Dimensional reduction breakdown and correction to scaling in the random-field Ising model
DDimensional reduction breakdown and correction to scaling in the random-field Isingmodel
Ivan Balog ∗ Institute of Physics, P.O.Box 304, Bijeniˇcka cesta 46, HR-10001 Zagreb, Croatia
Gilles Tarjus † and Matthieu Tissier ‡ LPTMC, CNRS-UMR 7600, Sorbonne Universit´e,boˆıte 121, 4 Pl. Jussieu, 75252 Paris cedex 05, France (Dated: September 1, 2020)We provide a theoretical analysis by means of the nonperturbative functional renormalizationgroup (NP-FRG) of the corrections to scaling in the critical behavior of the random-field Isingmodel (RFIM) near the dimension d DR ≈ . d → d − d > d DR ) from a region where both supersymmetry and dimensional reductionbreak down at criticality ( d < d DR ). We show that the NP-FRG results are in very good agreementwith recent large-scale lattice simulations of the RFIM in d = 5 and we detail the consequencesfor the leading correction-to-scaling exponent of the peculiar boundary-layer mechanism by whichthe dimensional-reduction fixed point disappears and the dimensional-reduction-broken fixed pointemerges in d DR . PACS numbers: 11.10.Hi, 75.40.Cx
I. INTRODUCTION
One of the puzzles raised by the critical behavior ofthe random-field Ising model is the way the underlyingsupersymmetry (SUSY) unveiled by Paris and Sourlas and the associated dimensional-reduction property (ac-cording to which the critical behavior of the RFIM indimension d is the same as the critical behavior of thepure Ising model in d −
2) break down as a function ofdecreasing dimension. Dimensional reduction has rigor-ously been shown to be wrong in dimensions d = 3 and d = 2 but is valid at the upper critical dimension d = 6(although it only has a weak meaning there). By using anonperturbative functional renormalization group (NP-FRG) approach, we showed that the solution of thepuzzle is the existence of a nontrivial critical dimension d DR ≈ . d DR both SUSY and dimensional reduction arebroken. Our predictions have since been supported bylarge-scale state-of-the-art lattice simulations in dimen-sions up to d = 5. One point however remains puzzling, which has beenrecently raised by Fytas and coworkers. If the criticaldimension is indeed near 5 .
1, the corrections to scalingaround the critical point of the RFIM in d = 5 shouldkeep track of an eigenvalue that vanishes for d = d DR and is then parametrically small in d DR − d ≈ .
1. Yet,this does not seem to be seen in the simulation study ofthe 5-d RFIM in which one instead finds a correction-to-scaling exponent ω = 0 . / − As arguedin Ref. [12] the correction-to-scaling exponent may thenbe the “smoking gun” to settle whether our scenario of acritical dimension in d DR ≈ . d = 6 but is exponentially small in 1 / (6 − d ).In this paper we revisit the critical behavior of theRFIM predicted by the NP-FRG in the vicinity of thecritical dimension d DR . We show that the peculiar wayby which the new fixed point emerges from the collapse ofa pair of dimensional-reduction fixed points (one stableand one unstable) in d = d DR allows one to escape thecurse of the small correction-to-scaling exponent ω . It in-deed leads to a rapid increase of ω below d DR , in the formof a square-root singularity, which results in a predictedvalue of the exponent ω that is compatible with the nu-merical finding of Refs. [10,12]. Furthermore we arguethat in such a scenario of disappearance and emergenceof fixed points, it would in any case be extremely difficultto detect the true asymptotic critical behavior if one hap-pens to get very close to the critical dimension d DR be-cause of the existence of a “Larkin length” that divergesextremely rapidly, as an exponential of 1 / √ d DR − d , asone approaches d DR from below.The rest of the paper is organized as follows. In thefollowing section, Sec. II, we recall the definition of themodel and of its field-theoretical version and we give abrief recap of the NP-FRG approach. In Sec. III, we dis-cuss the scenario found within the NP-FRG for the ap-pearance and disappearance of the critical fixed pointsnear the critical dimension d DR and its consequencesfor the correction-to-scaling exponent ω and the Larkinlength. In Sec. IV we present the results obtained withinour NP-FRG approach for the critical exponents in d = 5and compare them with recent lattice simulations. Wenext study in Sec. V the appearance of a Larkin lengthalong the NP-FRG flow for d < d DR and we discuss itsmeaning and implications. In Sec. VI we consider the a r X i v : . [ c ond - m a t . d i s - nn ] A ug corrections to scaling in the immediate vicinity of d DR and we discuss separately the region below and above d DR . Finally we provide some discussion and conclud-ing remarks in Sec. VII. Additional details are given inseveral appendices. II. THE RFIM
The RFIM is defined on a d -dimensional hypercubiclattice by the following Hamiltonian H [ S ; h ] = J (cid:88)
12 [ ∂ϕ ( x )] + r ϕ ( x ) + u ϕ ( x ) (cid:27) , (3)where (cid:82) x ≡ (cid:82) d d x and h ( x ) is a random “source” (a ran-dom magnetic field). This quenched random field h ( x )is taken with a Gaussian distribution characterized by azero mean and a variance h ( x ) h ( y ) = ∆ B δ ( d ) ( x − y ) (andsimilarly for the lattice version).We are interested in the equilibrium thermodynamicproperties of the model and, more precisely, in the crit-ical behavior. The critical point is controlled by a zero-temperature fixed point and its characteristics can bestudied directly at zero temperature by focusing on theproperties of the ground state which is what we do inthis paper. This is in this zero-temperature setting thatSUSY emerges in the theory. In the NP-FRG approach of the RFIM, the cen-tral quantities are the cumulants of the effective actionat a running infrared scale k , which generate all the 1-particle irreducible correlation functions. One focusesin particular on the effective potential, from which thethermodynamics is obtained, on the field renormaliza-tion, from which one derives the anomalous dimensions,and on the cumulants of the renormalized random field,which are field derivatives of the cumulants of the effec-tive action. In this framework, the breaking of SUSYat the fixed point and the breakdown of the dimensional-reduction predictions for scaling around the critical pointare related to the appearance of a singular behavior, inthe form of linear cusps, in the functional dependence ofthe dimensionless cumulants of the renormalized randomfield when two field arguments become equal, e.g. , for thesecond cumulant δ ∗ and uniform field configurations, δ ∗ ( ϕ − δϕ, ϕ + δϕ ) − δ ∗ ( ϕ, ϕ ) ∝ | δϕ | , when δϕ → . (4) The cusps generate a breakdown of the Ward-Takahashiidentities associated with SUSY, with ensuing spon-taneous SUSY breaking and dimensional-reductionbreakdown. These cusps have been shown to be inti-mately related to the presence of “static avalanches”or “shocks” in the variation of the ground state ofthe RFIM with the external source (magnetic field).Avalanches on all scales are always present in the crit-ical behavior of the RFIM at zero temperature but inthe region where dimensional reduction and SUSY arevalid, the effect of these avalanches is too weak to influ-ence the critical physics. (Note that this subdominantbehavior and the associated irrelevant eigenvalue aroundthe cuspless fixed point can be accessed through a pertur-bative but functional RG treatment of the RFIM in anexpansion in (cid:15) = 6 − d around the upper critical dimen-sion: see Ref. [17].) It is only below a critical dimension d DR that the avalanches and the resulting cusps appearat the level of the fixed point. A brief recap on our NP-FRG approach is given inAppendix A (see also Ref. [9]).
III. APPEARANCE AND DISAPPEARANCEOF THE FIXED POINTS NEAR THE CRITICALDIMENSION d DR The way the fixed point associated with SUSY anddimension reduction disappears when d decreases anda new fixed point emerges has been studied in a pre-vious paper. What is found within the NP-FRG is thatthe SUSY/dimensional-reduction fixed point that con-trols the critical behavior of the RFIM below the up-per dimension d = 6 annihilates with another (unstable)SUSY/dimensional-reduction fixed point when d = d DR .Then emerges from these collapsed fixed points a “cuspy”fixed point at which both SUSY and dimensional reduc-tion are broken. This fixed point appears below d DR through an unusual boundary-layer mechanism that canonly be captured through a functional approach. One of the consequences of this boundary-layer mech-anism is that the dependence of the lowest positive (irrel-evant) eigenvalues of the stability matrix at the criticalfixed point have a singular square-root behavior around d DR . If we denote by λ the lowest eigenvalue associ-ated with a “cuspy” perturbation around the fixed point(above d DR the fixed point itself is “cuspless” whereasit is “cuspy” below d DR ), we find that λ approaches asmall but nonzero value with a square-root singularity when d → d + DR , has a discontinuity in d DR , and below d DR increases from 0 with another square-root behavior: λ ( d ) ≈ λ ‡ + c + (cid:112) d − d DR , for d > d DR , ≈ c − (cid:112) d DR − d , for d < d DR , (5)with λ ‡ > c ± positive and of O(1).Above d DR , there are additional irrelevant eigenvalueswhich display a square-root, √ d − d DR , dependence and d λ ,d
13. The lowest eigenvalue λ is discontinuous in d DR and has square-root singularities on approaching d DR fromboth sides (dashed and full lines). (As the numerical reso-lution of the FRG equations is done by discretizing the fieldarguments on a grid, it is very difficult to properly approach d DR from below and the curve there is just a guide for theeye.) The eigenvalue denoted by ω DR is the correction-to-scaling exponent obtained by dimensional reduction for thepure φ theory in two dimensions less (it vanishes as (cid:15) = 6 − d near d = 6). This is a true eigenvalue of the RFIM fixedpoint above d DR but not below where dimensional reductionis broken. Above d DR the eigenvalue λ is associated with acuspy perturbation around the cuspless fixed point. The redcurve denoted λ is associated with a cuspless perturbation inthe 2-replica component of the second cumulant of the renor-malized random field (see the main text); it also follows asquare-root dependence when approaching d DR . are associated with cuspless (but possibly showing weakersingularities in the form of “subcusps”) perturbationsin the 2-replica sector. There is also the correction-to-scaling exponent ω DR obtained by dimensional reduction(and therefore characterizing the 1-replica sector). All ofthis will be again discussed in Sec. VI.This pattern of singular behavior is hard to analyt-ically extract from the full NP-FRG equations for theRFIM. (It is on the other hand easily obtained in a toymodel having a similar but simpler structure which weintroduced in Ref. [8] and is discussed in Appendix B.)Numerical evidence from the NP-FRG is shown in Fig. 1,where we plot the three lowest eigenvalues as a functionof d in the region near d DR . The square-root behavior ofthe lowest irrelevant eigenvalue, which is reasonably wellfollowed by the numerical solution (although the limit d → d − DR is very hard to access due to the boundary-layerstructure), implies a rapid increase of this eigenvalue be-low d DR ≈ .
13, at variance with the conventional behav-ior of a crossover exponent near the dimension at whichthere is an exchange of stability between two fixed points.Another specific feature of the boundary-layer mech-anism for the emergence of the cuspy fixed point below d DR is the existence of a large “Larkin length”, (cid:96) L , whichdiverges when d → d − DR . The notion of a Larkin length has been introduced in the context of elastic manifoldspinned by a random potential. There, it marks thecrossover between (small) lengths over which the behav-ior of the manifold is controlled by its elastic propertiesand (large) lengths over which the pinning potential dom-inates and induces a multiplicity of metastable states,avalanches between states, and a linear cusp in the func-tional dependence of the cumulants of the renormalizedrandom force.
In the case of the RFIM the phys-ical nature of the Larkin length is more elusive. It cannonetheless be operationally obtained as the lenghscalealong the functional RG flow at which a cusp first appearsin the cumulants of the renormalized random field whenstarting from a cuspless initial condition at the UV (mi-croscopic) scale (see Appendix A for more details). Theso-defined Larkin length diverges exponentially rapidlywhen approaching the critical dimension d DR from be-low, ln (cid:96) L ( d ) ∼ √ d DR − d , (6)where the square-root dependence is analytically found inthe toy model already mentioned (see Appendix B) andis also compatible with the numerical solution of the NP-FRG of the RFIM: see Fig. 2. There are prefactors in theabove relation which depend on the microscopic detailsof the theory (and the Larkin length itself is expressed interms of some microscopic length), but in any case thelength becomes extremely large in the immediate vicinityof the critical dimension. A more extended discussion ofthe Larkin length in the RFIM will be given in Sec. V. IV. NP-FRG RESULTS IN d = 5 As already mentioned, our FRG approach is nonper-turbative but approximate. We have developed a trunca-tion scheme of the running effective action that respectsthe symmetries and supersymmetries of the RFIM. Sincethe resulting functional flow equations require a numer-ical resolution, the best level of truncation of the NP-FRG that we have at present achieved consists in trun-cating the cumulant expansion after the second cumulantand truncating the expansion in spatial derivatives of thefield after the second order: see also Appendix A.The results for the critical exponents are in very goodagreement with the best estimates from simulations inall dimensions d . With this truncation we find thatSUSY and dimensional reduction are valid at the (cusp-less) fixed point above d DR ≈ . d = 5 we have obtained the fol-lowing results: The anomalous dimensions are found tobe η ≈ . η ≈ . ν ≈ . The anomalous dimensionsvery slightly break the SUSY related relation η = ¯ η .The exponents also slightly deviate from the dimensional-reduction property. (In the same approximation we in-deed obtain for the pure 3- d Ising model η ≈ .
045 and d | t L | -4 -3.5 -3 -2.5 -2 -1.5 -1 ln(d DR -d) l n ( | t L | ) FIG. 2: Dimensional dependence of the Larkin length (cid:96) L ( d )for d < d DR ≈ .
13 obtained from the NP-FRG solution ofthe RFIM at criticality. The flow starts with a cuspless ini-tial condition for the second cumulant of the random field, δ k ( ϕ , ϕ ). The cuspless solution is valid down to a valueof the IR cutoff k (or RG time t = ln( k/ Λ) where Λ is theUV scale) that can be easily found numerically from the di-vergence of the second derivative in the direction ϕ − ϕ evaluated for equal replica fields, δ k, ( ϕ ): This is our defini-tion of the Larkin length. Top panel: | t L | = ln( (cid:96) L Λ) versus d for d ≤ d DR = 5 . / √ d DR − d behav-ior. Bottom panel: log-log plot of | t L | versus ( d DR − d ); thedashed red line has a slope of − / ν ≈ . ω = λ ≈ . η = 0 . η =0 . ν = 0 . ω = 0 . / − One, of course, has to be careful before drawing defi-nite conclusions. Both the NP-FRG and the simulationresults are affected by uncertainties. They result fromthe truncation of the effective average action in the for-mer and to the limited system sizes and limited numberof samples in the latter. (Due to the uncertainties in thesimulations, the authors of Ref. [10] and of the more re-cent work in Ref. [12] for instance comment that theirresults are also compatible with dimensional reduction -0.2 -0.1 0 0.1 0.2 ϕ ∗ (ϕ)δ ∗ (ϕ,ϕ) FIG. 3: Test of the SUSY Ward identity at the NP-FRG fixedpoint in d = 5. SUSY implies that the second cumulant ofthe renormalized random field δ ∗ ( ϕ, ϕ ) is equal to the fieldrenormalization function z ∗ ( ϕ ); The observed violation is byless than 1%, as found for a related Ward identity in a latticesimulation . and with SUSY. ) In the case of the NP-FRG result itwould be interesting to be able to assess the uncertaintyassociated with the truncation. While expected to besmall for most critical exponents, as shown for a varietyof models studied by the FRG, this uncertainty may bequite significant in the case of the correction-to-scalingexponent because of the uncertainty on the precise valueof the critical dimension d DR . To give a rough estimate,let us assume that the uncertainty on the determinationof d DR is, say, ± . i.e. , 5 . ≤ d DR ≤ .
18, and thatwe can use the square-root expression in Eq. (5) withthe same parameter c − . We then find that ω would varyfrom 0 . . d DR and we are presently workingon this (difficult) issue.In any case, the overall agreement between NP-FRGand simulation results appears as a strong support forour scenario of dimensional-reduction and SUSY break-ing below d DR ≈ . δ ∗ ( ϕ, ϕ ), and the field renormalization function, z ∗ ( ϕ ), which is the derivative with respect to q of the q -dependent 2-point vertex obtained from the first cumu-lant of the effective action: δ ∗ ( ϕ, ϕ ) = z ∗ ( ϕ ) . (7)With the truncation that we use, this identity is exactlysatisfied above d DR . The comparison in d = 5 betweenthe left and right members of the above equation is shownin Fig. 3. One can see that the violation of the SUSYWard identity is indeed extremely small (less than 1%),as also found in Ref. [12] for a related Ward identity. V. THE LARKIN LENGTH WHEN d → d − DR We now come back to the issue of the physical interpre-tation of the Larkin length (cid:96) L in the vicinity, but below, d DR and on its consequences. We would like to betterunderstand the meaning of the Larkin scale as it appearsin the NP-FRG flow and relate it to some characteristiclength in a microscopic realization of the RFIM in a d -dimensional lattice at zero temperature. In principle, ifone wishes to make a direct comparison between nonuni-versal quantities obtained from the field-theoretical NP-FRG on the one hand and from lattice simulations onthe other, and not only focus on universal predictions,one should consider the hard-spin lattice version of theNP-FRG, as developed for instance in Ref. [26]. In thelatter approach, the initial condition of the NP-FRG flowis provided by the limit of decoupled sites in the originalsystem. In the present case this limit implies computingthe effective action and all the cumulants of the renor-malized random field for a single-site RFIM, which canbe done essentially exactly.One subtlety that appears in the case of the RFIM isthat two distinct IR cutoff functions are generically intro-duced to derive exact FRG equations (see Appendix A): One of them, which we called (cid:98) R k in Ref. [7], restricts thefluctuations of the fundamental field (the local magne-tization in the language of magnetic systems) with mo-menta less than the running scale k and is furthermorechosen to enforce site decoupling in the lattice NP-FRGat a UV wavevector k UV = 2 π/a where a is the latticespacing; the other one, which we called (cid:101) R k in Ref. [7], re-stricts the fluctuations associated with the random source(or field) and reduces the variance of the latter. In thecontinuum description, if one wishes to avoid an explicit breaking of the SUSY Ward identity, one must impose arelation between the two IR cutoff functions. In momen-tum space, the relation reads (cid:101) R k ( q ) = − ∆ k Z k ∂∂q (cid:98) R k ( q ) , (8)where q is the squared momentum, ∆ k the (squared)strength of the renormalized random field and Z k thefield renormalization constant (see Appendix A). How-ever, SUSY being broken at the fixed point in d < d DR ,there is some leeway in choosing the IR cutoff function (cid:101) R k in this region as the same long-distance physics shouldbe obtained when k → (cid:98) R k there is therefore a spectrumof possible (cid:101) R k ’s, from one that mimics the SUSY-inducedrelation and exactly cancels the bare variance of the ran-dom field at the UV scale to others that lead to onlypartial cancellation. Although the initial conditions ofthe FRG flow are different for each (cid:101) R k , the very same critical physics is attained in the limit where the IR cut-off k goes to zero. Of course, when approximations areintroduced, this is no longer exactly true and, e.g. , thecritical exponents obtained through the different choicesof (cid:101) R k may not exactly coincide.Without delving more into this question, we just notethat the cumulants of the effective random field obtainedfrom the solution of the single-site RFIM display a cusp intheir functional field dependence for choices of (cid:101) R k that donot exactly cancel the bare variance of the random fieldat the UV cutoff k UV . One may therefore start the NP-FRG flow with or without a cusp in the cumulants. Onlyin the latter case will a crisp Larkin scale be observedalong the flow; it will be replaced by a crossover in theformer cases.In the present work we do not attempt to study thefull-blown lattice NP-FRG for the RFIM. We defer thisto a future study. We rather use a proxy by consid-ering the continuum field-theoretical NP-FRG, as pre-viously investigated, in two different settings. In thefirst setting, we consider cuspless initial conditions andFRG flow equations obtained from the choice in Eq.(8). In the second one, we consider cuspy initial con-ditions and FRG flow equations derived with (cid:101) R k ( q ) = − α (∆ k /Z k )( ∂/∂q ) (cid:98) R k ( q ) with α <
1: for illustrationwe choose α = 0 .
6. (For completeness we also considera case with α = 1 but with cuspy initial conditions.) Inall of the cases, the initial condition for the first cumu-lant is a ϕ theory as in Eq. (3), with the bare potentialrewritten as u Λ ( ϕ ) = g Λ
4! ( ϕ − κ Λ ) (9)where we keep g Λ , the ϕ coupling constant, fixed andwe fine-tune, for each choice of the initial cusp amplitude,the location of the nontrivial minimum of the potential, κ Λ , in order for the flow to approach as close as possibleto the cuspy fixed point describing the critical behaviorof the RFIM. (We achieve this by dichotomy.) We illustrate the results for d = 5. We first show theNP-FRG flow of the amplitude of the cusp in the (di-mensionless) second cumulant of the renormalized ran-dom field δ k, cusp ( ϕ = 0) in the top panel of Fig. 4. Asanticipated, the Larkin length is not crisply defined fora nonzero initial cusp amplitude, but it remains as anoticeable crossover in the flow and its order of magni-tude does not vary significantly with the amplitude of thecusp. (Note that the curve shown for the cuspless caseis obtained from a full functional solution that requiresdiscretizing all the fields and therefore, due to numericalprecision, displays some systematic error in the vicinityof the Larkin time which leads to a spurious rounding;the Larkin scale can be accurately accessed by instead ex-panding the second cumulant in the difference betweenthe two fields, which corresponds to the results shownin Fig. 2 and to the vertical line in Fig. 4.) Quite no-tably, the flows corresponding to the 3 different settingsonly converge to their fixed-point value for RG times sig-nificantly beyond the Larkin one, i.e. , for length scalesbeyond the Larkin length. The IR cutoff k plays a rolesimilar to that of a finite system size L in restricting thespatial extent of the fluctuations and, with a grain of salt,one can associate k with 2 π/L . This means that onlywith system sizes much larger than the Larkin lengthcan one properly access the asymptotic critical behavior.We display in the bottom panel the second derivative ofthe dimensionless potential u (cid:48)(cid:48) k ( ϕ = 0), i.e. , the squaredmass. Again, the 3 curves corresponding to the differ-ent settings converge to their fixed-point value for timesmuch beyond the Larkin time.As we showed that when the dimension d approaches d DR from below the Larkin length increases exponen-tially fast in 1 / √ d DR − d , proper access to the asymp-totic critical behavior would require astronomically largesystem sizes in a simulation if the dimension happens tobe very close to d DR . VI. CORRECTIONS TO SCALING IN THEVICINITY OF d DR To complement the discussion of the critical behaviorof the RFIM in the vicinity of d DR we study the cor-rections to scaling. To do so, we consider the flows ofspecially tailored quantities built from running renor-malized observables, e.g. , the squared mass u (cid:48)(cid:48) k ( ϕ = 0),the running anomalous dimension η k obtained from thefield renormalization function z k ( ϕ ) (see Appendix A),or the amplitude of the cusp δ cusp ( ϕ = 0). For a genericrunning quantity m k ≡ m ( t ), with as before t being theRG time, we define f m ( t ) = ddt ln( | ddt m ( t ) | ) . (10)In practice, a very small positive quantity ∼ − isadded to the argument of the logarithm to avoid spuri-ous singularities in the numerics. The flow of f m displaysplateaus separated by transient regions. The values of theplateaus give a direct access to the exponents controllingthe corrections to scaling, i.e. , some irrelevant eigenval-ues around the fixed point, and, for very large RG time,to the relevant eigenvalue 1 /ν describing the escape fromthe fixed point due to imperfect numerical fine-tuning ofthe critical initial conditions at the UV scale [actually,the plateau for the relevant eigenvalue is at − /ν due tothe definition in Eq. (10)].We first discuss the case where d < d DR . We againconsider the different settings, with different choices ofIR cutoff functions associated with cuspy and cusplessinitial conditions (see above). In Fig. 5, we show the flowof f mass and f cusp for d = 5. (The behavior of the func-tion associated with the running anomalous field dimen-sion is very similar to that of f mass and is not displayedhere.). In d = 5, we see first some ill-characterized behav-ior before and around the Larkin length, where no realplateau is found. (There seems to be an indication for -20 -15 -10 -5 0 t -2-1.5-1-0.50 δ k , c u s p ( ) / | δ , c u s p ( ) | -20 -15 -10 -5 0 t -1.002-1.001-1-0.999 u ’’ k ( ) / | u ’’ ( ) | cuspless, α=1 cuspy, α=1 cuspy, α=0.6 FIG. 4: NP-FRG flow of the amplitude of the cusp in thedimensionless second cumulant of the renormalized randomfield, δ k, cusp ( ϕ = 0), (top panel) and of the second deriva-tive of the dimensionless potential u (cid:48)(cid:48) k ( ϕ = 0) (bottom panel)as a function of the RG time t = ln( k/ Λ), where Λ is theUV scale, in d = 5. The initial condition is fine-tuned (by di-chotomy) to asymptotically approach the (critical) fixed pointand δ k, cusp ( ϕ = 0) and u (cid:48)(cid:48) k ( ϕ = 0) are divided by the modulusof their fixed-point value. Different settings are shown, onewith the IR cutoff functions related by the SUSY-like form(Eq. (8) and a cuspless initial condition, one with the same IRcutoff functions ( i.e. , α = 1) but a cuspy initial condition, andone with a modified relation between the cutoff functions witha prefactor α = 0 . t L ) is indicated by a verticalline. It becomes a crossover when the initial amplitude of thecusp departs from 0. (However, due to numerical precision inthe full functional numerical solution, there is a small nonzerovalue near the Larkin time even when starting from the cus-pless condition, which also leads to a spurious rounding; theLarkin time can be crisply defined by expanding instead inone of the field arguments and detecting a divergence in theappropriate function, see the main text.) Note the scale ofthe vertical axis for u (cid:48)(cid:48) k ( ϕ = 0) / | u (cid:48)(cid:48) ( ϕ = 0) | : To zoom in onthe interesting region around and beyond the Larkin time, wedo not display the early stages of the RG flow. f mass that a short plateau around ω DR ≈ .
85 emerges;on the other hand the minimum of f cusp at a value near λ is accidental and is not found in other quantities orother dimensions.) This is followed by an approach tothe fixed-point value that is governed by the correction-to-scaling exponent ω = λ ≈ .
65 (more clearly reachedfor f cusp than for f mass ). Finally, because one cannot -20 -15 -10 -5 0 t -1.5-1-0.500.51 f m a ss cuspless, α=1 cuspy, α=0.6 -1/ νλω DR -20 -15 -10 -5 0 t -1.5-1-0.500.51 f c u s p FIG. 5: Corrections to scaling for d < ∼ d DR : NP-FRG flow ofthe quantities f mass , defined in Eq. (10)) from the (squared)mass u (cid:48)(cid:48) k ( ϕ = 0), and f cusp built from the amplitude of thecusp in the second cumulant δ k, cusp (0), versus the RG time t =ln( k/ Λ) in d = 5. We also show the asymptotic correction-to-scaling exponent ω = λ , the dimensional-reduction correction-to-scaling exponent ω DR , and the relevant eigenvalue − /ν .The RG time | t L | associated with the Larkin length is about7 for d = 5. The two different schemes, with cuspy or cusplessinitial conditions and the appropriate NP-FRG equations, areshown. In all cases the initial conditions are fine-tuned bydichotomy to approach the critical fixed point. start exactly at the critical initial condition with an in-finite precision, we also find a final regime governed bythe relevant eigenvalue − /ν ≈ − .
59 (with a differentapproach to the asymptote for f mass and f cusp ) as theflow is driven away from the vicinity of the fixed point.The situation is similar in d = 5 . d is close butabove d DR , d > ∼ d DR . The critical theory at the fixedpoint now satisfies SUSY and the main scaling behavior isdescribed by dimensional reduction. There are nonethe-less small irrelevant eigenvalues, significantly smallerthan ω DR (see Fig. 1), but they are associated with eigen-functions that depend on distinct copies (or replicas) andhave no projection on the -copy sector ( i.e. , when allfield arguments are equal). To be more specific, we con-sider the second cumulant of the renormalized randomfield, which is generically of function of two replica fields: δ k ( ϕ , ϕ ), or, after introducing the fields ϕ = ( ϕ + ϕ ) / y = ( ϕ − ϕ ) / δ k ( ϕ, y ). The function δ k ( ϕ, y ) is even in ϕ , as a result of the statistical Z symmetry ofthe model, and even in y , as a result of replica permu-tation invariance. When the two replica fields becomeclose, y →
0, one can expand the function at the cusplessfixed point as δ ∗ ( ϕ, y ) = δ ∗ , ( ϕ ) + 12 δ ∗ , ( ϕ ) y + · · · , (11)where the equal-field ( y = 0) component satisfies theSUSY Ward identity, δ ∗ , ( ϕ ) ≡ δ ∗ ( ϕ, ϕ ) = z ∗ ( ϕ ) [seealso Eq. (7)]. In the vicinity of this fixed point, for smallnonzero k , the second cumulant can be expanded as δ k ( ϕ, y ) = δ k, ( ϕ ) + δ k, cusp ( ϕ ) | y | + 12 δ k, ( ϕ ) y + O ( | y | ) , (12)where we have now included nonanalytic terms in oddpowers of | y | , which go to zero as k →
0. The ad-ditional irrelevant eigenvalues that are not part of the1-replica dimensional-reduction spectrum are associatedwith δ k, cusp ( ϕ ) which flows to zero at the fixed point withthe eigenvalue λ , with δ k, ( ϕ ) which flows to zero withthe eigenvalue λ , etc.We stress again that the small additional irrelevanteigenvalues λ , λ , etc., are only visible in the correc-tions to scaling of observables probing correlations be-tween distinct copies of the system. We illustrate thispoint by showing the NP-FRG flows of the two quan-tities f mass and f cusp introduced above in Eq. (10) inFig. 6 for the dimension d = 5 .
17 and we consider acuspy initial condition (otherwise δ k, cusp and f cusp stayidentically zero), but with IR cutoff functions that satisfythe SUSY Ward identity so that the proper fixed pointcan be reached. One can see that the flow of the quantity f mass associated with the squared mass is only sensitive tothe dimensional-reduction correction-to-scaling exponent ω DR ≈ .
70 and, at very long RG time where the systemfinally escapes from the vicinity of the fixed point, to therelevant eigenvalue − /ν ≈ − .
67. (The same is true forthe quantity associated with the field renormalization, f η but it is not displayed here.) On the other hand, the cor-rection to scaling of the amplitude of the cusp, which is agenuine characteristics of the 2-copy sector, is controlledby the small eigenvalue λ ≈ .
19, as seen in the flow of f cusp . At very long RG time f cusp does not follow thebehavior of f mass and does not seem to converge to therelevant eigenvalue − /ν ; this indicates that the projec-tion of the eigenvector associated with − /ν on the cuspamplitude in the second cumulant vanishes. VII. CONCLUSION
We have shown that the NP-FRG prediction for theRFIM of the critical dimension d DR ≈ . d > d DR ) where the critical behavior followsdimensional reduction (except in irrelevant multi-copycuspy directions ) and is controlled by a SUSY -15 -10 -5 0 t -1.5-1-0.500.51 f m a ss -1/ νλω=ω DR -15 -10 -5 0 t -1.5-1-0.500.51 f c u s p FIG. 6: Corrections to scaling: NP-FRG flow of the quan-tities [defined in Eq. (10)] f mass , built from the squaredmass u (cid:48)(cid:48) k ( ϕ = 0), (top) and f cusp , built from the amplitudeof the cusp in the second cumulant δ k, cusp (0), (bottom) ver-sus the RG time t = ln( k/ Λ) in d = 5 . > d DR . We alsoshow in both cases the dimensional-reduction correction-to-scaling exponent ω DR ≈ .
70, the smallest irrelevant eigen-value λ ≈ .
19 associated with a cuspy eigenfunction in the2-copy sector, and the dimensional-reduction relevant eigen-value − /ν ≈ − .
67. The initial conditions are fine-tunedby dichotomy to approach the critical fixed point, at whichSUSY is satisfied; they are chosen with a cusp in the secondcumulant For f mass , due to imperfect numerical fine-tuning ofthe critical initial conditions, the long-time RG time behav-ior displays a final plateau associated with relevant eigenvalue1 /ν shown as a dashed line. The behavior of f cusp at long RGtimes is different. Note that the flow of f mass is only sensitiveto ω DR , contrary to that of f cusp which is only sensitive to λ . fixed point from a region ( d < d DR ) where both di-mensional reduction and SUSY are broken at critical-ity is compatible with the recent large-scale numericalsimulations in d = 5. In particular, the peculiarmechanism through which the SUSY-broken fixed pointemerges from the SUSY fixed point at d DR explains whycorrection to scaling can be controlled by a rather largeexponent ω despite the fact that d = 5 is not far from d DR .We have also discussed the properties of the correc-tions to scaling in the immediate vicinity of d DR andtheir consequences for finite-size lattice simulations. In-deed, the NP-FRG prediction for the critical dimension, d DR ≈ .
1, is an approximate result only. (Work is nowin progress to assess the uncertainty in its determina- tion.) We have argued that for a dimension very closebut below d DR the existence of a very large Larkin lengththat diverges exponentially fast as one approaches d DR precludes any detection of the asymptotic large-scale be-havior in simulations with presently achievable systemsizes. On the other hand for a dimension very close butabove d DR , the usual observables considered in finite-sizeanalyses, masses or correlation lengths, main critical ex-ponents, etc., which are all properties of the 1-copy sec-tor, i.e. , quantities evaluated for equal replica fields, areonly affected by the dimensional-reduction correction-to-scaling exponent ω DR , which is large near d DR ; it is onlyby studying quantities that are genuinely associated with2 or more distinct copies that one would capture the smallirrelevant eigenvalues associated with singular eigenfunc-tions.If the correction to scaling in d = 5 turns out not tobe the “smoking gun” proposed in Ref. [12], can one pro-pose a crisper test of our prediction of a critical d DR that could be checked in computer simulations of theRFIM? It seems that the 1-d long-range RFIM model,in which the interactions decay as a power law with dis-tance, r − ( d + σ ) but the correlations of the bare randomfield are short-ranged, could be a good candidate.Dimension d is fixed but in some sense varying the rangeexponent σ has a similar effect to that of changing d inthe short-range RFIM: for σ ≤ / σ ≥ /
2; between the two valuesa critical point with nontrivial σ -dependent exponentsexists. Our prediction based on the NP-FRG is that acritical value σ c ≈ .
38 separates a regime where the fixedpoint is “cuspless” from a regime where it is “cuspy”,with the same boundary-layer mechanism governing theappearance of the cuspy fixed point for σ → σ + c . Thediscontinuity in the eigenvalue λ is now large in σ c (it goesfrom 0 . Acknowledgments
IB acknowledges the support of the Croatian ScienceFoundation Project No. IP-2016-6-3347 and the Quan-tiXLie Centre of Excellence, a project cofinanced by theCroatian Government and European Union through theEuropean Regional Development Fund - the Competi-tiveness and Cohesion Operational Programme (GrantKK.01.1.1.01.0004).
Appendix A: Nonperturbative FRG
We summarize here the main features of the NP-FRGdescription of the equilibrium critical behavior of RFIMdeveloped in Refs. [5–7,9]. The central quantity is theso-called “effective average action” Γ k , in which onlyfluctuations of modes with momentum larger than an in-frared cutoff k are effectively taken into account. In thelanguage of magnetic systems, Γ k is the Gibbs free-energyfunctional of the local order parameter field obtained af-ter a coarse-graining down to the (momentum) scale k .The effective average action obeys an exact RG equationunder the variation of the infrared cutoff k . In the presence of disorder, the generating (free-energy) functionals are sample-dependent, i.e. ran-dom, and should therefore be characterized either bytheir probability distribution or by their cumulants.The latter description is more convenient as it fo-cuses on quantities—cumulants and associated Green’sfunctions—which are translationally invariant and can begenerated through the introduction of copies (or “repli-cas”) of the original system that are submitted to dis-tinct external sources.
The effective average actionthat generates the cumulants of the renormalized disor-der, Γ k [ { φ a } ], then depends on the local order parameterfields associated with the various copies a . It satisfies thefollowing exact FRG flow equation: ∂ k Γ k [ { φ a } ] =12 (cid:90) d d q (2 π ) d (cid:88) ab ∂ k R abk ( q ) (cid:18) (cid:104) Γ (2) k + R k (cid:105) − (cid:19) abq, − q , (A1)where Γ (2) k is the matrix formed by the second functionalderivatives of Γ k with respect to the fields φ a ( q ) and R abk ( q ) = (cid:98) R k ( q ) δ ab + (cid:101) R k ( q ), where (cid:98) R k and (cid:101) R k are in-frared cutoff functions that enforce the decoupling of thelow- and high-momentum modes at the scale k and re-duce the fluctuations induced by the disorder: More pre-cisely, the function (cid:98) R k ( q ) adds a mass ∼ k − η (where η is a running anomalous dimension of the fields) to modeswith q < ∼ k and decays rapidly to zero for q > ∼ k ,whereas the function (cid:101) R k ( q ) reduces the fluctuations ofthe bare random field. The formalism can be upgradedto a superfield theory in order to describe the physicsdirectly at zero temperature ( i.e. , the ground-state prop-erties), which allows one to make the underlying SUSY explicit and describe its spontaneous breaking. From Eq. (A1), one can derive a hierarchy of coupledRG flow equations for the cumulants of the renormalizeddisorder, Γ k [ φ ], Γ k [ φ , φ ], etc., that are obtained fromΓ k [ { φ a } ] through an expansion in increasing number ofunrestricted sums over copies, Γ k [ { φ a } ] = (cid:80) a Γ k [ φ a ] − (cid:80) a,b Γ k [ φ a , φ b ] + · · · . Alternatively, one can considerthe exact hierarchy of RG equations for the field deriva-tives of these quantities, Γ (1) k ,x [ φ ], Γ (1 , k ,xy [ φ , φ ], etc.,which represents the cumulants of the renormalized ran-dom field and are the objects that are naturally obtainedfrom the superfield construction.At the UV scale, say k = Λ, the effective average ac-tion reduces to some “bare” action of the multy-copysystem, which generates the cumulants of the renormal-ized disorder at the mean-field level. At the end of theflow, when k = 0, all the fluctuations are incorporated and one recovers the full effective action Γ[ { φ a } ] whichis the generating functional of the 1-particle irreducible(1-PI) correlation functions that are associated with thecumulants of the fully renormalized disorder. The min-imal truncation that already contains the key featuresfor a nonperturbative study of the critical physics of theRFIM is the following:Γ (1) k ,x [ φ ] = U (cid:48) k ( φ ( x )) + 12 [ δδφ ( x ) Z k ( φ ( x ))[ ∂φ ( x )] Γ (1 , k ,x x [ φ , φ ] = δ ( d ) ( x − x )∆ k ( φ ( x ) , φ ( x ))Γ (1 , ··· , kp,x ··· x p [ φ , · · · , φ p ] = 0 for p ≥ , (A2)with three functions, the field renormalization function Z k ( φ ), the potential U k ( φ ) [or its derivative U (cid:48) k ( φ )], andthe second cumulant of the renormalized random field∆ k ( φ , φ ), to be determined. On the other hand, toensure that there is no explicit breaking of SUSY the IRcutoff functions must satisfy the relation (cid:101) R k ( q ) = − (∆ k /Z k ) ∂∂q (cid:98) R k ( q ) , (A3)with ∆ k the strength of the renormalized random fieldand Z k the field renormalization constant [defined, e.g. ,by ∆ k = ∆ k ( φ = 0 , φ = 0) and Z k = Z k ( φ = 0)]. In-serting the above ansatz in the exact RG equations for thecumulants leads to a set of coupled flow equations for thethree functions Z k ( φ ), U (cid:48) k ( φ ) and ∆ k ( φ , φ ). The RG isfunctional as its central objects are functions instead ofcoupling constants.To cast the NP-FRG flow equations in a form that issuitable for searching for zero-temperature fixed pointsdescribing the critical behavior of the RFIM, one hasto introduce appropriate scaling dimensions. This entailsdefining a renormalized temperature T k which flows tozero as k →
0. Near such a fixed point, one has thefollowing scaling dimensions: T k ∼ k θ , Z k ∼ k − η , φ a ∼ k ( d − η ) / , with θ and ¯ η related through θ = 2 + η − ¯ η ,as well as U (cid:48) k ∼ k d − θ − ( d − η ) / and ∆ k ∼ k − (2 η − ¯ η ) . Let-ting the dimensionless counterparts of U k , Z k , ∆ k , φ bedenoted by lower-case letters, u k , z k , δ k , ϕ , the resultingflow equations can be symbolically written as ∂ t u (cid:48) k ( ϕ ) = β u (cid:48) ( ϕ ) ,∂ t z k ( ϕ ) = β z ( ϕ ) ,∂ t δ k ( ϕ , ϕ ) = β δ ( ϕ , ϕ ) , (A4)where t = log( k/ Λ). The beta functions themselves de-pend on the functions u (cid:48) k , z k , δ k and their derivatives [inaddition, the running anomalous dimensions η k and ¯ η k are fixed by the conditions z k (0) = δ k (0 ,
0) = 1]. Theirexpressions are given in Ref. [7] for the zero (bare) tem-perature case.The beta functions also depend on the dimension-less cutoff functions, which are defined from (cid:98) R k ( q ) = Z k k (cid:98) r ( q /k ) and (cid:101) R k ( q ) = ∆ k (cid:101) r ( q /k ) with (cid:101) r ( x ) =0 − ∂ x (cid:98) r ( x ), where we chose for the function (cid:98) r ( x ) theform ( a + bx + cx ) e − x . The parameters a , b and c of the function can be further optimized by invoking theprinciple of minimum sensitivity and we find good sta-bility of the results for a ≈ . b ≈ .
81 and c ≈ . d DR does not signif-icantly depend on the values of the parameters.)In this work, we study either the fixed-point solutionof Eqs. (A4) (obtained by setting the left-hand sidesto zero) and the spectrum of eigenvalues obtained fromsolving the eigenvalue equations derived by linearizingthe beta functions around the fixed point or the flowof the various functions as a function of the RG time t for initial conditions that are fine-tuned by dichotomy toapproach as close as possible the fixed at large RG time.For the numerical resolution the fields are discretized ona grid. Appendix B: Toy model of NP-FRG flow equationsaround d DR We consider a toy model inspired by the 1-loop flowequation of the RF O ( N )M, which we introduced inRef. [8] to investigate in detail the scenario of appear-ance and disappearance of fixed points in the whole plane( N, d ). It consists in a partial differential equation whichmimics the RG flow of the second cumulant of the renor-malized random field, represented here by a function∆( z ) where z ∈ [ − , ∂ t ∆ k ( z ) =∆ k ( z ) − ∆ k ( z )∆ (cid:48) k ( z ) − (∆ k ( z ) − z ∆ (cid:48) k ( z ))∆ k (1)+ B (cid:2) ∆ k (1) − z ∆ k ( z ) (cid:3)(cid:2) ∆ k ( z ) + z ∆ (cid:48) k ( z ) (cid:3) + A − z )∆ (cid:48) k ( z ) (cid:2) z ∆ (cid:48) k ( z ) − (1 − z )∆ (cid:48)(cid:48) k ( z ) (cid:3) . (B1)The beta function depends on two parameters, A and B , which replace the two parameters d and N of theRF O ( N )M.We showed that for A > / B < /
20, the critical( i.e. , once unstable) cuspy fixed point emerges from thecollapse of two cuspless fixed points when B DR ( A ) =1 / [8(1+ A )] via a boundary-layer mechanism, in a mannerthat appears very similar to the appearance of the cuspycritical fixed point in the RFIM below d DR ≈ .
1. Morespecifically, one can show that within a boundary layerof width (cid:15) , where (cid:15) = B − B DR ( A ), the function ∆ k ( z )can be written as∆ k ( z ) = 1 − (cid:15)f (cid:32)(cid:114) − z(cid:15) (cid:33) , (B2)where (cid:15) → f ( y ) has a cusp at small y = √ − z/(cid:15) , f ( y ) = f (0) + ay + f ( y / y converges tothe outer solution that has a regular expansion in 1 − z , f ( y ) ∼ by + c , when y → ∞ . We find f (0) = − a (2 A +3) / [16( A + 1)], f = 2(8 A + 5) / [3(8 A + 16 A + 9)], b = 1 / (4 A + 3), and c = − (8 A + 3) /
3. The outer solution inthe region where (cid:15) (cid:28) − z (cid:28) − z ) and (cid:15) as∆ ∗ ( z ) = 1+(1 − z )[ − A + 3 + √ (cid:15) d ]+ (cid:15)f (0) 8 A + 33 )+ · · · (B3)where d has an (unilluminating) expression in terms of A and f (0) and where we have used constraints comingfrom matching the inner and outer solutions.To obtain the lowest irrelevant eigenvalue λ when B ≥ B DR ( A ) (which corresponds to d ≤ d DR ) we rewrite thecumulant function as ∆ k ( z ) = ∆ ∗ ( z ) + k λ δ ( z ) and welinearize the flow equation, Eq. (B1), around the fixedpoint. Both in the inner and in the outer region, thisleads to λ = 0 when B = B DR ( A ) and, by consideringthe linearized equation for the outer solution in the regionwhere (cid:15) (cid:28) − z (cid:28)
1, we immediately obtain that λ ∝ √ (cid:15) , i.e. , λ ∝ (cid:112) B − B DR ( A ) for a given A > / A − A DR at B < /
20 fixed).The irrelevant eigenvalue λ associated with a cuspyperturbation around the cuspless fixed point thereforehas a discontinuity in B DR , λ ( B − DR ) − λ ( B + DR ) = λ ( B − DR ) = 2 A − A + 3) = 1 − B DR − B DR )(B4)with A > / B DR < / B DR . In addition, it iseasily checked (see Eq. (20) in Ref. [8]) that for B < B DR there is an irrelevant eigenvalue λ , which is associatedwith a perturbation that has no cusp and which vanishesat B DR with a square-root singularity.We can also show in this toy model that the Larkinlength grows exponentially as one approaches the criticalvalue B DR ( A ) (or A DR ( B ) from the cuspy phase ( B >B DR ). We find that | t L | ∼ √ B − B DR (B5)for A > / A − A DR for B < /
20 fixed. The Larkin length can be obtained fromthe RG time at which the first derivative of the function∆ k ( z ) in z = 1 diverges when starting from cuspless ini-tial conditions and fixing ∆ k (1) = 1 which correspondsto the cuspless fixed point. The flow equation for ∆ (cid:48) k (1)is then easily obtained as ∂ t ∆ (cid:48) k (1) = − B (cid:2) (cid:48) k (1) (cid:3) + ∆ (cid:48) k (1) (cid:2) − (1 + 2 A )∆ (cid:48) k (1) (cid:3) . (B6)In the region where the true fixed point has a cusp, i.e. , for B > B DR ( A ) with A < /
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