ( N ) + w (cid:0) det F † + det F (cid:1) . (7.5)Nonvanishing quark masses can be accounted for by adding an external-field term Tr ( H F + h . c . ) .(iii) The critical behavior at a continuous transition is determined by the FPs of the RG flow: theabsence of a stable FP generally implies first-order transitions. Therefore, a necessary con-dition of consistency with the initial hypothesis (i) of a continuous transition is the existenceof stable FP in the corresponding LGW F theory. If no stable FP exists, the finite- T chiraltransition of QCD (aQCD) is predicted to be first order. If a stable FP exists, the transitioncan be continuous, and its critical behavior is determined by the FP; but this does not excludea first-order transition if the system is outside the attraction domain of the stable FP.The above RG arguments show that the nature of the finite- T transition in QCD and aQCDcan be investigated by studying the RG flow of the corresponding 3D LGW F field theories.RG studies based on high-order perturbative calculations in the MZM and MS are reported inRefs. [37, 41, 67]. Table 4 presents a summary of the predictions obtained by these RG analysesfor the finite- T chiral transitions in QCD and aQCD with N f massless quarks.The case relevant to N f = N =
1, reduces to the O(2)symmetric F theory, corresponding to the 3D XY universality class. The determinant term relatedto the axial anomaly, cf. Eq. (7.5), plays the role of an external field, thus no continuous transitionis expected, but a crossover.In the case relevant for N f = A anomaly, i.e. the F theory(7.4) with N =
2, the analyses of both MZM and 3D MS schemes provide a robust evidence of astable FP, see Fig. 7. A corresponding 3-D U(2) ⊗ U(2)/U(2) universality class exists, with criticalexponents n ≈ . h ≈ .
1. No stable FP is found close to d = e -expansioncalculations [66], but, as already remarked in Sec. 5, the extension to d = e -expansion resultsmay fail. A stable FP is also found when the field F is symmetric, which is relevant for aQCDwith one flavor, thus showing the existence of a universality class characterized by the simmetrybreaking U(2) → O(2). 16
G flow of multi-parameter F field theories Ettore Vicari -4 -2 0 2 4 y E ( y ) n=0n=1 An=1 B Figure 8:
The scaling function E ( y ) for the O(4) universality class, from Ref. [69]. The different linesrepresent different approximations. In two-flavor QCD, taking into account the U(1) A anomaly, the symmetry breaking (7.1) be-comes equivalent to the one of the O(4) vector universality class, i.e. O(4) → O(3). The symmetrybreaking (7.3) of N f = N f = N f = ~ M (cid:181) ~ H | H | ( − d ) / d E ( y ) , y (cid:181) t | H | − / ( b + d ) , (7.6)provides a scaling relation between the quark condensate h ¯ yy i , and the quark mass m f , whichcorrespond respectively to the magnetization M and the external field H . The critical equation ofstate of the 3D O(4) universality class has been accurately determined in the 3D O(4) vector model:the critical exponents are [68] d = . ( ) , b = . ( ) , and the universal scaling function E ( y ) is shown in Fig. 8.Actually, the LGW F theory corresponding to N f = L SU ( ) = Tr ( ¶ m F † )( ¶ m F ) + r Tr F † F + u (cid:0) Tr F † F (cid:1) + v (cid:0) F † F (cid:1) + (7.7) + w (cid:0) det F † + det F (cid:1) + x (cid:0) Tr F † F (cid:1) (cid:0) det F † + det F (cid:1) + y (cid:2) ( det F † ) + ( det F ) (cid:3) , where w , x , y ∼ g and g parametrizes the effective breaking of the U(1) A symmetry. If theanomaly is suppressed ( g = w = x = y = L SU ( ) contains two quadratic (mass)terms, therefore it describes several transition lines in the T - g plane, which meet at a multicriticalpoint for g =
0. In the case of QCD the multicritical behavior is controlled by the U(2) L ⊗ U(2) R symmetric theory. Possible phase diagrams in the T - g plane are shown in Fig. 9. When g = g , see Fig. 9. If | g | issmall (a partial suppression of anomaly effects around T c is suggested by MC simulations [62])we may observe crossover effects controlled by the U(2) ⊗ U(2) multicritical point at g =
0: if thetransition is continuous at g = F sing ≈ t n f ( gt − f ) , where t (cid:181) T − T c ( g = ) , n ≈ . f ≈ .
5. If the transition is first order at g =
0, then it is expected toremain first order for small g . A similar scenario applies also to N f = G flow of multi-parameter F field theories Ettore Vicari g gT TO(4) O(4)
U(2)xU(2)/U(2)
O(4) 1st order O(4)
Figure 9:
Possible phase diagrams in the T - g plane for the LGW theory (7.7) describing the transition of N f = g =
0, is continuous (left) or firstorder (right). Thick black lines indicate first-order transitions. At their end points, thus for particular valuesof g , the transition should be of mean-field type (apart from logarithms). No stable FPs are found for N > N f > N f > N f = n ≈ . h ≈ . u and d , are light. Thephysically interesting case is QCD with N f = s with m s ≈
100 Mev). Therefore, it is important to consider the effects of the quarkmasses in the above transition scenarios. According to the above RG arguments, if the transition iscontinuous in the chiral limit then an analytic crossover is expected for nonzero values of the quarkmasses m f , because the quark masses act as external fields in the corresponding LGW F theories.On the other hand, a first-order transition is generally robust against perturbations, and therefore itis expected to persist for m f >
0, up to an Ising end point. Actually, the presence of the massivequark s makes the above scenario more complicated, because the nature of the transition may besensitive to its mass m s . Since the transition is expected to be first order in the chiral limit of N f = m s . On the other hand, if the transition is continuous in the limit m s → ¥ corresponding to N f = m s = m ∗ s (where the critical behavior ismean field apart logarithms) separating the first-transition line from the O(4) critical line.The nature of the transition in QCD can be investigated by lattice MC simulations. For m f > T hadronic and high- T quark-gluon plasma regimes are not separated by a phase transition, but by an analytic crossover wherethe thermodynamic quantities change rapidly in a relatively narrow temperature interval, see, e.g.,Refs. [70, 71, 72, 73, 74]. Neverthless, the nature of the transition in the chiral limit is still ofinterest. Since the physical masses of the lightest quarks u and d are very small, some scalingrelations may still be valid at the physical values of the quark masses, such as, for example, theO(4) relation (7.6) between the quark condensate and masses in the case the phase transition in thechiral limit is continuous and belongs to the O(4) universality class.The numerical investigation of the transition in the chiral limit is a hard task because it mustbe studied in the infinite-volume limit ( V → ¥ ), in the continuum limit ( N t → ¥ where N t is the18 G flow of multi-parameter F field theories Ettore Vicari number of lattice spacings along the Euclidean time direction), and massless limit ( m f → D ≃ . N f = N f = N f ≥
3: MC simulations [77, 82, 83, 84, 73] show first-order transitions, in agreementwith the RG predictions. Finally, in the case of N f = References [1] A. Pelissetto, E. Vicari,
Phys. Rep. (2002) 549 [ arXiv:cond-mat/0012164 ].[2] J. Zinn-Justin,
Quantum Field Theory and Critical Phenomena , (Clarendon Press, Oxford, 1989),fourth edition Oxford 2002.[3] L.D. Landau, P hys. Z. Sowjetunion (1937) 26; (1937) 545.[4] K.G. Wilson, Phys. Rev.
B 4 (1971) 3174;
Phys. Rev.
B 4 (1971) 3184.[5] K.G. Wilson, J. Kogut, P hys. Rep. (1974) 77.[6] M.E. Fisher, R ev. Mod. Phys. (1974) 597.[7] K. G. Wilson, Phys. Rev.
D 10 (1974) 2445;
Quarks and Strings on a Lattice , in
New Phenomena inSubnuclear Physics , edited by A. Zichichi (Plenum Press, New York, 1975).[8] C. Bagnuls, C. Bervillier, P hys. Rev. B 32 (1985) 7209.[9] G. Parisi, Cargèse Lectures (1973),
J. Stat. Phys. (1980) 49.[10] G. A. Baker, Jr., B. G. Nickel, M. S. Green, D. I. Meiron, Phys. Rev. Lett. (1977) 1351;D. B. Murray, B. G. Nickel, Revised estimates for critical exponents for the continuum n-vector modelin 3 dimensions , unpublished Guelph University report (1991).[11] A. Pelissetto, E. Vicari,
Nucl. Phys.
B 519 (1998) 626 [ arXiv:cond-mat/9801098 ].[12] J.C. Le Guillou, J. Zinn-Justin,
Phys. Rev. Lett. (1977) 95; Phys. Rev.
B 21 (1980) 3976.[13] G. ’t Hooft, M.J.G. Veltman,
Nucl. Phys.
B 44 (1972) 189.[14] K. G. Wilson, M. E. Fisher,
Phys. Rev. Lett. (1972) 240.[15] V. Dohm, Z. Phys.
B 60 (1985) 61;
B 61 (1985) 193; R. Schloms, V. Dohm,
Nucl. Phys.
B 328 (1989)639. G flow of multi-parameter F field theories Ettore Vicari[16] R. Guida, J. Zinn-Justin,
J. Phys
A 31 (1998) 8103 [ arXiv:cond-mat/9803240 ].[17] M. Campostrini, A. Pelissetto, P. Rossi, E. Vicari,
Phys. Rev.
E 65 (2002) 066127[ arXiv:cond-mat/0201180 ].[18] Y. Deng, H.W.J. Blöte,
Phys. Rev.
E 68 (2003) 036125.[19] J.A. Lipa, D.R. Swanson, J.A. Nissen, T.C.P. Chui, and U.E. Israelsson,
Phys. Rev. Lett. (1996)944; J.A. Lipa, D.R. Swanson, J.A. Nissen, Z.K. Geng, P.R. Williamson, D.A. Stricker, T.C.P. Chui,U.E. Israelsson, and M. Larson, Phys. Rev. Lett. (2000) 4894; J.A. Lipa, J.A. Nissen, D.A. Stricker,D.R. Swanson, T.C.P. Chui, Phys. Rev. B B 68 (2003) 174518.[20] M. Campostrini, M. Hasenbusch, A. Pelissetto, E. Vicari,
Phys. Rev.
B 74 (2006) 144506[ arXiv:cond-mat/0605083 ].[21] E. Burovski, J. Machta, N. Prokof’ev, B. Svistunov,
Phys. Rev.
B 74 (2006) 132502[ arXiv:cond-mat/0507352 ].[22] J.A. Lipa, S Wang, J.A. Nissen, D. Avaloff,
Advances in Space Research (2005) 119.[23] A. Aharony, in Phase Transitions and Critical Phenomena , C. Domb and M.S. Green eds. (AcademicPress, New York, 1976), Vol. 6, p. 357.[24] E. Brézin, J.C. Le Guillou, J. Zinn-Justin, in
Phase Transitions and Critical Phenomena , C. Domband M.S. Green eds. (Academic Press, New York, 1976), Vol. 6, p. 125.[25] E. Brézin, J.C. Le Guillou, J. Zinn-Justin,
Phys. Rev.
B 10 (1974) 893.[26] J. Carmona, A. Pelissetto, E. Vicari,
Phys. Rev.
B 61 (2000) 15136 [ arXiv:cond-mat/9912115 ].[27] Y. Zhang, E. Demler, S. Sachdev,
Phys. Rev.
B 66 (2002) 094501 [ arXiv:cond-mat/0112343 ].[28] M. De Prato, A. Pelissetto, E. Vicari,
Phys. Rev.
B 74 (2006) 144507[ arXiv:cond-mat/0601404 ].[29] D.R. Nelson, J.M. Kosterlitz, M.E. Fisher,
Phys. Rev. Lett. (1974) 813; J.M. Kosterlitz, D.R.Nelson, M.E. Fisher, Phys. Rev.
B 13 (1976) 412.[30] P. Calabrese, A. Pelissetto, E. Vicari,
Phys. Rev.
B 67 (2003) 054505[ arXiv:cond-mat/0209580 ].[31] M. Hasenbusch, A. Pelissetto, E. Vicari,
Phys. Rev.
B 72 (2005) 014532[ arXiv:cond-mat/0502327 ].[32] A. Pelissetto, E. Vicari,
Phys. Rev.
B 76 (2007) 024436 [ arXiv:cond-mat/0702273 ].[33] H. Kleinert, V. Schulte-Frohlinde,
Phys. Lett.
B 342 (1995) 284.[34] D.V. Pakhnin, A.I. Sokolov,
Phys. Rev.
B 61 (2000) 15130 [ arXiv:cond-mat/9912071 ].[35] A. Pelissetto, E. Vicari,
Phys Rev.
B 62 (2000) 6393 [ arXiv:cond-mat/0002402 ].[36] A. Pelissetto, P. Rossi, E. Vicari,
Phys. Rev.
B 63 (2001) 140414(R)[ arXiv:cond-mat/0007389 ].[37] A. Butti, A. Pelissetto, E. Vicari,
JHEP (2003) 029 [ arXiv:hep-ph/0307036 ].[38] P. Calabrese, P. Parruccini, Nucl. Phys.
B 679 (2004) 568 [ arXiv:cond-mat/0308037 ].[39] P. Calabrese, P. Parruccini,
JHEP (2004) 018 [ arXiv:hep-ph/0403140 ]. G flow of multi-parameter F field theories Ettore Vicari[40] A. Pelissetto, E. Vicari,
Condensed Matter Physics (Ukraine) arXiv:hep-th/0409214 ].[41] F. Basile, A. Pelissetto, E. Vicari,
JHEP (2005) 044 [ arXiv:hep-th/041202 ].[42] A.B. Zamolodchikov, Pis’ma Zh. Eksp. Teor. Fiz. (1986) 565; JETP Lett. (1986) 730.[43] J. Cardy. Phys. Lett.
B 215 (1989) 749; S. Forte, J.I. Latorre,
Nucl. Phys.
B 535 (1998) 709[ arXiv:hep-th/9805015 ]; D. Anselmi,
Annals Phys. (1999) 361[ arXiv:hep-th/9903059 ]; A. Cappelli, G. D’Appollonio,
Phys. Lett.
B 487 (2000) 87[ arXiv:hep-th/0005115 ]; A. Cappelli, R. Guida, N. Magnoli
Nucl. Phys.
B 618 (2001) 371[ arXiv:hep-th/0103237 ].[44] E. Vicari, J. Zinn-Justin,
New Journal of Physics (2006) 321 [ arXiv:cond-mat/0611353 ].[45] H. Kawamura, J. Phys.: Condens. Matter (1998) 4707 [ arXiv:cond-mat/9805134 ].[46] B. Delamotte, D. Mouhanna, M. Tissier, Phys. Rev.
B 69 (2004) 134413[ arXiv:cond-mat/0309101 ].[47] P. Calabrese, P. Parruccini, A. Pelissetto, E. Vicari,
Phys. Rev.
B 70 (2004) 174439[ arXiv:cond-mat/0405667 ].[48] M. De Prato, A. Pelissetto, E. Vicari,
Phys. Rev.
B 70 (2004) 214519[ arXiv:cond-mat/0312362 ].[49] P. Calabrese, P. Parruccini, A.I. Sokolov,
Phys. Rev.
B 66 (2002) 180403[ arXiv:cond-mat/0205046 ].[50] A. Peles, B.W. Southern, P hys. Rev. B 67 (2003) 184407 [ arXiv:cond-mat/0209056 ].[51] A. Pelissetto, P. Rossi, E. Vicari,
Nucl. Phys.
B 607 (2001) 605 [ arXiv:hep-th/0104024 ].[52] A.B. Harris,
J. Phys.
C 7 (1974) 1671.[53] P. Calabrese, P. Parruccini, A. Pelissetto, E. Vicari,
Phys. Rev.
E 69 (2004) 036120[ arXiv:cond-mat/0307699 ].[54] D.P. Belanger,
Braz. J. Phys. (2000) 682 [ arXiv:cond-mat/0009029 ].[55] R. Folk, Yu. Holovatch, T. Yavors’kii, Uspekhi Fiz. Nauk (2003) 175 [
Phys. Usp. (2003) 175][ arXiv:cond-mat/0106468 ].[56] W. Janke, B. Berche, C. Chatelain, P.E. Berche, M. Hellmund, PoS (LAT2005)
J. Stat. Mech.: Theory Exp. (2007) P02016[ arXiv:cond-mat/0611707 ].[58] M. Hasenbusch, F. Parisen Toldin, A. Pelissetto, E. Vicari,
Phys. Rev.
B 76 (2007) 094402[ arXiv:0704.0427 ]; P hys. Rev. B in press [ arXiv:0707.2866 ].[59] S.F. Edwards, P.W. Anderson, J. Phys.
F 5 (1975) 965.[60] See, e.g., F. Wilczek,
QCD in extreme conditions , arXiv:hep-ph/0003183 ; F. Karsch, Lect.Notes Phys. (2002) 209 [ arXiv:hep-lat/0106019 ].[61] D.J. Gross, R.D. Pisarski, L.G. Yaffe,
Rev. Mod. Phys. (1981) 43.[62] C. Bernard, T. Blum, C. DeTar, S. Gottlieb, U.M. Heller, J.E. Hetrick, K. Rummukainen, R. Sugar,D. Toussaint, M. Wingate, Phys. Rev. Lett. (1997) 598 [ arXiv:hep-lat/9611031 ]; J. B.Kogut, J.-F. Lagaë, D. K. Sinclair, Phys. Rev.
D 58 (1998) 054504 [ arXiv:hep-lat/9801020 ];P. M. Vranas,
Nucl. Phys. (Proc. Suppl.) (2000) 414 [ arXiv:hep-lat/9911002 ]. G flow of multi-parameter F field theories Ettore Vicari[63] A. Smilga, J.J.M. Verbaarschot,
Phys. Rev.
D 51 (1995) 829.[64] F. Karsch, M. Lütgemeier,
Nucl. Phys.
B 550 (1999) 449 [ arXiv:hep-lat/9812023 ].[65] J. Engels, S. Holtmann, T. Schulze,
Nucl. Phys.
B 724 (2005) 357 [ arXiv:hep-lat/0505008 ]; PoS (LAT2005)
148 [ arXiv:hep-lat/0509010 ].[66] R.D. Pisarski, F. Wilczek,
Phys. Rev.
D 29 (1984) 338.[67] F. Basile, A. Pelissetto, E. Vicari,
PoS (LAT2005)
199 [ arXiv:hep-lat/0509018 ].[68] M. Hasenbusch,
J. Phys.
A 34 (2001) 8221 [ arXiv:cond-mat/0010463 ].[69] F. Parisen Toldin, A. Pelissetto, E. Vicari,
JHEP (2003) 029 [ arXiv:hep-ph/0305264 ].[70] C. Bernard e t al [MILC collaboration], Phys. Rev.
D 71 (2005) 034504[ arXiv:hep-lat/0405029 ].[71] M. Cheng, N. H. Christ, S. Datta, J. van der Heide, C. Jung, F. Karsch, O. Kaczmarek, E. Laermann,R. D. Mawhinney, C. Miao, P. Petreczky, K. Petrov, C. Schmidt, T. Umeda,
Phys. Rev.
D 74 (2006)054507 [ arXiv:hep-lat/0608013 ].[72] Y. Aoki, Z. Fodor, S.D. Katz, K.K. Szabo,
Phys. Lett.
B 643 (2006) 46[ arXiv:hep-lat/0609068 ]; Y. Aoki, G. Endrodi, Z. Fodor, S.D. Katz, K.K. Szabo,
Nature (2006) 675 [ arXiv:hep-lat/0611014 ].[73] P. de Forcrand, O. Philipsen,
JHEP (2007) 077 [ arXiv:hep-lat/0607017 ].[74] F. Karsch, talk at this conference; Z. Fodor, talk at this conference.[75] A. Ali Khan et al . (CP-PACS Collaboration), Phys. Rev.
D 63 (2001) 034502[ arXiv:hep-lat/0008011 ].[76] C.W. Bernard, e t. al [MILC collaboration], Phys. Rev.
D 61 (2000) 111502[ arXiv:hep-lat/9912018 ].[77] F. Karsch, E. Laermann, A. Peikert,
Nucl. Phys.
B 605 (2001) 579 [ arXiv:hep-lat/0012023 ].[78] J. B. Kogut, D. K. Sinclair,
Phys. Rev.
D 64 (2001) 034508 [ arXiv:hep-lat/0104011 ].[79] J. Engels, S. Holtmann, T. Mendes, T. Schulze,
Phys. Lett.
B 514 (2001) 299[ arXiv:hep-lat/0105028 ].[80] M. D’Elia, A. Di Giacomo, C. Pica, P hys. Rev. D 72 (2005) 114510 [ arXiv:hep-lat/0503030 ];G. Cossu, M. D’Elia, A. Di Giacomo, C. Pica, arXiv:0706.4470 .[81] J.B. Kogut, D.K. Sinclair,
Phys. Rev.
D 73 (2006) 074512 [ arXiv:hep-lat/0603021 ].[82] Y. Iwasaki, K. Kanaya, S. Sakai, T. Yoshié,
Z. Physik
C 71 (1996) 337[ arXiv:hep-lat/9504019 ].[83] P. de Forcrand, O. Philipsen,
Nucl. Phys.
B 673 (2003) 170 [ arXiv:hep-lat/0307020 ].[84] M. Cheng, N. H. Christ, M.A. Clark, J. van der Heide, C. Jung, F. Karsch, O. Kaczmarek, E.Laermann, R. D. Mawhinney, C. Miao, P. Petreczky, K. Petrov, C. Schmidt, W. Soeldner, T. Umeda,
Phys. Rev
D 75 (2007) 034506 [ arXiv:hep-lat/0612001 ].].