Potential flows in renormalization group transformation
Hsiu-Hau Lin1,3 and Sze-Bi Hsu2,31Department of Physics, National Tsing Hua University, Hsinchu 30013, Taiwan 2
Department of Mathematics, National Tsing Hua University, Hsinchu 30013, Taiwan
3
National Center for Theoretical Scienes, Hsinchu 300, Taiwan (Dated: October 11, 2013)
Renormalization group is a powerful technique to analyze instabilities in strongly correlated elec-tron systems where interactions between individual particles bring up surprising phases. Here we show that the existence of potential flows in renormalization group transformation leads to several important theorems dictating the ultimate fates of the flows. A systematic classification for the renormalization group flows is made possible and the recently proposed scaling Ansatz, found in numerical studies, is justified.
PACS numbers: 71.10.Fd, 71.10.Hf, 71.27.+a, 71.10.Pm
Finding the ground states of the strongly correlated systems remains one of the most challenging task in mod-ern sciences. Exact solutions for these strongly correlated systems are illuminating and helpful but scarcely any. By integrating out the degrees of freedom at the longer length scale, renormalization group (RG) analysis1–5
re-mains one of the most powerful techniques to determine the effective interactions in the low-energy limit. The RG approach has been successfully applied to a wide va-riety of physical phenomena, including the pairing mech-anism in iron-based superconductors,6–8 phase diagrams
for cold atoms,9,10 transport in topological insulators11 and other correlated systems.
To construct a RG scheme for a targeted correlated system, one needs to characterize the allowed interactions by a set of couplings gi. Integrating out the fluctuations
at shorter length scale, these couplings are renormalized accordingly and described by a set of the first-order differ-ential equations. In general, the perturbative RG equa-tions to the one-loop order take the following form,
dgi dl = X j ∆ijgj+ X jk Aijkgjgk, (1)
where the last two indices in the coefficient tensor can be made symmetric, Aijk = Aikj. In the perturbative
regime, 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.
How-ever, it is often encountered that ∆ij = 0 for many
in-teresting correlated systems12–19 and the RG equations become intrinsically non-linear.
A common resolution to analyze these non-linear RG flows is by numerical integration – messy but relatively easy with modern computing power. Because no analytic form is available, the difficulties lie in the subtlety of choosing an appropriate cutoff in the numerics and then reading off the relevant couplings. A recent breakthrough points out the existence of the RG potential and proposes a scaling Ansatz for the relevant couplings,19,20
gi(l) ∼
1 (ld− l)γi
, (2)
where ld is the divergent length scale in the RG flows.
The RG exponents γi appearing the the scaling Ansatz
determine the hierarchy of relevant couplings without ambiguity. However, the scaling Ansatz is assumed from numerics without proof and a systematic classification of the RG flows is still missing. In this Rapid Communica-tions, we fill in this important cornerstone and show how the RG flows can be classified into three different regimes by the RG potential.
To brought the RG transformation into potential flows,17 we rescale all couplings, x
i =
√
λigi and replace
the derivative to the logarithmic length l by that to the fictitious time t denoted as the dot,
˙ xi=
X
jk
Pijkxjxk, (3)
where Pijk = pλi/(λjλk)Aijk is the coefficient tensor
after rescaling. The RG equation can be viewed as a strongly over-damped particle moving in the multi-dimensional space under the influence of the external force on the right-band side of the equation. The ex-istence of the potential requires the curl of the force is zero, ∂jFi− ∂iFj = 0, leading to the constraint that Pijk
is totally symmetric. It has been shown explicitly how the coefficient tensor Pijk can be made totally symmetric17
and the presence of the RG potential is justified. The RG equations can be cast into the elegant form,
˙ xi = − ∂V ∂xi , where V = −1 3 X i,j,k Pijkxixjxk. (4)
Based on the explicit for of the RG potential V and the corresponding potential flows, we would like to prove sev-eral important theorems in the following paragraphs.
Theorem 1: the RG potential is a monotonically de-creasing function. Making use of the chain rule, one can compute the time derivative of the RG potential,
˙ V =X i ∂V ∂xi ˙ xi= − X i ∂V ∂xi 2 ≤ 0. (5)
2
Because the derivative of the potential is negative defi-nite, the RG potential decreases monotonically.
Lemma: Zero derivative means zero potential. In the regime where the potential derivative vanishes, ˙V = 0, it immediately leads to ∂V /∂xi= 0. Rewrite the potential
in terms of its derivatives,
V = −1 3 X i xi X jk Pijkxjxk = 1 3 X i xi ∂V ∂xi = 0. (6)
Thus, zero derivative ˙V = 0 means zero potential V = 0. Next, we would like to show that the RG flows of differ-ent initial couplings can be systematically classified into three categories: the divergent manifold D, the backward orbits β(Γ) and the stable basin S. To proceed, let us in-troduce the zero-potential surface V (xi) = 0, composed
of two parts with negative and zero derivatives,
Γ ≡ {(x1, x2, ..., xN)|V (xi) = 0, ˙V (xi) < 0}, (7)
F ≡ {(x1, x2, ..., xN)| ˙V (xi) = 0}. (8)
Furthermore, for each set r0 of initial couplings in Γ, a
backward orbit β(r0) can be constructed by back tracking
the RG flow. These backward-orbit manifold β(Γ) is thus defined as
β(Γ) ≡ {β(r0), r0∈ Γ}. (9)
Now we are set to classify the ultimate fate of the RG flows by the following theorems.
Theorem 2: Starting anywhere with V (0) < 0, RG flows diverge in the form of
r(t) ≡ v u u t n X i=1 x2 i(t) ∼ 1 td− t , (10)
at some finite time td. The detail mathematical
deriva-tions are rather involved and will be presented elsewhere. Here we just walk through the major steps leading to this important claim.
First of all, introduce the square sum of all couplings s(t) = r2(t). It is straightforward to show that
˙s = 2X i xix˙i= 2 X ijk Pijkxixjxk= −6V (xi) > 0, (11)
because we start from V (0) < 0 and the potential is monotonically decreasing. From the potential flows, one can show that the square sum of all couplings satisfies a useful inequality,
s0(t) ≥ C[s(t)]δ, (12) where C > 0 is a positive constant and the exponent sat-isfies 1 < δ < 3/2. Separate the variables and integrate on both sides, 1 δ − 1 1 [s(0)]δ−1 − 1 [s(t)]δ−1 ≥ Ct. (13)
After some algebra, we obtain the lower bound for s(t),
[s(t)]δ−1≥ 1 C(δ − 1) 1 td− t , (14)
where 1/td= C(δ − 1)[s(0)]δ−1. Therefore, s(t) diverges
as t → t−d. It means some of the couplings xi(t) must
diverge at the finite divergent time scale td.
Once the divergence is established, the divergent rate can be estimated conveniently in the polar coordinates in the multidimensional coupling space. It turns out that r(t) =ps(t) ∼ 1/(td− t). It means that the couplings
take the approximate form, xi ≈ Xi/(td− t), where Xi
are some constants. For couplings with Xi 6= 0, they
are captured by the scaling form 1/(td− t). For those
with Xi = 0, the subleading terms can be computed
from standard instability analysis17 and takes the form
as 1/(td− t) γ
i, where γi < 1. This theorem explains the
scaling Ansatz in Eq. (2) and provides the analytic form of the potential flows.
Theorem 3: Starting with V (0) > 0 and r0 ∈ β(Γ),
the RG flow will cross Γ and diverges as described in Theorem 2. For r0∈ β(Γ) (note that V (0) > 0 initially),
the flow crosses Γ with negative derivative ˙V (t0) < 0 at
some time t0 and the potential turns negative in later
time. Applying Theorem 2, it is evident that the RG flow becomes divergent at later time td> t0.
Lemma: The derivative ds/dt and the potential V have opposite signs. In the divergent manifold D where V (t) < 0, ds/dt > 0 i.e. the flow runs away from the ori-gin. On the other hand, in the backward-orbit manifold β(Γ) where V (t) > 0, ds/dt < 0 and the flow approaches the origin. This Lemma would be helpful to explain the interesting crossover behavior unique in the backward-orbit manifold β(Γ).
Theorem 4: Starting with V (0) > 0 but r0 ∈ β(Γ),/
then xi(t) goes to the maximally invariant manifold M
in F as t → ∞. Because V (0) > 0 but r0 ∈ β(Γ),/
the flow never cross the zero-potential surface in finite time. From Cauchy-Schwartz inequality, one can show that V (t) goes to zero in the asymptotic limit t → ∞. Be-cause the potential decreases monotonically, by LaSalle’s invariance principle,21 as time goes to infinity, x
i(t)
ap-proaches the maximally invariant manifold in F , i.e. limt→∞xi(t) = r∞∈ M .
The above theorems provide the solid ground for the emergence of the scaling Ansatz, previously found in nu-merical investigations. In addition, it also divides the coupling space into three regimes: the divergent mani-fold D with negative RG potential, the backward-orbit manifold β(Γ) with positive potential but crossing the zero-potential surface with negative derivative and the remaining stable basin S where the RG flows do not blow up and end up in the maximally invariant manifold M such as fixed points, fixed lined and so on. It is rather re-markable that all these results can be proven solely from the existence of the potential flows in RG transformation. To fully appreciate the above theorems, it is helpful to realize the classification of RG flows in simple yet
inspir-3
FIG. 1: RG flows in the reduced coupling space for a two-band model. The flows inside the stable basin S approach the stable fixed point at the origin, while those in β(Γ) and D flow away and eventually become divergent. Note that, for initial couplings inside β(Γ), it flows closer the fixed point initially before running away at the end – hinting a non-trivial crossover between the two phases.
ing examples. We would like to elaborate on the phase diagram of Fermi liquid, conventional and unconventional superconductivity in a two-band model as shown in Fig. 1. For a quasi-one-dimensional systems, the low-energy theory consists of NF pairs of Fermi points with mutual
interactions between them. To the one-loop order15,17,
the renormalized couplings flow according to the set of nonlinear differential equations:
dcσ ii dl = −(c σ ii) 2 −X k6=i γii,k 2 c ρ ikc σ ik+ (c σ ik) 2 (15) dcσ ij dl = − X k γij,k 4 h cρikcσkj+ cσikcρkj+ 2cσikcσkji +1 2(c ρ ijf σ ij+ c σ ijf ρ ij− 2c σ ijf σ ij) (16) dfijσ dl = −(f σ ij) 2 +1 2c ρ ijc σ ij− (c σ ij) 2 (17) dcρii dl = − X k6=i γii,k 4 (c ρ ik) 2+ 3(cσ ik)2 (18) dcρij dl = − X k γij,k 4 h cρikcρkj+ 3cσikcσkji +1 2(c ρ ijf ρ ij+ 3c σ ijf σ ij) (19) dfijρ dl = 1 4(c ρ ij) 2+ 3(cσ ij) 2 (20)
Fermi velocities are different in general and give rise to
the coefficient tensor
γij,k=
(vi+ vk)(vj+ vk)
2vk(vi+ vj)
. (21)
It should be clear from the definition that γij,i= γij,j= 1
even when Fermi velocities are different.
Even though the above RG equations are originally derived from the quasi-one-dimensional systems. It can be applied to some other correlated systems within rea-sonable approximation. For instance, the Fermi sur-faces for the iron-based superconductors are complicated. However, it turns out that picking out the symmetric points on the Fermi surfaces as representatives6–8is good
enough to reproduce the phase diagram. The RG equa-tions for these representative couplings are the same as the above.
As a demonstrating example, it is insightful to work on NF = 2 case.19 Furthermore, we set the Fermi velocities
equal, v1= v2, to simplify the algebra. Aiming at the
su-perconducting instabilities, the most relevant couplings are the intraband c11 = c22 and interband c12 = c21
Cooper scattering. The number of RG equations are greatly reduced to just two,
˙c11= −(c11)2− (c12)2, (22)
˙c12= −2c11c12. (23)
The RG flows in the reduced coupling space is shown in Fig. 1. Introduce the variables x1 = c11 and x2 = c12
and the coupled RG equations can be cast into the form of potential flows, ˙ xi= − ∂V ∂xi , (24)
and the RG potential can be written down explicitly,
V (x1, x2) =
1 3x
3
1+ x1x22. (25)
In this simple example, no rescaling of the couplings is needed to construct the potential because we set v1= v2
here. But, the major conclusions remain true even when the Fermi velocities are not the same. As shown in Fig. 1, the zero-potential surface is x1 = 0. The divergent
manifold D (in blue) with negative potential is x1 < 0
and the divergent flows are captured by the run-away lines x1− x2= 0 and x1+ x2= 0. The positive potential
regime consists of the backward orbits β(Γ) (in pink) and the stable basin S (in green).
For initial couplings in the stable basin S, it flows to-ward the origin where all couplings are zero. The cor-responding phase is Fermi liquid. On the other hand, for purely repulsive interactions (x1, x2 > 0) in the
backward-orbit manifold β(Γ), it approaches the Fermi-liquid fixed point first and then turns toward the run-away line x1+ x2 = 0 at the end. It means that,
4
becomes strongly attractive while the interband one be-comes strongly repulsive. It is not surprising that de-tail analysis shows that the run-away line x1+ x2 = 0
corresponds to superconducting instability with opposite signs of the gap functions in the two bands. Let’s re-fer this phase as d−wave superconductor. The RG flow starting from β(Γ) in the upper plane then represents an interesting crossover from Fermi liquid to the uncon-ventional d−wave superconductor. For purely attractive interactions (x1, x2< 0), it belongs to the divergent
man-ifold and flows toward the run-away line x1− x2 = 0.
Both intraband and interband Cooper scatterings are strongly attractive and correspond to the conventional pairing without sign flip in the gap functions. Summa-rizing the above analysis, the stable basin gives the usual Fermi liquid phase, while all instabilities can be found in the divergent manifold, representing different correlated ground states. The backward orbits exhibit interesting crossover from the Fermi liquid to one of the correlated
ground states. The classification scheme arisen from the RG potential helps us to understand the global topology of the phase diagram and also the interesting connections between different phases.
In conclusion, we show that the existence of RG po-tential leads to several important theorems dictating the ultimate fates of the RG flows. The space of initial cou-plings can be classified into three regimes by the RG po-tential: the divergent manifold, the backward orbits and the stable basin. In addition, these theorems provide the solid ground for the scaling Ansatz found in previous nu-merical works and the relevant couplings can be classified by the set of RG exponents.
We acknowledge supports from the National Science Council in Taiwan through grant NSC 100-2112-M-007-017-MY3. Financial supports and friendly environment provided by the National Center for Theoretical Sciences in Taiwan are also greatly appreciated.
1
K. G. Wilson, Rev. Mod. Phys. 55, 583 (1983).
2
N. Goldenfeld, Lectures on Phase Transitions and the Renormalization Group (Addison Wesley, 1992).
3
R. Shankar, Rev. Mod. Phys. 66, 129 (1994).
4
M. E. Fisher, Rev. Mod. Phys. 70, 653 (1998).
5 M. Salmhofer and C. Honerkamp, Progress of Theoretical
Physics 105, 1 (2001).
6
A. V. Chubukov, D. V. Efremov and I. Eremin, Phys. Rev. B 78, 134512 (2008).
7 F. Wang, H. Zhai, Y. Ran, A. Vishwanath and D.-H. Lee,
Phys. Rev. Lett. 102, 047005 (2009).
8
F. Wang and D.-H. Lee, Science 332, 200 (2011).
9 L. Mathey, S.-W. Tsai and A. H. Castro Neto, Phys. Rev.
Lett. 97, 030601 (2006).
10
K. B. Gubbels and H. T. C. Stoof, Phys. Rev. Lett. 100, 140407 (2008).
11
Q. Liu, C.-X. Liu, C. Xu, X.-L. Qi and S.-C. Zhang, Phys. Rev. Lett. 102, 156603 (2009).
12
M. Fabrizio, Phys. Rev. B 48, 15 838 (1993).
13
L. Balents and M. P. A. Fisher, Phys. Rev. B 53, 12133 (1996).
14 H. J. Schulz, Phys. Rev. B 53, R2959 (1996). 15
H.-H. Lin, L. Balents and M. P. A. Fisher, Phys. Rev. B 56, 6569 (1997).
16 H.-H. Lin, L. Balents and M. P. A. Fisher, Phys. Rev. B
58, 1794 (1998).
17
M.-H. Chang, W. Chen and H.-H. Lin, Prog. Theor. Phys. Suppl. 160, 79 (2005).
18
E. Szirmai and J. S`olyom, Phys. Rev. B 74, 155110 (2006).
19
H.-Y. Shih, W.-M. Huang, S.-B. Hsu and H.-H. Lin Phys. Rev. B 81, 121107(R) (2010).
20
Y. Cai, W.-M. Huang and H.-H. Lin, Phys. Rev. B 85, 134502 (2012).
21 S.-B. Hsu, Ordinary Differential Equations with