diff --git "a/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/load_file.txt" "b/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/AtAzT4oBgHgl3EQfF_uW/content/tmp_files/load_file.txt" @@ -0,0 +1,792 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf,len=791 +page_content='A fixed point can hide another one: the nonperturbative behavior of the tetracritical fixed point of the O(N) models at large N Shunsuke Yabunaka1, ∗ and Bertrand Delamotte2 1Advanced Science Research Center, Japan Atomic Energy Agency, Tokai, 319-1195, Japan 2Sorbonne Universit´e, CNRS, Laboratoire de Physique Th´eorique de la Mati`ere Condens´ee, LPTMC, F-75005 Paris, France.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (Dated: January 4, 2023) We show that at N = ∞ and below its upper critical dimension, d < dup, the critical and tetracritical behaviors of the O(N) models are associated with the same renormalization group fixed point (FP) potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Only their derivatives make them different with the subtleties that taking their N → ∞ limit and deriving them do not commute and that two relevant eigenperturbations show singularities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This invalidates both the ϵ− and the 1/N− expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We also show how the Bardeen-Moshe-Bander line of tetracritical FPs at N = ∞ and d = dup can be understood from a finite-N analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Field theories sometimes exhibit nonperturbative fea- tures such as confinement [1], presence of bound states [2] or exotic excitations [3], fixed points (FPs) of the renor- malization group (RG) flows that are nonperturbative as in the Kardar-Parisi-Zhang equation [4], divergence of the perturbative RG flow at a finite RG scale [5], pres- ence of a cusp in the FP potential as in the random field Ising model [6], to cite but a few.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Very often, these non- perturbative effects are assumed either to occur in rather complicated theories such as gauge and string theories or in highly nontrivial statistical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' O(N) models, which are the simplest scalar field theo- ries, are often implicitly considered to be immune to these complex phenomena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Perturbative methods are there- fore assumed to work almost all the time for these mod- els, the exception to the rule being the Bardeen-Moshe- Bander (BMB) phenomenon [7], related to the existence of a line of tricritical FPs at N = ∞ and d = 3, which re- quires nonperturbative FPs to be fully understood from a large-N analysis [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' From this viewpoint, the enormous success of the ϵ = 4 − d expansion for the perturbative calculation of the critical exponents associated with the Wilson-Fisher (WF) FP [11] could let us believe that the critical physics of the O(N) models is fully understood for any N and d, especially since it is corroborated by the 1/N and ϵ = d − 2 expansions [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Our goal in this Letter is to show instead that although the critical physics of the O(N) models, described by the WF FP, is fully under perturbative control at both finite and infinite N, the tetracritical physics of these models at N = ∞ –and probably of infinitely many multicritical behaviors– is not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show below (i) that at N = ∞, it is also associated with the WF FP, which is unexpected, and (ii) that it nonetheless shows non-perturbative fea- tures that are beyond the reach of the standard imple- mentation of both the large-N and ϵ- expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in particular a very intriguing phenomenon related to the large-N limit of the tetracritical FP of the O(N) models: from the second order, the derivatives of the ∗ yabunaka123@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='com N = ∞ tetracritical FP potential, that is, of the WF FP potential, are not identical to the limit of the derivatives of the finite-N tetracritical FP potentials when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This turns out to be crucial for understanding the large- N limit of tetracritical phenomena and shows that this limit is much less trivial than what is usually said [9–11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The perturbative tetracritical FP corresponds to the massless (ϕ2)4 theory, the upper critical dimension of which is dup = 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is found in perturbation theory in ϵ = 8/3 − d for all N ≥ 1 and it is three times infrared unstable [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Calling λ/(384N 3) the coupling in front of the dimensionless (ϕ2)4 term, the large-N perturbative flow equation for λ reads [13]: ∂tλ = −3ϵλ + 9λ2 4N + O(N −2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (1) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (1), we find that at leading order in N, the nontrivial FP solution is λ∗ = 4ϵN/3 from which follows that perturbation theory does not allow for a control of the large-N limit of the tetracritical FP at fixed ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Only the double limit N → ∞ and ϵ → 0 such that the product ϵN remains finite can possibly be under control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We come back on this point in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Let us recall that in generic dimensions d < 4, the only nontrivial FP found in the standard large-N anal- ysis of the O(N) models is the WF FP [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, no tetracritical FP is found at N = ∞ and d < 8/3 which is paradoxical considering that it is perturbatively found for all N < ∞ and ϵ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show below that the solution to the paradox above lies in the field dependence of the tetracritical FP poten- tial whereas it cannot be obtained from its field expansion and in particular from λ∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The recourse to functional RG methods is therefore mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The best way to implement functional RG is to con- sider Wilson’s RG, as it is inherently functional [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We recall below the take-away philosophy of the modern ver- sion of Wilson’s RG known as the nonperturbative – or functional – renormalization group (NPRG).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' NPRG is based on the idea of integrating fluctuations step by step [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is implemented on the Gibbs free energy Γ [17–23] of a model defined by an Hamiltonian arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01021v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='stat-mech] 3 Jan 2023 2 (or euclidean action) H and a partition function Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' To this model is associated a one-parameter family of models with Hamiltonians Hk = H + ∆Hk and partition func- tions Zk, where k is a momentum scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In Hk, ∆Hk is chosen such that only the rapid fluctuations in the origi- nal model, those with wavenumbers |q| > k, are summed over in the partition function Zk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, the slow modes (|q| < k) need to be decoupled in Zk and this is achieved by giving them a mass of order k, that is by taking for ∆Hk a quadratic (mass-like) term, which is nonvanishing only for the slow modes: Zk[J] = � Dϕi exp(−H[ϕ] − ∆Hk[ϕ] + J · ϕ) (2) with ∆Hk[ϕ] = 1 2 � q Rk(q2)ϕi(q)ϕi(−q), where, for in- stance, Rk(q2) = (k2 − q2)θ(k2 − q2) and J · ϕ = � x Ji(x)ϕi(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The k-dependent Gibbs free energy Γk[φ] is defined as the (slightly modified) Legendre transform of log Zk[J]: Γk[φ] + log Zk[J] = J · φ − 1 2 � q Rk(q2)φi(q)φi(−q) (3) with � q = � ddq/(2π)d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' With the choice of regulator function Rk above, Γk[φ] interpolates between the Hamil- tonian H when k is of order of the ultraviolet cut-off Λ of the theory: ΓΛ ∼ H, and the Gibbs free energy Γ of the original model when k = 0: Γk=0 = Γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The exact RG flow equation of Γk gives the evolution of Γk with k between these two limiting cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is known as the Wetterich equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It reads [18]: ∂tΓk[φ] = 1 2Tr[∂tRk(q2)(Γ(2) k [q, −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' φ] + Rk(q))−1], (4) where t = log(k/Λ), Tr stands for an integral over q and a trace over group indices and Γ(2) k [q, −q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' φ] is the matrix of the Fourier transforms of δ2Γk/δφi(x)δφj(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In most cases, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (4) cannot be solved exactly and approximations are mandatory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The best known approx- imation consists in expanding Γk in powers of the deriva- tives of φi and to truncate the expansion at a given fi- nite order[24–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The approximation at lowest order is dubbed the local potential approximation (LPA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' For the O(N) model it consists in approximating Γk by: Γk[φ] = � x �1 2(∇φi)2 + Uk(φ) � (5) where φ = √φiφi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Fixed points are found only for di- mensionless quantities and the standard large-N limit by rescaling the field and the potential by factors N −1/2 and N −1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, we define the dimen- sionless and rescaled field ¯φ and potential ¯Uk as ¯φ = v − 1 2 d k 2−d 2 N −1/2φ and ¯Uk(¯φ) = v−1 d k−dN −1Uk (φ) with v−1 d = 2d−1dπd/2Γ( d 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The LPA flow of ¯Uk then reads: ∂t ¯Uk(¯φ) = − d ¯Uk(¯φ) + 1 2(d − 2)¯φ ¯U ′ k(¯φ)+ � 1 − 1 N � ¯φ ¯φ + ¯U ′ k(¯φ) + 1 N 1 1 + ¯U ′′ k (¯φ) (6) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: ¯U(¯φ) for the T3 FP of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Green, red, blue and black curves correspond to N = 1500, 2250, 4500 and 42000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The orange dashed curve corresponds to the WF FP at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Inset: Close view of ¯U(¯φ) around ¯φi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' with ∂t = k∂k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The standard large-N limit of the LPA flow equation above is obtained by (i) replacing the fac- tor 1 − 1/N by 1, (ii) dropping the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) because it is assumed to be sub-leading [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As a con- sequence of the two steps above, the explicit dependence in N in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) disappears in the large-N limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The crucial point of the large-N limit is that assuming point (ii) above, the resulting LPA flow equation on ¯Uk can be shown to be exact in the limit N → ∞ [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Under this assumption, all FPs of the O(N) models have been found exactly at N = ∞ [14, 33–36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The result is the following: In a generic dimension d < 4 there is only one nongaussian FP at N = ∞ which is the usual Wilson- Fisher FP (WF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The exceptions to the rule above are the BMB lines of FPs [7, 14, 37–39] existing in dimensions d = 2 + 2/p with p an integer larger than 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that the procedure described above is too restrictive to study the large-N limit of the tetracritical FPs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As said above, the standard large-N analysis con- sists in neglecting the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, this term is negligible only if (1 + ¯U ′′ k (¯φ))−1 does not coun- terbalance at large N its 1/N prefactor for some finite values of ¯φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that because of singularities in the third derivative of ¯Uk(¯φ), the contribution of the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) cannot be neglected in the FP equation of ¯U ′′ k (¯φ) obtained by differentiating twice Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) (see footnote below Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) for more detail).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This turns out to be sufficient to invalidate the standard large-N limit in the tetracritical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have numerically solved Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) and have found for several values of N and d < 8/3 the perturbative tetracritical FP that we call T3(N, d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As expected, T3 bifurcates from the Gaussian FP in d = 8/3−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have followed it down to d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 3 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The FP potential of T3, (i) shows as expected two maxima, one of which being located at ¯φ = 0 and another one at ¯φ2 > 0, and two minima at ¯φ1 and ¯φ3 such that ¯φ3 > ¯φ2 > ¯φ1 > 0, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1, (ii) can be continuously followed up to arbitrarily large values of N at fixed d < 8/3, (iii) has its three extrema T() 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38466 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38464 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38462 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38460 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='80 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='85 $1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 ¯φ1, ¯φ2, ¯φ3 approaching each other when N is increased at fixed d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These extrema tend to a common value ¯φ0 when N → ∞ which is the minimum of the FP potential, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Point (ii) above is paradoxical because it seems to contradict the standard large-N approach where only the WF FP is found in a generic dimension d < 8/3 at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that the WF FP potential at N = ∞ is in fact the limit when N → ∞ of the potential of T3 for d < 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This solves the above paradox because it explains why on one hand there exists a nontrivial tetracritical FP at N = ∞ and d < 8/3 and on the other hand that there is no other nontrivial and smooth solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) at N = ∞ than the WF FP potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, this creates a new paradox since obviously the critical and tetracritical universal behaviors cannot be the same since the two FPs do not have the same number of unstable eigendirections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now explain in detail this new paradox.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We can see on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 that the FP potentials found in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 for large values of N are extremely flat in the region, ¯φ ∈ [¯φ1, ¯φ3] because the three extrema are very close and the height of the barrier between the two minima very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have numerically found that the height of the barrier scales as N −1 and the distance be- tween the two minima as N −1/2 so that the curvatures ¯U ′′(¯φi) at the three extrema approach constant values as N → ∞, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This suggests that ¯U ′′(¯φ) while being well-behaved everywhere but be- tween the three extrema, changes very rapidly within a boundary layer around ¯φ0 of typical width N −1/2, mak- ing divergent ¯U ′′′(¯φ0) when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is not common in physics to encounter this kind of situation where a series of functions fn(x) tends to a smooth function f∞(x) whereas from a certain order p, their derivatives f (p) n (x) do not tend to f (p) ∞ (x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, a simple toy model explains trivially how this can occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Consider the series of functions fn(x) = n−1 sin(n2x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Obviously, f∞(x) ≡ 0 which implies that f ′ ∞(x) ≡ 0 whereas limn→∞ f ′ n(0) = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In our case, at fixed d < 8/3, the limit of the T3 po- tentials when N → ∞ is a nontrivial and well-defined function that therefore must be the WF FP potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have checked that it is indeed the limit of T3 when N → ∞, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The difference between the critical and tetracritical behaviors is therefore not visible on the potentials themselves but only on their derivatives as we now show.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Let us study the boundary layer around ¯φ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is con- venient for what follows to change variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Following Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [40], we define: V (µ) = U(φ) + (φ − Φ)2/2 with µ = Φ2 and φ − Φ = −2ΦV ′(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As above, it is conve- nient to rescale µ and V (µ): ¯µ = µ/N, ¯V = V/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In terms of these quantities, the FP equation for ¯V (¯µ) reads 0 = 1 − d ¯V + (d − 2)¯µ ¯V ′ + 4¯µ ¯V ′2 − 2 ¯V ′ − 4 N ¯µ ¯V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) has two remarkable features: (i) it is much sim- pler than Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) because the nonlinearity comes only 0 2 4 6 8 10 12 14 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="08 V''[μ] N=6×103 N=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='7×104 N=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2×106 N=∞WF FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Second derivative of the WF and T3 FP potentials for different values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' from the ( ¯V ′)2 term, (ii) it is the LPA equation obtained from the Wilson-Polchinski (WP) version of the NPRG [15, 41, 42].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, ¯V (¯µ) is related to the potential ¯U(¯φ) of the Wetterich version of the RG by the Legendre trans- form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The standard large-N analysis per- formed in this version of the NPRG consists here again in neglecting the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) because it is sup- pressed by a 1/N factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Under the assumption that this term is indeed negligible, the resulting equation can be solved exactly in the large-N limit [14, 35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, at large N, it is clear on Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) that we have to deal with singular perturbation theory since the small parameter used in the 1/N expansion is in front of the term of high- est derivative, that is, ¯V ′′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In this case, it is well-known that at large N a boundary layer can exist for a partic- ular value of ¯µ that becomes a singularity at N = ∞, making this term non negligible [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The value of ¯µ corresponding to ¯φ0 is called ¯µ0 and is the minimum of ¯V (¯µ) at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We find for ¯V (¯µ) the same features about its three extrema ¯µi as for ¯U(¯φ) at ¯φi: The three extrema ¯µi approach each other and to ¯µ0 as N → ∞, the distances between them scale as N −1/2 and the curvatures ¯V ′′(¯µi) as N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Taking into account the scaling around ¯µ0 inside the boundary layer, we introduce another scaled variable ˜µ = N 1/2(¯µ − ¯µ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Since at N = ∞, ¯V ′(¯µ) vanishes at ¯µ = ¯µ0, ¯V (¯µ0) should approach 1/d at leading order in N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We therefore define a scaled boundary layer by ˜VN(˜µ) = N � ¯V (¯µ0 + N −1/2˜µ) − 1/d � which implies ˜V ′′ N(˜µ) = ¯V ′′(¯µ0 + N −1/2˜µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We plot ˜V ′′ N(˜µ) for several values of N in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' By substituting ˜VN(˜µ) by its value in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) and solving it at order O(N −1/2), we find that ¯µ0 = 2/(d−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At order O(N −1), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) becomes − 8 ˜V ′′ ∞(˜µ) d − 2 + 8 ˜V ′ ∞(˜µ)2 d − 2 +(d−2)˜µ ˜V ′ ∞(˜µ)−d ˜V∞(˜µ) = 0 (8) [44] which is clearly invariant under ˜µ → −˜µ from which it follows that ˜V ′ ∞(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At ˜µ = ∞, ˜V ′′ ∞(˜µ) should tend to a finite value that matches with ¯V ′′(µ) at ¯µ+ 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This implies that the solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) should be quadratic when ˜µ → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Substituting ˜V∞(˜µ) by ˜V ′′ ∞(˜µ = ∞)˜µ2/2 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) and balancing the leading terms as ˜µ → ∞, we find that ˜V ′′ ∞(˜µ = ∞) = (−d2 + 6d − 8)/16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Imposing 4 the two boundary conditions found above at ˜µ = 0 and ˜µ = ∞ selects a unique and globally defined solution ˜V ′′ ∞(˜µ) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We find ¯V ′′(¯µ+ 0 ) = ¯V ′′ WF(¯µ0) = ˜V ′′ ∞(˜µ = ∞) which proves the matching at N = ∞ between the boundary layer and the potential outside of the layer, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' the boundary layer for ¯U ′′(¯φ) analogous to that of ¯V ′′(¯µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' To conclude, we have proven that for d < 8/3, a boundary layer develops at large N for the second derivative of the T3 potential that becomes a singularity when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' What remains to be understood is its physical relevance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At first sight, what we have obtained for T3 looks para- doxical because we could think that its potential being identical to the WF potential at N = ∞, the linearized flow around these two FPs should also be identical and thus the same for all critical exponents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We now show that this naive argument is wrong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have computed in d < 8/3 the relevant eigenvalues of the RG flow around T3 and WF at finite and large N and as expected we have found three for T3 and one for WF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' When N → ∞, one of the three eigenvalues at T3 tends as expected to d − 2 which is the relevant eigen- value ν−1 of the critical WF FP at N = ∞ [11, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The nontrivial point is that the two other relevant eigenvalues at T3 have a well-defined limit when N → ∞ although they do not play any role for the critical behavior of the O(N = ∞) model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The solution to this paradox is that they are associated with eigenperturbations that become singular when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' That these two eigenperturba- tions become singular is clear for one of them, called δ ¯V2, on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 9 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' As for the other one, δ ¯V1, its slope at ¯µ0 diverges as N 1/3 which implies that at N = ∞, it becomes discontinuous at ¯µ0, see Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 9 and 10 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' For ordinary second order phase transitions, these eigenperturbations are excluded which explains that the associated relevant eigenvalues do not play any role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This solves all the paradoxes associated with the tetracritical FPs at N = ∞ and d < 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' What remains to be studied is the particular case N = ∞ and d = 8/3 where a line, called the BMB line, of smooth tetracritical FPs shows up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is obtained in the WP version of the RG by integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) in which the last term, proportional to 1/N, has been discarded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It is given by the following implicit expression [14]: ¯µ± = C ¯V ′ � 1 − 2 ¯V ′� � ±2 ¯V ′ 1 − 2 ¯V ′ �4/3 + 2f(4 ¯V ′), (9) where f(x), which is analytic for x < 2, is given by f(x) = 3 2 − x + 4x (2 − x)7/3 � 1 0 dz �2 − xz z �1/3 (10) and ¯µ± correspond to the two branches ¯µ > 3 and ¯µ < 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The derivative of the potential ¯V ′ is positive (negative) on the former (latter) branch and C is a non- negative integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯V (¯µ) is analytic at ¯µ = ¯µ0 = 3 and ¯V ′(¯µ = 3) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 7 of the Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' different ¯V ′(¯µ) corresponding to different FPs of the BMB line are shown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' All FPs along the BMB line share the same critical exponents, that is, the exponents of the Gaussian FP which is itself tetracritical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Notice that the WF FP which corresponds to C = 0, is the end point of this line and deserves special attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We come back on this point in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (1), we have seen that λ∗ remains constant at leading order in 1/N along the hyperbola of constant ϵN of the (d, N) plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This suggests that when the double limit d → 8/3 and N → ∞ is taken at fixed α = ϵN, T3 converges in d = 8/3 to one of the FPs of the BMB line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We have analytically and numerically checked this and have derived analytically the relation between α and C: α = 162/C3, see Suppl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Two extreme cases are worth studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' First, the Gaussian FP corresponds to the limit N → ∞ at fixed dimension d = 8/3, that is, at α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It corresponds to C = ∞ in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Second, α = ∞, which implies C = 0, corresponds to taking the limit ϵ → 0 at fixed N = ∞, that is, to following the WF FP at N = ∞ up to d = 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, at finite ϵ and N = ∞, we know from the analysis above that the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) can- not be neglected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Consistently, the same occurs for the BMB line: the WF FP potential is indeed the end point of the BMB line obtained by taking the limit C → 0 in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) but the derivatives of this potential can only be studied by retaining the last term in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Here again, this explains why the T3 FP in the C → 0 limit is three times unstable and not only once unstable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' To conclude, we have solved the paradox of the appar- ent absence of a nontrivial tetracritical FP at N = ∞ and d < 8/3 by showing that this FP does exist but is nothing else than the WF FP up to the subtlety that the derivatives of the tetracritical FP potential are not the derivatives of the WF FP potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This makes the large-N limit of the O(N) model much less trivial than is usually advocated at least for multicritical phenom- ena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The fact that the tetracritical FP has two more unstable infrared directions than the WF FP is related to this subtle point because they are associated with sin- gular eigenperturbations, a possibility which is usually not considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We conjecture that what has been found above at large N and for d ≤ 8/3 is valid for all mul- ticritical points with an odd number of eigendirections below or at their upper critical dimension because the BMB lines for all of them terminate at the WF FP [14], a fact that in itself is almost enough to imply everything else.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Let us finally point out that what we have found for the tetracritical FP is very different from what was found around d = 3 at large-N in the tricritical case which re- quired the existence of new FPs to be fully understood at finite N [45–48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We also conjecture that this phe- nomenon is not specific to the O(N) models but should rather be generic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We acknowledge A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Codello and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Defenu and for correspondence and discussions at an early stage of this 5 work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' was supported by Grant-in-Aid for Young Scientists (18K13516).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [1] M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Pel´aez, U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Reinosa, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Serreau, M.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Fields 58, 619 (1993).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [20] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Morris, Int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' A 09, 2411 (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [21] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Berges, N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Tetradis and C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Wetterich, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 363, 223 (2002) [22] B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mouhanna, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Tissier, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' B 69, 134413 (2004).' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Mouhanna, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Vidal, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' B 68, 064421 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [25] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Canet, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' B 71, 012418 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [26] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Kloss, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Canet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, and N.' metadata={'source': 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N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Wschebor, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' E 84, 061128 (2011);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' E 86, 019904(E) (2012).' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Chat´e, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 92, 255703 (2004);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 95, 100601 (2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [30] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Canet, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Wschebor, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} 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+page_content=' 119, 191602 (2017);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 121, 231601 (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [48] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Yabunaka, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Fleming, and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Delamotte, Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' E 106, 054105 (2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6 SUPPLEMENTAL MATERIALS I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' T3 FP POTENTIALS IN d < 8/3 We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 3 the tetracritical FP potential ¯U(¯φ) obtained with the LPA and solution of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) for small values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' They have the typical shape of a tetracritical potential showing two nontrivial minima.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 0 2 4 6 8 ϕ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='38 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='385 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='39 U(ϕ) N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 N=1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯U(¯φ) for the T3 FP for different values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' LARGE-N BEHAVIOR OF THE EXTREMA OF THE TETRACRITICAL POTENTIAL The three nontrivial extrema of the T3 FP potential in either the WP or Wetterich version of the RG, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 of the main text, behave the same way when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show on Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4 the scaling in N of the height of the barrier between the extrema ¯φi of the rescaled potential ¯U(¯φ) of the Wetterich version of the RG, as well as the distance between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These extrema are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 2500 3500 4500N 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0×10-6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0×10-6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0×10-5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2×10-5 U[ϕ2]-U[ϕ3] 2500 3500 4500N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0325 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0350 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0375 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0400 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0425 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0450 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0475 ϕ3-ϕ2 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Left: Height of the potential barrier for the T3 FP of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) for large values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The equation of the full line is y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0257/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Right: Distance between the maximum ¯φ2 and the minimum ¯φ3 for the T3 FP of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6) for large values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 (blue dots).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The equation of the full line is y = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='12506/N 1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Since the height of the barrier, ∆ ¯U, scales as N −1 and the distance between the extrema, ∆¯φ, as N −1/2, a simple dimensional argument shows that the curvatures at these extrema that goes as ∆ ¯U/(∆¯φ)2, do not scale with N, that is, tend to constants when N → ∞, a fact that we have numerically checked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Thus, for d < 8/3 and at large and finite N, the curvature of ¯U(¯φ) varies between a positive value at ¯φ1, a negative value at ¯φ2 and again a positive value at ¯φ3 on a distance of order N −1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' THE SCALED BOUNDARY LAYER ˜V ′′(˜µ) By translating and rescaling by a factor N 1/2 the po- sition and the width of the boundary layer of the second derivative of the potential ¯V , it is possible to obtain a finite limit for this scaled boundary layer when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We thus define the scaled variable ˜µ = N 1/2(¯µ − ¯µ0) where ¯µ0 is the location of the boundary layer and the scaled potential by ˜VN(˜µ) = N � ¯V (N −1/2˜µ + ¯µ0) − 1/d � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It follows from the definitions above that ˜V ′′ N(˜µ) = ¯V ′′(¯µ0+N 1/2˜µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5 this scaled boundary layer for different values of N at large N as well as its limit ˜V ′′ ∞(˜µ) at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1000 500 500 1000 μ˜ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='14 N=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5×104 N=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='7×105 N=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="2×106 ˜V''[μ] ˜ FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The scaled boundary layer for the second derivative of the T3 FP potential ˜V ′′ N(˜µ), Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) for large values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The dashed curve is the global solution ˜V ′′ ∞(˜µ) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (8) at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The red horizontal line is y = (−d2+6d−8)/16 for d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' It coincides with ˜V ′′ ∞(˜µ = ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Notice that a finite difference ¯µ − ¯µ0 translates into an infinite ˜µ when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The matching at N = ∞ be- tween the scaled boundary layer and the value of ¯V ′′(¯µ) outside of the layer therefore requires that ˜V ′′ ∞(∞) = ¯V ′′(¯µ+ 0 ) = (−d2 + 6d − 8)/16 which is the case with the solution for the scaled boundary layer given in the main text, see also Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' THE BOUNDARY LAYER OF ¯U ′′(¯φ) The boundary layer has been derived in the main text in WP version of the RG because it is simpler in this version than in Wetterich version.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' However, it can also be derived directly in this latter version or, once it is 7 obtained in one version, it can be translated in the other by performing the Legendre transform given in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="5 ϕ 5 10 15 U''[ϕ] N=1." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='5⨯104 N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2⨯105 N=4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='2⨯106 WF N=∞ FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The second derivative of the T3 FP potential ¯U ′′(¯φ) in the Wetterich version of the RG, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (6), for different values of N in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 6 the boundary layer of ¯U ′′(¯φ) for different values of N at large N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' DIFFERENT FP POTENTIALS OF THE BMB LINE We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 7 the first derivative of different FP potentials of the BMB line at N = ∞ and in d = 8/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These FP potentials, implicitly given by the exact ex- pression given in Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) and (10) of the main text, are indexed by the nonnegative constant C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The WF FP potential corresponds to C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="15 V'[μ] C=2 C=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='625 C=0 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯V ′(¯µ) for different FPs indexed by the constant C on the BMB line given by Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) and (10) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We emphasize that the limit of ¯V ′′(¯µ0 = 3), when C → 0 is not given by the second derivative of the WF FP potential which is however the limit when C → 0 of ¯V (¯µ) along the BMB line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This is consistent with what happens at fixed d < 8/3 when N → ∞ since the limit d → 8/3 at fixed α = ∞ consists in following the WF FP at N = ∞ up to d = 8/3, the derivatives of which are not the limit of the derivatives of the T3 potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' EIGENPERTURBATIONS AT THE TETRACRITICAL FP FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: Eigenperturbation δ ¯V3(¯µ) at the T3 FP corresponding, when N → ∞, to the relevant eigenvalue λ3 = d − 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 8 and 9 the relevant eigenperturba- tions δ ¯Vi of the T3 FP in d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6 for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Whereas δ ¯V3 tends to the relevant eigenperturbation of the critical WF FP –with eigenvalue d − 2 which is the inverse of the critical exponent νWF–, the two others become singular in the N → ∞ limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This is the rea- son why they play no role for the critical behavior of the O(N) model at N = ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 δV1[μ] N=5×102 N=3×103 N=1×106 0 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 δV2[μ] N=5×102 N=3×103 N=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='4×105 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: Eigenperturbations δ ¯Vn(¯µ) for n = 1, 2 at the T3 FP corresponding respectively to the relevant eigen- values λ1 ≃ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 and λ2 ≃ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='326 for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These eigenperturbations tend to singular functions of ¯µ when N → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 10 that the slope of δ ¯V1(¯µ) at ¯µ0 in- creases as N 1/3 which proves that this eigenperturbation becomes discontinuous at infinite N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' [μ] V[] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='000 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 35 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='0 N=3x103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='005 N=3x104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 WF N=8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00 μ 1 2 3 5 6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='028 1000 104 105 106 N 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="20 δV 1'[10/3] FIG." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' d = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='6: Slope of the eigenperturbation δ ¯V1(¯µ) of the T3 FP at its minimum ¯µ0 = 10/3 for different values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The equation of the full line is y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='00175N 1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' BMB LINE AND THE JOINED LIMIT ϵ → 0 AND N → ∞ AT FIXED α = ϵN When a T3 FP is followed along the hyperbola d = 8/3 − α/N, α ≥ 0, of the (d, N) plane, its potential con- verges when N → ∞ to the potential of one of the FPs of the BMB line, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We derive below the re- lationship α = 162/C3 between the parameter α of the hyperbola and the parameter C that indexes the FPs along the BMB line, see Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) and (10) of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This relationship can be derived as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' The FP potential of T3 is expanded as ¯V (¯µ) = ∞ � n=0 an(¯µ − 3)n (1) around the minimum ¯µ0 = 3 of the N = ∞ potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Then, the coefficients an are expanded as an = a(0) n + N −1a(1) n + O(N −2) (2) in power of 1/N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At order O(N 0), Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (7) yields a(0) n = 0 for n = 1, 2 and 3 and recursively determines a(0) n for n larger than 5 in terms of a(0) 4 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' Now, �∞ n=0 a(0) n (¯µ−3)n is the expansion of a FP poten- tial of the BMB line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' For this potential, ¯µ± behaves from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' (9) as ¯µ± ≃ 3 ± 24/3C| ¯V ′|1/3 for C ̸= 0 and | ¯V ′| ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' This implies that a(0) 4 is related to C by a(0) 4 = 3/(192C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' At order O(N −1), it can be shown that a(1) n for all n but n = 4 can be recursively determined in terms of a(0) 4 or C, if and only if the condition α = 162/C3 is satisfied which proves the relationship between these two parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' We show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11 different T3 FP potentials along an hyperbola d = 8/3 − α/N with increasing values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' These FP potentials converge to a potential corresponding to the FP on the BMB line indexed by C = (162/α)1/3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 0 1 2 3 4 5 6 7 μ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content="10 V'[μ] C=0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='672, N=∞ N=4000 α=1600/3 N=8000 α=1600/3 N=24000 α=1600/3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' ¯V ′(¯µ) of the T3 FP followed on the hyperbola d = 8/3 − α/N with fixed α = 1600/3 for increasing values of N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content=' In the double limit d → 8/3 and N → ∞, it converges to the FP potential of the BMB line corresponding to C = (162 × 3/1600)1/3 ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'} +page_content='672.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/AtAzT4oBgHgl3EQfF_uW/content/2301.01021v1.pdf'}