Critical exponents for higher order phase transitions: Landau theory and RG flow
CCritical exponents for higher order phasetransitions: Landau theory and RG flow
Joydeep Chakravarty, Diksha Jain International Centre for Theoretical Sciences (ICTS-TIFR)Tata Institute of Fundamental ResearchShivakote, Hesaraghatta, Bangalore 560089, India. International Centre for Theoretical PhysicsStrada Costiera 11, Trieste 34151 Italy
E-mail: [email protected] , [email protected] Abstract:
In this work, we define and calculate critical exponents associated with higher or-der thermodynamic phase transitions. Such phase transitions can be classified into two classes:with or without a local order parameter. For phase transitions involving a local order param-eter, we write down the Landau theory and calculate critical exponents using the saddle pointapproximation. Further, we investigate fluctuations about the saddle point and demarcatewhen such fluctuations dominate over saddle point calculations by introducing the generalizedGinzburg criteria. We use Wilsonian RG to derive scaling forms for observables near criticalityand obtain scaling relations between the critical exponents. Afterwards, we find out fixed pointsof the RG flow using the one-loop beta function and calculate critical exponents about the fixedpoints for third and fourth order phase transitions. a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b ontents r th order phase transitions 14 ( r = 2) ( r = 3) ( r = 4) ( h, t ) plane? 28B Kinetic fluctuations in d ≤ and loss of order 29 – 1 – Introduction
A thermodynamic phase transition is characterized by the non-analyticity of thermodynamicobservables with respect to temperature across two or more different phases. These phasetransitions were systematically classified by Ehrenfest [1, 2]. Ehrenfest’s classification is basedon the order of the phase transition, i.e. a r th order phase transition is characterized by the r thderivative of the partition function (or free energy) becoming discontinuous at the temperaturewhere the phase transition takes place. A thermodynamic function that takes different valuesacross different phases is called an order parameter.With regards to second order phase transitions, Landau put forward a phenomenologicalHamiltonian which captures essential aspects of the transition in the infrared limit of thesystem [3]. His formalism suggests that near the critical temperature T C (the temperature atwhich phase transition takes place), the Hamiltonian can be expanded in terms of a local orderparameter φ ( x ) on the basis of appropriate symmetries. For example, if a system has a Z symmetry i.e. it is symmetric under φ ( x ) → − φ ( x ) ; then for T ≈ T C and a small externalcoupling field h , the Hamiltonian H ( φ ) can be expressed as follows: βH = (cid:90) d d x (cid:18) K ∇ φ ) + tφ ( x ) + uφ ( x ) + · · · − h.φ ( x ) (cid:19) (1.1)where t and u are couplings which are functions of the temperature. Here we have restrictedourselves to only the lowest order terms. It can be readily demonstrated that the partitionfunction computed using this Hamiltonian gives rise to different values of the order parameterfor t < and t > using the saddle point approximation. The divergences of observables nearthe critical point are captured using critical exponents .To obtain the correct picture of fluctuations away from the saddle point, we need to incor-porate the field-theoretic machinery of the renormalization group (RG) flows of couplings in thetheory. We can use the behavior of couplings under RG flow to determine how thermodynamicobservables scale in the vicinity of the critical surface. Consequently, we can derive variousscaling relations between the theory’s critical exponents, which are robust under fluctuationsaway from the saddle point.In order to understand the RG flows in the coupling space, we start by finding out fixedpoints of the flow and then expand the couplings about the fixed points. This leads to aclassification of the couplings into relevant, marginal, and irrelevant depending on their behaviorin the infrared (IR). In the IR, relevant couplings diverge away from the IR fixed points as weintegrate out degrees of freedom while irrelevant couplings converge towards the IR fixed point.The subspace of relevant couplings is usually finite-dimensional, and hence theories in the IRare characterized by only a finite number of relevant couplings. Such IR behaviour defines thenotion of universality classes [4–7]. The set of relevant couplings can be used to determine someof the critical exponents, and scaling relations between them further allows us to determine allthe critical exponents [6–10]. – 2 – igher order Phase transitions Local order parameter Non-local order parameter: (To be discussed inforthcoming work) d > r d ≤ r Saddle Point approximationvalid, critical exponentsgiven in Table 1 Saddle Point valid,critical exponents aregiven in Table 1 Fluctuations dominate over saddlepoint, Scaling relations foundusing RG flow in §4 (Table 2),Critical exponents given in Table 3FollowsGinzburg Criteria (§3.3) Does not followGinzburg criteria
Figure 1 : A flowchart describing the general outline of the paper.We will now briefly describe our work, a summary of which is given in Fig. 1. In thefirst part of our work, we apply various aspects of our above discussion to the study of higherorder phase transitions described by local order parameters. In §2.1 we generalize Landau’sphenomenological Hamiltonian for higher-order phase transitions using a local order parametersuch that physical observables derived from the partition function have analytic properties inthe h − t plane apart from a branch cut ending at the critical point. Next, we define criticalexponents for higher order phase transitions in §2.2 and calculate them using saddle pointapproximation to get the leading order predictions.We treat fluctuations over the saddle point in §3, and show that saddle point analysis isvalid above the upper critical dimension d u = 2 r for higher order transitions. Afterwards, weformulate the Ginzburg criteria to classify when saddle point results are valid and when criticalexponents are modified due to fluctuations below the upper critical dimension. The argumentfor the lower critical dimension remains unchanged as compared to the second order case.In §4 we obtain scaling forms of physical observables for higher order phase transitionsfrom the renormalization group analysis of the system. We use this to derive scaling relationsbetween critical exponents beyond the saddle point. We show that the imposition of analyticityin the phenomenological Hamiltonian combined with the scaling implies that critical exponentsare the same above or below T = T C and h = 0 (Appendix A). We also show that the divergenceof the correlation length naturally leads to the hyperscaling relation.– 3 –e then proceed onto computing corrections to the critical exponents using the one-loopbeta functions in §5. We use this to calculate the critical exponents corresponding to r = 3 and r = 4 phase transitions below the upper critical dimension. We show that there are no relevantcouplings in the vicinity of the non-trivial fixed point, and consequently, the sole correctionsarise due to flows near the Gaussian fixed point. For r ≥ , the one-loop corrections to thebeta function vanish, and hence we need to go beyond one-loop calculations to find correctionsto the critical exponents for such cases.In general, there also exist phase transitions which are described by non-local order param-eters. An important class of such phase transitions include transitions characterized by a gapin the eigenvalue spectrum. These include the Gross-Witten-Wadia (GWW) model [11–13],Douglas-Kazakov model [14], Brownian walk models which map onto two-dimensional con-tinuum Yang-Mills with different gauge groups [15], constrained Coulomb gas [16], bipartiteentanglement [17–19] and various other examples [20–25]. Further examples of third orderphase transitions are given in the review [26]. Such phase transitions are outside the scope ofour present work and will be dealt with in a forthcoming work.Fourth order phase transitions with local order parameter have been proposed as the natureof superconducting transition in certain materials in [27–31] . Another example of fourth orderphase transition is the Ising model on the Cayley tree proposed in [32]. Divergences using zeroesof the partition function in higher order phase transitions were analyzed in [33].Apart from thermodynamic phase transitions, higher order phase transitions also occur inthe context of topological phase transitions. An example of this is the phase transition be-tween 2D Chern insulators, in which the third derivative of the free energy has a discontinuity.However, there are no local order parameters for such topological phase transitions, and conse-quently, Landau theory cannot be formulated to describe them. Consequently, we will not belooking at such phase transitions in our work. In this section, we will study a class of r th order phase transitions, which are described usinglocal order parameters. We argue that the phenomenological theory describing these phasetransitions in the infrared (IR) can be described using the Landau formalism. We will alsointroduce critical exponents corresponding to r th order phase transitions and calculate themusing the saddle point approximation. We feel [29, 31] have not received significant attention in the literature regarding higher order phase tran-sitions, and in particular, we found their work only after we had independently rederived some of their results,and consequently there is overlap between parts of §2.1 and §4.2 of our work with theirs. – 4 – .1 The Landau Hamiltonian for higher order phase transitions
We work in d spatial dimensions, which can be understood both as the non-relativistic spatiallimit of a d + 1 dimensional relativistic theory or as an analytic continuation of the Lorentziantheory to a Euclidean theory via Wick rotation. Let us now write down the most generalHamiltonian, which obeys the following assumptions:1. Analyticity assumption : We assume that the critical exponents are analytic every-where apart from a singular line which terminates at the critical point in the planespanned by the external field h i and temperature difference t (where t = T − T C T C ), withthe line given by h i = 0 and t < as shown in Figure 2. In other words, the free energy isanalytic everywhere in the plane apart from this line. Thus the free energy term contain-ing the external field in the phenomenological Hamiltonian should be chosen such thatthe associated critical exponents are analytic, i.e., the critical exponents have the samevalues when the critical point is approached from different directions in the ( h, t ) plane. h t Figure 2 : The order parameter φ is analytic everywhere except on the branch cut in ( h − t ) plane shown above.2. Assumption regarding higher order terms : We assume a small external field h suchthat only the leading order term in h contributes and higher order terms are suppressed.We also demand that terms involving higher derivatives of the order parameter are sup-pressed. These conditions are weaker than the first condition and serve to simplify ourcalculations and to clearly demonstrate the physics of the problem at hand.3. Fine tuning assumption : Let φ i ( x ) denote the order parameter of the phase transition.In hindsight, for r th order phase transitions, we assume that the terms | φ i | k − , k < r either do not appear in the Hamiltonian, or their coefficients have a very small magnitudeand do not change signs at the critical temperature of the phase transition. We enforcethis requirement so that these terms do not alter the order of the phase transitions, as– 5 – significant contribution from such terms can potentially change the order. Similarly,we also demand that the terms of the form | φ i | k , k > r die off for a r th order phasetransition.4. Rotational symmetry : We further assume that our system possesses rotational sym-metry and hence the Hamiltonian is invariant under O( N ) transformation ( N ≥ ) givenby φ i → φ (cid:48) i = R ji φ j , R ∈ O ( N ) (2.1)where the rotation matrices R belong to the matrix representation of O ( N ) .The most general phenomenological IR Hamiltonian obeying above assumptions whichdescribes r th order phase transition is given by: βH = (cid:90) d d x (cid:20) K | ∇ φ i | + t r | φ i ( x ) | r − + u r | φ i ( x ) | r − ( h i φ i ) | φ i ( x ) | r − (cid:21) , (2.2)where K is the coefficient of the kinetic term and the couplings t r and u r are expressed asfunctions of temperature as follows: t r ( T ) = c ( T − T C ) + O ( T − T C ) = c (cid:48) t + O ( t ) u r ( T ) = u + u ( T − T C ) + O ( T − T C ) = u + u (cid:48) t + O ( t ) (2.3)where T C is the temperature at which the transition takes place, c, c (cid:48) , u , u (cid:48) and u are unde-termined constants, and t = T − T C T C as defined previously. The constant u must be positive inorder to ensure that our Hamiltonian is positive-definite while c must be positive in order toobtain the correct description of phases above and below the critical temperature. The exactvalues of these constants depend on the details of the system.The mass dimensions of the couplings appearing in (2.2) are given by: [ t r ] = 2( r + d − − dr [ u r ] = r (2 − d ) + d [ h ] = d + (2 r − (cid:18) − d (cid:19) . (2.4)Notice that the couplings in the above Hamiltonian can be non-renormalizable depending on thedimension d , and hence the physical observables can potentially be plagued by UV divergencesuntameable by a finite number of counterterms. However our Hamiltonian is only defined inthe IR, and we can impose a UV cutoff upto which the Hamiltonian description is valid. Thuswe are not bothered by non-renormalizable interactions, analogous to the standard treatmentof second order phase transitions in d > .The (Gibbs) partition function of the system described by the Hamiltonian (2.2) is givenby: Z = (cid:90) [ Dφ ( x )] exp [ − βH ( φ ( x ) , h )] . (2.5)The (Gibbs) free energy is given by βF = − log Z ≈ V d min [ V ( φ )] φ where V d is the d -dimensional volume and the potential V ( φ ) is evaluated at the minima φ .– 6 – .2 Critical exponents In this subsection, we define various critical exponents and compute them using saddle pointapproximation. We will drop the index i in φ i for notational convenience at various places inthe text. Let us introduce the critical exponents corresponding to these phase transitions. These expo-nents are generalizations of the exponents which characterize the second order phase transitions.1. The exponent β is defined as the divergence of the order parameter at zero external field: φ ( t, h = 0) ∝ (cid:40) , T > T C ; | t | β , T < T C . (2.6)2. The exponent δ is defined as the divergence of the order parameter at the critical tem-perature: φ ( T = T C , h ) ∝ h δ . (2.7)3. Denoting Φ as the integral of the order parameter over the full space i.e. Φ = (cid:82) d d x φ ( x ) ,we define the generalized susceptibility (response function) as χ = ∂ (cid:104) Φ (cid:105) ∂h (cid:12)(cid:12)(cid:12) h =0 , (2.8)where (cid:104) Φ (cid:105) is the one-point function defined as: (cid:104) Φ (cid:105) = 1 Z (cid:90) [ Dφ ( x )] (cid:90) d d x φ ( x ) exp [ − βH ( φ ( x ) , h )] (2.9)and where Z is given in (2.5). The divergence of the susceptibility is given by χ ( T, h = 0) ∝ | t | − γ . (2.10)Using the definition of susceptibility in equation (2.8), we obtain it’s expression in termsof the order parameter by differentiating the partition function w.r.t h . χ ≡ β (cid:104) Φ( x ) Φ r − ( y ) (cid:105) c = β (cid:0) (cid:104) Φ( x ) Φ r − ( y ) (cid:105) − (cid:104) Φ( x ) (cid:105) (cid:104) Φ r − ( y ) (cid:105) (cid:1) ≡ β (cid:90) d d x d d y (cid:104) φ ( x ) φ r − ( y ) (cid:105) c = β (cid:90) d d x d d y (cid:0) (cid:104) φ ( x ) φ r − ( y ) (cid:105) − (cid:104) φ ( x ) (cid:105) (cid:104) φ r − ( y ) (cid:105) (cid:1) (2.11)where (cid:104) Φ( x ) Φ r − ( y ) (cid:105) c and (cid:104) φ ( x ) φ r − ( y ) (cid:105) c are the connected correlators. Note thatsetting a UV cuttoff removes divergences associated with the evaluation of product offields coinciding at a point y in (cid:104) φ r − ( y ) (cid:105) , thus making it a well-defined quantity. The critical exponent β should not be confused with β appearing in the partition function, which is givenby β = k B T . To avoid confusion, whenever we talk about the critical exponent β , we mention it explicitly. – 7 –. The (Gibbs) free energy is given by βF = − log Z . Let us denote the n th derivative of thefree energy as C n = ∂ n ∂t n (cid:0) βFV (cid:1) . Since we are interested in a r th order phase transition, weexpect the r th derivative of the free energy to diverge at t = 0 . We define the exponent α , which characterizes this divergence, as follows: C r ∝ | t | − α . (2.12)Note here that the specific heat is given by the second derivative of free energy i.e. C V = − C ∝ ∂ ∂t (cid:0) βFV (cid:1) . Hence for a second order phase transition, C V diverges. For r thorder phase transition, the specific heat is given by: C V ( T, h = 0) ∝ | t | r − α − . (2.13)5. We also define a two-point connected correlator given by G rc ( x, y ) = (cid:104) φ ( x ) φ r − ( y ) (cid:105) c = (cid:104) φ ( x ) φ r − ( y ) (cid:105) − (cid:104) φ ( x ) (cid:105) (cid:104) φ r − ( y ) (cid:105) (2.14)Note that G rc ( x, y ) only depends on the separation between the operator insertions. Thisis because we are working in Euclidean space which has translations and rotations as apart of its isometry group. If the separation is taken to be very large than the corelationlength ζ i.e. | x − y | (cid:29) ζ , then G rc ( x, y ) roughly decays as G rc ( x, y ) ∼ exp (cid:110) − | x − y | ζ (cid:111) .We can also write χ as given in equation (2.11) as an integral over the connected Greenfunction as defined in equation (2.14) χ = β (cid:90) d d x d d y G rc ( x, y ) = βV (cid:90) d d ρ G rc ( ρ, , (2.15)where we have defined ρ = | x − y | . Let the largest value of G rc ( ρ, be given by G when ρ < ζ , where G is always finite. Using the mean value theorem, we have the followinginequality, where V is the system’s volume (where V ≥ ζ d ): χβV < G ζ d . (2.16)Here we have assumed that the contributions from ρ (cid:29) ζ are negligible since the corre-lator dies off exponentially. Thus when χ diverges, the above inequality ensures that ζ necessarily diverges as well. This gives rise to the critical exponent ν defined by ζ ( T, h = 0) ∝ | t | − ν . (2.17)Note that here we have not labelled two different exponents γ ± , α ± etc. while approachingfrom the t < and t > directions. This is because we have already imposed analyticity inthe ( h, t ) plane while writing the phenomenological Hamiltonian, and consequently, we have γ = γ ± , α = α ± and so on. It can be shown that even for deviations away from the saddlepoint, we still get single-valued exponents in the ( h, t ) plane. We will prove this statement inAppendix A. – 8 – .2.2 Dilatation symmetry at critical temperature In the previous subsection, we discussed that at critical temperature, the correlation length ζ diverges as a consequence of the divergence of susceptibility, which follows from equation(2.16). At the critical temperature, the correlation length becomes infinite (or more precisely,it becomes of the order of system size). Since there are no other length scales in the prob-lem, G rc ( x, y ) should decay as a power law. This implies that there is an emergent dilatationsymmetry at the critical temperature, and the connected Green function transforms as follows G rc ( t = 0 , λx ) = λ κ G rc ( t = 0 , x ) . (2.18)Here κ is the scaling dimension under the scaling transformation. Thus the system at criticalityexhibits self-similarity at all intermediate scales between system size and the UV cutoff. In this subsection, we will compute the critical exponents defined above, using saddle pointapproximation. The saddle points can be found by minimizing the potential appearing in theHamiltonian (2.2). The potential is given by (as mentioned before, we have dropped the index i for notational convenience): V [ φ ≡ φ ( x )] = t r φ r − + u r φ r − ( h.φ ) φ r − . (2.19)1. Working in the saddle point approximation, we impose ∂V∂φ = 0 to get the values for theorder parameter above and below the critical temperature, φ = (cid:114) − t ( r − ru , t < , t > . (2.20)This reduces to the familiar value φ = − t u for second order phase transitions. Comparingthis with (2.6), we obtain β = 1 / .2. Upon substituting the value of φ from eqn (2.20), the free energy βFV = min [ V ( φ )] is givenby βFV = − (cid:18) ur − (cid:19) (cid:18) − t ( r − ru (cid:19) r , t < , t > . (2.21)Thus the r th derivative of the free energy with respect to the temperature is discontinuousat T = T C , which is given by C r ( T, h = 0) ∝ ( − r +1 (cid:18) r ! u ( r − r − ( ur ) r (cid:19) , t < , t > (2.22)– 9 –or r = 2 , this reduces to the familiar value for the specific heat, i.e. C = − C V ∝ u .Thus using (2.22) we conclude that α = 0 .3. Let us now look at the critical exponent associated with susceptibility. Using our Hamil-tonian and the definition of susceptibility from (2.8), we obtain the following expression. χ = − r − t (2 r − , t < − r − t (2 r − , t > (2.23)This gives us γ = 1 .4. The critical exponent δ can be obtained from setting t = 0 in the saddle point equationof state: h ( φ ) = 2 φ ru (2 r − . (2.24)Using (2.7), we obtain δ = 3 .Critical Exponents α β γ δ ν (See §4) ∆ (See §4 )Landau Theory 0 y t = 12( r + d − − dr ; y h y t = d + (2 r − (cid:0) − d (cid:1) r + d − − dr ;(Saddle point) if y t is relevant if y t is relevant Table 1 : Critical exponents from the saddle point expansion in Landau theory.We will introduce another exponent ∆ and ν in §4. The critical exponents from Landautheory are displayed in Table 1 for convenience. In this section, we will venture beyond the saddle point analysis by considering small fluctuationsabout the saddle point and taking their effects into account. Readers who are not interested inthe calculational aspects of fluctuations can read the following introductory discussion of thissection where we have stated the main results and skip directly to §3.3.We will show that incorporating fluctuations about the saddle point into the analysis of r th order phase transitions leads to the following features:1. Fluctuations which lead to powers of the order parameter in the Hamiltonian of the order k − , k < r destroy the delicate analytic properties of the critical exponents. We willshow this by looking at the case of quadratic fluctuations, where we find such fluctuationsdestroy analyticity for r > . – 10 –. In d < r , saddle point analysis does not always hold. The Ginzburg criteria as defined in§3.3 tells us when saddle point analysis holds in d < r , and when fluctuations contributesignificantly. In case fluctuations dominate, critical exponents are modified.3. Saddle point analysis is valid above the upper critical dimension given by d > d u = 2 r .Hence the critical exponents corresponding to observables like C r do not change for d > r .4. Fluctuations destroy all order in the system for d ≤ d l for second order phase transitions,where d l is the lower critical dimension [34, 35]. This argument remains unchanged for r thorder phase transition as shown in Appendix B, where we show that d l = 2 for systemswith continuous symmetry (in the phenomenological Lagrangian) and d l = 1 for systemswith discrete symmetry.We begin by looking at the effect of fluctuations on two-point correlators. Afterwards,we analyze the effects of fluctuations on C r . Specifically, we study how these observables’divergences near critical temperature are modified due to fluctuations. The computations inAppendix B also stress the importance of fluctuations, as we show that fluctuations lead topartial or complete destruction of order depending on the dimension of the system. We are interested in fluctuations of the form: φ ( x ) = (cid:0) φ + φ (cid:107) ( x ) , φ ⊥ , ( x ) , . . . , φ ⊥ ,n ( x ) (cid:1) (3.1)where φ (cid:107) ( x ) and φ ⊥ ,i denote the longitudinal and transverse fluctuations respectively. Thekinetic term goes as ( ∇ φ ) = (cid:0) ∇ φ (cid:107) (cid:1) + ( ∇ φ ⊥ ) , (3.2)and the square of the order parameter goes as φ = φ + 2 φ φ (cid:107) + φ (cid:107) + n (cid:88) i =2 φ ⊥ ,i (3.3)By using (3.2) and (3.3) in (2.2) and keeping upto only quadratic terms in the fluctuations, weobtain: βH = (cid:90) d d x (cid:16) t r φ r − + u r φ r (cid:17) + (cid:90) d d x (cid:34) K (cid:0) ∇ φ (cid:107) (cid:1) + (cid:32) K ξ (cid:107) (cid:33) φ (cid:107) (cid:35) + n (cid:88) i =2 (cid:90) d d x (cid:34) K ∇ φ ⊥ ,i ) + (cid:32) K ξ ⊥ ,i (cid:33) φ ⊥ ,i (cid:35) + O (cid:0) φ (cid:107) , φ ⊥ (cid:1) (3.4)where ξ (cid:107) and ξ ⊥ ,i are correlation scales of the fluctuations and the first term is the potentialevaluated at the saddle point. The correlation lengths ξ (cid:107) and ξ ⊥ ,i corresponding to the two– 11 –oint correlator are explicitly given by: ∂ V∂φ (cid:107) (cid:12)(cid:12)(cid:12)(cid:12) φ = 2 Kξ (cid:107) = ru r (cid:18) t − rtru (cid:19) r − , t < t ( r − r − δ r, t > ∂ V∂φ ⊥ ,i (cid:12)(cid:12)(cid:12)(cid:12) φ = 2 Kξ ⊥ ,i = (cid:40) , t < t ( r − δ r, t > (3.5)Note that the lengths ξ (cid:107) and ξ ⊥ ,i corresponding to the two point correlator display a differentanalytic behaviour as compared to ζ . The two point correlators ξ which were zero in thesaddle point approximation acquire a non-zero value due to fluctuations. It can be seen fromthe nature of the longitudinal polarization that the two point correlator ξ is not an analyticfunction, since quadratic terms in the Hamiltonian characterizing the fluctuations destroy thedelicate analytic features. As a result we have the scalings: ξ + (cid:107) ∼ | t | − δ r, , ξ −(cid:107) ∼ | t | − r ξ + ⊥ ,i ∼ | t | − δ r, , ξ −⊥ ,i ∼ (3.6)We note from equation (3.6) that only in the case of second order phase transitions with r = 2 we have ξ + (cid:107) = ξ −(cid:107) ≡ ξ (cid:107) = | t | − . This is because such fluctuations give rise to analyticterms in the Hamiltonian corresponding to the second order phase transitions.Here we have shown that quadratic fluctuations destroy the analyticity of the physicalobservables in the h − t plane. The same argument can be extended to fluctuations of the order k − , k < r . Let us now evaluate the effect of fluctuations on the full partition function. By including thequadratic fluctuations in the partition function we obtain: Z ≈ exp [ − V d V ( φ )] (cid:90) (cid:2) Dφ (cid:107) (cid:3) exp (cid:34) − K (cid:90) d d x (cid:34)(cid:0) ∇ φ (cid:107) (cid:1) + 2 φ (cid:107) ξ (cid:107) (cid:35)(cid:35) × (cid:90) (cid:34)(cid:89) i Dφ ⊥ ,i (cid:35) exp (cid:34) − K (cid:90) d d x (cid:34) ( ∇ φ ⊥ ,i ) + 2 φ ⊥ ,i ξ ⊥ ,i (cid:35)(cid:35) (3.7)where V d is the volume of the system. The free energy of the system is given as follows (wherewe have removed the label i indicating the transverse directions for convenience): f = − ln ZV d = V ( φ )+ 12 (cid:90) d d q (2 π ) d ln (cid:104) K (cid:16) q + 2 (cid:0) ξ (cid:107) (cid:1) − (cid:17)(cid:105) + n − (cid:90) d d q (2 π ) d ln (cid:2) K (cid:0) q + 2 ( ξ ⊥ ) − (cid:1)(cid:3) , (3.8)– 12 –here q is the momentum associated with the fluctuations. We will now look at the effect offluctuations on the singularity structure of the r th derivative of the free energy which is givenby C r ≡ d r fdt r . Using (3.6) and (3.8) we obtain the following corrections to C r : C r ∝ C + r + (cid:82) d d q (2 π ) d ( − r − ( r − δ r, (cid:16) q + 2( ξ + (cid:107) ) − (cid:17) r + ( n − (cid:82) d d q (2 π ) d ( − r − ( r − δ r, (cid:0) q + 2( ξ + ⊥ ) − (cid:1) r ; t > C − r + (cid:82) d d q (2 π ) d ( − r − ( r − r − r ( ξ −(cid:107) ) r (2 − r ) r − (cid:16) q + 2( ξ −(cid:107) ) − (cid:17) r + O (cid:16) q r − (cid:17) ; t < (3.9)where we have only written the most divergent correction to C − r explicitly and O (cid:16) q r − (cid:17) de-notes the subleading corrections. Notice that the following momentum-integral appears in thecorrection terms: I = (cid:90) d d q (2 π ) d (cid:18) q + 2( ξ ) − (cid:19) r (3.10)where we have suppressed the labels indicating longitudinal or transverse, and whether t isgreater or less than 0. The integral I has mass dimension d − r , and thus diverges for d ≥ r .Introduction of a UV regulator Λ = a cuts these divergences off. We can perform a similaranalysis for the integrals at t < . We can thus write C r in d > r as C r ∝ (cid:40) C + r + A n a r − d δ r, t > C − r + A a r − d + O ( a r − d − ) t < (3.11)while C r in d < r takes the following form upon rescaling q by ξ − C r ∝ C + r + A (cid:18)(cid:16) ξ + (cid:107) (cid:17) r − d + ( n − (cid:0) ξ + ⊥ (cid:1) r − d (cid:19) δ r, t > C − r + A (cid:16) ξ −(cid:107) (cid:17) r − d + r (2 − r ) r − t < (3.12)Note that in the equations above, A , A , A and A are constants, which do not influence thediscontinuity. Physical aspects of fluctuations:
We will now analyze the physical aspects of thesecomputations. As shown in equation (3.11), for d > r , C r gets an additive constant correctionproportional to a r − d which does not introduce any new singularity in the expression. However,we see from (3.12) that in d < r , there are corrections that have associated singular parts,as the correlation lengths diverge near critical points. The calculation above is done by takinginto account the first order corrections; inclusion of higher order terms can potentially introducemore singular terms to C r . Thus the saddle point analysis does not work correctly in d < r because fluctuation contributions tamper with the predicted singularity structure. Thereforewe denote d = 2 r as the upper critical dimension above which the saddle point approximation– 13 –olds, and fluctuations do not modify the critical exponents. The following question naturallyarises: Under what conditions does saddle point analysis robustly hold below the upper criticaldimension where fluctuations modify the singularities? We discuss this in the next subsection§3.3.The predicted corrections to C V match with the values for second order phase transitionswith r = 2 . By substituting the expressions for ξ ’s from equations (3.5) and (3.6) into (3.12),we also see that corrections to the saddle point critical exponent α of C r do not have thesimilar scaling for t > and t < . As stated earlier, this arises due to the fact that quadraticcontribution to the action destroys the analyticity of physical observables in the h − t plane. r th order phase transitions In this subsection, we quantify when the effects of fluctuations dominate over the saddle pointexpectations. We do this by generalizing the Ginzburg criteria to understand when saddle pointcalculations accurately predict critical exponents in the presence of fluctuations.In order that the saddle point prediction is the dominant contribution, we demand thatthe saddle point discontinuity ∆ C r ≡ C + r − C − r is larger as compared to the contributions fromfluctuations. Only in such situations, we can trust the saddle point computation. Using (3.6)in (3.12), we see that the the condition for saddle point analysis to dominate over fluctuationsis given by: ∆ C r (cid:29) ∆ C fluc. r = A (cid:18)(cid:16) ξ + (cid:107) (cid:17) r − d + ( n − (cid:0) ξ + ⊥ (cid:1) r − d (cid:19) δ r, − (cid:32) A (cid:16) ξ −(cid:107) (cid:17) r − d + r (2 − r ) r − (cid:33) = A (cid:48) δ r, t − ( r − d ) + A (cid:48)(cid:48) t ( d − r ) ( r − r ( r − (3.13)where A (cid:48) and A (cid:48)(cid:48) are linear combinations of proportionality constants A and A . Notice thatthe term A (cid:48) drops out for higher order phase transitions ( r > ), whereas for r = 2 both theterms have the same order of magnitude. Thus the requirement for saddle point singularitiesto dominate over the fluctuations is given by | t | (cid:29) t G ∼ (cid:18) A (cid:48)(cid:48) ∆ C r (cid:19) − d − r )( r − r ( r − . (3.14)For the case of r = 2 , we recover the standard Ginzburg criteria from equation (3.13) which isgiven by | t | (cid:29) t G ∼ (cid:18) − A (cid:48) + A (cid:48)(cid:48) ∆ C V (cid:19) − d . (3.15)where we have used the fact that C = − C V . The interpretation of this temperature scale isthat it is possible to recover the saddle point critical exponents if the system does not go withina distance t G near the critical temperature T C .– 14 – Scaling relations among critical exponents
In the previous section, we saw that critical exponents obtain corrections due to fluctuations. Wenow address the following question: Given that fluctuations modify the saddle-point calculatedvalues of the critical exponents, can we still find some robust relations between the criticalexponents that are valid beyond the saddle point? As we already know, there exist scalingrelations between the exponents for second-order phase transitions. Can we generalize them for r th order phase transitions?Let us begin by looking at the free energy obtained using saddle point description. It isgiven by f ( t, h ) ∝ t r u r − , h = 0; t < h r u r − h (cid:54) = 0; t = 0 . (4.1)Thus we can now write the above free energy (at saddle point) as a homogenous function of t and h . f ( t, h ) = | t | r g (cid:18) ht ∆ (cid:19) = | t | r g ( x ) , where x ≡ h | t | ∆ (4.2)Here ∆ is the generalization of the familiar gap exponent which appears for the case of secondorder phase transitions. Let us now look at the h → and the t → limits of the function g ( x ) such that the power of h is kept intact. These go as lim x → g ( x ) ∼ u r − , and lim x →∞ g ( x ) ∼ x r u r − . (4.3)Equation (4.3) implies that the free energy takes the following form near h (cid:54) = 0 and t = 0 : f ∼ | t | r h r t r ∆3 (cid:16) u r ∆3 − (cid:17) . (4.4)Since f remains finite at t = 0 , the gap exponent is given by setting the coefficient of | t | to 0,and we obtain ∆ = . In this subsection, we will argue that beyond the saddle point, the singular part of the freeenergy has the following homogenous form: f sing ( t, h ) = | t | r − α g (cid:18) ht ∆ (cid:19) (4.5)This form can be understood using Wilsonian Renormalization Group (RG) flow. Under theRG transformations, the partition function remains unchanged and hence the correspondingfree energies are related by: V f ( t, h ) = V (cid:48) f ( t b , h b ) (4.6)– 15 –here t b and h b are the rescaled couplings obtained by rescaling x → x/b , where b > . Noticethat we have suppressed the subscript r in t r , which we will keep doing for the rest of thissection as well. Hence, in d - dimensions, the rescaled volume gets smaller by a factor of b d andwe obtain: f ( t, h ) = b − d f ( b y t t, b y h h ) (4.7)where y t and y h are the mass dimensions of the couplings t and h as given in (2.4). In case thecoupling t is relevant i.e. y t > , we can choose b = t − /y t , such that b > . Notice that if theLagrangian does not contain any relevant coupling, this choice of b is not possible and hence wecannot write down the scaling relations derived below. With above choice of b , the free energytakes the following form: f ( t, h ) = t d/y t f (cid:18) , ht y h /y t (cid:19) ≡ t d/y t g f (cid:18) ht y h /y t (cid:19) (4.8)Thus we obtain the same form as (4.5) with r − α = d/y t and ∆ = y h /y t . Notice that the abovescaling is valid only when the coupling t is relevant.We can obtain similar homogeneous scaling forms for the specific heat and C r by differen-tiating f w.r.t. t . We have: dfdt = ( r − α ) | t | r − α − g (cid:18) ht ∆ (cid:19) − ∆ h | t | r − α − ∆ − g (cid:18) ht ∆ (cid:19) = | t | r − α − (cid:20) ( r − α ) g (cid:18) ht ∆ (cid:19) − ∆ ht ∆ g (cid:18) ht ∆ (cid:19)(cid:21) = | t | r − α − ˜ g (cid:18) h | t | ∆ (cid:19) (4.9)where ˜ g (cid:16) h | t | ∆ (cid:17) = ( r − α ) g (cid:0) ht ∆ (cid:1) − ∆ ht ∆ g (cid:0) ht ∆ (cid:1) . Hence the specific heat takes the form C sing V ∼ − ∂ f∂t ∼ | t | r − α − (cid:98) g (cid:18) ht ∆ (cid:19) , (4.10)while the r th derivative of the free energy takes the form C sing r ∼ − ∂ r f∂t r ∼ | t | − α ¯ g (cid:18) ht ∆ (cid:19) . (4.11) In this subsection, we write down similar homogeneous forms for other physical observablesnear the critical point. We will use these scaling forms to obtain relations between variouscritical exponents.1. Using the scaling of the free energy from eqn (4.5) and the relation between the orderparameter and the free energy, we conclude that φ scales as φ r − ∼ | t | r − α − ∆ g φ (cid:18) ht ∆ (cid:19) , (4.12)– 16 –sing which we can conclude from equation (2.6) that β is given by: β = r − α − ∆2 r − . (4.13)2. When x → ∞ , the dominant contribution in g φ scales as g φ ∼ x p where p is the lead-ing power in the expansion, and therefore φ r − ∼ | t | r − α − ∆ (cid:0) ht ∆ (cid:1) p . As this limit is t -independent, we have the following relation between the exponents ∆ p = r − α − ∆ , = ⇒ φ r − ( t = 0 , h ) ∼ h r − α − ∆∆ . (4.14)Using equation (2.7) the above identity implies the following relation δ = (2 r − r − α − ∆ = ∆ β . (4.15)3. Using the definition of generalized susceptibility as χ = d (cid:104) Φ (cid:105) dh , and using the scaling of φ from eqn (4.12), we see that χ scales as χ ( t, h ) = d (cid:104) Φ (cid:105) dh ∼ | t | r − α − ∆2 r − − ∆ g χ (cid:18) ht ∆ (cid:19) . (4.16)Using equation (2.10) we obtain: γ = ∆ − (cid:18) r − α − ∆2 r − (cid:19) . (4.17)Thus we see that all the critical exponents ( α, β, γ, δ, ∆) can be derived from two indepen-dent critical exponents (say from ∆ and α in the case of second order phase transition). As acheck for the above relations, we can set r = 2 , and use equations (4.13), (4.15) and (4.17) toobtain α + 2 β + γ = 2 & δ − γβ (4.18)which are the saturation of Rushbrooke’s inequality [8] (also known as Rushbrooke’s identity)and Widom’s scaling law [10] respectively for the second order phase transitions. In this subsection, we will use the two-point correlator given in equation (2.14) to understandit’s divergence near the critical point. We hereby derive the generalized Josephson identityfor r th order phase transitions. Following the main argument from our previous section, thecorrelation length ζ ( t, h ) has the following form: ζ ( t, h ) ∼ | t | − ν g ζ (cid:18) h | t | ∆ (cid:19) (4.19)– 17 –n the vicinity of the critical point, the most important length scale is ζ , and the singularcontribution to observables arise solely due to ζ ’s singularity. A similar singular behaviourwas demonstrated in the case of quadratic fluctuations in §3, where the correlation length ξ dictated the singular contributions in d < r , with the only significant difference being that thecorrections led to non-analytic behaviour. Hence the singular part of the free energy in termsof the correlation length ζ is given by: V d f ( t, h ) ∼ ln Z = B (cid:18) Lζ (cid:19) d + B (cid:18) La (cid:19) d , (4.20)where B and B denote non-singular quantities, L denotes the system size, and Λ = a is theUV cutoff. Since the singular part of the free energy arises from the first term in the aboveequation, we have f ( t, h ) ∼ ln ZL d ∼ ζ − d ∼ | t | dν g (cid:18) ht ∆ (cid:19) (4.21)This is the same scaling form as given in (4.8) with ν = 1 y t , ∆ = y h y t . (4.22)The scaling form in (4.21) has the following important properties:1. From (4.20), we observe that the free energy grows as (cid:16) Lζ (cid:17) d near the critical point. It canbe interpreted as a measure of the system size divided by the correlation length ζ , suchthat the system is divided into many small blocks. Each small block can be thought of asan independent degree of freedom which contributes a constant factor to the overall freeenergy. We also naturally obtain the homogeneity of the singular part of the free energy.2. We arrive at the generalized Josephson’s identity by comparing (4.21) with the homoge-nous form of free energy given in (4.5). The identity is given by r − α = dν. (4.23)Josephson’s identity is basically a hyperscaling relation that holds below and at the uppercritical dimension, but it does not hold above the upper critical dimension. This is becausethe scaling form for ζ as given in equations (4.19) and (4.20) does not hold above theupper critical dimension. Above the upper critical dimension g ζ ( x ) is singular in x [36]and can be shown to grow as g ζ ( x ) ∼ x using mean-field theory, which is singular when x → .We have summarized the four independent relations between the critical exponents α, β, γ, δ, ∆ and ν in Table 2. Thus all the critical exponents can be written in terms of two independentones, say in terms of γ and ∆ . – 18 –erial No. Exponents Relation1. β, α, ∆ β = r − α − ∆2 r − δ, α, ∆ δ = (2 r − r − α − ∆ γ, α, ∆ γ = ∆ − (cid:18) r − α − ∆2 r − (cid:19) α, ν r − α = dν Table 2 : Independent relations between different critical exponents
In the previous sections, we calculated the critical exponents using the saddle point approxi-mation and derived relations between critical exponents away from the saddle point. In thissection, we will use perturbative RG flow to find fixed points of the theory and use them tocalculate the critical exponents. Specifically, we will be looking at the case when fluctuationsdominate over saddle point contributions as quantified by the Ginzburg criteria in d ≤ r . Inthat case, the critical exponents are captured by relevant couplings about fixed points. We willperform linear stability analysis near these fixed points to calculate the critical exponents intheir vicinity.Let us denote the full coupling space of our theory by S ( t r , u r , . . . ) . A particular Hamil-tonian ( H ) is described by couplings at a point in this space. RG flow involves rescaling thecouplings and renormalizing the order parameter, thereby taking us from one point in thisparameter space to another, i.e., S b → R b S . Here we have denoted the action of RG flow by R b , with b being the rescaling parameter defined in §4.1. The fixed points of these flows aredefined as R b S ∗ = S ∗ . Since we rescale the couplings in the RG procedure, the correlationlength ζ ∗ should either go to zero or infinity at the fixed point. When ζ ∗ = 0 , the system is ina completely disordered phase, and when ζ ∗ → ∞ , the system is in the ordered phase.We can now study the stability of the fixed points by linearizing the beta function nearthem. The subspace of irrelevant couplings is known as the basin of attraction since the flowin this subspace converges towards the fixed point. Since the correlation length diverges atthe fixed point, it diverges at each point in the basin of attraction. This basin of attractionis basically the critical surface at which the phase transition takes place. The behavior ofcritical exponents near the phase transition is hence determined only by the set of relevantcouplings. We will use them to compute the value of critical exponents beyond the saddle pointapproximation. – 19 – .1 One-loop beta function In this subsection, we compute the one-loop beta functions for the couplings appearing in theHamiltonian (2.2). We follow the procedure described in [4, 37]. The Euclidean action for r thorder phase transition at zero external magnetic field is given by: S Λ [ φ i , h = 0] = (cid:90) d d x (cid:20)
12 ( ∂ µ φ i ∂ µ φ i ) + V ( φ i ) (cid:21) (5.1)where Λ is the UV cutoff and the potential is given by: V ( φ i ) = t r ( φ ) r − + u r ( φ ) r (5.2)with φ = (cid:80) Ni =1 φ i φ i . We work with dimensionless couplings which are defined as follows: t r = g r − (2 r − d − ( r − d − , u r = g r r ! Λ d − r ( d − . (5.3)In terms of these dimensionless couplings, the potential is given by: V ( φ i ) = g r − (2 r − d − ( r − d − ( φ ) r − + g r r ! Λ d − r ( d − ( φ ) r (5.4)We will now use this potential to study RG flows using the Wilsonian formalism. This involvessplitting the scalar field in terms of high energy and low energy modes and the subsequentprocedure of systematically integrating out the high energy modes. Therefore we split the field φ i into the low energy modes ϕ i and the high energy modes χ i i.e. φ i = ϕ i + χ i . Hence theaction can be expanded as: S [ φ i ] = S [ ϕ i ] + (cid:90) d d x (cid:20)
12 ( ∂ µ χ i ∂ µ χ i ) + 12 χ i ∂ V ( ϕ k ) ∂ϕ i ∂ϕ j χ j + .. (cid:21) (5.5)where we have chosen ϕ i such that it minimises the potential i.e. V (cid:48) ( ϕ i ) = 0 . We will nowintegrate out the high energy modes χ i in the partition function, by lowering the scale infinites-imally i.e. setting Λ (cid:48) = Λ − δ Λ . We will focus on the first order terms in δ Λ , which is similar tocomputing one-loop effects only. It can be argued that at this order, only the quadratic termsin χ i contribute to the effective action [37]. Upon integrating out the high energy modes, theone-loop correction to the effective action is given by: δ Λ S [ φ ] = Λ d − δ Λ(4 π ) d/ Γ( d/ (cid:90) d d x ln (cid:20) det (cid:18) Λ δ ij + ∂ V ( ϕ i ) ∂ϕ i ∂ϕ j (cid:19) (cid:21) (5.6)This process of integrating out the high energy modes changes the effective couplings dependingon the UV scale at hand. The change in the k th coupling is captured by the beta function ( β k = dg k /d log Λ) . We can use (5.6) to compute the beta functions of the couplings g r − and g r appearing in the potential (5.4), which are given by: Λ dg r − d Λ = [( r − d − − d ] g r − − a Λ ( r − d − ∂ r − ∂ϕ r − i (cid:20) ln (cid:20) det (cid:18) Λ δ ij + ∂ V ( ϕ i ) ∂ϕ i ∂ϕ j (cid:19) (cid:21)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ϕ i =0 Λ dg r d Λ = [ r ( d − − d ] g r − a Λ r ( d − ∂ r ∂ϕ ri (cid:20) ln (cid:20) det (cid:18) Λ δ ij + ∂ V ( ϕ i ) ∂ϕ i ∂ϕ j (cid:19) (cid:21)(cid:21)(cid:12)(cid:12)(cid:12)(cid:12) ϕ i =0 (5.7)– 20 –here we have defined a as follows a = 1(4 π ) d/ Γ( d/ . (5.8)Using (5.4) we compute the first derivative of potential which is given by: ∂V∂ϕ i = g r − (2 r − d − ( r − d − (cid:32)(cid:88) m ϕ m (cid:33) r − ϕ k δ ik + g r (2 r − d − r ( d − (cid:32)(cid:88) m ϕ m (cid:33) r − ϕ k δ ik , while the second derivative is given by: ∂ V∂ϕ i ϕ j = g r − (2 r − d − ( r − d − (cid:32)(cid:88) m ϕ m (cid:33) r − δ ij + g r (2 r − d − r ( d − (cid:32)(cid:88) m ϕ m (cid:33) r − δ ij +2( r − g r (2 r − d − r ( d − (cid:32)(cid:88) m ϕ m (cid:33) r − ϕ k ϕ l δ ik δ jl +2( r − g r − (2 r − d − ( r − d − (cid:32)(cid:88) m ϕ m (cid:33) r − ϕ k ϕ l δ ik δ jl . We can now compute the determinant appearing in the expression for beta function (5.7) whichtakes the following form:det (cid:18) Λ δ ij + ∂ V ( ϕ i ) ∂ϕ i ∂ϕ j (cid:19) = Λ + (cid:32)(cid:88) m ϕ m (cid:33) r − (cid:32) g r − Λ d − ( r − d − (2 r − g r Λ d − r ( d − (2 r − (cid:88) m ϕ m (cid:33) N − × Λ + (cid:32)(cid:88) m ϕ m (cid:33) r − (cid:32) g r − Λ d − ( r − d − (2 r − g r Λ d − r ( d − (2 r − (cid:88) m ϕ m (cid:33) (5.9)Substituting the expression for the determinant given by (5.9) in the beta function (5.7), wecan calculate the one-loop beta functions for various r . In the rest of this section, we computethe critical exponents for phase transitions of different orders. ( r = 2) In this subsection, we focus on second order phase transitions ( r = 2 ). We can see that for r = 2 the couplings g and g appear in the Hamiltonian (2.2). We use (5.7) to compute theone-loop beta-functions for these couplings and then evaluate the critical exponents for thiscase. The results in this subsection are a review of previously known results for second orderphase transitions. Using (5.7) and (5.9), we find the following one-loop beta functions for thecouplings g and g : Λ dg d Λ = − g − ( N + 2)(4 π ) d/ Γ( d/ g g ) (5.10) Λ dg d Λ = ( d − g + ( N + 8)3(4 π ) d/ Γ( d/ g (1 + g ) (5.11)– 21 –he fixed points of the RG flow can be found by setting β ( g ) and β ( g ) to zero. There are twofixed points. The first one is a trivial fixed point known as the Gaussian fixed point and it isgiven by: g ∗ = 0 , g ∗ = 0 (5.12)Near the Gaussian fixed point, the coupling g is relevant and g is marginal and their massdimensions are given in (2.4). There also exists a non-trivial Wilson-Fisher (WF) fixed point[38] given by: g ∗ = − ( d − N + 2) d ( N + 2) − N + 4) , g ∗ = − d − N + 8) a ( d ( N + 2) − N + 4)) (5.13)where a is defined in (5.8). We will now study flows about this fixed point, which appears "justbelow" four dimensions. Defining (cid:15) ≡ − d , the Wilson-Fisher fixed point is given by: g ∗ = − (cid:15) ( N + 2)2( N + 8) + O ( (cid:15) ) , g ∗ = 3 (cid:15)a ( N + 8) + + O ( (cid:15) ) (5.14)We can now perform (cid:15) -expansion of the beta functions near these fixed points: β ( g ∗ + δg ) = β ( δg ) = − g ∗ + δg ) − ( N + 2)3 a ( g ∗ + δg )(1 + g ∗ + δg )= − δg + (cid:15) ( N + 2)( N + 8) δg − a ( N + 2)3 δg (cid:18) (cid:15) ( N + 2)2( N + 8) (cid:19) + O ( (cid:15) ) β ( g ∗ + δg ) = β ( δg ) = − (cid:15) ( g ∗ + δg ) + a ( N + 8)3 ( g ∗ + δg ) (1 + g ∗ + δg ) = (cid:15)δg + O ( (cid:15) ) Hence we obtain the following matrix of the linearized beta functions about the Wilson-Fisherfixed point: (cid:18) β ( δg ) β ( δg ) (cid:19) = (cid:32) − (cid:15) ( N +2)( N +8) − a ( N +2)3 (cid:16) (cid:15) ( N +2)2( N +8) (cid:17) (cid:15) (cid:33) (cid:18) δg δg (cid:19) (5.15)The eigenvalues are − (cid:15) ( N +2)( N +8) and (cid:15) . The first eigenvalue is negative and hence the coupling g is relevant around the WF fixed point and the scaling dimension y t is given by: y t = 2 − (cid:15) ( N + 2)( N + 8) (5.16)On the other hand, the eigenvalue corresponding to the coupling g is positive for d < .Hence the coupling g is irrelevant (making the WF fixed point stable along the corresponding Note that our beta functions are negative as compared to [6, 7], who perform renormalization by studyingthe variation of couplings with respect to length scale, as opposed to momentum scale in our case. – 22 –igenvector) for d < . We can use y t to calculate the critical exponent ν using the relation(4.22) and it is given by: ν = 1 y t = (cid:18) (cid:18) − (cid:15) ( N + 2)2( N + 8) (cid:19)(cid:19) − = 12 + (cid:15) N + 2)( N + 8) + O ( (cid:15) ) (5.17)For r = 2 , the scaling dimension of the coupling h i is still given by (2.4). This is because inthe Lagrangian h i couples to the zero momentum mode (cid:82) d d x φ i ( x ) = ˜ φ i ( k = 0) , and henceintegrating out high energy modes using Wilsonian RG leaves h i unchanged. Therefore for thespecial case of second order phase transition ( r = 2 ) the scaling dimension of h i just below fourdimensions is given by: y h = 3 − (cid:15) O ( (cid:15) ) (5.18)We can now calculate the gap exponent ∆ from (4.22) which is given by: ∆ = y h y t = 32 + (cid:15) (cid:18) N + 2) N + 8 − (cid:19) + O ( (cid:15) ) (5.19)We can use the exponents ν and ∆ to compute all other critical exponents using relationsderived in §4.2. These exponents are given by: α = (cid:15) (cid:18) − N + 2 N + 8 (cid:19) + O ( (cid:15) ) β = 12 − (cid:15) (cid:18) − N + 2 N + 8 (cid:19) + O ( (cid:15) ) γ = 1 + (cid:15) (cid:18) N + 2 N + 8 (cid:19) + O ( (cid:15) ) δ = 3 + (cid:15) (cid:18) N + 2 N + 8 (cid:19) + O ( (cid:15) ) (5.20)The above results match with the standard results for second order phase transitions as givenin [5–7]. ( r = 3) In this subsection, we calculate the critical exponents for r = 3 . The calculation of exponentsfor r > is exactly the same as for r = 2 with an important distinction. For r = 2 , integratingout high energy modes does not renormalize the coupling h i because the term h i φ i is a zeromomentum mode. Hence the mass dimension of the coupling h i does not receive any quantumcorrections under Wilsonian renormalization. However we can see from going to the Fourierspace that h i φ r − i no longer has only zero-momentum contribution. This term can potentiallybecome relevant in the IR and give large contributions.In order to simplify the detailed calculation and qualitatively understand the underlyingphysics of the problem, we work in the limit when h i φ r − i is extremely small as compared to– 23 –he other terms. This assumption is reasonable because we want to capture details of higherorder transitions very close to the critical point, i.e., h i → , and also, the physics of differentphase transitions really depends on moving in the t direction. In other words, even though thecoupling may be relevant, we will restrict ourselves to integrating out modes with an IR cutoffsuch that quantum corrections to h i do not blow out of proportion and consequently changethe nature of the fixed points. In the first order approximation, the scaling dimension of h i canthus be taken to be its classical mass dimension.With this assumption, the stage is clear to calculate the critical exponents of r = 3 . Westart with writing the beta functions of the couplings g and g which appear in (2.2) for r = 3 .The beta functions are given by: Λ dg d Λ = [ d − g + a ( N + 8)3 g (5.21) Λ dg d Λ = [2 d − g + a ( N + 14) g g − a ( N + 26)9 g (5.22)where a is given in (5.8). As in the case of r = 2 , we obtain two fixed points in this case aswell. The trivial Gaussian fixed point is given by: g ∗ = 0 , g ∗ = 0 (5.23)We now perform linear stability analysis about this fixed point by expanding the beta functionnear the same. We again look at RG flows in the vicinity of this fixed point just below fourdimensions, where we have defined (cid:15) = 4 − d . We obtain the following linearized beta functions: (cid:18) β ( δg ) β ( δg ) (cid:19) = (cid:18) − (cid:15)
00 2(1 − (cid:15) ) (cid:19) (cid:18) δg δg (cid:19) (5.24)Since < (cid:15) (cid:28) , the coupling g is relevant while g is irrelevant near this fixed point. We usethe relevant coupling to compute the critical exponents which are given below: ν = 1 (cid:15) , ∆ = 12 + 1 (cid:15)α = 4 (cid:18) − (cid:15) (cid:19) , β = −
12 + 1 (cid:15)γ = 1 , δ = 2 + (cid:15) − (cid:15) (5.25)Notice that for d = 3 i.e. (cid:15) = 1 , the above values of critical exponents match the ones computedvia saddle point approximation in Table 1. This is because the one-loop corrections near theGaussian fixed point do not change the scaling dimensions of the couplings. Hence we do notget any corrections to the saddle point result at this order.Notice that there is no N -dependence in the critical exponents near the Gaussian fixedpoint. This is because about the Gaussian fixed point, the terms in (5.21) and (5.22) propor-tional to N are quadratic or higher order in δg and δg . However since we are only considering– 24 –inear terms in our analysis, these terms do not contribute and hence we do not get any N dependence.For d < , there exists another non-trivial fixed point of the beta functions in (5.21), whichis given by: g ∗ = − d − a ( N + 8) , g ∗ = − d − ( N + 26) a ( N + 8) (6( N + 20) − d ( N + 26)) (5.26)The fixed point takes the following form in d = 4 − (cid:15) dimensions: g ∗ = 3 (cid:15)a ( N + 8) , g ∗ = 15 (cid:15) ( N + 26) a ( N + 8) (5.27)Linearizing the beta functions about this fixed point, we obtain: (cid:18) β ( δg ) β ( δg ) (cid:19) = (cid:32) (cid:15) O ( (cid:15) ) 2 + (cid:15) N +26 N +8 (cid:33) (cid:18) δg δg (cid:19) (5.28)Notice that near this fixed point both the couplings are irrelevant. Hence the whole g − g plane is a basin of attraction. As we change the temperature i.e. vary the coupling g , thetheory always flows to the above non-trivial fixed point. As stressed in §4.1, we need atleastone relevant coupling to write down the scaling relations, hence we cannot use the RG flowto obtain critical exponents near this fixed point. Hence the critical exponents for r = 3 aredetermined by RG flows near the Gaussian fixed point.Note that we computed the critical exponents solely below four dimensions even though theupper critical dimension for r = 3 is d u = 6 . This is because the couplings g and g appearingin the phenomenological Hamiltonian for r = 3 become irrelevant for d > . Hence we cannotobtain critical exponents using linear stability analysis in the above fashion for d > . ( r = 4) We will now compute the critical exponents for fourth order phase transitions. The betafunctions for the couplings g and g appearing in (2.2) for r = 4 are given by: Λ dg d Λ = [2 d − g − a ( N + 6)7 g (5.29) Λ dg d Λ = [3 d − g + 7 a N + 24) g (5.30)Again we have a trivial Gaussian fixed point, i.e. g ∗ = 0 , g ∗ = 0 . (5.31)We perform the linear stability analysis about the Gaussian fixed point just below three di-mensions, since by naive power-counting we know that these couplings are irrelevant for d ≥ .– 25 –efining λ ≡ − d , the linearized expansion of beta functions near the Gaussian fixed point isgiven by: (cid:18) β ( δg ) β ( δg ) (cid:19) = (cid:18) − λ − a ( N + 6)0 1 − λ (cid:19) (cid:18) δg δg (cid:19) (5.32)We see that the coupling g is relevant while the coupling g can either be relevant, irrelevantor marginal depending on λ . For small λ , only g is relevant, which we use to determine thecritical exponents: ν = 12 λ , ∆ = 1 + 3 λ λα = 32 (cid:18) − λ (cid:19) , β = 1 − λ λγ = 1 , δ = 1 + 3 λ − λ (5.33)The beta functions given by (5.29) also admit a non-trivial fixed point which is given by: g ∗ = −
10 (3 d − d + 24) a ( N + 30 N + 144) , g ∗ = − d − (3 d − a ( N + 6) ( N + 24) , (5.34)Notice that this fixed point exists only for d < . For d > , g ∗ becomes negative, therebymaking the Hamiltonian unbounded. Just below three dimensions, the fixed point is given by: g ∗ = 10(1 − λ ) λa ( N + 30 N + 144) g ∗ = 140 λ (3 λ − a ( N + 6) ( N + 24) (5.35)Expanding the beta-function near the fixed point in − λ dimensions, we obtain the followinglinearized form: (cid:18) β ( δg ) β ( δg ) (cid:19) = (cid:32) − λ − a ( N + 6) λa ( N +6) − λ (cid:33) (cid:18) δg δg (cid:19) (5.36)The eigenvalues of above matrix are given by: (cid:16) − λ − (cid:112) λ − λ + 1) (cid:17) and (cid:16) − λ + √ λ − λ + 1 (cid:17) . Since λ is small, we can linearly expand the above eigenvalues while ignoring higher order termsin λ . At the first order in λ , the eigenvalues are given by: λ + O ( λ ) , − λ + O ( λ ) We find that both the eigenvalues are positive i.e. both the couplings g and g are irrelevantnear the non-trivial fixed point. Hence the critical exponents are given by the values obtainednear the Gaussian fixed point, similar to our discussion for r = 3 . Notice that just like the casefor r = 3 , the critical exponents are independent of N here as well.For r > , the one-loop corrections to the beta functions of the corresponding couplingsvanish. Hence we need to look at higher loop corrections to the beta function in order toobtain the values of critical exponents beyond saddle point. The main results of this sectionare summarized below: – 26 – α β γ δ ν ∆ (cid:15) (cid:0) − N +2 N +8 (cid:1) − (cid:15) (cid:0) − N +2 N +8 (cid:1) (cid:15) (cid:0) N +2 N +8 (cid:1) (cid:15) (cid:0) N +2 N +8 (cid:1) + (cid:15) N +2)( N +8) 32 + (cid:15) (cid:16) N +2) N +8 − (cid:17) (cid:0) − (cid:15) (cid:1) − + (cid:15) (cid:15) − (cid:15) (cid:15) + (cid:15) (cid:0) − λ (cid:1) − λ λ λ − λ λ λ λ Table 3 : One-loop corrections to the Critical exponents. Here (cid:15) = 4 − d and λ = 3 − d . We briefly summarize our work and discuss our conclusions here. In our work, we look at higherorder thermodynamic phase transitions. For phase transitions involving a local order parameter,we write a generalized phenomenological Landau Hamiltonian, which describes r th order phasetransitions. Near criticality, these phase transitions are characterized by divergences in thephysical observables. We capture such divergences by introducing critical exponents. As a firststep, we calculate these critical exponents by using the saddle point approximation.Next, we investigate the role of fluctuations. We consider fluctuations giving rise to poly-nomials in the local field whose order is smaller than the order of terms in the Lagrangian.We show that such fluctuations lead to non-analyticity in the critical exponents. We also showthat there is a generalization of the notion of upper and lower critical dimensions, which is astraightforward extension of the r = 2 case. Above the upper critical dimension d = 2 r , thesaddle point calculation is valid since fluctuations do not introduce new singularities. Belowthe upper critical dimension, we generalize the Ginzburg criteria to precisely quantify whenfluctuations dominate over the saddle point calculation.Next, we introduce scaling forms for physical observables derived from the partition func-tion. These scaling forms can be conveniently derived using Wilsonian renormalization. We usethese forms to obtain scaling relations between the critical exponents, which continue to holdbeyond the saddle point approximation. We show that given relevant couplings t and h , onecan use their mass dimensions and these scaling relations to determine all the critical exponentsfor these transitions.We further use the renormalization group to compute corrections to the critical exponentsby calculating the one-loop beta functions. The one-loop beta functions are used to identifyfixed points of the RG flow. We calculate the scaling dimensions of the couplings about thesefixed points and use them to write the corrections to the critical exponents beyond the saddle– 27 –oint. For r = 3 and r = 4 , we determine the corrected critical exponents of the couplings nearthe Gaussian fixed point. We also find a non-trivial fixed point in both cases. However, thereare no relevant couplings in the vicinity of the non-trivial fixed point, and consequently, thesole corrections arise due to flows near the Gaussian fixed point. For r ≥ , the one-loop betafunction vanishes, and hence we need to go beyond one-loop calculations to find corrections tothe critical exponents.We now briefly discuss why we rarely observe higher order local phase transitions in physicalsystems. Higher order phase transitions with a local order parameter require a delicate fine-tuning which sets lower order terms in the Hamiltonian to zero. Such a fine-tuning is uncommonin most systems of physical interest, where such terms can arise due to fluctuations. These lowerorder terms can be avoided due to the existence of a symmetry, however we presently do notknow any symmetry argument which explicitly rules out such terms.We will list out possible avenues that branch out from our work. An immediate direction isto construct a guiding principle that classifies higher order non-local phase transitions. Usingsuch a principle, we hope to write a phenomenological Hamiltonian describing higher-ordernon-local phase transitions. An important class of non-local third order phase transitions arisein Gross-Witten-Wadia type models. In a forthcoming work, we will investigate such phasetransitions and calculate critical exponents for the same. In general, one can investigate suchphase transitions in the context of finite-range Coulomb gas models. We visualize our presentwork to be a starting basis for looking at these problems and hope to uncover some of the abovemysteries in future works. Acknowledgements
We thank Abhishek Dhar and Akhil Sivakumar for their comments on the draft. We are alsograteful to Spenta Wadia for going through the earlier version of the draft meticulously andfor suggesting corrections. The authors acknowledge gratitude to the people of India for theirsteady and generous support to research in basic sciences.
A Why critical exponents are single valued in the ( h, t ) plane? In this appendix, we will see that single valuedness of critical exponents in the ( h, t ) plane followsfrom the analyticity assumption combined with our scaling conjecture. Let us consider a moregeneral scaling form such that C ± V , α ± , g ± , ∆ ± are different for t > and t < respectively. Asan example, we will consider that the singular part of specific heat is given by the followingfunction C ± V ∼ | t | r − α ± − (cid:98) g ± (cid:18) ht ∆ ± (cid:19) . (A.1)– 28 –owever we argue that this is ruled out by free energy being analytic everywhere apart from h = 0 and t < . To understand this, consider a point in the ( h, t ) plane at t = 0 and finite h .Since the function C is analytic in the vicinity of this point, it can be expanded in terms of aTaylor series C V ( t (cid:28) h ∆ ) = A ( h ) + tB ( h ) + O ( t ) + . . . (A.2)Since we already have a power series description, C can be obtained from the two power seriesexpansions given by C ± V = | t | r − α ± − (cid:20) A ± (cid:18) ht ∆ ± (cid:19) p ± + B ± (cid:18) ht ∆ ± (cid:19) q ± + . . . (cid:21) , (A.3)where ( p ± , q ± ) are powers of the largest terms of (cid:98) g ± . We will now match the expressions for C ± V with the power series given in (A.2), which leads to the identification r − α ± − − p ± ∆ ± & r − α ± − − q ± ∆ ± (A.4)Thus the series expansions can be written as C ± V ( t (cid:28) h ∆ ) = A ± h − ( r − α ±− ± + B ± | t | h − ( r − α ±− ± + . . . (A.5)As a consequence of continuity at t = 0 , we can use the equation above to obtain the followingrelations between the critical exponents α + = α − ≡ α, ∆ + = ∆ − ≡ ∆ , (A.6)and so on for other critical exponents as well since these arguments generalize for any scalingfunction X ( t, h ) which possess analyticity everywhere apart from the line h = 0 and t < . B Kinetic fluctuations in d ≤ and loss of order Consider the Hamiltonian (2.2) where the order parameter given by (cid:126)φ = φ ˆ φ is n -dimensional,i.e. the target space is n -dimensional. Taking the saddle point of the partition function fixesthe magnitude of the order parameter, thereby spontaneously breaking the O( n ) symmetry toO ( n − , with the n − angles remain unconstrained. Denoting these angles by Θ α and themetric on the target space as g αβ , we can rewrite the kinetic part of the Hamiltonian, whichdescribes the Goldstone modes parametrized by Θ α as βH = (cid:90) d d x (cid:20) K φ g αβ ∂ Θ α ∂x i ∂ Θ β ∂x i + t r φ r − ( x ) + u r φ r ( x ) − ( h.φ ) φ r − (cid:21) . (B.1)Our analysis can be made easier by considering physical target spaces where g αβ has nooff-diagonal terms. In this case, the two point correlator can be found out by solving the Greenfunction equation and is given by (cid:104) Θ α ( x )Θ β ( x (cid:48) ) (cid:105) = − C ( x − x (cid:48) ) Kφ g αβ , (B.2)– 29 –here C ( x ) in the long distance limit is given by lim x →∞ C ( x ) = d > x − d (2 − d ) S d d < x π d = 2 (B.3)Thus we see that kinetic fluctuations or Goldstone modes destroy long range order in d ≤ , ascorrelators in these dimensions grow with distance. This leads to a destruction of order in d ≤ for systems possessing continuous symmetry, which defines the lower critical dimension. Theanalysis in this subsection is essentially the content of the Mermin Wagner theorem [34, 35],since Goldstone modes in the system would have infrared divergent two-point correlator asgiven by (B.3). For discrete systems, the lower critical dimension is d l = 1 , the correspondingargument remains unchanged as well for the case of higher order phase transitions. References [1] P. Ehrenfest,
Phasenumwandlungen im ueblichen und erweiterten Sinn, classifiziert nach dementsprechenden Singularitaeten des thermodynamischen Potentiales. , Verhandlingen derKoninklijke Akademie van Wetenschappen (Amsterdam) 36: 153–157; Communications from thePhysical Laboratory of the University of Leiden, Supplement No. 75b (1933).[2] G. Jaeger,
The ehrenfest classification of phase transitions: Introduction and evolution , Archivefor History of Exact Sciences (05, 1998) 51–81.[3] L. Landau, On the theory of phase transitions , Zh. Eksp. Teor. Fiz. (1937) 19–32.[4] T. J. Hollowood, , in , 9, 2009. arXiv:0909.0859 .[5] M. E. Peskin and D. V. Schroeder, An Introduction to quantum field theory . Addison-Wesley,Reading, USA, 1995.[6] M. Kardar, .[7] M. Kardar,
Statistical Physics of Fields . Cambridge University Press, 2007.[8] G. S. Rushbrooke,
On the thermodynamics of the critical region for the ising problem , TheJournal of Chemical Physics (1963), no. 3 842–843, [ https://doi.org/10.1063/1.1734338 ].[9] B. Josephson, Relation between the superfluid density and order parameter for superfluid he neartc , Physics Letters (1966), no. 6 608 – 609.[10] B. Widom, Equation of state in the neighborhood of the critical point , The Journal of ChemicalPhysics (1965), no. 11 3898–3905, [ https://doi.org/10.1063/1.1696618 ].[11] D. Gross and E. Witten, Possible Third Order Phase Transition in the Large N Lattice GaugeTheory , Phys. Rev. D (1980) 446–453. – 30 –
12] S. R. Wadia,
N = ∞ phase transition in a class of exactly soluble model lattice gauge theories , Physics Letters B (1980), no. 4 403 – 410.[13] S. R. Wadia, A study of u(n) lattice gauge theory in 2-dimensions , arXiv:1212.2906 .[14] M. R. Douglas and V. A. Kazakov, Large N phase transition in continuum QCD intwo-dimensions , Phys. Lett. B (1993) 219–230, [ hep-th/9305047 ].[15] P. J. Forrester, S. N. Majumdar, and G. Schehr,
Non-intersecting brownian walkers andyang–mills theory on the sphere , Nuclear Physics B (2011), no. 3 500 – 526.[16] F. D. Cunden, P. Facchi, M. Ligabò, and P. Vivo,
Universality of the third-order phase transitionin the constrained coulomb gas , Journal of Statistical Mechanics: Theory and Experiment (may, 2017) 053303.[17] A. De Pasquale, P. Facchi, G. Parisi, S. Pascazio, and A. Scardicchio,
Phase transitions andmetastability in the distribution of the bipartite entanglement of a large quantum system , Phys.Rev. A (May, 2010) 052324.[18] P. Facchi, U. Marzolino, G. Parisi, S. Pascazio, and A. Scardicchio, Phase transitions of bipartiteentanglement , Phys. Rev. Lett. (Jul, 2008) 050502.[19] C. Nadal, S. N. Majumdar, and M. Vergassola,
Phase transitions in the distribution of bipartiteentanglement of a random pure state , Phys. Rev. Lett. (Mar, 2010) 110501.[20] A. Dhar, A. Kundu, S. N. Majumdar, S. Sabhapandit, and G. Schehr,
Exact extremal statisticsin the classical 1d coulomb gas , Phys. Rev. Lett. (Aug, 2017) 060601.[21] K. Damle, S. N. Majumdar, V. Tripathi, and P. Vivo,
Phase transitions in the distribution of theandreev conductance of superconductor-metal junctions with multiple transverse modes , Phys.Rev. Lett. (Oct, 2011) 177206.[22] F. Colomo and A. G. Pronko,
Third-order phase transition in random tilings , Phys. Rev. E (Oct, 2013) 042125.[23] Y. V. Fyodorov and C. Nadal, Critical behavior of the number of minima of a random landscapeat the glass transition point and the tracy-widom distribution , Phys. Rev. Lett. (Oct, 2012)167203.[24] P. Kazakopoulos, P. Mertikopoulos, A. L. Moustakas, and G. Caire,
Living at the edge: A largedeviations approach to the outage mimo capacity , IEEE Transactions on Information Theory (2011), no. 4 1984–2007.[25] P. Vivo, S. N. Majumdar, and O. Bohigas, Distributions of conductance and shot noise andassociated phase transitions , Phys. Rev. Lett. (Nov, 2008) 216809.[26] S. N. Majumdar and G. Schehr,
Top eigenvalue of a random matrix: large deviations and thirdorder phase transition , Journal of Statistical Mechanics: Theory and Experiment (jan,2014) P01012.[27] P. Kumar, D. Hall, and R. G. Goodrich
Phys. Rev. Lett. , 4532 (1999).[28] D. Hall, R. G. Goodrich, C. G. Grenier, P. Kumar, M. Chaparala, and M. Norton Phil. Mag. B
B80 (2000). – 31 –
29] P. Kumar,
Theory of a higher-order phase transition: The superconducting transition in ba . k . bio , Phys. Rev. B (07, 2002).[30] P. Kumar, Complex phase diagrams , Philosophical Magazine (2009), no. 22-24 1771–1777,[ https://doi.org/10.1080/14786430802585158 ].[31] P. Kumar and A. Saxena, Thermodynamics of a higher-order phase transition: Scaling exponentsand scaling laws , Philosophical Magazine B (2002), no. 10 1201–1209,[ https://doi.org/10.1080/13642810208223158 ].[32] B. D. Stošić, T. Stošić, and I. P. Fittipaldi, Third and fourth order phase transitions: Exactsolution for the ising model on the cayley tree , Physica A: Statistical Mechanics and itsApplications (2009), no. 7 1074 – 1078.[33] W. Janke, D. Johnston, and R. Kenna,
Properties of higher-order phase transitions , NuclearPhysics B (2006), no. 3 319 – 328.[34] N. D. Mermin and H. Wagner,
Absence of ferromagnetism or antiferromagnetism in one- ortwo-dimensional isotropic heisenberg models , Phys. Rev. Lett. (Nov, 1966) 1133–1136.[35] S. R. Coleman, There are no Goldstone bosons in two-dimensions , Commun. Math. Phys. (1973) 259–264.[36] “Why does josephson’s identity dν = 2 − α only hold for mean field theory in dimension Critical exponents in 3.99 dimensions , Phys. Rev. Lett. (Jan,1972) 240–243.(Jan,1972) 240–243.