ITEP-TH- 02/14

RG limit cycles

K.Bulycheva$\,{}^{a}$, A.Gorsky$\,{}^{a,b}$

a Institute of Theoretical and Experimental Physics, Moscow 117218, Russia
b Moscow Institute of Physics and Technology, Dolgoprudny 141700, Russia

<EMAIL_ADDRESS>
<EMAIL_ADDRESS>
Contribution to the ”Pomeranchuk-100” Volume

Abstract

In this review we consider the concept of limit cycles in the renormalization
group flows. The examples of this phenomena in the quantum mechanics and field
theory will be presented.

##  1 Generalities

It is usually assumed that the RG flow connects fixed points, starting at a UV
repelling point and terminating at a IR attracting point. However it turned
out that this open RG trajectory does not exhaust all possibilities and the
clear-cut quantum mechanical example of the nontrivial RG limit cycle has been
found in [1] confirming the earlier expectations. This example triggered the
search for patterns of this phenomena which was quite successful. They have
been identified both in the systems with finite number of degrees of freedom
[2, 3, 4, 5] and in the field theory framework [6, 7, 8]. Now the cyclic RG
takes its prominent place in the world of RG phenomena however the subject
certainly deserves much more study.

The appearance of critical points corresponds to phase transitions of the
second kind, hence there exists a natural question concerning the connection
of RG cycles with phase transitions. The very phenomenon of the cyclic RG flow
has been interpreted in the important paper [7] as a kind of generalization of
the BKT phase transitions in two dimensions. One can start from a usual
example of an RG flow connecting UV and IR fixed points and then consider a
motion in a parameter space which results in a merging of the fixed points.In
[7] it was argued that when the parameter goes into the complex region the
cyclic behaviour of the RG flow gets manifested and a gap in the spectrum
arises. This happens similar to the BKT transition case when a deconfinement
of vortices occurs at the critical temperature and the conformal symmetry is
restored at lower temperatures. The appearance of the RG cycles can be also
interpreted as the peculiar anomaly in the classical conformal group [9]. This
anomaly has the origin in some ”falling to the center” UV phenomena which
could have quite different reincarnations. We would like to emphasize one more
generic feature of the phenomena — the cyclic RG usually occurs in the system
with at least two couplings. One of them undergoes the RG cyclic flow while
the second determines the period of the cycle.

The collision of the UV and IR fixed points can be illustrated in a quite
general manner as follows. Assume that there are two couplings $(\alpha,g)$ in
the theory and we focus at the renormalization of the coupling which enjoys
the following $\beta$-function

$\beta_{g}=(\alpha-\alpha_{0})-(g-g_{0})^{2},$ (1)

which vanishes at the hypersurface in the parameter space

$g=g_{0}\pm\sqrt{\alpha-\alpha_{0}}.$ (2)

It was argued in [7] that the collision of two roots at $\alpha=\alpha_{0}$
can be interpreted as the collision of UV and IR fixed points. Upon the
collision the points move into the complex $g$ plane and an RG cycle emerges.
The period of the cycle can be immediately estimated as

$T\propto\int_{g_{UV}}^{g_{IR}}\frac{dg}{\beta(\alpha;g)}\propto\frac{1}{\sqrt{\alpha-\alpha_{0}}}.$
(3)

The phenomena is believed to be generic once the beta–function has the form
(1). Note that is was shown that the RG cycles are consistent with the
c-theorem [10].

Breaking of the conformal symmetry results in the generation of the mass scale
which has non-perturbative nature. Due to the RG cycles the scale is not
unique and the whole tower with the Efimov-like scaling gets manifested

$E_{n+1}=\lambda E_{n},$ (4)

where $\lambda$ is fixed by the period of RG cycle.

In the examples available we could attempt to trace the physical picture
behind. It turns out that the origin of two couplings is quite general. One
coupling does not break the conformal symmetry which is exact in some subspace
of the parameter space. The second coupling plays the role of UV
regularization which can be imposed in one or another manner. It breaks the
conformal symmetry however some discrete version of the scale symmetry
survives which is manifested in the cycle structure. The UV regularization
will have different reincarnations in the examples considered: the account of
the finite size of the nuclei, contact interaction in the model of
superconductivity or the brane splitting in the supersymmetric models.

Historically the first example of this phenomena has been found long time ago
by Efimov [11] in the context of nuclear physics. He considered the three-body
system when two particles are near threshold and have attractive potential
with the third particle. It was shown that two–particle bound states are
absent in the spectrum, but there is a tower of the three–particle bound
states with the geometrical scaling corresponding to $\lambda\approx 22,7$.
The review of the RG interpretation of the Efimov phenomena can be found in
[12].

When considering the system with finite number of degrees of freedom the
meaning of the RG flows has to be clarified. To this aim some UV cutoff should
be introduced. In the first example in [1] the step of the RG corresponds to
the integrating out the highest energy level taking into account its
correlation with the rest of the spectrum. This approach has a lot in common
with the renormalization procedure in the matrix models considered in [13].
The same UV cutoff for formulation of RG procedure has been used for the
Russian Doll (RD) model describing the restricted BCS model of
superconductivity [14]. In that case the coupling providing the Cooper pairing
undergoes the RG cycle while the CP-violating parameter defines the period.

In the second class of examples the UV cutoff is introduced not at high energy
scale but at small distances. The RG cycles have been found in the non-
relativistic Calogero-like models with $\frac{1}{r^{2}}$ potential which
enjoys naive conformal symmetry [3, 4, 5]. The RG flow is formulated in terms
of the short distance regularization of the model. It is assumed that the wave
function with $E=0$ at large $r$ does not depend on the UV cutoff at small
$r$. This condition yields the equation for the parameter of a cutoff in the
regularization potential. This equation has multiple solutions which can be
interpreted as the manifestation of the tower of shallow bound states with the
Efimov scaling in the regularized Calogero model with attraction. The scaling
factor in the tower is determined by the Calogero coupling constant which
reflects the remnant of the conformal group upon the regularization.

The list of the field theory examples in different dimensions with the cyclic
RG flows is short but quite representative. In two dimensions the explicit
example with the RG cycle has been found in some range of parameters in the
sin-Gordon model. The cycle manifests itself in the pole structure of the
$S$-matrix. Efimov-like tower of states corresponds to the specific poles with
the Regge-like behavior of the resonance masses [6]

$m_{n}=m_{s}e^{\frac{n\pi}{h}},$ (5)

where $h$ is a certain parameter of the model. Moreover it was argued that the
$S$-matrix behaves universally under the cyclic RG flows. The tower of Efimov
states scales in the same manner as in the quantum mechanical case.

The origination of the cyclic RG behavior in the sine–Gordon model is not
surprising. Indeed it was argued in [7] that the famous
Berezinsky–Kosterlitz–Thouless (BKT) phase transition in XY system belongs to
this universality class. On the other hand one can map the XY system at the
$T$ temperature into the sin-Gordon theory with the parameters:

$L_{SG}=T(\partial\phi)^{2}-4z\cos\phi,$ (6)

and look at the renormalization of the interaction coupling. The
$\beta$–functions read as

$\beta_{u}=-2v^{2},\qquad\beta_{v}=-2uv,$ (7)

where

$u=1-\frac{1}{8\pi T},\qquad v=\frac{2z}{T\Lambda^{2}},$ (8)

and $\Lambda$ is the UV cutoff introduced to regularize the vortex core. The
form of $\beta$ functions implies the existence of the limit cycle with the
following expression for the correlation length:

$\xi_{BKT}\Lambda\propto\exp\left(\frac{c}{\sqrt{|T-T_{c}|}}\right),$ (9)

above the phase transition. This RG behavior gets mapped into the RG cycle in
the sine–Gordon model.

The example of the Efimov tower in 2+1 dimensions has been found in [15] in
the holographic representation. The model is based on the $D3-D5$ brane
configuration and corresponds to the large $N$ 3d gauge theory with
fundamentals enjoying $\mathcal{N}=4$ supersymmetry. In addition the magnetic
field and the finite density of conserved charge are present. At strong
coupling the gauge theory is described in terms of the probe $N_{f}$ flavor
branes in the nontrivial $AdS_{5}\times S^{5}$ geometry when the $U(1)$ bulk
gauge field is added providing the magnetic field in the boundary theory.

The generation of the tower of the Efimov states happens at some value of the
“filling fraction” $\nu$ in external magnetic field. The phase transition
corresponds to the change of the minimal embedding of the probe $D5$ branes in
the bulk geometry with the BKT critical behavior of the order parameter. In
that case the order parameter gets identified with the condensate $\sigma$
which behaves as:

$\sigma\propto\exp\left(-\frac{1}{\nu}\right).$ (10)

Above the phase transition the embedding gets changed and the brane becomes
extended in one more coordinate. The scale associated with this extension into
new dimension is nothing but the nonperturbative scale amounting to the mass
gap. The phenomena of the cyclic RG flow in this case has the Breitenlohner-
Freedman instability as the gravitational counterpart.

In four dimensions the most famous example of the Efimov tower is the so-
called Miransky scaling for the condensate in the magnetic field. In [16] was
argued that the chiral condensate is generated in the external magnetic field
in the abelian theory with the following behavior:

$\langle\bar{\Psi}\Psi\rangle\propto\Lambda^{3}\exp(-\frac{c}{\sqrt{\alpha-\alpha_{crit}}}),$
(11)

where $\alpha$ is the fine–structure constant, and $c$ is some parameter of
the model.

More recent example [17] of the Efimov tower in four dimensions concerns the
Veneziano limit of QCD when $N_{f},N_{c}\rightarrow\infty$ while the ratio
$x=\frac{N_{f}}{N_{c}}$ is fixed. It turns out that this parameter can be
considered as the variable in the RG flow which reminds the finite-dimensional
examples. At some value of RG scale the tower of condensates gets generated
with geometrical Efimov scaling. The period of the RG cycle reads as:

$T\propto\frac{\kappa}{\sqrt{x_{c}-x}},$ (12)

where $x_{c}$ is the critical value of the $x$ parameter. Finally the 4d
example with the RG cycle has been found in the $\mathcal{N}=2$ SUSY gauge
theory in the $\Omega$-background [8]. In that case the gauge coupling
undergoes the RG cycle whose period is determined by the parameter of the
$\Omega$-background,

$T\propto\epsilon^{-1}.$ (13)

The appearance of the RG cycle in this model can be traced from its relation
with the quantum integrable systems of the spin chain type.

In this review we provide the reader with the examples of this phenomenon. The
list of the systems with finite number of degrees of freedom involves the
Calogero model and the relativistic model with the classical conformal
symmetry describing the external charge in graphene. Another finite-
dimensional example concerns the RD model of the restricted BCS
superconductivity. The field theory examples concern the $3d$ and $4d$
theories in external fields. We shall focus on their brane representations and
use their relations to the finite dimensional integrable systems.

##  2 RG cycles in non-relativistic quantum mechanics

In this Section we consider the example of the limit cycle in RG in the non-
relativistic system with the inverse-square potential, or the Calogero system:

$H=\frac{\partial^{2}}{\partial r^{2}}-\frac{\mu(\mu-1)}{r^{2}}.$ (14)

The distinctive feature of the system described by the Hamiltonian (14) is its
conformality. Namely, the operators $(H,D,K)$, where $D$ is the dilation
generator and $K$ is the conformal boost, generate the conformal
$\mathfrak{sl}_{2}$ algebra (see Section 4).

The eigenfunctions of (14) having finite energy immediately break this
symmetry; more non-trivial is the fact that even the ground state breaks
conformality. Namely, the solution to the $H\psi=0$ equation is the following:

$\psi_{0}=c_{+}r^{\mu}+c_{-}r^{1-\mu}.$ (15)

This solution is scale-invariant only if one of the coefficients $c_{\pm}$ is
zero. If both the coefficients are present, they define an intrinsic length
scale $L=\left(c_{+}/c_{-}\right)^{1/(-2\mu+1)}$. Requiring that the quantity
$c_{+}/c_{-}$ which describes the ground-state solution be invariant under the
change of scale,

$\frac{c_{+}}{c_{-}}=-r_{0}^{-2\mu+1}\frac{\gamma-\mu+1}{\gamma+\mu},$ (16)

we arrive at the beta-function for the $\gamma$ parameter,

$\beta_{\gamma}=\frac{\partial\gamma}{\partial\log
r_{0}}=-\left(\gamma+\mu\right)\left(\gamma-\mu+1\right)=\left(\mu-\frac{1}{2}\right)^{2}-\left(\gamma-\frac{1}{2}\right)^{2},$
(17)

where $r_{0}$ is the RG scale. We can identify $\gamma=\mu-1,\gamma=-\mu$
points, i.e. solutions with $c_{+}=0$, $c_{-}=0$, with UV and IR attractive
points of the renormalization group flow [7].

If $\mu=i\nu$ is imaginary, i.e. the potential is attractive, then the
equation (17) allows us to determine the period of the renormalization group:

$T=-\int_{-\nu+1}^{\nu}\frac{d\gamma}{\beta_{\gamma}}=\frac{\pi}{\nu-\frac{1}{2}}.$
(18)

This means that an infinite number of scales is generated, differing by a
factor of $\exp\left(-\frac{\pi}{\nu-\frac{1}{2}}\right)$. To see this
explicitly, we find the solutions to the Schrödinger equation at finite
energies. In the attractive potential the solution (15) can be written as:

$\psi_{0}\propto\sqrt{r}\sin\left(\left(\nu-\frac{1}{2}\right)\log\left(\frac{r}{r_{0}}\right)+\alpha\right).$
(19)

We observe that this solution oscillates indeterminately in the vicinity of
the origin and there is no way to fix the $\alpha$ constant. To regularize
this behaviour, we can break the scale invariance at the level of the
Hamiltonian and introduce a regularizing potential. Two most popular
regularizations involve the square-well potential [4, 5] or the delta-shell
potential [3]. One more choice is to introduce a $\delta$-function at the
origin [7].

Choosing the square-well regularization,

$V(r)=\left\\{\begin{array}[]{l}-\frac{\nu(\nu-1)}{r^{2}},r>R,\\\
-\frac{\lambda}{R^{2}},r\leq R,\end{array}\right.$ (20)

we require that the action of the dilatation operator on the wavefunction
inside the well and outside it be equal at $r=R$. This condition amounts to
the equation on $\lambda$,

$\sqrt{\lambda}\cot\sqrt{\lambda}=\frac{1}{2}+\nu\cot\left(\nu\log\left(\frac{R}{r_{0}}\right)\right).$
(21)

The multivalued function $\lambda(R)$ can be chosen to be continuous [5].

The wavefunction regular at infinity is given as a combination of the Bessel
functions [5],

$\psi\left(r,\kappa_{m}\right)=\sqrt{r}(-1)^{m}\left(ie^{-i\nu\frac{\pi}{2}}J_{i\nu}\left(\kappa_{m}r\right)-ie^{i\nu\frac{\pi}{2}}J_{-i\nu}\left(\kappa_{m}r\right),\right),$
(22)

where $\kappa$ is the energy of the state. The spectrum consists of infinitely
many shallow bound states with adjacent energies differing by an exponential
factor,

$\frac{\kappa_{m+1}}{\kappa_{m}}=e^{-\frac{\pi}{\nu}}.$ (23)

Note that the coordinate enters the wavefunction (22) only in combination with
energy, and the spectrum is generated by the dilation operator:

$\psi_{m+1}=\exp\left(-\frac{\pi}{\nu}r\partial_{r}\right)\psi_{m}.$ (24)

One can think of that relation as that the action of the dilatation operator
shifts zeroes of the wave function from the area of $r<R$ to the area with the
inverse square potential, and one step of (24) evolution corresponds to
elimination of a single zero in the area with the square-well potential. Since
the wave function oscillates infinitely at the origin, the elimination of all
the zeroes would require an infinite number of steps, and in this way a whole
tower of states gets generated.

##  3 RG cycle in graphene

In this Section we shall consider the similar problem in 2+1 dimensions which
physically corresponds to the external charge in the planar graphene layer.
The problem has the classical conformal symmetry and is the relativistic
analogue of the conformal non-relativistic Calogero-like system. Due to
conformal symmetry we could expect the RG cycles and Efimov-like states in
this problem upon imposing the short distance cutoff. The issue of the charge
in the graphene plane has been discussed theoretically [19, 20, 21] and
experimentally [22, 23]. It was argued that indeed there is the tower of
”quasi-Rydberg” states with the exponential scaling [24]. The situation can be
interpreted as an atomic collapse phenomena similar to the instability of
$Z>137$ superheavy atoms in QED [25].

Turn now to the consideration of an electron in graphene which interacts with
an external charge. The two-dimensional Hamiltonian reads as,

$H_{D}=v_{F}\sigma_{i}p^{i}+V(r),\qquad i=1,2.$ (25)

The external charge creates a Coulomb potential,

$V(r)=-\frac{\alpha}{r},\qquad r\geq R.$ (26)

As we shall see, the solution in presence of the potential (26) oscillates
indefinitely at the origin and needs to be regularized by some cutoff $R$.
Hence close enough to the origin $r\leq R$ the potential (26) gets replaced by
some constant potential $V_{reg}(r,\lambda(R))$. The renormalization condition
for the $\lambda$ parameter is that the zero-energy wave function is not
dependent on the short-distance regularization. This condition is chosen
similarly to that of the renormalization of the Calogero system (see Section
2). Hence our primary task is to find the zero-energy solution to the Dirac
equation,

$H_{D}\psi_{0}=0.$ (27)

Since the Hamiltonian commutes with the $J_{3}$ operator,

$J_{3}=i\frac{\partial}{\partial\varphi}+\sigma_{3},\qquad\left[H_{D},J_{3}\right]=0,$
(28)

we can look for the solutions of (27) in the form:

$\psi_{0}=\begin{pmatrix}\chi_{0}(r)\\\
\xi_{0}(r)e^{i\varphi}\end{pmatrix},\qquad J_{3}\psi_{0}=\psi_{0}.$ (29)

Then in polar coordinates the equation (27) reads as:

$\left\\{\begin{array}[]{l}-i\hbar
v_{F}\left(\partial_{r}+\frac{1}{r}\right)\xi_{0}=-V(r)\chi_{0},\\\ -i\hbar
v_{F}\partial_{r}\chi_{0}=-V(r)\xi_{0},\end{array}\right.$ (30)

which is equivalent to:

$\left\\{\begin{array}[]{l}\xi_{0}(r)=i\hbar
v_{F}(V(r))^{-1}{\partial_{r}\chi_{0}},\\\
\partial_{r}^{2}\chi_{0}+\left(\frac{1}{r}-\frac{V^{\prime}(r)}{V(r)}\right)\partial_{r}\chi_{0}+\frac{V^{2}(r)}{\hbar^{2}v_{F}^{2}}\chi_{0}=0.\end{array}\right.$
(31)

For the potential $V=-\frac{\alpha}{r}$ we get the following equation on
$\chi_{0}(r)$:

$\partial_{r}^{2}\chi_{0}+\frac{2}{r}\partial_{r}\chi_{0}+\frac{\beta^{2}}{r^{2}}\chi_{0}=0,\qquad\beta=\frac{\alpha}{\hbar
v_{F}}.$ (32)

Supposing that $\beta^{2}=\frac{1}{4}+\nu^{2}$ we write the solution as:

$\chi_{0}=\sqrt{r}\left(c_{-}\left(\frac{r}{r_{0}}\right)^{-i\nu}+c_{+}\left(\frac{r}{r_{0}}\right)^{i\nu}\right)\propto\sqrt{r}\sin\left(\nu\log\frac{r}{r_{0}}+\varphi\right).$
(33)

We see that this solution shares the properties of the ground-state Calogero
wavefunction (15), namely at nonzero $c_{\pm}$ it generates its own intrinsic
length scale and it oscillates indeterminately at the origin. In order to fix
the $\varphi$ constant we need to introduce a cut-off potential. Hence we
consider the solution in the potential:

$V(r)=\left\\{\begin{array}[]{l}-\frac{\alpha}{r},\qquad r>R,\\\
V_{reg}=-\hbar v_{F}\frac{\lambda}{R},\qquad r\leq R.\end{array}\right.$ (34)

The dilatation operator acts on $\chi$ as following:

$r\partial_{r}\chi_{0}=\left(\frac{1}{2}+\nu\cot\left(\nu\log\frac{r}{r_{0}}\right)\right)\chi_{0}.$
(35)

For the constant potential $V_{reg}$ we get from $(\ref{Dechi})$:

$\partial^{2}_{r}\chi_{0}^{reg}+\frac{1}{r}\partial_{r}\chi_{0}^{reg}+\frac{\lambda^{2}}{R^{2}}\chi_{0}^{reg}=0.$
(36)

Choosing the solution of (36) which is regular at the origin we obtain,

$\chi_{0}^{reg}\propto J_{0}\left(\lambda\frac{r}{R}\right).$ (37)

Computing the action of the dilation operator on the solution in the area of
constant potential and equating it to the action of the dilation operator (35)
we get the equation on the $\lambda$ regulator parameter:

$\frac{1}{2}+\nu\cot\left(\nu\log\left(\frac{R}{r_{0}}\right)\right)=-\lambda\frac{J_{1}(\lambda)}{J_{0}(\lambda)}.$
(38)

The equation (38) defines $\lambda$ as a multi-valued function of $R$. The
period of the RG flow corresponds to jump from one branch of the $\lambda(R)$
function to another.

Now we proceed to find the bound states in the (26) potential. We consider
again the Dirac equation,

$H_{D}\psi_{\kappa}=-\hbar v_{F}\kappa\psi_{\kappa}.$ (39)

Then the equation on $\chi$ analogous to (31) is as following:

$\partial^{2}_{r}\chi_{\kappa}+\frac{2\beta-\kappa r}{\beta-\kappa
r}\frac{1}{r}\partial_{r}\chi_{\kappa}+\left(\frac{\beta}{r}-\kappa\right)^{2}\chi_{\kappa}=0.$
(40)

Asymptotically when $r\gg\frac{\beta}{\kappa}$ the solution of (40) regular at
infinity is given by the Hankel function,

$\chi_{\kappa}\propto H_{0}^{(1)}(i\kappa r).$ (41)

At small $r\ll\frac{\beta}{\kappa}$ the solution is not regular at the origin,

$\chi_{\kappa}\propto\sqrt{r}\sin\left(\nu\log\frac{r}{r_{0}}\right),$ (42)

and we are again in need for the regulator potential. Solving again the Dirac
equation (39) in presence of the constant potential $V_{reg}$ and computing
the action of the dilatation operator,

$r\partial_{r}\chi_{k}^{reg}=-\left(\lambda-\kappa
R\right)\frac{J_{1}\left(\lambda-\kappa R\right)}{J_{0}\left(\lambda-\kappa
R\right)}\chi_{\kappa}^{reg},$ (43)

we can equate (43) to the action of the dilatation operator on (42) and get
the equation on the spectrum of the bound states,

$\frac{1}{2}+\nu\cot\left(\nu\log\left(\kappa
R\right)\right)=-\left(\lambda-\kappa R\right)\frac{J_{1}\left(\lambda-\kappa
R\right)}{J_{0}\left(\lambda-\kappa R\right)}.$ (44)

This condition gives the spectrum of infinitely many shallow bound states,

$\kappa_{n}=\kappa_{*}\exp\left(-\frac{\pi
n}{\nu}\right),\qquad\kappa\to\infty.$ (45)

##  4 Anomaly in the ${\bf so}(2,1)$ algebra

Let us make some comments on the algebraic counterpart of the phenomena
considered following [9]. As we have mentioned the conformal symmetry is the
main player since Hamiltonians under consideration are scale invariant before
regularization. Actually this group can be thought of as the example of
spectrum generating algebra when the Hamiltonian is one of the generators or
is expressed in terms of the generators in a simple manner. This is familiar
from the exactly or quasi-exactly solvable problems when the dimension of the
representation selects the size of the algebraic part of the spectrum.

Let us introduce the generators of the ${\bf so}(2,1)$ conformal algebra
$J_{1},J_{2},J_{3}$: the Calogero Hamiltonian,

$J_{1}=H=p^{2}+V(r),$ (46)

the dilatation generator,

$J_{2}=D=tH-\frac{1}{4}(pr+rp),$ (47)

and the generator of special conformal transformation,

$J_{3}=K=t^{2}H-\frac{t}{2}(pr+rp)+\frac{1}{2}r^{2}.$ (48)

They satisfy the relations of the ${\bf so}(2,1)$ algebra:

$[J_{2},J_{1}]=-iJ_{1},\qquad[J_{3},J_{1}]=-2iJ_{2},\qquad[J_{2},J_{3}]=iJ_{3}.$
(49)

The singular behavior of the potential at the origin amounts to a kind of
anomaly in the ${\bf so}(2,1)$ algebra,

$A(r)=-i[D,H]+H,$ (50)

which in $d$ space dimensions can be presented in the following form:

$A(r)=-\frac{d-2}{2}V(r)+(r^{i}\nabla_{i})V(r).$ (51)

The simple arguments imply the following relation

$\frac{d}{dt}\langle D\rangle_{\mathrm{ground}}=E_{\mathrm{ground}},$ (52)

where the matrix element is taken over the ground state.

It turns out that (52) is fulfilled for the singular potentials in Calogero-
like model or in models with contact potential, $V(r)=g\delta(r)$. The
expression for anomaly does not depend on the regularization chosen. Moreover
more detailed analysis demonstrates that the anomaly is proportional to the
$\beta$-function of the coupling providing the UV regularization as can be
expected.

A similar calculation of the anomaly for the graphene case can be performed
for arbitrary state,

$\left\langle{\frac{dD}{dt}}\right\rangle_{\psi}=\braket{\Xi}_{\psi}=-\int
d^{2}x\psi^{*}(V(x)+x_{i}\partial_{i}V(x))\psi,$ (53)

which yields using square-well regularization:

$\braket{\Xi}_{\psi}=\hbar
v_{F}\frac{\lambda(R)}{R}\frac{\int\limits^{R}_{0}r|\psi|^{2}dr}{\int\limits^{\infty}_{0}r|\psi|^{2}dr}.$
(54)

It is convenient to use the two-dimensional identity in (51),

$\nabla\frac{\vec{r}}{r}=2\pi\delta(\vec{r}),$ (55)

which simplifies the calculation of the anomaly for any normalized bound
state,

$\frac{d}{dt}\langle D\rangle_{\Psi}=-g\pi\int d^{2}r\delta(r)|\Psi(r)|^{2}.$
(56)

##  5 RG cycles in models of superconductivity

In this Section we explain how the cyclic RG flows emerge in truncated models
of superconductivity. To this aim we shall first describe the Richardson model
and then consider its generalization to the RD model which enjoys the cyclic
RG flow. These models are distinguished by the finiteness of the number of
fermionic levels. The relation with the integrable many-body systems proves to
be quite useful.

###  5.1 Richardson model versus Gaudin model

Let us recall the truncated BCS-like Richardson model of superconductivity
[26] with some number of doubly degenerated fermionic levels with the energies
$\epsilon_{j\sigma},j=1,\dots,N$. It describes the system of a fixed number of
the Cooper pairs. It is assumed that several energy levels are populated by
Cooper pairs while levels with the single fermions are blocked. The
Hamiltonian reads as

$H_{BCS}=\sum_{j,\sigma=\pm}^{N}\epsilon_{j\sigma}c^{+}_{j\sigma}c_{j\sigma}-G\sum_{jk}c^{\dagger}_{j+}c^{\dagger}_{j-}c_{k-}c_{k+},$
(57)

where $c_{j\sigma}$ are the fermion operators and $G$ is the coupling constant
providing the attraction leading to the formation of the Cooper pairs. In
terms of the hard-core boson operators it reads as

$H_{BCS}=\sum_{j}\epsilon_{j}b^{\dagger}_{j}b_{j}-G\sum_{jk}b^{\dagger}_{j}b_{k},$
(58)

where

$[b^{\dagger}_{j},b_{k}]=\delta_{jk}(2N_{j}-1),\qquad
b_{j}=c_{j-}c_{j+},\qquad N_{j}=b^{\dagger}_{j}b_{j}.$ (59)

The eigenfunctions of the Hamiltonian can be written as,

$|M\rangle=\prod_{i}^{M}B_{i}(E_{i})|\mathrm{vac}\rangle,\qquad
B_{i}=\sum_{j}^{N}\frac{1}{\epsilon_{j}-E_{i}}b^{\dagger}_{j},$ (60)

provided the Bethe ansatz equations are fulfilled,

$G^{-1}=-\sum_{j}^{N}\frac{1}{\epsilon_{j}-E_{i}}+\sum_{j}^{M}\frac{2}{E_{j}-E_{i}}.$
(61)

The energy of the corresponding states reads as:

$E(M)=\sum_{i}E_{i}.$ (62)

It was shown in [27] that the Richardson model is exactly solvable and closely
related to the particular generalization of the Gaudin model. To describe this
relation it is convenient to introduce the so-called pseudospin ${\bf sl}(2)$
algebra in terms of the creation-annihilation operators for the Cooper pairs,

$t_{j}^{-}=b_{j},\qquad t_{j}^{+}=b^{\dagger}_{j},\qquad t^{0}_{j}=N_{j}-1/2.$
(63)

The Richardson Hamiltonian commutes with the set of operators $R_{i}$,

$R_{i}=-t^{0}_{i}-2G\sum^{N}_{j\neq
i}\frac{t_{i}t_{j}}{\epsilon_{i}-\epsilon_{j}},$ (64)

which are identified as the Gaudin Hamiltonians,

$[H_{BCS},R_{j}]=[R_{i},R_{j}]=0.$ (65)

Moreover the Richardson Hamiltonian itself can be expressed in terms of the
operators $R_{i}$ as:

$H_{BCS}=\sum_{i}\epsilon_{i}R_{i}+G\left(\sum
R_{i}\right)^{2}+\mathrm{const}.$ (66)

The number $N$ of the fermionic levels coincides with the number of sites in
the Gaudin model and the coupling constant in the Richardson Hamiltonian
corresponds to the ”twisted boundary condition” in the Gaudin model. The Bethe
ansatz equations for the Richardson model (61) exactly coincide with the ones
for the generalized Gaudin model. It was argued in [2] that the Bethe roots
correspond to the excited Cooper pairs that is natural to think about the
solution to the Baxter equation as the wave function of the condensate. In
terms of the conformal field theory Cooper pairs correspond to the screening
operators [28].

For the nontrivial degeneracies of the energy levels $d_{j}$ the BA equations
read as:

$G^{-1}=-\sum_{j}^{N}\frac{d_{j}}{\epsilon_{j}-E_{i}}+\sum_{j\neq
i}^{M}\frac{2}{E_{j}-E_{i}}.$ (67)

###  5.2 Russian Doll model of superconductivity and twisted XXX spin chains

The important generalization of the Richardson model describing
superconductivity is the so-called RD model [2]. It involves the additional
dimensionless parameter $\alpha$ and the RD Hamiltonian reads as:

$H_{RD}=2\sum_{i}^{N}(\epsilon_{i}-G)N_{i}-\bar{G}\sum_{j<k}(e^{i\alpha}b^{+}_{k}b_{j}+e^{-i\alpha}b^{+}_{j}b_{k}),$
(68)

with two dimensionful parameters $G,\eta$ and $\bar{G}=\sqrt{G^{2}+\eta^{2}}$.
In terms of these variables the dimensionless parameter $\alpha$ has the
following form:

$\alpha=\arctan\left(\frac{\eta}{G}\right).$ (69)

It is also useful to consider two dimensionless parameters $g,\theta$ defined
as $G=gd$ and $\eta=\theta d$ where $d$ is the level spacing. The RD model
reduces to the Richardson model in the limit $\eta\rightarrow 0$.

The RD model turns out to be integrable as well. Now instead of the Gaudin
model the proper counterpart is the generic quantum twisted XXX spin chain
[29]. The transfer matrix of such spin chain model $t(u)$ commutes with the
$H_{RD}$ which itself can be expressed in terms of the spin chain
Hamiltonians.

The equation defining the spectrum of the RD model reads as:

$e^{2i\alpha}\prod_{l=1}^{N}\frac{E_{i}-\varepsilon_{l}+i\eta}{E_{i}-\varepsilon_{l}-i\eta}=\prod_{j\neq
i}^{M}\frac{E_{i}-E_{j}+2i\eta}{E_{i}-E_{j}-2i\eta},$ (70)

and coincides with the BA equations for the spin chain.

Taking the logarithm of the both sides of the equation (70) we obtain:

$\alpha+\pi
Q_{i}+\sum_{l=1}^{N}\arctan\left(\frac{\eta}{E_{i}-\varepsilon_{l}}\right)-\sum_{j=1}^{M}\arctan\left(\frac{2\eta}{E_{i}-E_{j}}\right)=0.$
(71)

Note that here we have added an arbitrary integer term to account for
generically multivalued arctangent function.

The RG step amounts to integrating out the $N$-th degree of freedom in the RD
model, or equivalently to integrating out the $N$-th inhomogeneity in the XXX
chain. This results into renormalization of the twist. From (71) it is easy to
see that:

$\arctan\left(\frac{\eta}{G_{N}}\right)-\arctan\left(\frac{\eta}{G_{N-1}}\right)=\sum_{i=1}^{M}\arctan\left(\frac{2\eta}{E_{i}-\varepsilon_{N}}\right).$
(72)

When $M=1$ it implies that:

$G_{N-1}-G_{N}=\frac{G_{N}^{2}+\eta^{2}}{\varepsilon_{N}-G_{N}-E},$ (73)

which is a discrete version of the (1) equation. Of course the same relation
can be derived from the RD Hamiltonian (68). If we consider the wavefunction
$\psi=\sum_{i}^{N}\psi_{i}b_{i}^{\dagger}|0\rangle$, the Schrödinger equation
for a state with one Cooper pair amounts to:

$\left(\varepsilon_{i}-G-E\right)\psi_{i}=(G+i\eta)\sum_{j=1}^{i-1}\psi_{j}+(G-i\eta)\sum_{j=i+1}^{N}\psi_{j}.$
(74)

Integration out the $N$-th degree of freedom amounts to expressing $\psi_{M}$
in terms of the other modes,

$\psi_{N}=\frac{G+i\eta}{\varepsilon_{N}-G-E}\sum_{j=1}^{N-1}\psi_{j},$ (75)

and substituting it back into the Schrödinger equation (74). The $G_{N-1}$
constant in the resulting equation will differ from the initial $G_{N}$ value
as in (73).

The key feature of the RD model is the multiple solutions to the gap equation.
The gaps are parameterized as follows:

$\Delta_{n}=\frac{\omega}{\sinh t_{n}},\qquad t_{n}=t_{0}+\frac{\pi
n}{\theta},\qquad n=0,1,\dots,$ (76)

where $t_{0}$ is the solution to the following equation:

$\tan(\theta t_{0})=\frac{\theta}{g},\qquad 0<t_{0}<\frac{\pi}{\theta}.$ (77)

and $\omega=dN$ for equal level spacing. Here
$E^{2}=\varepsilon^{2}+|\Delta|^{2}$. This behavior can be derived via the
mean field approximation [14]. The gap with minimal energy defines the ground
state, and the other values of the gap describe excitations. In the limit
$\theta\rightarrow 0$ the gaps $\Delta_{n>0}\rightarrow 0$ and

$t_{0}=\frac{1}{g},\qquad\Delta_{0}=2\omega\exp\left(-\frac{1}{g}\right),$
(78)

therefore the standard BCS expression for the gap is recovered. At the weak
coupling limit the gaps behave as:

$\Delta_{n}\propto\Delta_{0}e^{-\frac{n\pi}{\theta}}.$ (79)

In terms of the solutions to the BA equations the multiple gaps correspond to
the choices of the different branches of the logarithms, i.e. to different
choices of the integer $Q$ parameter in (70).

If the degeneracy of the levels is $d_{n}$ then the RD model gets modified a
little bit and is related to the higher spin XXX spin chain. The local spins
$s_{i}$ are determined by the corresponding higher pair degeneracy $d_{i}$ of
the $i$-th level,

$s_{i}=d_{i}/2,$ (80)

and the corresponding BA equations read as:

$e^{2i\alpha}\prod_{l=1}^{N}\frac{E_{i}-\varepsilon_{l}+id_{l}+i\eta}{E_{i}-\varepsilon_{l}-id_{l}-i\eta}=\prod_{j\neq
i}^{M}\frac{E_{i}-E_{j}+2i\eta}{E_{i}-E_{j}-2i\eta}.$ (81)

###  5.3 Cyclic RG flows in the RD model

The RD model of truncated superconductivity enjoys the cyclic RG behavior [2].
The RG flows can be treated as the integrating out the highest fermionic level
with appropriate scaling of the parameters using the procedure developed in
[1]. The RG equations read as (73):

$g_{N-1}=g_{N}+\frac{1}{N}(g_{N}^{2}+\theta^{2}),\qquad\theta_{N-1}=\theta_{N}.$
(82)

At large $N$ limit the natural RG variable is identified with
$s=\log(N/N_{0})$ and the solution to the RG equation is:

$g(s)=\theta\tan\left(\theta
s+\tan^{-1}\left(\frac{g_{0}}{\theta}\right)\right).$ (83)

Hence the running coupling is cyclic,

$g(s+\lambda)=g(s),\qquad g(e^{-\lambda}N)=g(N),$ (84)

with the RG period,

$\lambda=\frac{\pi}{\theta},$ (85)

and the total number of the independent gaps in the model is:

$N_{cond}\propto\frac{\theta}{\pi}\log N.$ (86)

The multiple gaps are the manifestations of the Efimov-like states. The sizes
of the Cooper pairs in the $N$-th condensates also have the RD scaling. The
cyclic RG can be derived even for the single Cooper pair.

What is going on with the spectrum of the model during the period? It was
shown in [14] that it gets reorganized. The RG flow experiences
discontinuities from $g=+\infty$ to $g=-\infty$ when a new cycle gets started.
At each jump the lowest condensate disappears from the spectrum,

$\Delta_{N+1}(g=+\infty)=\Delta_{N}(g=-\infty),$ (87)

indicating that the $(N+1)$-th state wave function plays the role of $N$-th
state wave function at the next cycle (see (75)).

The same behavior can be derived from the BA equation [14]. To identify the
multiple gaps it is necessary to remind that the solutions to the BA equations
are classified by the integers $Q_{i},i=1,\dots,M$ parameterizing the branches
of the logarithms. If one assumes that $Q_{i}=Q$ for all Bethe roots then this
quantum number gets shifted by one at each RG cycle and was identified with
the integer parameterizing the solution to the gap equations,

$\Delta_{Q}\propto\Delta_{0}\exp^{-\lambda Q}.$ (88)

At the large $N$ limit the BA equations of the RD model reduce to the BA
equation of the Richardson-Gaudin model with the rescaled coupling,

$G_{Q}^{-1}=\eta^{-1}({\alpha}+\pi Q),$ (89)

which can be treated as the shifted boundary condition in the generalized
Gaudin model parameterized by an integer. Let us emphasize that the unusual
cyclic RG behavior is due to the presence of two couplings in the RD model.

##  6 Triality in the integrable models and RG cycles

|  |  |
---|---|---|---
RS modelQC duality$\scriptstyle{\begin{matrix}\text{non-relativistic}\\\
\text{limit}\end{matrix}}$XXX
chain$\scriptstyle{\begin{matrix}\text{semiclassical}\\\
\text{limit}\end{matrix}}$RD modelCalogero systemQC dualityGaudin
modelRichardson model Figure 1: Besides the triality shown on the picture, a
bispectral duality acts on RS/Calogero and XXX/Gaudin sides of the
correspondence. Being originated from three-dimensional mirror-symmetry [30],
this duality interchanges coordinates with Lax eigenvalues in the classical
systems, and inhomogeneities with twists in quantum ones.

In this Section we summarize several dualities between the integrable models
and consider the realizations of the cyclic RG flows in these systems. The
question is motivated by the close relationship between the restricted BCS
models and spin chains. Actually there are three different families of models
related with each other by the particular identifications of phase spaces and
parameters. The first family concerns the system of fermions (Richardson-
Russian Doll) which develop superconducting gap. The second family involves
the spin systems of twisted inhomogeneous Gaudin-XXX-XXZ type and their
generalizations. The third family involves the Calogero-Ruijsenaars (CR) chain
of the integrable many body systems.

We look for the answers on the following questions

  * •

What is the condition yielding the RG equation for some coupling in each
family?

  * •

What is the RG variable?

  * •

What determines the period of the cycle?

In the superconducting system at RG step one decouples the highest energy
level and looks at the renormalization of the interaction coupling constant.
The RG time is identified with the number of energy levels $t=\log N$. The
period of RG is defined by the T-asymmetric parameter of RD model.

In the spin chain model the RG step corresponds to the ”integrating out” one
”highest” inhomogeneity with the corresponding renormalization of the twist.
The period of the RG flow is fixed by the Planck constant in the quantum spin
chain. In the bispectral dual spin chain [33] one now ”integrates out” one of
the twists and ”renormalizes” the inhomogeneity. Since the Planck constant
gets inverted upon bispectrality $\hbar_{spin}\rightarrow\hbar_{spin}^{-1}$
the period of the RG cycle gets inverted as well. Note that the RG equation in
the superconducting model can be mapped into BAE in the spin chain [14]. The
condition yealding the RG equation corresponds to the independence of the
Bethe root on the RG step.

For two-body system with attractive rational potential one can define the RG
condition as the continuity of the zero-energy wave function under the
changing the cutoff scale at small $r$. This condition imposes the RG equation
at the cutoff UV coupling constant. This RG equation has the cycle with the
period

$T_{Cal}=\frac{\pi}{\nu-\frac{1}{2}}.$ (90)

as was shown in Section 2.

The Quantum-Classical (QC) duality [30, 31] relates the quantum spin chain
systems and the classical Calogero–type systems. Through the QC
correspondence, the rational Gaudin model can be linked with the rational
Calogero system spin chain inhomogeneities being the Calogero coordinates, and
the twist in the spin chain (which is a single variable in our case) being the
Lax matrix eigenvalue. It is also possible to make a bispectrality
transformation of rational Calogero model, which interchanges Lax eigenvalues
with coordinates. This means that now the Calogero coordinates correspond to
the twists at the spin chain side. In this case the Calogero coupling gets
inverted which means that the period of the RG cycle gets inverted as well.

To consider the mapping of RG cycles in the Calogero system and the spin chain
we need the generalization of QC duality to the quantum-quantum case. The
spectral problem in Calogero model has been identified with the KZ equation
involving the Gaudin Hamiltonian,

$\frac{d}{dz_{i}}\Psi=H_{gaud}\Psi+\lambda\Psi.$ (91)

Since we formulate RG condition on the Calogero side for the $E=0$ state, the
inhomogeneous term in the KZ equation is absent. The simplest test of the
mapping of the RG cycles under QQ duality concerns the identifications of the
periods. On the spin chain side it is identified with the Planck constant
while at the Calogero side the period is defined by the coupling constant. The
following identification holds for QC duality [31]:

$\hbar_{spin}=\nu,$ (92)

which implies that the periods of the cycles at the Calogero and spin chain
sides match.

The Efimov-like tower in these families have the following interpretations. In
the superconducting system it corresponds to the family of the gaps
$\Delta_{n}$ with the Efimov scaling responsible of the scale symmetry broken
down to the discrete subgroup. In the spin chain it corresponds to the
different branches of the solutions to the BAE which can be also interpreted
in terms of the allowed set of twists. Finally in the CR family it corresponds
to the family of the shallow bound states near the continuum threshold.

##  7  RG cycles in $\Omega$-deformed SUSY gauge theories

In this Section we shall explain how the RG flows in $\Omega$\- deformed SUSY
gauge theories can be reformulated in terms of the brane moves. Why the very
RG cycles could be expected in the deformed gauge theories? The answer is
based on the identification of the quantum spin chains in one or another
context in the SUSY gauge theory. Once such quantum spin chain has been found
we can apply the results of the previous sections where the place of the RG
cycles in the spin chain framework has been clarified.

First, we shall briefly review the $\Omega$-deformation of the SUSY gauge
theories. Then we make some general comments concerning the realization of the
gauge theories as the worldvolume theories on $D$-branes to explain how the
parameters of the gauge theory are identified with the brane coordinates.

###  7.1 Four-dimensional $\Omega$-deformed gauge theory

The Bethe ansatz equations can be encountered not only in the models of
superconductivity, but also in gauge theories. The quantum XXX spin chain
governs the moduli space of vacua of an $\Omega$-deformed four-dimensional
theory in the Nekrasov–Shatashvili limit, i.e. when one of the deformation is
chosen to be zero: $\epsilon_{2}=0,\epsilon_{1}=\epsilon$ [35]. Since the
quantum XXX spin chain displays a cyclic RG behaviour, as we have seen in the
Section 5, it is interesting to identify this phenomenon in the four-
dimensional gauge theory.

Consider a four-dimensional $\mathcal{N}=2$ theory with matter hypermultiplet,
which has a vanishing $\beta$-function, i.e. when $N_{f}=2N_{c}$. This theory
is dual to a classical inhomogeneous twisted XXX chain, in a sense that the
Seiberg-Witten curve for the gauge theory coincides with the spectral curve
for the spin chain. The twist of the spin chain is identified with the modular
parameter of the curve and with the complexified coupling of the gauge theory,
the inhomogeneities of the spin chain get mapped into masses of the
hypermultiplets. For more information on the correspondence between classical
integrable systems and gauge theories the reader can consult [36].

The $\Omega$-deformation is introduced to regularize the instanton divergence
in the partition function of the gauge theory [37]. We can consider the four-
dimensional theory as a reduction of the six-dimensional $\mathcal{N}=1$
theory with metric:

$ds^{2}=2dzd\bar{z}+\left(dx^{m}+\Omega^{mn}x_{n}d\bar{z}+\bar{\Omega}^{mn}x_{n}dz\right)^{2},\qquad
m=1,\ldots,4,$ (93)

i.e. we can consider the theory on a four-dimensional space, fibered over a
two-dimensional torus. One can imagine the $\epsilon_{1,2}$ deformation
parameters as chemical potentials for the rotations in two orthogonal planes
in four-dimensional Euclidean space. One can also think that the Euclidean
$\mathbb{R}^{4}$ space gets substituted by a sphere $S^{4}$ with finite
volume.

The non-trivial $\Omega$-deformation modifies the correspondence between gauge
theories and integrable systems. Namely, in the Nekrasov-Shatashvili limit the
$\Omega$-deformed gauge theory corresponds to a quantum XXX spin chain with
$\epsilon$ playing the role of the Planck constant [35]. This deformed gauge
theory also appears to be dual to the two-dimensional effective theory on a
worldsheet of a non-abelian string [39].

Consider $\Omega$-deformed $\mathcal{N}=2$ SQCD with $SU(L)$ gauge group, $L$
fundamental hypermultiplets with masses $m^{f}_{i}$ and $L$ antifundamental
hypermultiplets with masses $m^{af}_{i}$. Let us denote the set of the
eigenvalues of the adjoint scalar in the vector multiplet by $\vec{a}$. We can
expand the deformed partition function around $\epsilon=0$ to identify the
prepotential and effective twisted superpotential,

$\log\mathcal{Z}\left(\vec{a},\epsilon_{1},\epsilon_{2}\right)\sim\frac{1}{\epsilon_{1}\epsilon_{2}}\mathcal{F}(\vec{a},\epsilon)+\frac{1}{\epsilon_{2}}\mathcal{W}(\vec{a},\epsilon).$
(94)

The effective twisted superpotential is a multivalued function, with the
branch fixed by the set of integers $\vec{k}$:

$\mathcal{W}(\vec{a},\epsilon)=\frac{1}{\epsilon}\mathcal{F}(\vec{a},\epsilon)-2\pi
i\vec{k}\cdot\vec{a},\qquad\vec{k}\in\mathbb{Z}^{L}.$ (95)

The equation on vacua,

$\frac{\partial\mathcal{W}(\vec{a},\epsilon)}{\partial\vec{a}}=\vec{n},\qquad\vec{n}\in\mathbb{Z}^{L},$
(96)

provides the condition on $\vec{a}$,

$\vec{a}=\vec{m}^{f}-\epsilon\vec{n}.$ (97)

This theory admits the existence of non-abelian strings probing the four-
dimensional space-time. The two-dimensional worldsheet theory of the non-
abelian string involves $L$ fundamental chiral multiplets with twisted masses
$M^{F}_{i}$ and $L$ antifundamental multiplets with twisted masses
$M^{AF}_{i}$, which are identified as:

$M^{F}_{i}=m^{f}_{i}-\frac{3}{2}\epsilon,\qquad
M^{AF}_{i}=m^{af}_{i}+\frac{1}{2}\epsilon.$ (98)

The two-dimensional theory also contains an adjoint chiral multiplet with mass
$\epsilon$. The rank of the gauge group $N$ (or equivalently the number of
non-abelian strings) is given in terms of $\vec{n}$ vector by the relation:

$N+L=\sum_{l=1}^{L}n_{l}.$ (99)

The modular parameters of the four-dimensional and the two-dimensional
theories are related as:

$\tau_{2d}=\tau_{4d}+\frac{1}{2}(N+1).$ (100)

The effective twisted worldsheet superpotential is given in terms of the four-
dimensional superpotential:

$\mathcal{W}_{4d}\left(a_{i}=m^{f}_{i}-\epsilon
n_{i},\epsilon\right)-\mathcal{W}_{4d}\left(a_{i}=m^{f}_{i}-\epsilon,\epsilon\right)=\mathcal{W}_{2d}\left(\\{n_{i}\\}\right).$
(101)

The two-dimensional superpotential depends on the set of eigenvalues of the
adjoint scalar in vector representation $\lambda_{i}$, $i=1,\ldots,N$. The set
of equations $\partial\mathcal{W}_{2d}/\partial\lambda=0$ appears to be
equivalent to the Bethe ansatz equations for the XXX spin chain:

$\prod_{l=1}^{L}\left(\frac{\lambda_{j}-M^{F}_{l}}{\lambda_{j}-M^{AF}_{l}}\right)=\exp\left(2\pi
i\tau_{4d}\right)\prod_{k\neq
j}^{N}\left(\frac{\lambda_{j}-\lambda_{k}-\epsilon}{\lambda_{j}-\lambda_{k}+\epsilon}\right).$
(102)

The Planck constant in the spin chain is identified with the $\epsilon$
deformation parameter. The complexified coupling parameter plays the role of
twist in the spin chain. The renormalization of the spin chain amounts to
decoupling of one fundamental and one anti-fundamental chiral multiplet. In
the four-dimensional theory it corresponds to the decrease of the number of
flavors $N_{f}\to N_{f}-2$ simultaneously with reducing the rank of the gauge
group $N_{c}\to N_{c}-1$. Therefore the theory remains conformal. The
renormalization of the coupling constant analogous to (73) derived from the
relation (102) for $N=1$ is:

$\exp\left(2\pi
i(\tau_{L}-\tau_{L-1})\right)=\frac{\lambda-M^{F}_{L}}{\lambda-M^{AF}_{L}}.$
(103)

If we choose the masses to be equidistant with spacing $\delta m$, the change
in the coupling constant during one step of RG flow is:

$\exp\left(2\pi i(\tau_{L}-\tau_{L-1})\right)\propto\frac{\epsilon}{\delta
m}.$ (104)

Hence a number of nonperturbative scales emerges in a theory, analogously to
the generation of the Efimov scaling in the Calogero model. These scales
correspond to multiple gaps in the superconducting model:

$\Delta_{n}\propto\Delta_{0}\exp\left(-\frac{\pi n\delta m}{\epsilon}\right).$
(105)

Note that the emergence of cyclic RG evolution is a feature caused by the
$\Omega$-deformation, since in a non-deformed theory a decoupling of the heavy
flavor does not lead to any cyclic dynamics.

###  7.2 $3d$ gauge theories and theories on the brane worldvolumes

Let us briefly explain the main points concerning the geometrical engineering
of the gauge theories on the $D$-branes suggesting the reader to consult the
details in the review paper [40]. The $Dp$ brane is the $(p+1)$-dimensional
hypersurface in the ten-dimensional space-time which supports the $U(1)$ gauge
field. This feature provides the possibility to built up the gauge theories
with the desired properties. Let us summarize the key elements of the
”building procedure”.

  * •

A stack of coinciding $N$ $D$-branes supports $U(N)$ gauge theory with the
maximal supersymmetry.

  * •

Displacing some branes from the stack in the transverse direction corresponds
to the Higgs mechanism in the $U(N)$ gauge theory and the distance between
branes corresponds to the Higgs vev.

  * •

To reduce the SUSY one imposes some boundary conditions at some coordinates
using other types of branes or rotates some branes.

  * •

All geometrical characteristics of the brane configurations have the meaning
of parameters of the gauge theory like couplings or vevs of some operators in
the gauge theory on their worldvolumes.

  * •

If we move some brane through another one the brane of smaller dimension could
be created. The Hanany-Witten move is the simplest example (see fig. 2).

  * •

Since generically we have branes of different dimensions in the configuration,
for example, $N$ $D2$ branes and $M$ $D4$ branes we have simultaneously $U(N)$
2+1 dimensional gauge theory and $U(M)$ dimensional 4+1 dimensional theory on
the brane worldvolumes. These theories coexist simultaneously hence there is
highly nontrivial interplay between two gauge theories.

Figure 2: Hanany-Witten move. Here vertical lines are $NS5$ branes, horizontal
lines are $D3$ branes, and circles are $D5$ branes. When a $D5$ brane is moved
through a sequence of $NS5$ branes the linking number between them is
conserved hence additional $D3$ branes appear.

Let us explain now how these brane rules can be used to engineer the gauge
theories which are related with the quantum spin chains. Our main example is a
$3d$ $\mathcal{N}=2$ quiver gauge theory.

The brane configuration relevant for this theory is built as follows. We have
$M$ parallel $NS5$ branes extended in $(012456)$, $N_{i}$ $D3$ branes extended
in $(0123)$ between $i$-th and $(i+1)$-th $NS5$ branes, and $M_{i}$ $D5$
branes extended in $(012789)$ between $i$-th and $(i+1)$-th $NS5$ branes (see
table 3). From this brane configuration we obtain the $\prod_{i}^{M}U(N_{i})$
gauge group on the $D3$ branes worldvolume with $M_{i}$ fundamentals for the
$i$-th gauge group. The distances between the $i$-th and $(i+1)$-th $NS5$
branes yield the complexified gauge coupling for $U(N_{i})$ gauge group while
the coordinates of the $D5$ branes in the $(45)$ plane correspond to the
masses of fundamentals. The positions of the $D3$ branes on $(45)$ plane
correspond to the coordinates on the Coulomb branch in the quiver theory. The
additional $\Omega$ deformation reduces the theory with $\mathcal{N}=4$ SUSY
to the $\mathcal{N}=2^{*}$ theory, i.e. an $\mathcal{N}=2$ theory with massive
adjoint. It is identified as $3d$ gauge theory when the distance between $NS5$
is assumed to be small enough. We assume that one coordinate is compact that
is the theory lives on $\mathbb{R}^{2}\times S^{1}$.

| 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9
---|---|---|---|---|---|---|---|---|---|---
D3 | $\times$ | $\times$ | $\times$ | $\times$ |  |  |  |  |  |
NS5 | $\times$ | $\times$ | $\times$ |  | $\times$ | $\times$ | $\times$ |  |  |
D5 | $\times$ | $\times$ | $\times$ |  |  |  |  | $\times$ | $\times$ | $\times$
Figure 3: Brane construction of the $3d$ quiver theory.

The mapping of the gauge theory data into the integrability framework goes as
follows. In the NS limit of the $\Omega$-deformation the twisted
superpotential in $3d$ gauge theory on the $D3$ branes gets mapped into the
Yang-Yang function for the $XXZ$ chain [35]. The minimization of the
superpotential yields the equations describing the supersymmetric vacua and in
the same time they are the Bethe ansatz equations for the $XXZ$ spin chain,
generally speaking the nested Bethe ansatz equations. That is $D3$ branes are
identified with the Bethe roots which are distributed according to the ranks
of the gauge groups at each of $M$ steps of nesting $\prod_{i}^{M}U(N_{i})$.
The positions of the $D5$ branes in the $(45)$ plane correspond to the
inhomogeneities in the $XXZ$ spin chain. The anisotropy of the $XXZ$ chain is
defined by the radius of the compact dimensions while the parameter of the
$\Omega$ deformation plays the role of the Planck constant in the $XXZ$ spin
chain. At small radius the XXZ spin chain turns to the XXX spin chain. The
twists in the spin chain correspond to the coordinates of the $NS5$ branes in
the (78) plane, and the Fayet–Iliopoulos parameters in the three-dimensional
theory [30].

One step of the RG flow corresponds to elimination of one inhomogeneity in the
spin chain resulting in renormalization of the twists. In the three-
dimensional theory this means that the integration of one massive flavor leads
to the renormalization of the FI parameters. In terms of the transformations
of the brane configurations this process receives transparent geometrical
interpretation:

  * •

The RG step is the removing of one $D5$ brane which amounts to the
renormalization of the position of $NS5$ branes or twists.

  * •

The period of the RG cycle is fixed by the number of $NS5$ branes [34], since
it was identified with the Planck constant in the spin chain.

  * •

At some scale the twists flow from $+\infty$ to $-\infty$.

##  8 Conclusion

Are there any general lessons which we could learn for the quantum field
theory from the very existence of the cyclic RG flows? The most important
point is that there is some fine structure at the UV scale which is reflected
in the Efimov tower with the BKT scaling behavior. Moreover the cyclic flows
imply the interplay between the UV and IR cutoffs in the theory which usually
was attributed to the noncommutative theory. This mixing presumably could shed
the additional light on the dimensional transmutation phenomena in the field
theory and provide the examples for the simultaneous generation of the
multiple scales.

The presence of two parameters in RG is quite common however probably some
additional properties of these parameters are required. In particular in many
(although not all) examples the period of the cycle is fixed by the “filling
fraction” in some external field which could be magnetic field or parameter of
$\Omega$ background. The latter has the meaning of the Planck constant in the
auxiliary finite dimensional integrable model. This could suggest that the
very issue can be formulated purely in terms of the quantum phase space since
the Planck constant can be interpreted as the external field applied to the
classical phase space.

Actually we could expect the relation of RG cycles with some refinement of the
path integral in quantum mechanics. As an aside remark note that the attempt
to get the rigorous mathematical formulation of the renormalization of the QFT
leaded to the motivic generalization of the path integral. It corresponds to
some fine structure at the regulator scale which has some similarities with
the discussion above. The RG cycle in the quantum rational Calogero model
implies the intimate relation with the knot theory since the knot invariants
at the rational Calogero coupling are the characteristics of the Calogero
spectrum (cf. [34]).

As we already mentioned, cyclic renormalization dynamics is connected with
BKT–pairing of partons in two-dimensional model. One could wonder whether this
connection is universal. One four-dimensional example of such pairing has to
be mentioned. It is bion condensation in 3+1 dimensions. The RG analysis of
the model involving the gas of bions and electrically charged W-bosons has
been considered in [42] where the RG flows involves the fugacities for
electric and magnetic components and the coupling constant. The coupled set of
the RG equations has been solved explicitly in the self-dual case and the
solution to the RG equations for the fugacities obtained in [42] is identical
to the solution for the coupling in the RD model upon the analytic
continuation. The period of the RG in the solution above is fixed by the RG
invariant which has been identified with the product of the UV values of the
electric and magnetic fugacities $y_{e}\times y_{m}$. The similarity between
the RG behavior is not accidental since the mapping of the gauge theory and
the perturbed XY model has been found in [42].

We would like to emphasize that the investigation of various aspects of limit
cycles in RG dynamics still remains on its early stage and there is a
considerable number of open questions. The RG cycles can have numerous
applications to different aspects of mathematical physics. In this case the RG
dynamics is considered as an example of non-trivial dynamical system.

The work of A.G. and K.B. was supported in part by grants RFBR-12-02-00284 and
PICS-12-02-91052. The work of K.B. was also supported by the “Dynasty”
fellowship. A.G. thanks FTPI at University of Minnesota where the part of this
work has been done for the hospitality and support. We would like to thank N.
Nekrasov and F. Popov for useful discussions and comments.

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