JHEP07(2016)041 Springer July 8, 2016 May 20, 2016 : June 22, 2016 : : = 3 Dimensions and Published D Received Accepted 10.1007/JHEP07(2016)041 6 Euclidean Dimensions

JHEP07(2016)041 Springer July 8, 2016 May 20, 2016 : June 22, 2016 : : = 3 dimensions and Published d Received Accepted 10.1007/JHEP07(2016)041 6 Euclidean dimensions. ≤ doi: d ≤ Published for SISSA by [email protected] , a functions is discussed and we estimate the error β [email protected] and M.A. Stephanov , b . 3 ) truncations of the scale-dependent eﬀective action. 4 ∂ ( 1605.06039 O The Authors. c Field Theories in Higher Dimensions, Renormalization Group, Nonperturba-

We determine the scaling properties of the Yang-Lee edge singularity as de- , [email protected] expansion and the conformal bootstrap. The relevance of operator insertions at D. Mesterházy d a − Department of Physics, University845 of W. Illinois Taylor at St., Chicago, Chicago,Albert IL Einstein 60607, Center U.S.A. forSidlerstrasse Fundamental Physics, 5, University 3012 of Bern, Bern, Switzerland E-mail: = 6 b a Open Access Article funded by SCOAP Keywords: tive Effects ArXiv ePrint: compare our results to recent estimates of criticalthe exponents corresponding obtained fixed point with of theassociated the with four-loop RG Abstract: scribed by a one-componentnonperturbative scalar functional field renormalization theory withWe group find imaginary in very cubic good 3 coupling, agreement using with the high-temperature series data in X. An, Functional renormalization group approach toYang-Lee edge the singularity JHEP07(2016)041 , ). α 28 z , 1 − z ( )], where 13 α θ 12 ( Q z − = z Z 6 ) ln [ θ ( ) is known, in principle, θg θ ( 9 d , when they coalesce along g R ) = 1) on a curve parametrized βV θ → ∞ ( − V θg = = 1). Clearly, once the location of d Z B R ) of a finite system can in general be k z ln ( β Z − = ]. Besides providing a rigorous basis to study – 1 – Z ]. 49 ), and the distribution 3 , θ 7 ( 55 , z 11 , 26 , 10 in the complex plane, i.e., we may write α 19 z ]. Expressed in terms of an external parameter, which we 63 , ) for a wide range of lattice models via numerical methods [ 37 θ is the inverse temperature ( ( g /T 1 , the partition function 14 z = 1 β These curves can be viewed as cuts that distinguish different branches of the 1 ) and ] and also experimentally [ θ ( In principle, the zeros may accumulate on a dense set in parameter space, which must not necessarily ) corresponds to the normalized density of zeros ( z 56 1 , θ ( elucidate features of fundamentalhave theories. led Drawing to on importantnonvanishing the baryon insights principle densities into of [ the universality they phase diagram of strongly-interactingbe matter one dimensional. at However, such a scenario is not relevant to this work. the zeros, or cuts theyall coalesce thermodynamic into, properties ofefforts the to system determine can be33 calculated. This hasthe led thermodynamic to properties numerous of finite lattice systems, such attempts have also helped to Their significance appears in theone-dimensional thermodynamic curves limit, that separatefunction. different infinite volumefree behaviors energy of (or grand the canonical potential) partition g Ω = as With the pioneering work of Yangsystems and Lee was a new established perspective bythe on the pointing partition properties of function out statistical [ theshall denote importance by ofexpressed the in distribution terms of of its zeros roots of 1 Introduction 6 Relevance of composite operators7 and quality of Residual finite regulator truncations dependence and8 principle of minimal Conclusions sensitivity 3 Critical equation of state4 and mean-field scaling Solving prediction the RG flow5 equations Critical scaling exponents and hyperscaling relations Contents 1 Introduction 2 Nonperturbative functional RG JHEP07(2016)041 3 6, φ / = 1 1 − z ). The ]. , where = 1) = values of 1 ) = 0 for ], as well θ − βH 34 6 relies on d ( , σ ( 2 the natural | g 30 σ ) , − 20 = 2) = 7 T H ]. Note that in -axis at ( d ]. Furthermore, c ( z ]. Depending on < d < 22 ], strong-coupling σ ]. As pointed out H 11 imaginary expansion were de- , 33 58 = exp ( − [ 23 , , 10 z σ H 6. In contrast to the 57 17 , , ∼ | ≤ ]. This allowed to exploit 16 52 , d T 15 ) = 6 and therefore, fluctuations 37 ≤ , 5 [ c and external field / d the free energy is analytic along the 36 T 22 , c theory), the upper critical dimension − , corresponding to ∂M/∂H expansion [ g 29 4 , θ = d φ = ( (or vice versa). On the other hand, in the T > T c − 24 , χ = theory) is ], and conformal bootstrap [ H 5 θ , 3 = 6 4 – 2 – φ 35 , 27 ] for dimensions 3 = 4. is imaginary. This has been proven rigorously and field theory that describes the critical point of the d 4 61 ]. Thus, for H , φ , characterized by the exponent , the distribution of zeros exhibits nonanalytic behavior, g | 60 56 ) one observes a finite gap in the distribution θ , ) functions to four-loop order in the c = 0, the field theory at the Yang-Lee edge point, the 1 and [ T & ( β c H | ), the set of zeros crosses the positive real + c c θ | H T | iβH T > T i , with central charge and 5 → , 2 c = 2 , for T = T T < T σ M θ ) ] and the corresponding critical exponents (obtained from constrained g H = θ 23 T ], Monte Carlo methods [ | − θ ] this behavior can be identified with a thermodynamic singularity that yields ) are commonly referred to as Yang-Lee or Lee-Yang zeros. In particular, | ]. On the other hand, most of our knowledge in the region 2 13 ( z ), where ( 20 34 ' iθ Z , that closes as ) axis. However, at the edge of the gap 20 g θ = 0 axis from positive to negative real ( g H Recently, there has been renewed interest in the Yang-Lee edge point for which the In contrast to the well-known Typically, for lattice spin models at temperature 2 [ is the magnetization. Thus, the Yang-Lee edge singularity at nonvanishing imaginary H < θ / = 0), which indicates the presence of a first-order phase transition as one traverses the = exp( 1 | θ θ Padéapproximants) were compared todevelopments, estimates we from examine other thenonperturbative methods. critical functional scaling RG In properties [ lightexpansion, of of the the these functional Yang-Lee RGis edge does therefore with not ideally the suited rely to on investigate the the expansion critical in behavior a of small the parameter Yang-Lee and edge away contrast to the Isingof critical the point Yang-Lee (described edge by pointare important (described even by above dimension renormalization group (RG) termined in ref. [ as by comparing withusing experimental integral high-field kernel magnetization techniques data− it [ is possibleappropriately resummed to results establish from the theexpansions [ exact result independent (relevant) exponent. Inhas two been dimensions identified the withthe corresponding that universality minimal class of model theconformal symmetry simplest in two nonunitary dimensions conformal towhich calculate field has the been theory scaling exponent confirmed (CFT), with remarkable accuracy by series expansions [ M values of the field is similar to a conventionalIsing second model order phase at transitiontheory, [ admits no discrete reflection symmetry and is therefore characterized by only one real the magnetic field i.e., by Fisher [ a divergence in the isothermal susceptibility the temperature one mayof distinguish the different Ising scenarios: model( ( in theRe low-temperature region high-temperature region ( | variable in terms of whichzeros the of partition function isfor a the polynomial ferromagnetic is Isingz model one findsis these known zeros as distributed the along Yang-Lee circle the theorem unit [ circle JHEP07(2016)041 ) we 2.2 (2.4) (2.1) (2.2) (2.3) 3 in eq. ( S k ] and conclude 22 ] as well as other we summarize our , 7 – 17 ) , 6 x ( ] and [ 16 ϕ S, ) 23 , k x ) ( , we give an overview of the y ∆ , is the scalar field expectation ( = 6 [ 2 ϕ − x J ) d d ) d ϕ /δJ k ( k x, y Λ Z ( W k U . The additional term ∆ δW + denotes the RG scale parameter. The − R 3 + ) ) S = k x 2 x k ( ) ( φ ∆ φ ) of a single-component scalar field ∂ϕ y ϕ ); − x ( d we consider the general properties of RG flow x ( S – 3 – 1 2 ( d S 4 J x xJ − d ]), where expansion around d x = d d d = 3 dimensions using high-temperature series expan- d J 50 Z , Z d 1 2 Z ] exp 46 , = = J is specified in section dϕ [ S S 18 k Λ , = sup Z U ∆ 8 k , Γ 6 ]. We observe that derivative interactions have an important effect = ln k 35 , W 27 , 22 , ], i.e., the generating functional of one-particle irreducible (1PI) diagrams (for 13 ], the three- and four-loop = 6. However, care must be taken to address possible systematic errors that arise 61 c 34 , d The outline of this article is as follows: first, in section 60 [ k and the classical potential is a quadratic functional with respect to thevalue. external Here, source we consider a classical action of the scale-dependent generating functional of connected correlation functions equations derived fromΓ an exact flowreviews equation see, for e.g., the refs.scale-dependent scale-dependent [ effective action effective is action obtained from the functional Legendre transform with an outlook on future work. 2 Nonperturbative functional RG In this work, we employ a RG scheme that relies on a truncation of a hierarchy of flow the Yang-Lee edge singularity. Intrajectories and section in particular their infraredresults (IR) for behavior. the In critical sections exponents atsystematic the errors Yang-Lee edge for singularity the andestimates analyze truncations the for employed expected the in critical this work. exponents to We recent close data by comparing from our refs. [ methods [ on the stability offramework the of scaling the solution nonperturbative and functional need RG. to be takennonperturbative into functional account RG and properly the indiscuss truncations the the employed in scaling this properties work. of In the section critical equation of state and the mean-field theory at from the truncation oferrors the are infinite under hierarchy of controlmary, flow we and equations. find comment that We onwith the show the previous that obtained results quality that values obtained of for these sions in different the [ truncations. critical exponents In are sum- in good agreement from JHEP07(2016)041 , k 2 ) Z ) = (2.5) (2.6) ∂φ ( x, y ). In ( n k p ( ... R + δφ that includes k 2 k / ··· Λ), which inter- δφ ) 2)] 2 ≤ , since the canonical 2 (1) k p 2 2 k ) for Γ ( k ¯ − / / Z , φ , where Λ is a charac- 2 d 1 ≤ δφ 2) 2) ( + , (up to total derivative / ) − ∞ 2 n 1 i − − 4 )( 2) / ) p 2 (0) k = ∂ φ ( d d q ¯ ansatz ( − 2) ( Z ( ( 2 ) k ,k n n d induces a functional RG flow =Λ − 1 ( /δφ R k + 2 + ] = k − functions. ∂φ − n d W φ ( R d ( 2 [ R 2 [ d β 1 2 ) − n , which is assumed to be homoge- k ) ) + − − − k q Γ φ d − + ¯ φ ; ( ∂φ →∞ n = = = 2 Λ δ φ = ,k = )( ) , and the full 1PI effective action, Γ = ( ) ) ) d 3 ) ) φ k n n n 1) n ( ( ( ( n 1 2 3 (2) k W Γ ∂φ ( ( k ). Here, we comment on the nature of our − ¯ ¯ ¯ 1 2 Γ ¯ ¯ Λ n )( U Z W W W Z ( h φ ) → 2.5 + ) Λ) is a dimensionless scale parameter, and ( k π q k φ ( dim dim dim dim dim k/ – 4 – Z (2 k . ∂s 1 2 = 0 and lim 7 2 – ) = lim k ∂R ] ≡ d 4 R ) + , is chosen in such a way that it leads to a decoupling S = ln( ) q ∂φ 0 ) µ 45 φ d π , 1 ( ∂ )( s → d k − µ φ ) ) ) k (2 44 ) ) n ( ∂ U n n n n )]. Note that we drop the RG scale index ( ( ( n 1 2 3 ( ,k ( Z 2 ≡ ¯ ¯ ¯ ¯ ¯ W W W U Z 1 2 2.6 x ]. In effect, this corresponds to an d W d 1 2 , . . . , p = 4 around a field configuration 2 k φ 2 + Z k Γ , p ¯ φ 2 1 2 [cf. eq. ( = ) 2 2 ) ∂ p 3, parametrize the contributions to order ∂s ) ) − ; φ k , k ), where ∂φ φ 2 φ Γ ( ( ∂φ ∂φ y , ( ) ( ( n = n n n n n − k ( k x Γ = 1 Operator Coupling Canonical dimension δφ δφ δφ δφ δφ ( ) i δφ d a p ( is the scale-dependent effective potential, and the scale-dependent functions . Thus, the scale-dependent regulator function , δ k ) k =1 . Operators and canonical dimension of associated parameters and couplings that appear n i Γ x a,k U 0 P W Clearly, an exact solution for the full functional flow is not feasible in practice, so one → − k ( ) d k ( eter or coupling that is field independent, e.g., and terms). Furthermore, for each offluctuation these functions, we employ aneous finite in series space expansion in [cf. the section only a finite set of independent operators, each of which is parametrized by a single param- where on the regulator function see sections has to rely on suitabletruncation approximations and of discuss eq. ( its limitations.scale-dependent We effective use action a [ truncated expansion in derivatives for the between these twoδ limits, where principle, we maylimiting properties. choose For any details of (sufficiently our implementation smooth) and necessary regulator requirements imposed that satisfies the above teristic scale that regularizes theinfinity. theory in In the effect, ultraviolet this (UV) definespolates and a can between one-parameter formally the family be classical of sent action, lim to theories (0 dimensions are defined at the Gaussian fixed point of theand RG serves to regularize the theoryR in the IR; in particular, theof regulator function IR modes. We require that lim Table 1 in the expansion of Γ JHEP07(2016)041 = 2, (2.8) (2.7e) (2.7c) (2.7a) (2.7d) (2.7b) d . etc. We ,k of the Ising 3 φ W . The canonical are expanded to ) in the presence ↔ − and a,k ,k 2.6 1 φ ,k W , 2 ) at the second level, that enters the series ) W 3 W k 2.8 arXiv:1605.06039v2 ≡ ¯ , p and φ 2 , k k ) ) by suitable differentiation 2 , p ) U W ..., 1 2 , p p p · 1 ; 2.7e ∂ 1 p φ ; } ← p ( . Clearly, above dimension ( )–( ] to establish the critical exponents , and φ ,k 3, are derived from the exact func- ( ∂ k 3 (4) k 1 , , ) 2 Z Γ 1 2 62 (3) k , p , 2.7a , , ,W ; ∂ Γ ) − k ∂s φ ,k p 54 2 1 U ( 2 ) ∂ , = 1 ; ∂s , 3 ∂ φ 2 p W a k (2) ∂p ( ) · 42 , Γ , 2 ∂ 2 p (2) k p ) , but is independent of ∂ – 5 – · 14 a,k ( 3 ∂s ∂ Γ =const. 1 ,k ∂ p 2 } ← { φ 1 p W ), while · ∂ | ) ( ∂s ] ∂ ,k 1 2 )] by applying functional derivatives and projecting 0 W 1 ∂ φ 2 p ∂ p [ 2.6 ( ( → 0 ∂ k i 2.5 ∂ lim ∂p ∂ p ,W → Γ , and i 1 2 k 0 0 and lim k 1 4 p Z → → ∂ k Z − ∂s 1 2 p p − Z , k = = lim = lim = = ← { , and similar expansions apply to U [cf. eq. ( k k k ,k ,k ,k N k 1 2 3 = 0 in eq. ( 2 Z U can be derived from eqs. ( U ∈ . Note that the order of the employed expansion might be different ) W W W ∂ ∂ ,k k ∂s ∂s n 3 n ( a,k ∂ ∂ ∂ ∂s ∂s ∂s δφ ¯ W ), W . The corresponding RG flow equations for the field-independent pa- k µ functions. Indeed, one of the objectives of this paper is to investigate their ¯ = φ q ( µ β ) are irrelevant as far as a counting of canonical dimensions goes, but this is ). That is, the RG flow equation for the scale-dependent effective potential ,k and p n 2 ) ( k ) n 2.6 ( n ≡ Z a,k W ( k ¯ ¯ q W Z ≡ · ) n p , eq. ( ( k and ¯ Z The RG flow equations display the following chain of dependencies The flow equations for ) Mathematica file available online on the website of the arXiv submission depends on the quantities 2 n ( k k ¯ U exploit this structure explicitlyi.e., by we truncating set the hierarchysome eq. finite order ( in where the ellipsis denotes higheransatz order contributions that we have chosen to neglect in our where rameters and successive projection ontoexpansion. the We reference do fieldmaterial not configuration available display online. them at this point but refer the reader to supplementary tional flow equation for Γ them onto the appropriate momentum contributions, i.e., solution in the IR. Wefective should action point were out considered also that in similarat refs. the truncations [ Ising of critical the point. scale-dependentof ef- Here, a we nonvanishing external study field, the when scalingmodel the properties is discrete of explicitly reflection eq. symmetry broken ( and the system is tuned to the Yang-Lee edge critical point. dimensions of these parametersZ are displayed in table not sufficient to conclude thatpoint this of is the also RG the caseeffect at at a the nontrivial Yang-Lee (i.e., edge non-Gaussian) point fixed as well as on RG trajectories that approach this scaling and JHEP07(2016)041 c = ≡ . T δh k field (3.2) (3.3) (3.1) c U (1) ), we h 0 ¯ ) = U ]. Near → φ k , with 2.7c ( 0 c T 63 U , )–( − 37 = lim , measures the imaginary T 2.7a characterize the c for the range of U c H ∼ δ 0, for which Λ − t > and H ) T > T for vanishing β Λ ∼ , and h ] and we expect that this is ( , Λ δφ ,c h δh 42 Λ , and , t | /β ϕ, ) 1 Λ 14 = − T . h | ( c 3 ,c / + around a reference field configuration δφ Λ 2 H | 4 | t ) ¯ ϕ Λ i ϕ δt Λ − h/λ λ Λ will typically include a large number of ϕ 3 = f 1 i 4! 1 = c − + – 6 – δ H k < | 2 ( 2 δϕ = 0 in the IR limit. At the Yang-Lee edge critical ϕ ≤ δφ λ/ | Λ t (2) k = 1 2 δφ ¯ U c ) = 0. According to the strategy outlined above, we fix t 0 for 0 = ϕ ) is a universal, dimensionless scaling function, which is ( ¯ → ) = k 0 Λ x k to its critical value φ ( U U ( U 0 f Λ lim t U = ≡ 0 we see that the corresponding critical field value f (2) (or Λ) index on all parameters. It is useful to express the potential in , respectively. Here, the parameter ¯ > U ,c and k . In the critical domain, the equation of state satisfies the scaling form Λ ¯ φ c t ¯ ) down from the cutoff scale Λ to the IR, the parameters and couplings is imaginary in accordance with the Yang-Lee theorem [ φ t 2 − 0, and tune − / 3 2.7c studied in this work. Λ is sent to zero and all modes have been integrated out, i.e., φ ) > t = 0 and Λ | k )–( /λ ) = 0. We derive two independent scaling solutions, which we identify as = Λ c δt = (1) k ϕ t h ( ¯ | ¯ U δφ 00 Λ 2.7a , 0 and inquire about possible critical points, by imposing in addition the condition 0 3 (2 U δt Λ Before we go on to consider the solution of the RG flow equations ( In order to arrive at a critical point in the IR the relevant parameters of the classical → > λ k | iλ/ , which is defined such that h ¯ that Assuming that discuss the mean-field scalingwe prediction. simply drop Since the thereterms is of no an scale expansion dependence inϕ in field differences this case, | uniquely defined up toasymptotic scaling normalization. behavior of Theand the critical (residual) exponents magnetization deviation from the criticalvalues of field strength expectation value where action need to becouplings tuned are to kept their constant.fix respective That critical is, values, in whilelim the all case other of parameters thepoint, or Yang-Lee edge the critical first point, and we second derivative are evaluated at a nonvanishing, tions ( of the classicaldependent potential effective acquire potential afluctuation-induced scale interactions. dependence. The full effective potentialparameter In is obtained fact, only when the the corresponding scale scale- with a nonvanishing coupling tothe critical a temperature symmetry-breaking at field the Ising critical point. Upon integration of the RG flow equa- the critical scaling exponents atalso the the Ising case critical for point the Yang-Lee [ edge critical point. 3 Critical equation ofHere, state we and consider a mean-field classical scaling potential prediction of the following form for each of these coefficients. Similar approximations have led to reasonable estimates of JHEP07(2016)041 , , 9

as for ) ] as k ϕ (4.1) d , and ( U 43 6 τ Λ , U 20 ) = + 5.5 Λ). Such a Λ 2 ) ϕ exp. exp. ( ¯ ≤ ε ε (3,1,0) (4,2,0) (4,2,1) (5,3,0) (6,4,0) , and is defined 00 5 Λ ∂ϕ k k ( U 1 2 high−T exp. = 1. Other critical ≤ = 0. 2−loop 1−loop expansion [ d 4.5 β Λ x (0 d W d , satisfies k 4 k is negative and therefore, R W functional RG χ ) and the coefficients that α = + ¯ . 3.5 = 2 and 3.1 ¯ φ and S δ ]. We observe that the numerical = 1 and , as specified by the set of integers is a solution to = 0 , and integrate the flow equations k = 1) lies almost exactly on top of that 3 k Λ , Λ Z Λ 34 k k , 2 s U Z ¯ χ ϕ φ , d 20 d¯ 0.4 0.3 ) with = ¯ 0.28 0.26 0.24 0.38 0.36 0.34 0.32

(3)

c 3.2 ν ¯ U =Λ ]. That is, in the absence of ﬂuctuations as a function of Euclidean dimension k = 3) [ ¯ + φ c 48 d ν (2) k – 7 – ¯ U and ∂ ∂s ), evaluated at 6 = 0, and we obtain the following scaling exponents: k /δ ¯ φ 4. Note that the exponent = η ( / , which depends on the scale parameter 00 k k = 1 5.5 (2) ¯ U k φ = 1 ¯ U exp. exp. σ ε ε ≡ c (5,3,0) (6,4,0) (3,1,0) (4,2,0) (4,2,1) s 5 ν d d (2) k high−T exp. ¯ 2−loop 1−loop U d 4.5 ]. The data for the truncation (4 4 2, and ) satisﬁes the scaling form ( / ϕ 4 ( 0 . The classical potential is given in eq. ( functional RG = 1 U ν 3.5 → −∞ 3 ) [cf. section = 1, ]. The employed expansion is organized around a nonvanishing, imaginary, and . Critical exponents s W γ 0). Shown in comparison are results from the one- and two-loop 0 42 , , n 0.1 0.3 0.2 0.5 0.4 1,

2 –

Z , σ − We use a truncated series expansion for the scale-dependent effective potential , n 40 , = U n homogeneous field configuration in the following way: 1)2) At the the cutoff scale scale derivative of Λ, down to parametrize the kinetic contribution to the action are well as for thestrategy field-dependent is renormalization often sufficient factors to14 extract the leading or subleading critical scaling behavior [ 4 Solving the RGTo flow solve equations the RGwhich equations is defined we in specify terms of the the short-distance classical potential action exponents that characterize the power-lawties singularities can of be various thermodynamic determinedthe quanti- via anomalous scaling dimension vanishes, relationsα [ at the mean-field level, the specific heat does not diverge at the Yang-Lee edge point. for (4 well as high-temperature seriesaccuracy expansion of data the ( functionalderivative and RG field results expansion, improves respectively. significantly as one goesthe to critical point higher orders in the Figure 1 different truncations of the( scale-dependent effective action Γ JHEP07(2016)041 , ¯ φ in Λ W Λ. n ) ϕ = 0 = 0, ≤ U (4.2) (4.3) 3 ¯ n Z (4.4a) (4.4b) / ( k k ) n and ¯ +1) U τ ≤ n Z + (2 n 0 if Λ ¯ = 0 and the U dim t τ ≡ ≤ k ) +(2 Z = Re Λ W 0 for all 0 n ) h ( k n > ¯ = . (2 W | n k ¯ U (1) (3) Λ k δφ ¯ ) , only when ¯ dim U U n ¯ | φ ( k ], W ¯ . We define n U Λ = ! ϕ N 1 4. Similarly, the expansions for k n ) we obtain a finite set of flow ) ¯ . ¯ φ τ , ∈ U 0 n k , =3 (2) k n is scale independent and therefore + n n k X n 2.7c → n > ¯ , which defines our truncation for the U δφ k Λ = 0, while Im ¯ ) φ U + t and max δφ )–( n ∂ ) ∂s n ( 2 k ) k n 2), which yields a coupled set of partial (2) 1 ( k U , ¯ ¯ W ¯ − n 3 U Z ( 2.7a ! 12 (5 k , ! τ δφ = 0 for 1 / ¯ n 1 U = n = 0, for 1 2 ) (3) k 1 n 1 ) ¯ ( – 8 – − U Λ + 1 + n − (1) =0 ( =0 Λ dim ¯ Z W X n k ¯ Λ X n W = 0, i.e., lim U ¯ n n U − is considered as a free parameter, while h ≤ k [ δφ ¯ = U = = ) χ 0) and (5 Λ Z , n (1) 0 k k ϕ k k n 5 s ¯ ( U → k χ , Z ) back into ( ). d 0, and W k ¯ = ¯ d¯ Z , while = 0). + W Λ (0) Λ λ 4.4b H , n (0) k n > ¯ dim U Z ¯ ) = (7 U = Z )–( n W , n = (4) U 4.3 Λ , n k ¯ n U Z U = 0 for , , n Λ ) = 0. In the following, we denote these type of series truncations in short n . Note that lim U ϕ ( Λ s , for which we obtain n 6 ¯ W ¯ , at the expense of introducing another scale-dependent quantity, the field k Z d / n τ / χ Λ k symmetry axis ( ) fixes the second derivative of the scale-dependent effective potential at all λ , are related to the couplings and parameters of the classical potential at the read = χ φ 0 if N = 1, k 4.1 = . Note that all of these coefficients are defined at a reference field configuration ). (2) k ≡ ∈ W = d¯ N ¯ (0) 6 Λ U k (3) s Λ ↔ − n ¯ ∈ Z ¯ d U , W Substituting eqs. ( Eq. ( φ / ) n and k n = 6 dimensions. This choice defines what we consider to be consistent truncations (see ( ¯ k k φ ¯ of order up todifferential ( equations of upsolution to is 12 identified and by 10effective inspecting potential, parameters, the which respectively. behavior should of The satisfy for the Yang-Lee scaling first and second derivatives of the by the set of integersare ( chosen such that max d section equations for the coefficients of the series expansion. In this work, we consider expansions with and and Z Here, the sum runs upscale-dependent to some effective finite potential integer with value U the prescribed expansion point.short-distance cutoff Λ, The i.e., coefficients scales and therefore the expansion of the scale-dependent effective potential reads Note that the corresponding setThis of flow does equations not requires hold that trueexpansion in point the is vicinity not ofthe adequate the to Ising investigate critical the point scaling and properties therefore, for the critical chosen points on d system has been tuned tomodel criticality. Clearly, conditions 1) andconfiguration 2) ¯ fix one parameter of the Of course, the imaginary field expectation value JHEP07(2016)041 , ≡ n λ k (5.5) (5.2) (5.3) (5.4) (4.5) (5.1) ¯ Z of the n λ , i.e., /∂s m,k ¯ g ] and can be written as , which are given by 2 ∂ } . Of course, as the eigen- ≡ W n m ... β + ≥ theory [ Z 3 χ n φ = 6, this ordering might change. , , . d . + . 2 2 I satisfies the following scaling relation k / / , etc. We order the eigenvalues ∈ ¯ ) c l U ) Z η } = 0 functions, η k /ν ,l d, ln ¯ χ ∗ 1 − β n,k 2 I ¯ g ¯ g = / ∂ ∈ 2 + { ∂s 1 ≡ = 6) = dim ¯ k ∂ n 2 , . . . , n ¯ 0 + 2 } 1 Z 2 − λ d m 2 ( , λ → d = 6 dimensions, where they are identical to the ,n – 9 – / d 2 lim 1 k ∗ + ) ∂β d { λ d ¯ g 1 at the critical point. Starting from a set of initial − { − = ( = ( λ is determined completely in terms of the anomalous ≥ = η c (2 = = c m − k /ν ν /ν β η (1) mn 1 I 1 Λ) ¯ γ = U k/ ∈ ( ,k 2 . g n c ∼ , ¯ , k ¯ Z (1) k n,k ¯ g U T > T ). That is, for any finite truncation of the scale-dependent effective 2 = 6) = dim / 1 d ( 2.6 − k 1 = 0), which is imaginary, corresponding to the imaginary magnetic field 6. ¯ Z λ k , with 2 | ¯ / χ ) eq. ( , from which we obtain d < 0 T . The Yang-Lee edge critical point is known to exhibit another hyperscaling +2) ( /ν → , according to their values in ≤ d η c k 1 ( , e.g., k H − | ≡ functions are derived for the dimensionless, renormalized parameters and cou- k ,... i ), which satisfies ansatz 2 2 k , ¯ λ β φ = We observe that the largest eigenvalue We introduce the following short-hand notation for the renormalization factor ( = = 1 ,k c (0) k 1 ¯ with and therefore, the critical exponent dimension relation, which follows from the equation of motion of the n canonical dimension of the parameters andpear couplings in associated with Γ the operatorsvalues that are ap- analytically continued to dimensions below The plings of theg model, ¯ action, we obtain a finitestability set matrix, critical exponents corresponding to the eigenvalues which is evaluated at the fixed point of the RG 5 Critical scaling exponentsThe and critical hyperscaling relations exponentsanalysis at of the the scaling Yang-Lee solution edgein with respect our critical to point perturbations with are those operators extracted included by a stability value in the IR: Note that the anomalousvalues dimension of at 1 the Yang-Lee edge critical point is negative for all H Z values for the parameterscritical and values, couplings we in may the therefore classical action, define which the are anomalous tuned dimension to by their the corresponding (where lim JHEP07(2016)041 , η c ). ν and 2] = 5.6 (5.6) / U ) n and η σ ) increases ) in the fol- 2 + 5.5 − 5.5 d 2) truncation, [( , / 3 2 ) and to eq. ( , . These values were λ in lower dimensions. 2 5.5 k

(5,3,2) we show the values for 2 ) to calculate the critical

(4,2,1) where we show the overall 5.6 1 6 at the example of 2) truncations, we obtain a 15% ≤ , 3 d , (7,5,0) , ) and ( η η ≤ 5.3 (6,4,0) − = 4 dimensions etc. This is an indication 2 + are reported in table d

(5,3,0) − + 2 σ 0) and (5 d d Truncation expansion. In ﬁgure ,

(4,2,0) 5 = – 10 – ]. Note, however, that for any finite truncation , , and δ 1 16 (3,1,0) c ν = , η , keeping in mind that the corresponding estimates will σ ν 70% error in for different truncations of the scale-dependent effective action d = 5 d = 4 d = 3

(3,0,0) − η 0 0) truncation for which, in contrast to the (5

, −0.7 −0.8 −0.1 −0.2 −0.3 −0.4 −0.5 −0.6

5 η , . Our results are summarized in figure σ . Therefore, we may use eqs. ( η 1 as an indicator for the quality of the employed truncation at the Yang-Lee ) for example, one may define the relative difference ∆ − and ) 5.5 c η ν = 5 dimensions, a 60 . Anomalous dimension 2+ d = 3, 4, and 5 dimensions for all truncations employed in this work, and our best = 1, independent of dimension [ − scaling relations between critical exponents need not necessarily be satisfied and d d ( β k The scaling properties of the Yang-Lee edge are completely determined by the anoma- = 3, 4, and 5 dimensions. / 2 d in λ contrasted against the one-η and two-loop estimates for the critical exponents obtained with the (7 seems to be reasonably close to its asymptotic value, which is reached in the infinite are considered. In particular,one the will above need numbers to suggest account for that an to increasing reach numberlous a of given dimension operators precision, in Γ exponents performance of different truncations in the range 3 error in that the considered seriesexpect expansions these are scaling relations not to fullylowing hold to for converged determine high yet. the enough exponent orders, Nevertheless,be we since employ associated eq. we with ( an error that is likely to decrease only when higher-order truncations therefore should beTaking checked eq. explicitly. ( This2 applies to bothedge eq. fixed point. ( Wewith observe smaller that dimensions. the relative For error both in the the (7 scaling relation ( and of Γ Figure 2 in Furthermore, from scaling and hyperscaling relations, one can derive JHEP07(2016)041 , ] 13 , we 22 3 = 3, 4, 4105(5) 2685(1) 085(1) ], Monte d . . . = 1) [cf. 13 in α c ] Ref. [ ν , 27 σ ], cf. table ]) including results 40(2) 0 261(12) 0 080(7) 0 23 . . . 23 0), lie within error bars , = 5 5 ] Ref. [ , d 2807(2) 4033(12) 126(6) . . . 35 0 − 402(5) 0 2648(15) 0 0877(25) 0 ) that yield a smaller error for . . . , which is reflected in the errors c = 6, which is consistent with our 5.6 ν ]. Considering the fact, that our c to a recent compilation of available ] Ref. [ d 22 and critical exponents σ = 4 13 ], strong-coupling expansion [ and η (as compiled in ref. [ d 3167(8) 0 2667(32) 0 316(16) . . . 23 σ ) and ( 0 σ 401(9) 0 258(5) 0 076(2) 0 ]. The values obtained from the functional . . . − 22 5.3 ], but are in general smaller than those values ], but are smaller than those obtained by conformal 0). – 11 – ] Ref. [ 23 , ]). 3981 0 2584 0 0747 0 expansion [ . . . 5 23 = 3 23 , 23 d 3581(19) 0 0742(56) 0 586(29) . . . 0 0 0 = 1) and truncation of the type (7 − Ref. [ α 3989 0 2616 0 0785 0 . . . ). This effect has also been observed with other methods 0 0 0 2 c (see, e.g., ref. [ η σ ν expansion results [ expansion results [ σ ], and conformal bootstrap [ ]. we account for a systematic bias due to our choice of the IR regulator ]. That is, we observe that larger orders of the finite field expansion are ). These values were obtained with an exponential regulator ( 22 4033(12) 2667(32) 0742(56) 3 2 . . . 35 and 7 , 0 0 0 ], and are slightly larger the values provided by constrained Padéapproximants c Critical exponent 27 ν functional RG and 35 for an in depth discussion of this issue). We remark that the difference in , 2 7 27 , . Different estimates for the critical exponent . Numerical values for the anomalous dimension 13 )] and the truncation of the type (7 ]. They lie slightly above the values obtained from constrained of Padéapproximants = 5 = 4 = 3 Comparing our estimates for the critical exponent In table 7.2 d d d limit [cf. figure 35 , Z Dimension three- and four-loop obtained from a recentnumerical conformal implementation bootstrap of analysis the [ the RG truncations that flow can equations be isremarkable considered that to not our a overly relatively present small sophisticated results number (limiting are of competitive operators) with it is other quite data in the literature. and may be attributed tothe the exponents scaling relations ( data on the Yang-Leefind edge that critical our scaling values exponents lie27 provided within in the ref. error [ bounds provided by other methods, e.g., refs. [ (see section the values of the anomalous dimensionlarger between than different that high-order obtained truncations for is thefor typically critical these exponents quantities (cf. table n necessary to reach the asymptoticfor scaling dimensions exponents well and below it theprevious seems upper observation that critical on dimension this the order validity increases of scaling relations. Carlo methods [ RG, with an exponential regulatorof ( refs. [ of three- and four-loop bootstrap methods [ Table 3 from the constrained three- and four-loop and 5 dimensions. Here,effects we show (see our section besteq. estimates ( with errors to account for possible systematic Table 2 JHEP07(2016)041 2, − − is ill- (6.1c) (6.1a) (6.1b) = comes ,k 0)-type 4 4 1 = 6 , λ λ 0 A d , turn out to U ,k n 3 . Interestingly, A ,k ]. In This is certainly 3 ]. However, this 12 3 A and 35 6. . Indeed, from a one- , ). The ( . ,k ,k 5 2 and 1 27 A A ≈ , 6.1c ,k , while truncations of the 2 d ,k A 3 )–( 23 ]. , A 31 6.1a that contributes the most. Thus, , 22 0) to stabilize the Yang-Lee edge , , 3 , 2 and ,k / Z 1 13 3, fail to produce a Yang-Lee edge = 1), this is not the case for the optimized 2 ,k A ) . , n 2 ]. In general, we expect that the scale- α > k 2 U A 1 / )( n U it is 2 ) n ∂δφ 6 7.2 k ( λ , k δφ 4! δφ / 2 – 12 – ( 4 k / δφ k k ) on an equal footing, both 0) truncations allow us to identify the corresponding , and for = 3 [cf. figure , , respectively, around the homogeneous field expecta- = = = ,k Z 2 . d 2 ) 6.1c ,k ,k ,k ,k , which mix under renormalization [ φ , n 3 A 2 3 1 9. Each of them corresponds to a different linear combina- 4 k U )–( A A A A / n δφ )( ). One can show that the dominant contribution to 19 φ 6.1a ( − 0)-type truncations, k 2 , 6.1c 0 W − 6 dimensions than the quartic interaction , 6, the ( expansion, as considered in refs. [ . )–( U 5 = ], we obtain the following eigenvalues of the stability matrix: d n 6 d < ≈ ]. However, the latter is not immediately applicable at higher orders in the derivative − λ 31 , and 6.1a , d 2 3 it is the operator ) 39 3, are inadequate to investigate the Yang-Lee scaling behavior. In fact, for , 5 = 6 38 ∂φ λ > 0) do not include , . Different truncations of the scale-dependent effective action are distinguished 9, and )( U k Z φ ¯ n φ / ( , for , n k − ,k U 2 Z 0), 3 n , − A ), 0 Treating the operators ( In particular, we consider the renormalization of quartic operators at the Yang-Lee , φ We remark that this observation depends on the choice of the IR regulator. While the Yang-Lee fixed = ( 3 U k 5 n fine consistent truncations that are adequate to describe the Yang-Lee edgepoint critical is point. unstable forLitim the regulator smooth [ exponential regulatorexpansion. ( fixed point. Whilefixed ( point below scaling solution all the way downdependent to effective action needs to respectrenormalization the of properties operators of with the the theory same under simultaneous canonical dimension. This is important to de- λ tion of operators ( from one might conclude that any truncationdefined, that as includes it only the neglects quartic the interaction it more is relevant contributions, sufficient namely to consider truncations of the type ( truncations, for instance dotype not ( include operators be more relevant in loop calculation [ Note that theyU correspond to particulartion contributions value in theby either finite including series or neglecting expansion some of of these operators, ( behavior can be understoodthe by one-loop examining the effect of operator insertions atfixed the point. level This of requiressame the simultaneous canonical renormalization dimension of all as dimensions operators these that operators carry can the be listed as follows (up to total derivative contributions) We observe that certain( truncations of thethese scale-dependent truncations, effective the action, Yang-Leesurprising fixed of point and the is in type unstable conflict below with other available data [ 6 Relevance of composite operators and quality of finite truncations JHEP07(2016)041 0. (7.3) (7.1) (7.2) ). α > opt α ). Largely with dimensions, ( in different 1 η d ]. The shown opt − defined by the opt α 3 1 ( k α − /η = ) )] 2 α /k opt 2 = 1 seems to be slightly α q = 5 1498 1500 ( ) [cf. figure . . α d η d 0 0 ( , we apply this criterion to − exp( − − η opt , which is determined by the 2 (1) α q α k η [ ¯ = Z , = 1 as well as ≡ α 1 ) α opt = 4 . 2542 2587 − α . . ) = opt d ) 0 0 q 2 , ]). α ( 2 q k ( − − 14 k ) = 0 /k exp α,k ¯ Z 2 αR R η/η q α opt = α – 13 – = exp( = 3 α,k 3270 3340 ) at the Yang-Lee edge critical point in . . α ) evaluated for R . Indeed, if an exact solution to the functional flow d = ( 0 0 α 0 α ( ∞ ( η − − ]. It states that the value of any given observable that η η exp α,k can be considered the best estimate for that quantity. dependence of the critical exponents. We employ the = ) as long as it satisfies the appropriate limiting behavior 14 = R q α α ( η =Λ k k ) R R opt = α k = 1) →∞ R Λ = depends on the dimension, i.e., α ( α α η ( One may identify an optimal value of η ]. To investigate this effect, we define a one-parameter family of functions Relative error 2.1% 1.7% 1.3% 4 47 were available, the calculated observables should not depend on the way we k 0). , 2 0, and consider the , = 0 and lim we compare the values of = 1) is typically already a good approximation to the optimal value k . Anomalous dimension ), i.e., to find the optimal value, we require that 4 R α α ( 0 ( α > η η → Note that the regulator should be sufficiently smooth in momentum space if higher order approximations k 4 = independent of dimension, the anomalous dimensionoverestimated evaluated with at a relative error of approximatelythat 3%. From this comparison we conclude η In table dimensions and determine the relative error ∆ for this analysis. principle of minimum sensitivity [ is least sensitive toSince changes by in virtuepoint of can scaling be relations expressed in all terms critical of exponents the anomalous at dimension the Yang-Lee edge critical with following set of exponential regulators However, in practice, we are boundequations. to consider This truncations yields of a the finite coupledtor set infinite dependence of set [ of RG flow equations for which one observes a residual regula- To determine the criticalany scaling regulator properties function of alim given model, we mayequation in for Γ principle choose choose to regularize the theory in the IR and therefore must be independent of the regulator. 7 Residual regulator dependence and principle of minimal sensitivity The optimal value of values were obtained usingindex a set truncation (4 of the scale-dependent effective action Γ in the derivative expansion are considered (see, e.g., ref. [ Table 4 evaluated for the (deformed) exponential regulator JHEP07(2016)041 , , η ], 3 opt φ 23 27 α , , the 13 3 . = 1) at the 0). , 3 2 α ]. However, we , ( . Different curves η 3 α and ]. We expect that 2 22 5 expansion results [ is lowered and eventually = 4 an ambiguity appears: d of the type (4 d 1 [cf. figure k 4 ) as the analytically continued d d = 5 d = 3 d = 4 ( α < increases, the relative error in opt is lowered. The displayed values were α opt d functions becomes quite demanding 3 shown as a function of α | β ) α opt α are slightly larger than the values obtained ( 2 η σ | / ) – 14 – α ( η 1 6. We find our results in good agreement with available = 3, 4, and 5 dimensions, respectively. The optimal value . 0 d ≤ d − −1 7. Although the value of −0.8 −0.9 . ≤ −1.05 −0.75 −0.85 −0.95

= 6

might depend on the dimension. Indeed, as shown in ﬁgure

opt

α η α η ( | / ) ( )| d ≈ shifts to larger values when the dimension opt α α α = 1 and estimate the corresponding systematic error in α ) = 0, shifts to larger values as the dimension . Rescaled anomalous dimension opt α = Since the search for fixed points of the RG In principle, 5% level (within the considered one-parameter family of regulators). This systematic ) develops a second extremum, a local maximum, for α ], our estimates for the critical exponent α expansion, strong coupling expansion, and Monte Carlo methods. While our results ( − ( 0 are consistent with the strong35 coupling expansion and Montefrom Carlo constrained methods Padéapproximants refs. [ and for generally three- lie and belowtruncations four-loop at those higher from orders in a the conformal derivative and bootstrap field expansion analysis will improve [ our estimates 8 Conclusions In this work we havetheory examined in the dimensions critical scaling 3 data properties in of the the literature, Yang-Lee which edge, includes or high-temperature series expansions, results from the 3 effect in the estimation of theexplicitly anomalous as dimension a has been systematic accounted error for in and the is indicated summary of our results in table η do not consider this solutionlocal to be minimum physical from and define numerically for higher-order truncations, weto use the this case information to limit our calculations optimal value of stabilizes around remains roughly constant. At this point, we remark that below Figure 3 correspond to data obtainedfor in which the criticalη exponent i sobtained for least a sensitive truncation to of changes the scale-dependent in effective the action deformation Γ parameter, i.e., JHEP07(2016)041 . ]. We . (1967) 72 51 , 21 38 . , World Scientific, (1968) 1328 40 . (1968) 1220 3) of the scale-dependent > 25 U Prog. Theor. Phys. n , (1977) 4657 Prog. Theor. Phys. B 15 , ]. It would be interesting to understand 59 , J. Phys. Soc. Jpn. , Universality in the percolation problem — 53 – 15 – , Phys. Rev. , 32 , 0)-type truncations ( , 25 0 , U ), which permits any use, distribution and reproduction in n operators 4 φ ) with exactly the same critical exponents as those of the Yang-Lee c CC-BY 4.0 This article is distributed under the terms of the Creative Commons T < T Field theory, the renormalization group, and critical phenomena Generalization of the Lee-Yang theorem Generalized Lee-Yang’s theorem Note on the critical behavior of Ising ferromagnets expansion provides a qualitative explanation for the observed lack of stability of the d − Singapore (1984). anomalous dimensions of Finally, we remark on possible applications of this work. Based on mean-field argu- We have shown that the stability of nontrivial fixed point associated to the Yang-Lee D.J. Amit, D.J. Wallace and R.K.P. Zia, T. Asano, T. Asano, R. Abe, D.J. Amit, = 6 [3] [4] [5] [1] [2] References DE-FG0201ER41195. Open Access. Attribution License ( any medium, provided the original author(s) and source are credited. We thank B.merical Delamotte implementation for of valuable theropean insights RG and Research flow Council for equations.FP7/2007-2013, under helpful 339220. the advice This European regarding research Thispartment the Union’s is material of nu- Seventh funded is Energy, Framework by Office based Programme the upon of Eu- Science, work supported Office by of the Nuclear U.S. Physics De- under Award Number the relation between theintend Yang-Lee to edge address these point issues and in the a spinodal future publication. singularity [ Acknowledgments the Ising model ( edge point — themetastable spinodal branch of singularity. the Themetastability. free corresponding However, energy its critical and existence point ishave (beyond appears been usually mean-field) subject associated on as well to with the as some the its debate classical scaling [ limit properties of comparing our results with a stability analysis atYang-Lee the edge fixed fixed point point to foreffective one-loop action. ( order in the ments, one expects another thermodynamic singularity in the low-temperature phase of study such truncations. edge singularity is sensitiveThis to the might insertion seem ofthe surprising operators functional that since RG mix a to under similar establish renormalization. the behavior scaling is behavior not at observed the in Ising critical applications point. of However, for the critical exponents. However, more elaborate numerical treatment is necessary to JHEP07(2016)041 , ]. ]. 3 ]. 10 , SPIRE (2012) SPIRE ]. IN D 92 IN (2003) B 33 [ D 73 ][ . (1978) 1610 E 86 Lect. Notes SPIRE ]. 40 B 68 , IN ][ (1980) L247 J. Math. Phys. Phys. Rev. , Phys. Rev. Phys. Rev. (1981) 2391 SPIRE , , cond-mat/9811157 , IN (2001) L811 [ Phys. Rev. ][ A 13 7, 13 Phys. Rev. A 14 hep-ph/0005122 , ≤ [ 4 ∂ Critical exponents to order Critical exponents for the d ]. Phys. Rev. Lett. ≤ , J. Phys. hep-th/0701172 , vol. 6, C. Domb and M.S. Green 2 (1998) 5644 , [ J. Phys. ]. 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