CLASSIFICATION OF POTENTIAL FLOWS UNDER RENORMALIZATION GROUP TRANSFORMATION

. Competitions between diﬀerent interactions in strongly correlated electron systems often lead to exotic phases. Renormalization group is one of the powerful techniques to analyze the competing interactions without pre-sumed bias. It was recently shown that the renormalization group transforma- tions to the one-loop order in many correlated electron systems are described by potential ﬂows. Here we prove several rigorous theorems in the presence of renormalization-group potential and ﬁnd the complete classiﬁcation for the potential ﬂows. In addition, we show that the relevant interactions blow up at the maximal scaling exponent of unity, explaining the puzzling power-law Ansatz found in previous studies. The above ﬁndings are of great importance in building up the hierarchy for relevant couplings and the complete classiﬁcation for correlated ground states in the presence of generic interactions.

1. Introduction. Interactions in many-particle systems often lead to novel collective phenomena, inaccessible by the mean-field theory based on the single-particle picture. Exact solutions for these strongly correlated systems are illuminating and helpful, but scarce. Renormalization groups (RG) ([1]- [3]) are one of the powerful techniques to address these challenging issues in all branches of science. By integrating out the degrees of freedom at the longer length scale, the couplings of the effective Hamiltonian flow are described according to a set of RG equations. This approach is particularly helpful when the quantum fluctuations in the system are strong so that the mean-field description is invalid or hard to justify. For instance, it has been demonstrated that, despite of the repulsive interactions at the short length scale, electrons form pairs with unconventional d-wave symmetry in the low-energy limit in the two-leg ladder or the doped carbon nanotubes. RG analysis predicts that the ground state is a spin liquid beautifully realized in recent experiments.

SZE-BI HSU, BERNOLD FIEDLER AND HSIU-HAU LIN
To be precise, interactions of a correlated system are characterized by a set of couplings g i . By integrating out the fluctuations at shorter length scale, these couplings are renormalized accordingly and described by a set of first-order differential equations. In general, the perturbative RG equations to the one-loop order take the following form, where the last two indices in the coefficient tensor can be made symmetric, While the derivation of the RG equations are rather standard and can be found in the literature, interpreting the RG flows obtained in numerics can be subtle and tricky. At the tree-level, classification of relevance for the couplings with non-vanishing scaling dimensions is often straightforward. The fates of the RG flows are dictated by the scaling dimensions obtained by diagonalizing the matrix ∆ ij and the phases can be classified rather easily.
However, it is often encountered that ∆ ij = 0 for many interesting correlated systems ([4]- [11]) and the RG equations become homogeneously quadratic. Analytic solutions are not known ( [12]- [13]) and, at the current stage, numerical integration is the common approach to analyze the RG flows. It is rather subtle to classify the relevant couplings g i in that case because they often flow to fixed points outside of the scope of perturbative regime. The typical recipe is to integrate the RG equations up to the cutoff length scale l c where the maximal coupling is of order one. At l = l c , we identify the couplings g i (l c ) ∼ O(1) to be relevant while those couplings g i (l c ) 1 are irrelevant. But, this face-value classification scheme does not always work and some couplings are ambiguous to be classified as either relevant or irrelevant.
A recent breakthrough ( [11]) points out the existence of the RG potential and renders a better analytic understanding of the RG flows in these correlated systems. To show the existence of an RG potential, we can apply the rescaling transformation to all couplings, x i = √ λ i g i , and replace the derivative with respective to the logarithmic length l by the derivative with respective to some fictitious time t, denoted as the dot,ẋ Here jk is the coefficient tensor after rescaling. The RG equation can be viewed as a strongly over-damped particle moving in the multidimensional space under the influence of the external force on the right-band side of the equation. The existence of a potential requires the curl of the force to vanish, ∂ j F i − ∂ i F j = 0. This leads to the following constraint on the rescaled coefficient tensor, Because x k is arbitrary and P ijk = P jik , this requires P ijk to be totally symmetric. It has been shown explicitly how the coefficient tensor P ijk can be made totally symmetric under appropriate rescaling transformation. The remaining part of the paper is organized as follows. In section 2 we consider the dynamics of the model equations in the presence of a renormalization-group potential and give a sufficient condition on the initial conditions for the blow up behavior of the solutions. In section 3, we derive the polar equation for the governing system. In section 4, we prove that the blow up rate of the solution x i is (t − T ) −1 where T is the blow-up time. Section 5 is the discussion section.
2. The model. Consider the following governing system of equations jk ) symmetric n × n matrix satisfying the totally symmetric conditions: ij for all i, j, k = 1, 2, ..., n. (H) Suppose the initial condition x(0) satisfies where E is the set of equilibria of (4). Let Then from (H) Hence we have and (4) can be rewritten as Obviously (4) is a gradient system satisfyinġ Theorem 2.1. Let x(t) be any solution of the initial value problem (4) satisfying (5) and (H). If V (x(0)) < 0 then the solution x(t) blows up in finite time. Claim.
3. Polar coordinates. In this section we derive the polar equation for the gradient system (8). Let x = rφ where r = |x| 2 , φ = x r ∈ S n−1 . Then We note that φ T φ = 1 and φ T φ = 0. Let be the orthogonal projection onto the tangent space T φ S n−1 = φ T to the unit sphere S n−1 at φ. Then from (14) Thus we obtain the radical part Similarly applying P φ to both sides of (14) and using P φ φ = 0 yields the tangential part We now express ∇ x V (x) in polar coordinates. From (2.3), V (x) is a homogeneous polynomial with degree p = 3. Therefore Differentiating both sides of (17) with respect to x i yields

SZE-BI HSU, BERNOLD FIEDLER AND HSIU-HAU LIN
Since ∂r ∂xi = φ i and ∂ ∂xi ( 1 r ) = − φi r 2 , it follows that Differentiating (17) with respect to r, we have Hence (15) implies Thus with p = 3 the polar equation of (8) is We discuss the resulting dynamics of system (20), (21) in the next section.
4. Blow up rate. In this section we discuss the global dynamics of system (20), (21). In particular we summarize blow up rates versus convergence behavior in Theorem 4.1.
The trivial equilibrium x ≡ 0 of (4), alias r ≡ 0 of (20), (21) implies that the set {r > 0} of positive radii r = |x| 2 is forward and backward invariant. Moreover, systems (4) Here we used that φ ∈ S n−1 is compact, and hence cannot blow up. Common terminology speaks of grow up if (22) holds with T = ∞, i.e. in case of global existence of solutions (r(t), φ(t)).
To study the global behavior of system (20), (21), it is useful to rescale time t to a new time τ . Writing t = t(τ ) for the inverse transformation, introducing the new variables R(τ ) := r(t), Φ(τ ) := φ(t) we rescale system (20), (21) to become with initial condition t = 0 at τ = 0 and R = r, Φ = φ there. Because Φ ∈ S n−1 is compact and V is bounded on S n−1 , all solutions of (24)-(25) exist globally for all times τ ∈ R. Moreover t = t(τ ) is strictly increasing and is the maximal time of existence in the original time variable 0 ≤ t < T . In particular, finite time blow-up T < ∞ is equivalent to a finite integral in (26).
We proceed with care and study the autonomous equation (25) on Φ ∈ S n−1 first.
Proof. Since P φ projects orthogonally onto the tangent space T φ S n−1 , the system (25) leaves S n−1 invariant in forward and backward time τ .
Since V (Φ) is a Lyapunov function of (25) on the compact sphere Φ ∈ S n−1 , we have a finite monotone limit We can now formulate our main blow up and convergence result for system (20), (21) alias (4).

SZE-BI HSU, BERNOLD FIEDLER AND HSIU-HAU LIN
Then T = +∞ and r(t) ↓ r * decreases monotonically to r * for t → ∞. The limit r * is strictly positive if and only if If the integral is infinite, however, then r * = 0 and x(t) converges to the trivial equilibrium for t → ∞. The latter case occurs, in particular, whenever V * > 0.
Then T < ∞ and r(t) blows up in finite time, as in (22). The blow up rate is In either case, the angular direction φ(t) = x(t) |x(t)|2 = Φ(τ ) of blow up or of convergence converges to the equilibrium set E of (25) for t ↑ T ≤ ∞.
We remark that dichotomy of the two cases of Theorem 4.1 does not allow for infinite time grow up r(t) → ∞ for t → T = ∞. The case V * = 0 of an equilibrium φ = Φ of the rescaled angular equation (25) is of course degenerate: by homogeneity of V , it implies ∇ x V = 0 at x = rφ for all r. Thus we may consider convergence to finite r * > 0 as exceptional nongeneric or degenerate , and we arrive at the dichotomy that either r(t) → 0 or else r(t) → ∞ in forward time t → +∞. By reversibility, the same holds true in backward time. Trajectories which start at the zero level set {V = 0} with r(0) > 0, then blow up in, both, forward and backward finite time t = T ± , for some maximal time interval of existence T − < t < T + . Trajectories which do not cross {V = 0} blow up in precisely one time direction, depending on their sign of V , and converge to zero in the opposite direction.
In case the rescaled angular equation (25) possesses only a discrete, and hence finite, set of equilibria, the asymptotic direction Φ = Φ * of blow up or convergence is also well-defined and Φ * = lim exists, for t → T . If the equilibrium Φ * of (25) is a nondegenerate critical point of V on S n−1 , i.e. hyperbolic, then the convergence of Φ(τ ) to Φ * , and of V (Φ(τ )) to V * , is exponential. This further refines the blow up result (30). We neither pursue this equation any further here, nor do we address the general case of p-homogeneous potentials V , p > 3 which can be treated in a similar spirit.

5.
Discussion. The rigorous results in the previous sections make a fundamental advance in renormalization group analysis. One common challenge in studying strongly correlated electron systems is the huge dimension of the couplings x i . For instance, renormalization-group analysis ([16]- [18]) for iron-based superconductors, easily involves hundreds or even thousands of couplings. With modern computation powers, these coupled differential equations can be solved numerically within negligible errors. However, because a huge number of couplings diverges under RG transformations, the interpretation of the numerical outcome remains a difficult task.
Our theorems provide a systematic classification for the ultimate fates of the RG flows. Depending on the sign of the RG potential V , the flows can be classified into three regimes: divergent, backward-orbit and stable. One can introduce the distance to the trivial fixed point x = 0 as r(t) = |x| 2 . In the divergent regime, r(t) grows monotonically and the ground state is driven away from the trivial fixed point under RG transformations. On the other hand, in the stable regime, r(t) decreases monotonically and all interactions become irrelevant, i.e. no instability develops within the stable basin. The most interesting case is the backward-orbit regime. The distance r(t) decreases initially and starts to grow upon hitting the zero-potential surface. Because the fictitious time t corresponds to cooling down the temperature (logarithmically) in realistic situations, the backward-orbit regime implies a nontrivial crossover from the trivial fixed point (Fermi liquid) to the correlated ground state. This peculiar crossover behavior in unconventional superconductivity is the realization of the backward-orbit regime. The complete classification presented here answers the long-standing puzzle why limit cycles, chaos and so on are rarely seen in RG transformations while they often occur in generic coupled differential equations.
The presence of the RG potential enables the complete classification of the phase diagram. In addition, it also provides a powerful scheme to quantify the relevance of runaway couplings. Because the blow-up rate for the radius is r(t) ∼ (t − T ) −1 , where T is the blow-up time, it implies the couplings can at most diverge at the same exponent of unity. Linear analysis near the blow-up regime shows that the relevant couplings scales as x i ∼ (t−T ) −γi , with exponents 0 ≤ γ i ≤ 1. This inverse power law was proposed before in previous studies, but can now be derived from Theorem 4.2. The robustness of these exponents renders the unique way to build up the hierarchy of all relevant couplings and will be extremely helpful to identify the dominant interactions in numerical RG studies.
In summary, the presence of a homogeneous renormalization-group potential V allows us to prove several rigorous theorems and, in consequence, find the complete classification for the potential flows. Meanwhile, the relevant couplings follow the scaling form described by a set of RG exponents. The above findings are of great importance in interpreting numerical RG studies and also revealing the generic structure of the phase diagrams for strongly correlated electron systems.