Charting the scaling region of the Ising universality class in two and three dimensions
CCharting the scaling region of the Isinguniversality class in two and three dimensions
Michele Caselle and Marianna Sorba , Department of Physics, University of Turin & INFN, TurinVia Pietro Giuria 1, I-10125 Turin, Italy SISSA & INFN Sezione di TriesteVia Bonomea 265, 34136 Trieste, Italy
Abstract
We study the behaviour of a universal combination of susceptibilityand correlation length in the Ising model in two and three dimensions,in presence of both magnetic and thermal perturbations, in the neigh-bourhood of the critical point. In three dimensions we address theproblem using a parametric representation of the equation of state. Intwo dimensions we make use of the exact integrability of the modelalong the thermal and the magnetic axes. Our results can be used asa sort of “reference frame” to chart the critical region of the model.While our results can be applied in principle to any possible re-alization of the Ising universality class, we address in particular, asspecific examples, three instances of Ising behaviour in finite tempera-ture QCD related in various ways to the deconfinement transition. Inparticular, in the last of these examples, we study the critical endingpoint in the finite density, finite temperature phase diagram of QCD.In this finite density framework, due to well know sign problem, Mon-tecarlo simulations are not possible and thus a direct comparison ofexperimental results with QFT/Statmech predictions like the one wediscuss in this paper may be important. Moreover in this example itis particularly difficult to disentangle “magnetic-like” from “thermal-like” observables and thus an explicit charting of the neighbourhoodof the critical point can be particularly useful. a r X i v : . [ h e p - l a t ] J u l Introduction
Despite its apparent simplicity the Ising model is one of the cornerstones ofmodern Statistical Mechanics. Over the years it has become a theoreticallaboratory to test new ideas, ranging from Symmetry Breaking to ConformalField Theories. Moreover, thanks to its exact solvability in two dimensions[1, 2] and the ease with which it can be simulated in three dimensions, ithas been widely used as a benchmark to test new numerical approaches andinnovative approximations.The corresponding universality class, in the Renormalization Group sense[3], is of central importance in theoretical physics due to its many exper-imental realizations in different physical contexts, ranging from condensedmatter to high energy physics. At the same time it describes the criticalbehaviour of a lot of different spin models and, in the limit of high temper-atures, also of gauge theories.From a Statistical Mechanics point of view it represents the simplestway to describe systems with short range interactions and a scalar orderparameter (density or uniaxial magnetization) which undergo a symmetrybreaking phase transition. From a Quantum Field Theory (QFT) point ofview it is the simplest example of a unitary Conformal Field Theory (CFT)[4] perturbed by only two relevant operators: the “spin” operator (which is Z odd) and the “energy” operator ( Z even) [5].Thanks to integrability, conformal perturbation and bootstrap [6, 7] lotsof results are known, both in two and in three dimensions, on the behaviourof the model at the critical point, or when only one of the two perturbingoperators is present. However, typically, the interesting regime for mostof the experimental realizations of the model is when both the perturbingoperators are present and much less is known in this situation.The aim of this paper is to partially fill this gap by studying a suit-able universal combination of thermodynamic quantities (see below for theprecise definition) in presence of both perturbing operators. In three dimen-sions we shall address the problem using a parametric representation of theequation of state [8], while in two dimensions we shall make use of the exactintegrability of the model in presence of a single perturbation [5]. Usingthese tools we shall be able to predict the value of this quantity in the wholephase space of the model in the neighbourhood of the critical point. Thesevalues can be used as a sort of “reference frame” to chart the critical regionof the model.The universal combination that we shall study involves the magnetic sus-ceptibility and thus our proposal is particularly effective when the model is1haracterized by an explicit Z symmetry. When this is not the case, likefor the liquid-vapour transition or for the finite density QCD example thatwe shall discuss below, the explicit knowledge of our universal combinationmay help to identify the exact directions in the phase space of the modelwith respect to which the magnetic susceptibility must be evaluated.Thanks to universality, our results hold not only for the standard near-est neighbour Ising model, but also for any possible realization of the Isinguniversality class and in fact we shall use the high precision Montecarlo es-timates obtained from an improved version of the Ising model to benchmarkand test our results [9, 10, 11, 12, 13, 14, 15].In particular, we shall concentrate in the second part of the paper onrealizations in the context of high energy physics, suggested by the latticeregularization of QCD. We shall discuss three instances of Ising behaviourin finite temperature QCD related in various ways to the deconfinementtransition. In the last of these examples, we shall address the critical endingpoint of finite density QCD. In this case, due to well know sign problem,Montecarlo simulations are not possible and thus a direct comparison of ex-perimental results with QFT/Statmech predictions like the one we discussin this paper may be important.This paper is organized as follows. Sect. 2 is devoted to a general intro-duction to the model and to the universal combination of thermodynamicquantities which is the main subject of the paper. In sect. 3 we shall addressthe problem in three dimensions using a suitable parametric representationof the equation of state of the model. We shall also show that the same ap-proach cannot be used in two dimensions. In sect. 4 we shall then addressthe two dimensional case using appropriate expansions around the exact so-lutions of the model. Finally sect. 5 will be devoted to the discussion of aset of examples in high temperature QCD. We collected in the appendicessome additional material which may be useful to reproduce our numericalanalysis. The Ising model has a global Z symmetry and is characterized by two rele-vant operators which encode the Z odd ( σ ) and Z even ( (cid:15) ) perturbationsof the critical point.From a QFT point of view, the model in the vicinity of the critical point2an be written as a perturbed Conformal Field Theory S = S CF T + t (cid:90) d d x (cid:15) ( x ) + H (cid:90) d d x σ ( x ) (1)where (cid:15) ( x ) and σ ( x ) are the energy and spin operators and represent thecontinuum limits of the lattice operators (cid:80) (cid:104) ij (cid:105) σ i σ j and (cid:80) i σ i respectively.These operators are conjugated to the reduced temperature t = T c ( T − T c )and magnetic field H , which measure the deviation from the critical point.The action S CF T is the conformal-invariant action of the model at the criticalpoint. In two dimensions this is the action of a free massless Majoranafermion with central charge c = 1 / H = 0 (pure thermalperturbation) and for t = 0 (pure magnetic perturbation) much is known ofthis QFT in two dimensions. In particular all the critical exponents and theuniversal ratios are know exactly and, as we shall see below, reliable expan-sions around the integrable lines can be constructed for several observables.In three dimensions there are not exact results, but from the recent progressof the bootstrap approach and the improvement of Montecarlo methods sev-eral universal quantities can be evaluated with very high precision.The most important realization of this QFT is the spin Ising model ona cubic (in d = 3) or square (in d = 2) lattice, which we shall use in thefollowing to fix notations. As it is well known, the model is defined by thefollowing energy function E ( { σ i } ) = − J (cid:88) (cid:104) ij (cid:105) σ i σ j − ˆ H N (cid:88) i =1 σ i (2)where the spins σ i can take the values σ i = ±
1, the index i labels the sitesof the lattice, the symbol (cid:104) ij (cid:105) means that the sum is performed over pairs ofnearest neighbour sites, J is the coupling strength between spins (we assumea positive isotropic interaction so that for all pairs of nearest neighbour spins J ij = J > H is the external magnetic field.The partition function of the model is Z = (cid:88) { σ i } e − kBT E ( { σ i } ) (3)let us define β = J/k B T and H = ˆ H/k B T . For H = 0 the model is explic-itly Z symmetric and is characterized by two phases, a low temperaturephase in which the Z symmetry is spontaneously broken and a spontaneous3agnetization is present and a high temperature phase in which the Z sym-metry is restored. The two phases are separated by a critical point. If oneswitches on the magnetic field it becomes apparent that the low T phase isactually a line of first order phase transitions which ends with the criticalpoint. In the following we shall be interested in the scaling region in thevicinity of this critical point. Standard Renormalization Group argumentstell us that in this limit the irrelevant operators of the model can be ne-glected and the behaviour is completely described only by the two relevantoperators (cid:15), σ and one can perform a continuum limit of the model whichleads exactly to the QFT described by eq. 1.From the partition function defined above it is easy to obtain all thethermodynamic observables. In particular, following the standard notationwe have for the magnetization and the magnetic susceptibility M = − ∂ log( Z ) ∂H , χ = − ∂ log( Z ) ∂H = ∂M∂H (4)The exponential correlation length can be extracted from the large distancedecay of the spin-spin connected correlator as (cid:104) σ ( x ) σ (0) (cid:105) c ∼ e −| x | /ξ | x | → + ∞ (5)where (cid:104) σ ( x ) σ (0) (cid:105) c ≡ (cid:104) σ ( x ) σ (0) (cid:105) − (cid:104) σ (0) (cid:105) .In several practical applications it is also useful the so called “second-moment” correlation length which is defined through the second momentof the spin-spin correlation function as ξ ≡ (cid:20) d (cid:82) d d x | x | (cid:104) σ ( x ) σ (0) (cid:105) c (cid:82) d d x (cid:104) σ ( x ) σ (0) (cid:105) c (cid:21) / (6)and it is simpler to evaluate than ξ both in numerical simulations and inexperiments.In the scaling limit the critical behaviour of all thermodynamic quantitiesis controlled by the two “scaling exponents” x (cid:15) and x σ which are universaland are shared by all physical realizations of the Ising universality class. Thecorresponding amplitudes are not universal, but one can construct suitablecombinations in which the non-universal features of the model cancel out(see appendix A) and represent testable predictions of the Ising QFT to becompared with any possible realization of the Ising universality class.While this is a well studied subject when only a single perturbation ispresent, its extension to the whole scaling region of the model, where boththe H and t perturbations are present is not straightforward.4he main goal of this paper is to show that such an extension can beeasily obtained making use of a parametric representation of the model andthat the resulting universal quantities can be used as a natural “referenceframe” to chart the scaling region of the Ising universality class.While the parametric approach is completely general and could be ap-plied in principle to any universal combination of thermodynamic quantities,in this paper we shall study in particular the following ratioΩ = (cid:18) χ ( t, H )Γ − (cid:19) (cid:18) ξ − ξ ( t, H ) (cid:19) γ/ν (7)and its natural extension to the second moment correlation lengthΩ = (cid:18) χ ( t, H )Γ − (cid:19) (cid:18) ξ , − ξ ( t, H ) (cid:19) γ/ν (8)where Γ − , ξ − and ξ , − denote the amplitudes of χ , ξ and ξ along the t < , H = 0 axis (see appendix A for detailed definitions and normalizations).This choice is motivated by the fact that the two observables whichappear in the ratio are rather easy to evaluate, both in numerical simulationsand in experiments, since they only involve derivatives or correlations of theorder parameter and are normalized with respect to the values they havealong the critical line of first order phase transitions, which is easy to identify(again, both numerically and experimentally).The main drawback of this choice is that it assumes an explicit realiza-tion of the Z symmetry. While in many interesting applications, like forthe liquid-vapour transition in which the role of the perturbing parameteris played by the density, this is not the case and the Z symmetry is justan “emergent” symmetry. The typical approach in these cases, followingRehr and Mermin [16], is to realize the t, H perturbations as suitable linearcombinations of the actual variables of the model.In the scaling region, when both the relevant perturbations are present,all the thermodynamic observables depend on the scaling combination η ≡ t | H | d − x(cid:15)d − xσ = t | H | βδ (9)The three limits in which only one of the two perturbations is present( H = 0 , t < H (cid:54) = 0 , t = 0) and ( H = 0 , t >
0) correspond respectively Notice that our definition of η differs from that of ref. [5] by a factor 2 π . η = −∞ , η = 0 and η = + ∞ . In these limits Ω can be written interms of the standard universal amplitude ratios Q , Γ + / Γ − and ξ − /ξ + (seeappendix A) as followsΩ( η ) = 1 η = −∞ Ω( η ) = 1 Q (cid:18) Γ + Γ − (cid:19) (cid:18) ξ − ξ + (cid:19) γ/ν η = 0Ω( η ) = (cid:18) Γ + Γ − (cid:19) (cid:18) ξ − ξ + (cid:19) γ/ν η = + ∞ (10)These values can be used as benchmarks to test the reliability of ourestimates and as “anchors” of the reference frame we are constructing. It is useful to introduce a parametric representation of the critical equationof state, that not only satisfies the scaling hypothesis but additionally al-lows a simpler implementation of the analytic properties of the equation ofstate itself. Following [17], we express the thermodynamic variables t, H interms of a couple of parameters
R, θ both positive. Intuitively, the first mea-sures the distance form the critical point in the ( t, H ) plane while the lattercorresponds to the angular displacement along lines of constant R aroundthe critical point. The parametrization of t and H results in a parametricexpression of M as well, explicitly M = m R β θ,t = R (1 − θ ) ,H = h R βδ h ( θ ) (11)Calling θ > h ( θ ), we seefrom the system (eq. 11) that the domain of interest in the ( R, θ ) plane is0 ≤ θ ≤ θ for every R ≥ θ = θ corresponds to the H = 0, t < θ = 1 to the H (cid:54) = 0, t = 0 axis and θ = 0 to the H = 0, t > h ( θ ). There are some general properties which h ( θ ) must satisfy. It must beanalytic in this physical domain in order to satisfy the regularity propertiesof the critical equation of state, i.e. the so-called Griffiths’ analyticity [18].6oreover, it must be an odd function of θ because of the Z symmetry ofthe system. The most general choice is thus a polynomial of the type h ( θ ) = θ + k (cid:88) n =1 h n +1 θ n +1 (12)Using standard QFT methods [19, 8, 20, 21] one can extract the coefficients h n +1 from a high temperature expansion of the free energy of the model.Using a variational method it is possible to obtain reliable and stable es-timates of the coefficients up to h in three dimensions and to h in twodimensions. We can use the known amplitude ratios as benchmarks to evalu-ate the reliability of these parametric representations. As we will see h willbe enough to obtain estimates which in three dimensions agree within theerrors with the amplitude ratios. The situation is worse in two dimensions,and this will prompt us to address the 2d case with a different approach.A similar parametric representation can be introduced also for the cor-relation length. Following [22] we parametrize the square mass of the un-derlying QFT i.e. ξ − as follows ξ − = R ν a (1 + cθ ) (13)and similarly, for the second moment correlation length ξ − = R ν ( a ) nd (1 + c θ ) (14)The constants c and c can be fixed using the universal ratios ξ + /ξ − and ξ , + /ξ , − respectively and we use then the Q and ( Q ) nd ratios to test theparametric representation.Truncating at the quadratic order eq.s 13 and 14 could seem a too drasticapproximation, but we will see below that in d = 3 it gives quite good results.The same is not true in d = 2 where however, as we anticipated above, weshall use a different approach to evaluate Ω( η ).Using the above results we may construct a parametric representationof Ω as a function of θ Ω( θ ) = Ω (1 − θ + 2 βθ )(1 + cθ ) γ ν βδθh ( θ ) + (1 − θ ) h (cid:48) ( θ ) (15)with Ω = (1 − θ ) h (cid:48) ( θ )(1 − θ + 2 βθ )(1 + cθ ) γ ν (16)7nd a similar expression for Ω ( θ ) with c → c .There are a few universal features of Ω that we may deduce from thisexpression and hold for any possible realization of the Ising universality class • Ω is a monotonic decreasing function of θ (see fig. 1 for a plot of Ω( θ )in d = 3), therefore it can be inverted. From any given experimentalestimate of Ω a precise value of θ may be extracted. • Using Ω we may identify the critical isothermal line, which correspondsto Ω iso ≡ Ω( θ = 1) (Ω iso = 1 . ... in d = 3). • There is a maximum value of Ω which corresponds to Ω max = Ω( θ =0) = Ω (Ω max = 1 . ... in d = 3).In practical applications one may be interested in the expression of Ω asa function of η = t/ | H | βδ . It is useful to introduce a “universal” version ofthe scaling variable defined as ˜ η = ( h ) βδ η (17)Using the parametric representation of eq. 11 it is possible to write theexpansion of θ as a function of ˜ η in the neighbourhood of the three singularpoints ˜ η = ( ±∞ ,
0) and then Ω as a function of ˜ η , which we plot in fig. 2below in the d = 3 case (see appendix B for more details on this expansion). d = 3 In three dimensions we have [8, 20] h ( θ ) = θ + h θ + h θ + h θ + O ( θ )= θ − . θ + 0 . θ − . θ + O ( θ ) (18)and the resulting value for the smallest positive root is θ = 1 . + / Γ − ∼ .
76 which is in good agree-ment (with a difference of the order of 1%) with the Montecarlo estimateΓ + / Γ − = 4 . ξ + /ξ − and ξ , + /ξ , − we then obtain c = 0 . c = 0 . Q and ( Q ) nd : we end up with Q ∼ . Q ) nd ∼ .
209 which, again, are in agreement (with a difference of theorder of 3%) with the best numerical estimates reported in eq.s 28, 29.This tells us that we can trust the parametric representation of Ω( θ )discussed above. We plot the result in fig. 1. θ θ Ω ( θ ) Figure 1: Result for Ω( θ ) according to eq. 15 with 0 ≤ θ ≤ θ in d = 3.We find in particular Ω iso = 1 . ... and Ω max = 1 . ... In fig. 2 we plot the expression of Ω as a function of ˜ η . - - η Ω ( η ) Figure 2: Result for Ω(˜ η ) in the three limits of ˜ η given in eq.s 39, 37, 38 in d = 3. 9 .2 Results in d = 2 In two dimensions the situation is not as good as in d = 3. We have [21] h ( θ ) = θ + h θ + h θ + h θ + h θ + h θ + O ( θ )= θ − . θ + 0 . θ + 0 . θ + 0 . θ + 0 . θ + O ( θ ) (19)and the smallest positive zero is θ = 1 . + / Γ − ∼ .
63 which is 5% away from the exact value:Γ + / Γ − = 37 . .. [23]. For the correlation length we get, from the exactvalues of ξ + /ξ − and ξ , + /ξ , − , the following estimates: c = − . .. and c = − . .. . Plugging these values in the expression for Q and ( Q ) nd we find for instance Q = 5 . .. and ( Q ) nd = 3 . Q = 3 . .. and ( Q ) nd = 2 . .. reported in [23]. This prompted us to address the study of the behaviour ofΩ( η ) in d = 2 with a different approach. Ω( η ) in the d = 2 case In d = 2, one can obtain much more precise results performing a perturba-tive expansion around the exact solutions along the two axes H = 0 and t = 0. A powerful tool to study the free energy of a perturbed CFT is thewell known Truncated Conformal Space Approach (TCSA) [24, 25]. In ourcase, thanks to the exact mapping of the H = 0 model to the QFT of a freeMajorana fermion it is possible to construct a more effective version of theTCSA which uses the free fermions as a basis, the “Truncated Free-FermionSpace Approach” [5]. With this approach it is possible to evaluate the freeenergy for almost all values of η [5, 26, 27, 28] and from that of the sus-ceptibility. With similar methods it is also possible to study the perturbedmass spectrum of the theory [26, 27, 28] and hence the correlation length ξ which is the inverse of the lowest mass of the spectrum M . The resultingexpansions for χ and M , in the three singular limits, are reported in ap-pendix C.Combining these quantities we obtain a precise estimate for Ω, which weplot in fig. 3. The resulting polynomial expansions in the three regions ofinterest are 10( η ) = (cid:88) n Ω − n ( − η ) n η → −∞ Ω( η ) = (cid:88) n Ω n η n η ∼ η ) = (cid:88) n Ω + n η n η → + ∞ (20)The coefficients are reported in tab. 1. n Ω − n Ω n Ω + n . . .
078 —2 0 . .
23 —3 − . . − . . − . − . . − . η ) in the three regimes of interest,according to eq.s 20.As expected, the three limiting cases Ω( ±∞ ) , Ω(0) (which correspondto the n = 0 values in the table) agree with the universal values obtainedplugging in eq. 10 the universal ratios quoted in eq. 25. Among the many physical realizations of the Ising universality class in thispaper we decided to focus on three examples taken from high energy physicsand in particular from the lattice regularization of QCD at finite tempera-ture. 11 - - η Ω ( η ) Figure 3: Result for Ω( η ) in the three limits of η given in eq.s 20 in d = 2. SU (2) pure gaugetheory The most direct realization of the Ising universality class in Lattice GaugeTheories (LGTs) is given by the deconfinement transition of pure gaugetheories with a symmetry group G which has Z as center. This result isbased on the Svetitsky Yaffe approach [29] to the study of finite temperature( d +1)-dimensional pure gauge theories. The main idea of [29] is to constructa d -dimensional effective theory from the original one by integrating outthe spacelike links and keeping as only remaining degrees of freedom thePolyakov loops. These loops are then treated as spins of an effective d -dimensional model whose global symmetry must coincide with the center ofthe original gauge group. If both the phase transitions of the original gaugetheory and that of the effective spin model are continuous, they must belongto the same universality class and one can use the effective model (which isusually much simpler than the original gauge theory) to extract informationson the deconfinement transition of the original model. These results arevery general: if in particular we focus on gauge theories with a gauge group G whose center is Z (like for instance, Z itself, SU (2) or Sp (2 N )), thedeconfinement transition will belong to the Ising universality class. A wellstudied example of this correspondence is the SU (2) model [30, 31, 32]. Evenif the gauge group is not SU (3) and the model only contains gluonic degreesof freedom this simplified model shares a lot of properties with QCD: thepresence of a confining flux tube at low temperatures, a deconfined phaseat high temperature, a rich glueball spectrum, asymptotic freedom. For12hese reasons it has been studied a lot in the past both in (2+1) [33, 34, 35]and in (3+1) [36] dimensions. Thanks to the Svetitsky-Yaffe construction,several gauge invariant observables of the SU (2) model can be mapped toequivalent Ising observables • The Polyakov loop is mapped to the spin ( Z odd) operator and thusthe Polyakov loop susceptibility is mapped to the magnetic suscepti-bility χ of the Ising model. • The deconfining transition of the gauge model corresponds to the mag-netization transition of the Ising model. In particular, the confiningphase (low temperature of the gauge model) is mapped to the Z sym-metric phase (high temperature phase) of the Ising model, while thedeconfined phase is mapped to the broken symmetry phase of the Isingmodel. • The Wilson action (i.e. the trace of the ordered product of the gaugefield along the links of a plaquette) is mapped to the energy operator of the Ising model. • The screening mass of the gauge model in the deconfined phase ismapped to the mass of the Ising model in the low temperature phase,while N t σ , where σ is the string tension and N t is the inverse tem-perature of the gauge model (i.e. the size N t of the lattice in thecompactified direction which defines the finite temperature setting inLGTs) is mapped to the inverse of the high temperature correlationlength of the Ising model.This mapping has been widely used in the past years to predict the behaviourof various physical observables of the gauge model near the deconfinementtransition, like for instance the short distance behaviour of the Polyakov loopcorrelator [37], the width of the flux tube [38], the Hagedorn-like behaviour ofthe glueball spectrum [39] or the behaviour of the universal ξ/ξ ratio [40].More generally the mapping allows one to relate all the thermodynamicobservables of the two models and in particular also Ω( η ), which can thusbe evaluated in the SU (2) LGT and then compared with the QFT predictiondiscussed in the previous section. Actually it is mapped to the most general Z even Ising observable, which is a mixtureof the identity and energy operators, but the identity operator plays no role in this context. .2 QCD with dynamical quarks and the Columbia plot The situation is different if we study full QCD, i.e. if we include dynamicalquarks in the model. In this case the center symmetry is explicitly brokenby the Dirac operator and all the above considerations do not hold anymore.For physical values of the quark masses there is no phase transition betweenthe high temperature quark-gluon plasma phase and the low temperatureconfined phase which are only separated by a smooth crossover [41, 42].However in the phase diagram of the model one finds a rich structure ofphase transitions as the values of the masses of the quarks are varied [43].This pattern of phase transitions is summarized in the well known Columbiaplot, which we report here in fig. 4. The plot describes the nature of thephase transitions as a function of the quark masses. On the two axes arerespectively the masses of up-down quarks (x axis) and the mass of thestrange quark (y axis). We see in the central part of the plot a wide region,where the physical point lies, in which there is no phase transition but onlya crossover between the two phases. In the top right and in the bottom leftcorners there are two regions where the transition is of the first order. Theseregions end with two lines of second order transitions which are expected toboth be of the Ising type.The phase diagram reported in the Columbia Plot can be studied withstandard Montecarlo simulations and in the vicinity of the critical lines theresults of these simulations could be mapped using our tools to the Isingphase diagram and then compared with the Ising predictions as we discussedabove for the simpler case of the pure gauge SU (2) model.It is important to notice that the two critical regions are of differentnature. The one in the top-right corner is a deconfinement transition similarto the one discussed in the previous section. In fact in the limit of infinitemass quarks the model becomes a pure gauge theory. In this limit the SU (3)gauge model has a first order deconfinement transition (differently from the SU (2) one discussed above which is of second order). As the mass of thequarks decreases the gap in the order parameter decreases and the first orderregion ends into a critical line of Ising-like phase transitions.The critical region in the bottom left portion of the Columbia Plot hasa completely different origin. It describes the restoration of the chiral sym-metry in QCD at finite temperature and small quark masses. In QCD withthree massless quark flavours the chiral phase transition is expected to befirst order and to remain of first order even for small but non-zero valuesof the quark masses. As the quark masses increase, the gap in the orderparameter decreases and the first order region terminates in a critical line of14he Ising type. Above this line chiral symmetry is restored through a smoothcrossover. In this case, the Z symmetry is an “emergent” symmetry andthe identification of the H and t axes of the equivalent 3d Ising model is nottrivial (see a discussion on this issue in [44, 45] and in sect. 5.3 below) andthus a universal charting of the scaling region could indeed be useful.Even if the precise location of the critical line is still debated, it seemsthat the physical point is not too far from this bottom left Ising line. If thisis the case, then our analysis could be applied, maybe with some degree ofapproximation, also to the physical point.Figure 4: Columbia Plot. The most interesting application of our results is for QCD at finite baryondensity, which is realized by adding a finite chemical potential µ to the QCDlagrangian. This regime is particularly interesting since it can be exploredexperimentally in heavy-ion collisions [46, 47, 48, 49, 50, 51, 52, 53] and atthe same time it cannot be studied using Montecarlo simulations due to thewell known sign problem (see for instance [54, 55] for a discussion of the signproblem in this context).In this regime the QCD phase diagram is expected to reveal interestingnovel phases [56, 57]. In particular it is widely expected that the hadronic15hase (low T , low µ ) should be separated from the quark-gluon plasma phase(high T , high µ ) by a line of first order transitions with a critical endpointat finite critical values of T and µ (see fig. 5) which should again belong tothe Ising universality class [58, 59, 60, 61, 62, 63, 64, 65].Also for this model, as for the liquid-vapour transition (or the chiraltransition discussed above), the Z symmetry is not realized explicitly but itis just an “emergent” symmetry. This class of models is typically addressedwith the “mixing-of-coordinates” scheme proposed in [16]. The approachwas pursued for the finite density QCD case in [61] and [63], leading tovery interesting results. In both cases the mapping between Ising and QCDcoordinates was performed via the parametric representation of the equationof state. In this respect, the explicit expression of Ω in terms of θ that wediscussed in this paper could be used as a shortcut in the process and couldfacilitate the identification of Ising-like behaviours in the experimental data.As more and more experimental results are obtained, it will become pos-sible to directly test them with universal predictions from the Ising modeland it will be important to have a precise charting of the Ising phase di-agram to organize results and drive our understanding of strongly coupledQCD in this regime. Our paper is a first step in this direction. We pro-posed and studied one particular combination, chosen for its simplicity froma theoretical point of view and its accessibility from an experimental andnumerical point of view, but other combinations are possible and could bestudied using, as we suggest here, parametric representations in d = 3 and/orexpansion around the exactly integrable solutions in d = 2. Critical endpoint ?Crossover First - orderPhase TransitionColor SuperconductingPhases ?Quark - Gluon PlasmaHadron Gas Nuclear Liquid - GasTransition μ [ MeV ] T [ M e V ] Figure 5: QCD phase diagram at finite chemical potential.16 cknowledgments
We thank C. Bonati, M. Hasenbusch and M. Panero for useful commentsand suggestions.
A Critical amplitudes and universal amplitude ra-tios
We list below the scaling behaviour of the observables used in the main text ξ ≈ ξ + t − ν ξ ≈ ξ , + t − ν t > , H = 0 ξ ≈ ξ − ( − t ) − ν ξ ≈ ξ , − ( − t ) − ν t < , H = 0 ξ ≈ ξ c | H | − ν c ξ ≈ ξ ,c | H | − ν c t = 0 , H (cid:54) = 0 χ ≈ Γ + t − γ t > , H = 0 χ ≈ Γ − ( − t ) − γ M ≈ B ( − t ) β t < , H = 0 χ ≈ Γ c | H | − γ c M ≈ B c | H | δ t = 0 , H (cid:54) = 0where the critical indices are defined in terms of the scaling exponentsas follows β = x σ ( d − x (cid:15) ) δ = ( d − x σ ) x σ γ = ( d − x σ ) d − x (cid:15) ν = 1 d − x (cid:15) γ c = ( d − x σ )( d − x σ ) ν c = 1( d − x σ ) (21)From these definitions it is possible to construct the following universalamplitude ratios Γ + Γ − , ξ + ξ − , Q = (cid:18) Γ + Γ c (cid:19) (cid:18) ξ c ξ + (cid:19) γ/ν (22) ξ , + ξ , − , ( Q ) nd = (cid:18) Γ + Γ c (cid:19) (cid:18) ξ ,c ξ , + (cid:19) γ/ν (23) A.1 Exact results for the amplitude ratios in d = 2 In d = 2, thanks to the exact integrability of the two relevant perturbationsall the above universal quantities are known exactly [23]17 (cid:15) = 1 x σ = 18 (24)Γ + Γ − = 37 . .., ξ + ξ − = 2 , Q = 3 . ... (25) ξ , + ξ , − = 3 . .., ( Q ) nd = 2 . ... (26) A.2 Numerical estimates of the amplitude ratios in d = 3 In three dimensions there are no exact results but, thanks to the recentimprovement of the bootstrap approach [6, 7, 66] and to the remarkableprecision of recent Montecarlo simulations [11, 13, 15], reliable numericalestimates for all these quantities exist x (cid:15) = 1 . x σ = 0 . + Γ − = 4 . , ξ + ξ − = 1 . , Q = 1 . ξ , + ξ , − = 1 . , ( Q ) nd = 1 . B Useful results in the parametric representation
B.1 Fixing the non-universal constant h The scaling parameter η is defined modulo a non-universal constant h whichsets its scale and depends on the specific model at hand, i.e. on the specificrealization of the Ising universality class in which one is interested. However,given such a realization, it is rather easy to fix h . The simplest option isto measure (numerically or experimentally) the magnetization M and thesusceptibility χ along two directions (or in the same direction) in the ( t, H )plane, for instance along the critical line of first order phase transitions.18hen from the ratio of the two amplitudes one can easily extract h . Wereport here for completeness the corresponding expressions in the case inwhich one measures, besides the amplitude B of the magnetization, thevalue of Γ + or that of Γ − h = B Γ + ( θ − β θ (30)or h = − B Γ − ( θ − γ + β − (1 − θ + 2 βθ ) θ h (cid:48) ( θ ) (31)For instance, in the case of the 3d Ising model defined by eq.s 2, 3 we have[67] B = 1 . − = 0 . h ∼ . B = 1 . + = 0 . h ∼ . B.2 Parametric representation of the magnetic susceptibility
From the parametric representation of the critical equation of state (eq. 11)we may obtain the magnetic susceptibility as follows χ ( R, θ ) = (cid:18) m h (cid:19) R − γ − θ + 2 βθ βδθh ( θ ) + (1 − θ ) h (cid:48) ( θ ) (32)and from this expression it is easy to extract the amplitude ratioΓ + Γ − = − ( θ − − γ h (cid:48) ( θ )(1 − θ + 2 βθ ) (33) B.3 Expansion of θ as a function of ˜ η in the neighbourhoodof the three singular points ˜ η = ( ±∞ , We report here the first few terms: in the θ → θ limit θ (˜ η ) = θ + ( θ − βδ h (cid:48) ( θ ) (cid:18) − ˜ η (cid:19) βδ + (cid:20) βδθ ( θ − βδ ( θ − h (cid:48) ( θ )) + − ( θ − βδ h (cid:48)(cid:48) ( θ )2( h (cid:48) ( θ )) (cid:18) − ˜ η (cid:19) βδ + O (cid:34)(cid:18) − ˜ η (cid:19) βδ (cid:35) , (34)in the θ → θ (˜ η ) = 1 −
12 ( h (1)) βδ ˜ η + 2 h (cid:48) (1) − βδh (1)8 βδh (1) ( h (1)) βδ ˜ η + O (cid:0) ˜ η (cid:1) , (35)19nd finally in the θ → θ (˜ η ) = 0 + (cid:18) η (cid:19) βδ − h (cid:48)(cid:48) (0)2 (cid:18) η (cid:19) βδ + O (cid:34)(cid:18) η (cid:19) βδ (cid:35) (36) B.4 Expansions of Ω as a function of ˜ η Plugging the above expansions into the expression for Ω( θ ) we find for Ω(˜ η ):in the ˜ η → −∞ limitΩ(˜ η ) = 1 + θ ( θ − βδ h (cid:48) ( θ ) (cid:20) cγν (1 + cθ ) − β (cid:2) θ (2 β − (cid:3) ( θ −
1) ++ 2 βδ ( θ − − h (cid:48)(cid:48) ( θ ) θ h (cid:48) ( θ ) (cid:21)(cid:18) − ˜ η (cid:19) βδ + O (cid:34)(cid:18) − ˜ η (cid:19) βδ (cid:35) , (37)in the ˜ η → η ) = Ω (1 + c ) γ ν δh (1) + Ω (1 + c ) γ ν δh (1) (cid:20) h (cid:48) (1) ( βδ − − δh (1) ( β − βδh (1) + − cγν (1 + c ) (cid:21) ( h (1)) βδ ˜ η + O (˜ η ) , (38)and finally in the ˜ η → + ∞ limitΩ(˜ η ) = Ω − Ω h (cid:48)(cid:48) (0) (cid:18) η (cid:19) βδ + O (cid:34)(cid:18) η (cid:19) βδ (cid:35) (39)Upon substitution c → c the same results are valid for Ω (˜ η ). C Exact results in d = 2 We report here the expansions, in the three regimes of interest, for χ andfor M ≡ /ξ , obtained using the results reported in [5, 26, 27, 28]. χ ( η ) = ( − t ) − (cid:88) n χ − n ( − η ) n η → −∞ χ ( η ) = | H | − (cid:88) n χ n η n η ∼ χ ( η ) = t − (cid:88) n χ + n η n η → + ∞ (40) We report here only the first term for the three expansions to avoid too complexformulas, it is straightforward to obtain the next orders. χ − n χ n χ + n . . . − . × − . . × − . − . − . × − . . × − . . × − − . × − − .
939 —6 3 . × − − . − . × − − . × − − . . × − − .
32 9 . × − − . × − . × − − . × − − . × − — —12 3 . × − — 8 . × − Table 2: Expansion coefficients of the magnetic susceptibility in the threeregimes of interest, according to eq.s 40.1 ξ ( η ) = ( − t ) (cid:88) n m − n ( − η ) n η → −∞ ξ ( η ) = | H | (cid:88) n m n η n η ∼ ξ ( η ) = t (cid:88) n m + n η n η → + ∞ (41)where we have already substituted the known values of the critical indicesof the model. The coefficients are reported in tab.s 2, 3. n m − n m n m + n π . π . − . − . .
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