We find that the multicritical fixed point structure of the O(N) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d=3) as well as at N=. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N. Many features found for the O(N) models are shared by the O(N) - O(2) models relevant to frustrated magnetic systems.
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TY - JOUR
T1 - Surprises in O (N) Models
T2 - Nonperturbative Fixed Points, Large N Limits, and Multicriticality
AU - Yabunaka, Shunsuke
AU - Delamotte, Bertrand
N1 - Funding Information: O ( N ) Models: Nonperturbative Fixed Points, Large N Limits, and Multicriticality Yabunaka Shunsuke 1 Delamotte Bertrand 2 Fukui Institute for Fundamental Chemistry, 1 Kyoto University , Kyoto 606-8103, Japan Laboratoire de Physique Théorique de la Matière Condensée, UPMC, CNRS UMR 7600, 2 Sorbonne Universités , 4, place Jussieu, 75252 Paris Cedex 05, France 7 November 2017 10 November 2017 119 19 191602 23 July 2017 25 September 2017 © 2017 American Physical Society 2017 American Physical Society We find that the multicritical fixed point structure of the O ( N ) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions ( d = 3 ) as well as at N = ∞ . These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N . Many features found for the O ( N ) models are shared by the O ( N ) ⊗ O ( 2 ) models relevant to frustrated magnetic systems. Japan Society for the Promotion of Science http://dx.doi.org/10.13039/501100001691 JSPS http://sws.geonames.org/1861060/ 263111 Grant-in-Aid for Young Scientists (B) The O ( N ) -symmetric and Ising statistical models have played an extremely important role in our understanding of second order phase transitions both because many experimental systems show this symmetry and because they have been the playground on which almost all the theoretical formalisms aiming at describing these phase transitions have been developed and tested: Integrability  , large- N [2,3] , 4 - ε  and 2 + ε  expansions, conformal field theory  , high and low temperature expansions  , and bootstrap program  , all these methods were born here. It is by now widely believed that everything is known about the criticality of the O ( N ) models either exactly or with an accuracy that is limited only by our finite computational ability. Let us summarize the common wisdom about criticality of the O ( N ) models, see Fig. 1 , because this is what we want to challenge in this Letter  . We start in infinite dimension where the mean-field approximation is exact. Lowering the dimension d down to d = 4 , the critical exponents remain those of the mean-field approximation because large scale fluctuations are Gaussian-like. This means that the only infrared fixed point (FP) of the renormalization group (RG) flow with one unstable eigendirection (1-unstable) is the Gaussian FP ( G ) for d ≥ 4 . Since the potential part of the Hamiltonian of the O ( N ) model can only involve ( φ 2 ) n terms, each time the dimension decreases enough for such a term to become relevant around G , that is, becomes perturbatively renormalizable, a new nontrivial FP emerges from G . For instance, in d = 4 - ε , the ( φ 2 ) 2 term becomes relevant at G and a new FP, called the Wilson-Fisher FP (WF), appears. It drives the second-order phase transition of the O ( N ) models in d < 4 and is 1-unstable while G becomes 2-unstable. The ( φ 2 ) 3 term becomes relevant in d = 3 - ε and a nontrivial 2-unstable FP emerges from G that becomes 3-unstable. This scenario repeats in each critical dimension d n = 2 + 2 / n below which a new n -unstable multicritical FP appears that we call T n . The FP T 2 is tricritical because it lies in the coupling constant space on the border separating the domain of second order and first order phase transitions. The common wisdom is that all the T n FPs can be followed by continuity in d down to d = 2 for all values of N . This is corroborated by the fact that in the Ising case ( N = 1 ), it has been rigorously proven that indeed all the T n exist in d = 2 and are nontrivial  . Because of Mermin-Wagner theorem, the situation is physically different for N ≥ 2 but at least T 2 can be followed smoothly from d = 3 - ε down to d = 2 for N = 2 , 3, and 4  . Notice that the N = 2 , d = 2 case is peculiar because topological defects can trigger in this case a finite-temperature phase transition. 1 10.1103/PhysRevLett.119.191602.f1 FIG. 1. Summary of the common wisdom: Below each critical dimension d n = 2 + 2 / n a new FP emerges from the Gaussian FP G . T 2 and T 3 stand or the tricritical and tetracritical FPs. The BMB FPs exist only at N = ∞ and in the critical dimensions d n > 1 . At N = ∞ , exact results can be derived such as a closed and exact RG flow equation for the Gibbs effective potential  . The common wisdom is that at N = ∞ and in generic dimensions 2 < d < 4 , the only nontrivial and nonsingular FP is WF, which is simple to obtain after an appropriate rescaling by a factor N  . Its nonsingular character means that it is a regular function of the field. The limit N = ∞ is in fact peculiar because in all the d n with n ≥ 2 , and only in these dimensions, there also exists a line of FPs. In d = 3 , this line corresponds to tricritical FPs sharing all the same (trivial) critical exponents. This line starts at G and terminates at the Bardeen-Moshe-Bander (BMB) FP whose effective potential is nonanalytic at vanishing field, see Fig. 1 [13–16] . It is surprising that this common wisdom about the O ( N ) models raises a simple paradox that, to the best of our knowledge, has remained unnoticed up to now. Let us first assume that for the O ( N ) models, the exact RG flow equation of the Gibbs free energy Γ —also called effective action—is continuous in d and N . Then, assuming moreover that the FPs Γ * of these flows are well-defined functions of d and N , they must also be continuous functions of these parameters and can therefore be followed smoothly in the ( d , N ) plane. For constant fields, the functional Γ * [ ϕ ] reduces to the effective potential U * ( ϕ ) . If U * can be Taylor expanded: U * ( ϕ ) = ∑ m g m * ( ϕ 2 ) m with ϕ = ⟨ φ ⟩ , the smoothness of Γ * as a function of d and N implies that of the g m * , which can, therefore, be followed continuously along a given path of the ( d , N ) plane. Notice that we do not need in the following to expand U * and we indeed do not expand it. However, the same continuity argument can be used on the function U * itself rather than on its couplings. Let us now consider, for instance, the tricritical FP T 2 . The paradox appears when we try to follow smoothly T 2 from a point in the ( d , N ) plane where we know from perturbation theory that it exists to a point where, according to the common wisdom, it is believed not to exist. We consider, for instance, the path shown in Fig. 2 starting at Q in d = 3 - and N = 40 and going to N = ∞ in d = 2.8 . How can we solve the apparent contradiction that T 2 should evolve continuously and that it exists at one end of the path, that is, in Q , and not at the other end? The simplest solution is that either T 2 disappears before reaching N = ∞ or it becomes singular at N = ∞ . We shall see in the following that both these possibilities are indeed realized depending on the path followed to reach N = ∞ . In particular, we shall see that there exists a line N c ( d ) [or, equivalently, d c ( N ) ], see Fig. 2 , such that when T 2 is followed along a path that crosses this line—such as the path shown in Fig. 2 that starts in Q —it collapses with another FP on the line N c ( d ) and disappears. This is why T 2 is not found at N = ∞ for d < 3 . And the paradox is now clear: According to the common wisdom, no known FP is available for collapsing with T 2 . We must therefore conclude that the common wisdom yields an incomplete picture and that there is a new FP—that we indeed find and call C 3 —with which T 2 collapses on N c ( d ) . Part of the solution to the paradox above is that C 3 is nonperturbative: It cannot emerge from G in any upper critical dimension because the stability of G in the O ( N ) models is well known for all d and N from perturbation theory. This is why C 3 has never been found previously. Some natural questions are then: What is the stability of C 3 ? Does it exist in d = 3 for some values of N ? Is it the only nonperturbative FP of the O ( N ) models? Since, most probably, it does not appear alone, where does it appear and together with which other FP? Does it exist in the large- N limit and why is it not found in the usual 1 / N expansion [2,3,12] ? It is the aim of this Letter to provide a first study of these different questions. 2 10.1103/PhysRevLett.119.191602.f2 FIG. 2. The two curves N c ( d ) and N c ′ ( d ) , respectively, defined by T 2 = C 3 and C 2 = C 3 and the curve 3.6 / ( 3 - d ) . N c ( d ) is calculated with the LPA (red circles) and at order 2 of the derivative expansion (blue squares). We show a path joining the point Q located at ( d = 3 - , N = 40 ) to the point at N = ∞ and d = 2.8 . The method of choice for studying FPs beyond perturbation theory is the nonperturbative (also called functional) renormalization group (NPRG) which is the modern implementation of Wilson’s RG. It allows us to devise accurate approximate RG flows. The NPRG is based on the idea of integrating fluctuations step by step  . In its modern version, it is implemented on the Gibbs free energy Γ [18–21] . A one-parameter family of models indexed by a scale k is thus defined such that only the rapid fluctuations, with wave numbers | q | > k , are summed over in the partition function Z k . The decoupling of the slow modes ( | q | < k ) in Z k is performed by adding to the original O ( N ) -invariant ( φ 2 ) 2 Hamiltonian H a quadratic (masslike) term which is nonvanishing only for these modes, Z k [ J ] = ∫ D φ i exp ( - H [ φ ] - Δ H k [ φ ] + J · φ ) , (1) with Δ H k [ φ ] = 1 2 ∫ q R k ( q 2 ) φ i ( q ) φ i ( - q ) —where, for instance, R k ( q 2 ) = α Z ¯ k q 2 [ exp ( q 2 / k 2 ) - 1 ] - 1 with α a real parameter and Z ¯ k the field renormalization—and J · φ = ∫ x J i ( x ) φ i ( x ) . The k -dependent Gibbs free energy Γ k [ ϕ ] is defined as the (slightly modified) Legendre transform of log Z k [ J ] : Γ k [ ϕ ] + log Z k [ J ] = J · ϕ - 1 2 ∫ q R k ( q 2 ) ϕ i ( q ) ϕ i ( - q ) , (2) with ∫ q = ∫ d d q / ( 2 π ) d . The exact RG flow equation of Γ k reads  ∂ t Γ k [ ϕ ] = 1 2 Tr [ ∂ t R k ( q 2 ) ( Γ k ( 2 ) [ q , - q ; ϕ ] + R k ( q ) ) - 1 ] , (3) where t = log ( k / Λ ) , Tr stands for an integral over q and a trace over group indices, and Γ k ( 2 ) [ q , - q ; ϕ ] is the matrix of the Fourier transforms of the second functional derivatives of Γ k [ ϕ ] with respect to ϕ i ( x ) and ϕ j ( y ) . For the systems we are interested in, it is impossible to solve Eq. (3) exactly and we therefore have recourse to approximations. The most appropriate nonperturbative approximation consists in expanding Γ k [ ϕ ] in powers of ∇ ϕ [22–31] . At order two of the derivative expansion, Γ k reads Γ k [ ϕ ] = ∫ x ( 1 2 Z k ( ρ ) ( ∇ ϕ i ) 2 + 1 4 Y k ( ρ ) ( ϕ i ∇ ϕ i ) 2 + U k ( ρ ) + O ( ∇ 4 ) ) , (4) where ρ = ϕ i ϕ i / 2 . Within this approximation, all critical exponents are accurately computed for all d and N . The LPA’ (Local Potential Approximation’) is a simpler approximation consisting in setting in Eq. (4) , Y k ( ρ ) = 0 and Z k ( ρ ) = Z ¯ k , a field-independent field renormalization. From Z ¯ k is derived the running anomalous dimension η t = - ∂ t log Z ¯ k that converges at the FP to the anomalous dimension η . The LPA consists in setting Z ¯ k = 1 , which implies η = 0 . The RG flow is one-loop exact in the ε = 4 - d (or ε = 3 - d for T 2 ) expansion for both the LPA and LPA’ and is also one-loop exact for the LPA’ for N > 1 , in the ε ′ = d - 2 expansion. Most importantly for what follows, even within the LPA, the flow of the effective potential U k is exact at N = ∞ . We give the flow of the effective potential U k for any N at the LPA in the Supplemental Material  . We have numerically integrated the fixed point equation for the effective potential ∂ t U ˜ * = 0 , Eq. (S.3) in the Supplemental Material  , at the LPA and LPA’. As expected, we find T 2 for any N emerging from G in d = 3 - . For sufficiently small values of N , typically N < 19 , we find that we can follow this FP down to d = 2 using the LPA’. For N > 19 , we find that by decreasing d at fixed N , T 2 disappears in a dimension d c ( N ) by collapsing with a 3-unstable FP that we call C 3 as already explained above, see Figs. 2 and 3 . We find that the line N c ( d ) is asymptotic to the d = 3 axis, see Fig. 2 , as expected for the disappearance of T 2 just below d = 3 at large N . A very good fit of the N c ( d ) curve is 3.6 / ( 3 - d ) , see Fig. 2 . We note that this result is fully consistent with six-loop calculations performed within the ε = 3 - d expansion, see Pisarski  and Osborn and Stergiou  . Within this ε expansion, these authors found that at leading order in 1 / N , T 2 can exist only when N ε < 36 / π 2 ≃ 3.65 , which is our bound N c ( d ) up to the numerical uncertainty on the prefactor 3.6 of our fit above. While this bound has been interpreted as the radius of convergence of the ε expansion at large N  , our results show that it is the location of the coalescence of T 2 with C 3 . 3 10.1103/PhysRevLett.119.191602.f3 FIG. 3. Singular point S and the two lines N c ( d ) (red squares) and N c ′ ( d ) (blue stars). Starting from P , the FP T 2 is followed along a clockwise (left) or anticlockwise (right) closed path surrounding S . On the clockwise path, T 2 becomes C 2 after a full rotation. On the anticlockwise path, T 2 collides with C 3 on N c ( d ) and disappears. It actually becomes complex-valued and remains so all along the dashed path. On N c ′ ( d ) it becomes real again but is now C 2 . The path joining N c ( d ) and N c ′ ( d ) at fixed N = 33 is also shown in panel (a). In panel (b), we indicate which (real) multicritical FPs exist in each region. In the white region, there is only one multicritical FP with two directions of instability that can be continuously followed from both T 2 and C 2 depending on the path followed. We have checked that the picture above is quantitatively stable when we go from the LPA to the order two of the derivative expansion, Eq. (4) , see Fig. 2 . This is completely consistent with the fact that η is very small on the curve N c ( d ) for N sufficiently large and decreases at large N which makes the LPA flow of U k exact at N = ∞ . For instance, for N = 40 , we find d c ( 40 ) = 2.924 and in this dimension, η = 1.7 × 10 - 3 . Thus, although we have no rigorous proof, we can safely claim that the existence of C 3 is doubtless and that the curve N c ( d ) approaches N = ∞ when d → 3 . We show the T 2 = C 3 FP potential shape on N = N c ( d ) in the Supplemental Material  . It is a regular function of ρ at N = ∞ , which is not the case for the BMB FP, which shows a cusp. Let us now follow C 3 by increasing d . We choose for instance N = 33 and we follow the path shown in Fig. 3(a) starting at d c ( N = 33 ) = 2.90 . We find that C 3 exists on this path up to d = 3.09 which shows that a nonperturbative FP can exist in d = 3 . In d = 3.09 , it collapses with a 2-unstable FP, that we call C 2 and both these FPs do not exist for d > 3.09 . The FP C 2 cannot be T 2 because T 2 does not exist above d = 3 . By changing the value of N , we generate a line where C 3 = C 2 that we call N c ′ ( d ) , see Figs. 2 and 3 . We find two interesting features of the curve N c ′ ( d ) . First, the two curves N c ( d ) and N c ′ ( d ) meet in a point, that we call S , located at ( d = 2.81 , N = 19 ), see Figs. 2 and 3 . This means that right at S : T 2 = C 3 = C 2 . We also find that S is a singular point: If we follow smoothly T 2 around a closed loop containing S starting, for instance, at P = ( d = 2.94 , N = 30 ) , see Fig. 3 , we do not come back at T 2 . More precisely, starting from P and following an anticlockwise closed path as in Fig. 3(b) , T 2 collides on the line N c ( d ) with C 3 and disappears. More precisely, it becomes complex. On the contrary, following the same path clockwise, T 2 does not collide with any FP but becomes C 2 after a full rotation around S . This is why we have claimed above that the fate of T 2 when N → ∞ depends on the path followed. In the Supplemental Material  , we give a toy model in terms of the roots of a cubic equation that shows how T 2 can become C 2 when it is continuously followed along a closed path surrounding S . From a purely mathematical point of view, the continuity argument for following smoothly the FPs everywhere in the ( d , N ) plane and exhibiting the double-valued structure of T 2 and C 2 makes sense only after allowing the FPs to be complex valued (or, in a Taylor expansion, the g m * to be complex). For instance, let us again consider Fig. 3(b) . We start at P with T 2 , which is very close to G . Beyond the line N c ( d ) , T 2 becomes complex. It becomes real again when the path crosses N c ′ ( d ) and it is then C 2 which is far from G. If we go on following the same path, C 2 remains real all the way but after the second full rotation, it is T 2 again. The second interesting feature of the curve N c ′ ( d ) is that it also becomes vertical at large N while being this time asymptotic to the d = 4 axis, see Fig. 2 . We therefore conclude that most probably C 3 exists at N = ∞ everywhere for d ∈ ] 3 , 4 [ and C 2 for d ∈ ] 2 , 4 [ . However, we also find that for larger and larger N in d > 3 , the FP potentials of C 2 and C 3 become steeper and steeper at ρ = 0 which indicates the presence of a singularity at the origin in their FP potential or its derivatives. The second derivative of the two potentials with respect to ρ becomes also discontinuous at a point ρ ≠ 0 in the large N limit. These singularities are a possible explanation of the fact that these two fixed points were not found previously in large N analyses [11–14,16] . Using the LPA’, we have checked that the line N c ′ ( d ) is only slightly modified compared to the LPA results because η is small all along this line. It makes us confident that the overall picture above is not an artifact of our truncations. The double-valued character of the FPs exhibited above concerns only C 2 and T 2 and we could wonder whether the same thing occurs for C 3 . We have indeed found two other nonperturbative FPs that are 3- and 4-unstable, two analogues of the curves N c ( d ) and N c ′ ( d ) , where these FPs show up and annihilate as well as a singular point S ′ where the two curves meet and that shares many similarities with S . It is of course tempting to imagine that this kind of structure repeats for the 4-unstable FP found that itself involves a 5-unstable FP and so on and so forth. A natural question is whether the intricate FP structure presented above is specific to the O ( N ) models or is generic. To shed some light on this question, we have therefore considered the O ( N ) ⊗ O ( 2 ) model which is relevant for frustrated antiferromagnetic systems [35–37] . The order parameter of this model is the N × 2 matrix Φ = ( φ 1 , φ 2 )  and the Hamiltonian is the sum of the usual kinetic terms and of the potential U = r ( φ 1 2 + φ 2 2 ) + u ( φ 1 2 + φ 2 2 ) 2 + v [ φ 1 2 φ 2 2 - ( φ 1 · φ 2 ) 2 ] . By a suitable choice of r , u , and v the symmetry is spontaneously broken down to O ( N - 2 ) ⊗ O ( 2 ) . For N typically larger than 21.8, two FPs are found in d = 4 - ε , a critical one, C + , that can be followed smoothly down to d = 2 and another one, C - , which is tricritical [39,40] . These FPs are also found in the large N expansion in all dimensions between 2 and 4 [40–42] . However, using the LPA’, we find for C - a picture which is very much similar to the O ( N ) case, see Fig. 4 : (i) There exists a line where C - collapses with a 3-unstable FP, that we call M 3 ; (ii) this line is asymptotic to the d = 3 axis, and (iii) M 3 appears on another line together with a 2-unstable FP that we call M 2  . 4 10.1103/PhysRevLett.119.191602.f4 FIG. 4. O ( N ) ⊗ O ( 2 ) model. In the gray region, starting in d = 4 at N = 21.8 , no FP at all is found. Above this region and for d close to 4, both the critical C + and the tricritical C - FPs are found. The line on the right joining the squares indicates the region where two nonperturbative FPs, M 2 and M 3 , appear. On the line joining the crosses, C - and M 3 collapse. In each region, we indicate the FPs that are present. To conclude, we have found that the multicritical FP structure of both the O ( N ) and O ( N ) ⊗ O ( 2 ) models is much more complicated than usually believed. In particular, we have shown that several nonpertubative FPs exist in d = 3 that were not previously found. Although they also exist at N = ∞ on a finite interval of dimensions they were not found by previous direct studies of this case and this is clearly a subject that must be further studied, see, however, Ref.  . The existence and role of possible singularities of the FP potential of C 2 and C 3 should be studied in the future as well. It would also be interesting to study the d = 3 case and figure out what the basins of attraction of both C 2 and C 3 are to know whether the multicriticality of some lattice models could be described by these FPs. The NPRG, here again, is a method of choice for this study but the conformal bootstrap program could probably definitively prove or disprove the existence of the C 2 and C 3 FPs in d = 3 . We can also expect that there are other nonperturbative FPs that collide with T n ( n = 3 , 4 , … ) as C 2 does with T 2 . They are also left for future study. Finally, an intriguing question is whether it could be that what we have found above has, for a known physical system, an impact on its criticality or multicriticality? We acknowledge H. Chaté, N. Defenu, C. Duclut, N. Dupuis, J.-M. Maillard, H. Osborn, R. Pisarski, G. Tarjus, and M. Tissier for discussions and/or advice about the Letter. S. 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PY - 2017/11/7
Y1 - 2017/11/7
N2 - We find that the multicritical fixed point structure of the O(N) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d=3) as well as at N=. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N. Many features found for the O(N) models are shared by the O(N) - O(2) models relevant to frustrated magnetic systems.
AB - We find that the multicritical fixed point structure of the O(N) models is much more complicated than widely believed. In particular, we find new nonperturbative fixed points in three dimensions (d=3) as well as at N=. These fixed points come together with an intricate double-valued structure when they are considered as functions of d and N. Many features found for the O(N) models are shared by the O(N) - O(2) models relevant to frustrated magnetic systems.
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U2 - 10.1103/PhysRevLett.119.191602
DO - 10.1103/PhysRevLett.119.191602
M3 - Article
C2 - 29219526
AN - SCOPUS:85033586335
VL - 119
JO - Physical Review Letters
JF - Physical Review Letters
SN - 0031-9007
IS - 19
M1 - 191602