11institutetext: International Centre for Theoretical Sciences (ICTS-TIFR)
Tata Institute of Fundamental Research
Shivakote, Hesaraghatta, Bangalore 560089, India.22institutetext:
International Centre for Theoretical Physics
Strada Costiera 11, Trieste 34151 Italy

# Critical exponents for higher order phase transitions: Landau theory and RG
flow

Joydeep Chakravarty<EMAIL_ADDRESS>2 Diksha Jain
<EMAIL_ADDRESS>
###### Abstract

In this work, we define and calculate critical exponents associated with
higher order thermodynamic phase transitions. Such phase transitions can be
classified into two classes: with or without a local order parameter. For
phase transitions involving a local order parameter, we write down the Landau
theory and calculate critical exponents using the saddle point approximation.
Further, we investigate fluctuations about the saddle point and demarcate when
such fluctuations dominate over saddle point calculations by introducing the
generalized Ginzburg criteria. We use Wilsonian RG to derive scaling forms for
observables near criticality and obtain scaling relations between the critical
exponents. Afterwards, we find out fixed points of the RG flow using the one-
loop beta function and calculate critical exponents about the fixed points for
third and fourth order phase transitions.

##  1 Introduction

A thermodynamic phase transition is characterized by the non-analyticity of
thermodynamic observables with respect to temperature across two or more
different phases. These phase transitions were systematically classified by
Ehrenfest Ehrenfest ; articlegj . Ehrenfest’s classification is based on the
order of the phase transition, i.e. a $r$th order phase transition is
characterized by the $r$th derivative of the partition function (or free
energy) becoming discontinuous at the temperature where the phase transition
takes place. A thermodynamic function that takes different values across
different phases is called an order parameter.

With regards to second order phase transitions, Landau put forward a
phenomenological Hamiltonian which captures essential aspects of the
transition in the infrared limit of the system Landau:1937obd . His formalism
suggests that near the critical temperature $T_{C}$ (the temperature at which
phase transition takes place), the Hamiltonian can be expanded in terms of a
local order parameter $\phi(x)$ on the basis of appropriate symmetries. For
example, if a system has a $Z_{2}$ symmetry i.e. it is symmetric under
$\phi(x)\to-\phi(x)$; then for $T\approx T_{C}$ and a small external coupling
field $h$, the Hamiltonian $H(\phi)$ can be expressed as follows:

$\beta H=\int
d^{d}x\left(\frac{K}{2}(\gradient\phi)^{2}+t\phi^{2}(x)+u\phi^{4}(x)+\dots-h.\phi(x)\right)$
(1)

where $t$ and $u$ are couplings which are functions of the temperature. Here
we have restricted ourselves to only the lowest order terms. It can be readily
demonstrated that the partition function computed using this Hamiltonian gives
rise to different values of the order parameter for $t<0$ and $t>0$ using the
saddle point approximation. The divergences of observables near the critical
point are captured using critical exponents.

To obtain the correct picture of fluctuations away from the saddle point, we
need to incorporate the field-theoretic machinery of the renormalization group
(RG) flows of couplings in the theory. We can use the behavior of couplings
under RG flow to determine how thermodynamic observables scale in the vicinity
of the critical surface. Consequently, we can derive various scaling relations
between the theory’s critical exponents, which are robust under fluctuations
away from the saddle point.

In order to understand the RG flows in the coupling space, we start by finding
out fixed points of the flow and then expand the couplings about the fixed
points. This leads to a classification of the couplings into relevant,
marginal, and irrelevant depending on their behavior in the infrared (IR). In
the IR, relevant couplings diverge away from the IR fixed points as we
integrate out degrees of freedom while irrelevant couplings converge towards
the IR fixed point. The subspace of relevant couplings is usually finite-
dimensional, and hence theories in the IR are characterized by only a finite
number of relevant couplings. Such IR behaviour defines the notion of
universality classes Hollowood:2009eh ; Peskin:1995ev ; kardarcourse ;
kardar_2007 . The set of relevant couplings can be used to determine some of
the critical exponents, and scaling relations between them further allows us
to determine all the critical exponents doi:10.1063/1.1734338 ;
JOSEPHSON1966608 ; doi:10.1063/1.1696618 ; kardarcourse ; kardar_2007 .

Higher order Phase transitionsLocal order parameterNon-local order
parameter:(To be discussed inforthcoming work)$d>2r$$d\leq 2r$Saddle Point
approximation valid, critical exponentsgiven in Table 1Saddle Point
valid,critical exponents are given in Table 1Fluctuations dominate over
saddlepoint, Scaling relations found using RG flow in §4 (Table 2),Critical
exponents given in Table 3Follows Ginzburg Criteria (§3.3)Does not follow
Ginzburg criteria Figure 1: A flowchart describing the general outline of the
paper.

We will now briefly describe our work, a summary of which is given in Fig. 1.
In the first part of our work, we apply various aspects of our above
discussion to the study of higher order phase transitions described by local
order parameters. In §2.1 we generalize Landau’s phenomenological Hamiltonian
for higher-order phase transitions using a local order parameter such that
physical observables derived from the partition function have analytic
properties in the $h-t$ plane apart from a branch cut ending at the critical
point. Next, we define critical exponents for higher order phase transitions
in §2.2 and calculate them using saddle point approximation to get the leading
order predictions.

We treat fluctuations over the saddle point in §3, and show that saddle point
analysis is valid above the upper critical dimension $d_{u}=2r$ for higher
order transitions. Afterwards, we formulate the Ginzburg criteria to classify
when saddle point results are valid and when critical exponents are modified
due to fluctuations below the upper critical dimension. The argument for the
lower critical dimension remains unchanged as compared to the second order
case.

In §4 we obtain scaling forms of physical observables for higher order phase
transitions from the renormalization group analysis of the system. We use this
to derive scaling relations between critical exponents beyond the saddle
point. We show that the imposition of analyticity in the phenomenological
Hamiltonian combined with the scaling implies that critical exponents are the
same above or below $T=T_{C}$ and $h=0$ (Appendix A). We also show that the
divergence of the correlation length naturally leads to the hyperscaling
relation.

We then proceed onto computing corrections to the critical exponents using the
one-loop beta functions in §5. We use this to calculate the critical exponents
corresponding to $r=3$ and $r=4$ phase transitions below the upper critical
dimension. We show that there are no relevant couplings in the vicinity of the
non-trivial fixed point, and consequently, the sole corrections arise due to
flows near the Gaussian fixed point. For $r\geq 5$, the one-loop corrections
to the beta function vanish, and hence we need to go beyond one-loop
calculations to find corrections to the critical exponents for such cases.

In general, there also exist phase transitions which are described by non-
local order parameters. An important class of such phase transitions include
transitions characterized by a gap in the eigenvalue spectrum. These include
the Gross-Witten-Wadia (GWW) model Gross:1980he ; WADIA1980403 ;
wadia2012study , Douglas-Kazakov model Douglas:1993iia , Brownian walk models
which map onto two-dimensional continuum Yang-Mills with different gauge
groups FORRESTER2011500 , constrained Coulomb gas Cunden_2017 , bipartite
entanglement PhysRevA.81.052324 ; PhysRevLett.101.050502 ;
PhysRevLett.104.110501  and various other examples PhysRevLett.119.060601 ;
PhysRevLett.107.177206 ; PhysRevE.88.042125 ; PhysRevLett.109.167203 ; 5730574
; PhysRevLett.101.216809 . Further examples of third order phase transitions
are given in the review Majumdar_2014 . Such phase transitions are outside the
scope of our present work and will be dealt with in a forthcoming work.

Fourth order phase transitions with local order parameter have been proposed
as the nature of superconducting transition in certain materials in pardeep97
; articlePK99 ; articlePK00 ; articlePK02 ; doi:10.1080/14786430802585158 ;
doi:10.1080/13642810208223158 ; PhysRevB.71.104509 111We feel pardeep97 ;
articlePK02 ; doi:10.1080/13642810208223158  have not received significant
attention in the literature regarding higher order phase transitions, and in
particular, we found their work only after we had independently rederived some
of their results, and consequently there is overlap between parts of §2.1,
§2.3 and §4.1 of our work with theirs, which we have clearly outlined..
Another example of fourth order phase transition is the Ising model on the
Cayley tree proposed in STOSIC20091074 . Divergences using zeroes of the
partition function in higher order phase transitions were analyzed in
JANKE2006319 .

Apart from thermodynamic phase transitions, higher order phase transitions
also occur in the context of topological phase transitions. An example of this
is the phase transition between 2D Chern insulators, in which the third
derivative of the free energy has a discontinuity. However, there are no local
order parameters for such topological phase transitions, and consequently,
Landau theory cannot be formulated to describe them. Consequently, we will not
be looking at such phase transitions in our work.

##  2 Higher order phase transitions with local order parameters: Landau
theory

In this section, we will study a class of $r$th order phase transitions, which
are described using local order parameters. We argue that the phenomenological
theory describing these phase transitions in the infrared (IR) can be
described using the Landau formalism. We will also introduce critical
exponents corresponding to $r$th order phase transitions and calculate them
using the saddle point approximation.

###  2.1 The Landau Hamiltonian for higher order phase transitions

We work in $d$ spatial dimensions, which can be understood both as the non-
relativistic spatial limit of a $d+1$ dimensional relativistic theory or as an
analytic continuation of the Lorentzian theory to a Euclidean theory via Wick
rotation. Let us now write down the most general Hamiltonian, which obeys the
following assumptions:

  1. 1.

Analyticity assumption: We assume that the critical exponents are analytic
everywhere apart from a singular line which terminates at the critical point
in the plane spanned by the external field $h^{i}$ and temperature difference
$t$ (where $t=\frac{T-T_{C}}{T_{C}}$), with the line given by $h^{i}=0$ and
$t<0$ as shown in Figure 2. In other words, the free energy is analytic
everywhere in the plane apart from this line. Thus the free energy term
containing the external field in the phenomenological Hamiltonian should be
chosen such that the associated critical exponents are analytic, i.e., the
critical exponents have the same values when the critical point is approached
from different directions in the $(h,t)$ plane.

$h$$t$ Figure 2: The order parameter $\phi$ is analytic everywhere except on
the branch cut in $(h-t)$ plane shown above.

  2. 2.

Assumption regarding higher order terms: We assume a small external field $h$
such that only the leading order term in $h$ contributes and higher order
terms are suppressed. We also demand that terms involving higher derivatives
of the order parameter are suppressed. These conditions are weaker than the
first condition and serve to simplify our calculations and to clearly
demonstrate the physics of the problem at hand.

  3. 3.

Fine tuning assumption: Let $\phi_{i}(x)$ denote the order parameter of the
phase transition. In hindsight, for $r$th order phase transitions, we assume
that the terms $\absolutevalue{\phi_{i}}^{2(k-1)},\,k<r$ either do not appear
in the Hamiltonian, or their coefficients have a very small magnitude and do
not change signs at the critical temperature of the phase transition. We
enforce this requirement so that these terms do not alter the order of the
phase transitions, as a significant contribution from such terms can
potentially change the order. Similarly, we also demand that the terms of the
form $\absolutevalue{\phi_{i}}^{2k},\,k>r$ die off for a $r$th order phase
transition.

  4. 4.

Rotational symmetry: We further assume that our system possesses rotational
symmetry and hence the Hamiltonian is invariant under O($N$) transformation
($N\geq 1$) given by

$\phi_{i}\to\phi_{i}^{\prime}=R_{i}^{j}\,\phi_{j},\quad R\in O(N)$ (2)

where the rotation matrices $R$ belong to the matrix representation of $O(N)$.

The most general phenomenological IR Hamiltonian obeying above assumptions
which describes $r$th order phase transition is given by:

$\beta H=\int
d^{d}x\left[\frac{K}{2}\absolutevalue{\gradient\phi_{i}}^{2}+t_{r}\,|\phi_{i}(x)|^{2(r-1)}+u_{r}\,|\phi_{i}(x)|^{2r}-(h_{i}\phi^{i})\,|\phi_{i}(x)|^{2(r-2)}\right],$
(3)

where we are using Einstein summation notation to sum over the index $i$ i.e.
$|\phi_{i}|^{2}=\sum_{i=1}^{n}\phi^{i}\phi_{i}$. Here $K$ is the coefficient
of the kinetic term and the couplings $t_{r}$ and $u_{r}$ are expressed as
functions of temperature as follows:

$\begin{split}t_{r}(T)&=c(T-T_{C})+\text{O}(T-T_{C})^{2}=c^{\prime}t+\text{O}(t^{2})\\\
u_{r}(T)&=u_{0}+u_{1}(T-T_{C})+\text{O}(T-T_{C})^{2}=u_{0}+u_{1}^{\prime}t+\text{O}(t^{2})\end{split}$
(4)

where $T_{C}$ is the temperature at which the transition takes place,
$c,c^{\prime},u_{1},u_{1}^{\prime}$ and $u_{0}$ are undetermined constants,
and $t=\frac{T-T_{C}}{T_{C}}$ as defined previously. The constant $u_{0}$ must
be positive in order to ensure that our Hamiltonian is positive-definite while
$c$ must be positive in order to obtain the correct description of phases
above and below the critical temperature. The exact values of these constants
depend on the details of the system. The Hamiltonian in (3) was also discussed
for the specific case of third and fourth order superconducting phase
transitions in pardeep97 ; articlePK99 ; articlePK00 ; articlePK02 ;
doi:10.1080/14786430802585158 ; doi:10.1080/13642810208223158 ;
PhysRevB.71.104509 .

The mass dimensions of the couplings appearing in (3) are given by:

$\begin{split}[t_{r}]&=2(r+d-1)-dr\\\ [u_{r}]&=r(2-d)+d\\\
[h]&=d+(2r-3)\left(1-\frac{d}{2}\right).\end{split}$ (5)

Notice that the couplings in the above Hamiltonian can be non-renormalizable
depending on the dimension $d$, and hence the physical observables can
potentially be plagued by UV divergences untameable by a finite number of
counterterms. However our Hamiltonian is only defined in the IR, and we can
impose a UV cutoff upto which the Hamiltonian description is valid. Thus we
are not bothered by non-renormalizable interactions, analogous to the standard
treatment of second order phase transitions in $d>4$.

The (Gibbs) partition function of the system described by the Hamiltonian (3)
is given by:

$Z=\int[D\phi(x)]\,\exp\left[-\beta H(\phi(x),\,h)\right].$ (6)

The (Gibbs) free energy is given by $\beta F=-\log Z\approx V_{d}\,\text{
min}\left[V(\phi)\right]_{\phi_{0}}$ where $V_{d}$ is the $d$-dimensional
volume and the potential $V(\phi)$ is evaluated at the minima $\phi_{0}$.

###  2.2 Critical exponents

In this subsection, we define various critical exponents and compute them
using saddle point approximation. We will drop the index $i$ in $\phi_{i}$ for
notational convenience at various places in the text.

####  2.2.1 Definition

Let us introduce the critical exponents corresponding to these phase
transitions. These exponents are generalizations of the exponents which
characterize the second order phase transitions.

  1. 1.

The exponent222The critical exponent $\beta$ should not be confused with
$\beta$ appearing in the partition function, which is given by
$\beta=\frac{1}{k_{B}T}$. To avoid confusion, whenever we talk about the
critical exponent $\beta$, we mention it explicitly. $\beta$ is defined as the
divergence of the order parameter at zero external field:

$\phi(t,h=0)\propto\begin{cases}0,&\quad T>T_{C};\\\
\absolutevalue{t}^{\beta},&\quad T<T_{C}.\\\ \end{cases}$ (7)

  2. 2.

The exponent $\delta$ is defined as the divergence of the order parameter at
the critical temperature:

$\phi(T=T_{C},h)\propto h^{\frac{1}{\delta}}.$ (8)

  3. 3.

Denoting $\Phi$ as the integral of the order parameter over the full space
i.e. $\Phi=\int d^{d}x\,\phi(x)$, we define the generalized susceptibility
(response function) as

$\chi=\frac{\partial\braket{\Phi}}{\partial h}\Big{|}_{h=0},$ (9)

where $\braket{\Phi}$ is the one-point function defined as:

$\braket{\Phi}=\frac{1}{Z}\int[D\phi(x)]\,\int
d^{d}x\,\phi(x)\,\exp\left[-\beta H(\phi(x),\,h)\right]$ (10)

and where $Z$ is given in (6). The divergence of the susceptibility is given
by

$\chi(T,h=0)\propto\absolutevalue{t}^{-\gamma}.$ (11)

Using the definition of susceptibility in equation (9), we obtain it’s
expression in terms of the order parameter by differentiating the partition
function w.r.t $h$.

$\begin{split}\chi&\equiv\beta\braket{\Phi(x)\,\Phi^{2r-3}(y)}_{c}=\beta\left(\braket{\Phi(x)\,\Phi^{2r-3}(y)}-\braket{\Phi(x)}\,\braket{\Phi^{2r-3}(y)}\right)\\\
&\equiv\beta\int d^{d}x\,d^{d}y\braket{\phi(x)\,\phi^{2r-3}(y)}_{c}=\beta\int
d^{d}x\,d^{d}y\left(\braket{\phi(x)\,\phi^{2r-3}(y)}-\braket{\phi(x)}\,\braket{\phi^{2r-3}(y)}\right)\end{split}$
(12)

where $\braket{\Phi(x)\,\Phi^{2r-3}(y)}_{c}$ and
$\braket{\phi(x)\,\phi^{2r-3}(y)}_{c}$ are the connected correlators. Note
that setting a UV cuttoff removes divergences associated with the evaluation
of product of fields coinciding at a point $y$ in $\braket{\phi^{2r-3}(y)}$,
thus making it a well-defined quantity.

  4. 4.

The (Gibbs) free energy is given by $\beta F=-\log Z$. Let us denote the $n$th
derivative of the free energy as $C_{n}=\frac{\partial^{n}}{\partial
t^{n}}\left(\frac{\beta F}{V}\right)$. Since we are interested in a $r$th
order phase transition, we expect the $r$th derivative of the free energy to
diverge at $t=0$. We define the exponent $\alpha$, which characterizes this
divergence, as follows:

$C_{r}\propto\absolutevalue{t}^{-\alpha}.$ (13)

Note here that the specific heat is given by the second derivative of free
energy i.e. $C_{V}=-C_{2}\propto\frac{\partial^{2}}{\partial
t^{2}}\left(\frac{\beta F}{V}\right)$. Hence for a second order phase
transition, $C_{V}$ diverges. For $r$th order phase transition, the specific
heat is given by:

$C_{V}(T,h=0)\propto\absolutevalue{t}^{r-\alpha-2}.$ (14)

  5. 5.

We also define a two-point connected correlator given by

$G^{r}_{c}(x,y)=\braket{\phi(x)\,\phi^{2r-3}(y)}_{c}=\braket{\phi(x)\,\phi^{2r-3}(y)}-\braket{\phi(x)}\,\braket{\phi^{2r-3}(y)}$
(15)

Note that $G^{r}_{c}(x,y)$ only depends on the separation between the operator
insertions. This is because we are working in Euclidean space which has
translations and rotations as a part of its isometry group. If the separation
is taken to be very large than the corelation length $\zeta$ i.e.
$\absolutevalue{x-y}\gg\zeta$, then $G^{r}_{c}(x,y)$ roughly decays as
$G^{r}_{c}(x,y)\sim\exp{-\frac{\absolutevalue{x-y}}{\zeta}}$.

We can also write $\chi$ as given in equation (12) as an integral over the
connected Green function as defined in equation (15)

$\chi=\beta\int d^{d}x\,d^{d}y\,G^{r}_{c}(x,y)=\beta V\int
d^{d}\rho\,G^{r}_{c}(\rho,0),$ (16)

where we have defined $\rho=\absolutevalue{x-y}$. Let the largest value of
$G^{r}_{c}(\rho,0)$ be given by $G_{0}$ when $\rho<\zeta$, where $G_{0}$ is
always finite. Using the mean value theorem, we have the following inequality,
where $V$ is the system’s volume (where $V\geq\zeta^{d}$):

$\frac{\chi}{\beta V}<G_{0}\,\zeta^{d}.$ (17)

Here we have assumed that the contributions from $\rho\gg\zeta$ are negligible
since the correlator dies off exponentially. Thus when $\chi$ diverges, the
above inequality ensures that $\zeta$ necessarily diverges as well. This gives
rise to the critical exponent $\nu$ defined by

$\zeta(T,h=0)\propto\absolutevalue{t}^{-\nu}.$ (18)

Note that here we have not labelled two different exponents
$\gamma_{\pm},\,\alpha_{\pm}$ etc. while approaching from the $t<0$ and $t>0$
directions. This is because we have already imposed analyticity in the $(h,t)$
plane while writing the phenomenological Hamiltonian, and consequently, we
have $\gamma=\gamma_{\pm},\,\alpha=\alpha_{\pm}$ and so on. It can be shown
that even for deviations away from the saddle point, we still get single-
valued exponents in the $(h,t)$ plane. We will prove this statement in
Appendix A.

####  2.2.2 Dilatation symmetry at critical temperature

In the previous subsection, we discussed that at critical temperature, the
correlation length $\zeta$ diverges as a consequence of the divergence of
susceptibility, which follows from equation (17). At the critical temperature,
the correlation length becomes infinite (or more precisely, it becomes of the
order of system size). Since there are no other length scales in the problem,
$G^{r}_{c}(x,y)$ should decay as a power law. This implies that there is an
emergent dilatation symmetry at the critical temperature, and the connected
Green function transforms as follows

$G_{c}^{r}\left(t=0,\lambda
x\right)=\lambda^{\kappa}\,G_{c}^{r}\left(t=0,x\right).$ (19)

Here $\kappa$ is the scaling dimension under the scaling transformation. Thus
the system at criticality exhibits self-similarity at all intermediate scales
between system size and the UV cutoff.

###  2.3 Critical exponents from saddle point approximation

In this subsection, we will compute the critical exponents defined above,
using saddle point approximation. The saddle points can be found by minimizing
the potential appearing in the Hamiltonian (3). The potential is given by (as
mentioned before, we have dropped the index $i$ for notational convenience):

$V\left[\phi\equiv\phi(x)\right]=t_{r}\,\phi^{2(r-1)}+u_{r}\,\phi^{2r}-(h.\phi)\,\phi^{2(r-2)}.$
(20)

  1. 1.

Working in the saddle point approximation, we impose $\frac{\partial
V}{\partial\phi}=0$ to get the values for the order parameter above and below
the critical temperature,

$\phi=\begin{cases}\sqrt{\dfrac{-t(r-1)}{ru}},\quad&t<0\\\
0,\quad&t>0.\end{cases}$ (21)

This reduces to the familiar value $\phi^{2}=\frac{-t}{2u}$ for second order
phase transitions. For the special case of $r=3,4$, equation (21) matches with
the expressions derived in pardeep97 . Comparing this with (7), we obtain
$\beta=1/2$.

  2. 2.

Upon substituting the value of $\phi$ from eqn (21), the free energy
$\frac{\beta F}{V}=\text{min}\left[V(\phi)\right]$ is given by

$\frac{\beta
F}{V}=\begin{cases}-\left(\dfrac{u}{r-1}\right)\left(\dfrac{-t(r-1)}{ru}\right)^{r},\quad&t<0\\\
0,\quad&t>0.\\\ \end{cases}$ (22)

For the special case of $r=3,4$; this matches with the expression of free
energy derived in pardeep97 . Thus the $r$th derivative of the free energy
with respect to the temperature is discontinuous at $T=T_{C}$, which is given
by

$C_{r}(T,h=0)\propto\begin{cases}(-1)^{r+1}\left(\dfrac{r!\,u\,(r-1)^{r-1}}{(ur)^{r}}\right),\quad&t<0\\\
0,\quad&t>0\\\ \end{cases}$ (23)

For $r=2$, this reduces to the familiar value for the specific heat, i.e.
$C_{2}=-C_{V}\propto\frac{1}{2u}$. Thus using (23) we conclude that
$\alpha=0$.

  3. 3.

Let us now look at the critical exponent associated with susceptibility. Using
our Hamiltonian and the definition of susceptibility from (9), we obtain the
following expression.

$\frac{1}{\chi}=\begin{cases}-\dfrac{4(r-1)t}{(2r-3)},\quad&t<0\\\
-\dfrac{2(r-1)t}{(2r-3)},\quad&t>0\end{cases}$ (24)

This gives us $\gamma=1$.

  4. 4.

The critical exponent $\delta$ can be obtained from setting $t=0$ in the
saddle point equation of state:

$h(\phi)=\frac{2\phi^{3}ru}{(2r-3)}.$ (25)

Using (8), we obtain $\delta=3$.

Critical Exponents | $\alpha$ | $\beta$ | $\gamma$ | $\delta$ |  $\nu$ (See §4) |  $\Delta$ (See §4 )
---|---|---|---|---|---|---
Landau Theory | 0 | $\dfrac{1}{2}$ | 1 | 3 |  $\dfrac{1}{y_{t}}=\dfrac{1}{2(r+d-1)-dr}$; |  $\dfrac{y_{h}}{y_{t}}=\dfrac{d+(2r-3)\left(1-\frac{d}{2}\right)}{2(r+d-1)-dr}$;
(Saddle point) |  |  |  |  | if $y_{t}$ is relevant | if $y_{t}$ is relevant
Table 1: Critical exponents from the saddle point expansion in Landau theory.

We will introduce another exponent $\Delta$ and $\nu$ in §4. The critical
exponents from Landau theory are displayed in Table 1 for convenience.

##  3 Fluctuations over saddle point approximation

In this section, we will venture beyond the saddle point analysis by
considering small fluctuations about the saddle point and taking their effects
into account. Readers who are not interested in the calculational aspects of
fluctuations can read the following introductory discussion of this section
where we have stated the main results and skip directly to §3.3.

We will show that incorporating fluctuations about the saddle point into the
analysis of $r$th order phase transitions leads to the following features:

  1. 1.

Fluctuations which lead to powers of the order parameter in the Hamiltonian of
the order ${2(k-1)},\,k<r$ destroy the delicate analytic properties of the
critical exponents. We will show this by looking at the case of quadratic
fluctuations, where we find such fluctuations destroy analyticity for $r>2$.

  2. 2.

In $d<2r$, saddle point analysis does not always hold. The Ginzburg criteria
as defined in §3.3 tells us when saddle point analysis holds in $d<2r$, and
when fluctuations contribute significantly. In case fluctuations dominate,
critical exponents are modified.

  3. 3.

Saddle point analysis is valid above the upper critical dimension given by
$d>d_{u}=2r$. Hence the critical exponents corresponding to observables like
$C_{r}$ do not change for $d>2r$.

  4. 4.

Fluctuations destroy all order in the system for $d\leq d_{l}$ for second
order phase transitions, where $d_{l}$ is the lower critical dimension
PhysRevLett.17.1133 ; Coleman:1973ci . This argument remains unchanged for
$r$th order phase transition as shown in Appendix B, where we show that
$d_{l}=2$ for systems with continuous symmetry (in the phenomenological
Lagrangian) and $d_{l}=1$ for systems with discrete symmetry.

We begin by looking at the effect of fluctuations on two-point correlators.
Afterwards, we analyze the effects of fluctuations on $C_{r}$. Specifically,
we study how these observables’ divergences near critical temperature are
modified due to fluctuations. The computations in Appendix B also stress the
importance of fluctuations, as we show that fluctuations lead to partial or
complete destruction of order depending on the dimension of the system.

###  3.1 Two point correlator and exponents

We are interested in fluctuations of the form:

$\phi(x)=\left(\phi_{0}+\phi_{\parallel}(x),\phi_{\bot,2}(x),\,\dots\,,\phi_{\bot,n}(x)\right)$
(26)

where $\phi_{\parallel}(x)$ and $\phi_{\bot,i}$ denote the longitudinal and
transverse fluctuations respectively. The kinetic term goes as

$\left(\gradient\phi\right)^{2}=\left(\gradient\phi_{\parallel}\right)^{2}+\left(\gradient\phi_{\bot}\right)^{2},$
(27)

and the square of the order parameter goes as

$\phi^{2}=\phi_{0}^{2}+2\phi_{0}\phi_{\parallel}+\phi_{\parallel}^{2}+\sum_{i=2}^{n}\phi_{\bot,i}^{2}$
(28)

By using (27) and (28) in (3) and keeping upto only quadratic terms in the
fluctuations, we obtain:

$\begin{split}\beta H=&\int
d^{d}x\left(t_{r}\phi_{0}^{2(r-1)}+u_{r}\phi_{0}^{2r}\right)+\int
d^{d}x\left[\frac{K}{2}\left(\gradient\phi_{\parallel}\right)^{2}+\left(\frac{2K}{2\xi_{\parallel}^{2}}\right)\phi_{\parallel}^{2}\right]\\\
&+\sum_{i=2}^{n}\int
d^{d}x\left[\frac{K}{2}\left(\gradient\phi_{\bot,i}\right)^{2}+\left(\frac{2K}{2\xi_{\bot,i}^{2}}\right)\phi_{\bot,i}^{2}\right]+\text{O}\left(\phi_{\parallel}^{3},\phi_{\bot}^{3}\right)\\\
\end{split}$ (29)

where $\xi_{\parallel}$ and $\xi_{\bot,i}$ are correlation scales of the
fluctuations and the first term is the potential evaluated at the saddle
point. The correlation lengths $\xi_{\parallel}$ and $\xi_{\bot,i}$
corresponding to the two point correlator are explicitly given by:

$\begin{split}\frac{\partial^{2}V}{\partial\phi_{\parallel}^{2}}\bigg{|}_{\phi_{0}}&=\dfrac{2K}{\xi_{\parallel}^{2}}=\begin{cases}2ru_{r}\,\left(\dfrac{t-rt}{ru}\right)^{r-1},\quad&t<0\\\
t(r-1)(2r-3)\,\delta_{r,2}\quad&t>0\\\ \end{cases}\\\
\frac{\partial^{2}V}{\partial\phi_{\bot,i}^{2}}\bigg{|}_{\phi_{0}}&=\frac{2K}{\xi_{\bot,i}^{2}}=\begin{cases}0,\quad&t<0\\\
t(r-1)\,\delta_{r,2}\quad&t>0\\\ \end{cases}\end{split}$ (30)

Note that the lengths $\xi_{\parallel}$ and $\xi_{\bot,i}$ corresponding to
the two point correlator display a different analytic behaviour as compared to
$\zeta$. The two point correlators $\xi$ which were zero in the saddle point
approximation acquire a non-zero value due to fluctuations. It can be seen
from the nature of the longitudinal polarization that the two point correlator
$\xi$ is not an analytic function, since quadratic terms in the Hamiltonian
characterizing the fluctuations destroy the delicate analytic features. As a
result we have the scalings:

$\begin{split}\xi_{\parallel}^{+}\sim\absolutevalue{t}^{-\frac{1}{2}}\delta_{r,2},\quad\quad&\xi_{\parallel}^{-}\sim\absolutevalue{t}^{\frac{1-r}{2}}\\\
\xi_{\bot,i}^{+}\sim\absolutevalue{t}^{-\frac{1}{2}}\delta_{r,2},\quad\quad&\xi_{\bot,i}^{-}\sim
0\\\ \end{split}$ (31)

We note from equation (31) that only in the case of second order phase
transitions with $r=2$ we have
$\xi_{\parallel}^{+}=\xi_{\parallel}^{-}\equiv\xi_{\parallel}=\absolutevalue{t}^{-\frac{1}{2}}$.
This is because such fluctuations give rise to analytic terms in the
Hamiltonian corresponding to the second order phase transitions.

Here we have shown that quadratic fluctuations destroy the analyticity of the
physical observables in the $h-t$ plane. The same argument can be extended to
fluctuations of the order ${2(k-1)},\,k<r$.

###  3.2 Fluctuation corrections to the saddle point

Let us now evaluate the effect of fluctuations on the full partition function.
By including the quadratic fluctuations in the partition function we obtain:

$\begin{split}Z\approx\exp&\left[-V_{d}V(\phi_{0})\right]\int\left[D\phi_{\parallel}\right]\exp\left[-\frac{K}{2}\int
d^{d}x\left[\left(\gradient\phi_{\parallel}\right)^{2}+\frac{2\phi_{\parallel}^{2}}{\xi_{\parallel}^{2}}\right]\right]\\\
\times&\int\left[\prod_{i}D\phi_{\bot,i}\right]\exp\left[-\frac{K}{2}\int
d^{d}x\left[\left(\gradient\phi_{\bot,i}\right)^{2}+\frac{2\phi_{\bot,i}^{2}}{\xi_{\bot,i}^{2}}\right]\right]\end{split}$
(32)

where $V_{d}$ is the volume of the system. The free energy of the system is
given as follows (where we have removed the label $i$ indicating the
transverse directions for convenience):

$f=-\frac{\ln
Z}{V_{d}}=V(\phi_{0})+\frac{1}{2}\int\frac{d^{d}q}{(2\pi)^{d}}\ln\left[K\left(q^{2}+2\left(\xi_{\parallel}\right)^{-2}\right)\right]+\,\frac{n-1}{2}\int\frac{d^{d}q}{(2\pi)^{d}}\ln\left[K\left(q^{2}+2\left(\xi_{\bot}\right)^{-2}\right)\right],$
(33)

where $q$ is the momentum associated with the fluctuations. We will now look
at the effect of fluctuations on the singularity structure of the $r$th
derivative of the free energy which is given by
$C_{r}\equiv\frac{d^{r}f}{dt^{r}}$. Using (31) and (33) we obtain the
following corrections to $C_{r}$:

$C_{r}\propto\begin{cases}C_{r}^{+}+\bigintss\dfrac{d^{d}q}{(2\pi)^{d}}\dfrac{(-2)^{r-1}(r-1)!\delta_{r,2}}{\left(q^{2}+2(\xi_{\parallel}^{+})^{-2}\right)^{r}}+(n-1)\bigintss\dfrac{d^{d}q}{(2\pi)^{d}}\dfrac{(-2)^{r-1}(r-1)!\delta_{r,2}}{\left(q^{2}+2(\xi_{\bot}^{+})^{-2}\right)^{r}};\quad\quad&t>0\\\
C_{r}^{-}+\bigintss\dfrac{d^{d}q}{(2\pi)^{d}}\left(\dfrac{(-2)^{r-1}(r-1)!(r-1)^{r}(\xi_{\parallel}^{-})^{\frac{2r(2-r)}{r-1}}}{\left(q^{2}+2(\xi_{\parallel}^{-})^{-2}\right)^{r}}+O\left(\frac{1}{q^{2r-1}}\right)\right);\quad\quad&t<0\end{cases}$
(34)

where we have only written the most divergent correction to $C_{r}^{-}$
explicitly and $O\left(\frac{1}{q^{2r-1}}\right)$ denotes the subleading
corrections. Notice that the following momentum-integral appears in the
correction terms:

$I=\int\dfrac{d^{d}q}{(2\pi)^{d}}\left(\dfrac{1}{q^{2}+2(\xi)^{-2}}\right)^{r}$
(35)

where we have suppressed the labels indicating longitudinal or transverse, and
whether $t$ is greater or less than 0. The integral $I$ has mass dimension
$d-2r$, and thus diverges for $d\geq 2r$. Introduction of a UV regulator
$\Lambda=\frac{1}{a}$ cuts these divergences off. We can perform a similar
analysis for the integrals at $t<0$. We can thus write $C_{r}$ in $d>2r$ as

$C_{r}\propto\begin{cases}C_{r}^{+}+A_{1}\,n\,a^{2r-d}\,\delta_{r,2}&t>0\\\
C_{r}^{-}+A_{2}\,a^{2r-d}+O(a^{2r-d-1})\quad&t<0\end{cases}$ (36)

while $C_{r}$ in $d<2r$ takes the following form upon rescaling $q$ by
$\xi^{-1}$

$C_{r}\propto\begin{cases}C_{r}^{+}+A_{3}\left(\left(\xi^{+}_{\parallel}\right)^{2r-d}+(n-1)\left(\xi^{+}_{\bot}\right)^{2r-d}\right)\delta_{r,2}&t>0\\\
C_{r}^{-}+A_{4}\left(\xi^{-}_{\parallel}\right)^{2r-d+\frac{2r(2-r)}{r-1}}\quad&t<0\end{cases}$
(37)

Note that in the equations above, $A_{1},A_{2},A_{3}$ and $A_{4}$ are
constants, which do not influence the discontinuity.

Physical aspects of fluctuations: We will now analyze the physical aspects of
these computations. As shown in equation (36), for $d>2r$, $C_{r}$ gets an
additive constant correction proportional to $a^{2r-d}$ which does not
introduce any new singularity in the expression. However, we see from (37)
that in $d<2r$, there are corrections that have associated singular parts, as
the correlation lengths diverge near critical points. The calculation above is
done by taking into account the first order corrections; inclusion of higher
order terms can potentially introduce more singular terms to $C_{r}$. Thus the
saddle point analysis does not work correctly in $d<2r$ because fluctuation
contributions tamper with the predicted singularity structure. Therefore we
denote $d=2r$ as the upper critical dimension above which the saddle point
approximation holds, and fluctuations do not modify the critical exponents.
The following question naturally arises: Under what conditions does saddle
point analysis robustly hold below the upper critical dimension where
fluctuations modify the singularities? We discuss this in the next subsection
§3.3.

The predicted corrections to $C_{V}$ match with the values for second order
phase transitions with $r=2$. By substituting the expressions for $\xi$’s from
equations (30) and (31) into (37), we also see that corrections to the saddle
point critical exponent $\alpha$ of $C_{r}$ do not have the similar scaling
for $t>0$ and $t<0$. As stated earlier, this arises due to the fact that
quadratic contribution to the action destroys the analyticity of physical
observables in the $h-t$ plane. Note that fluctuations for third order phase
transitions were also studied in pardeep97  for a different Hamiltonian as
compared to (3), for which they concluded the upper critical dimension is
given by $d=2$.

###  3.3 The Ginzburg criterion for $r$th order phase transitions

In this subsection, we quantify when the effects of fluctuations dominate over
the saddle point expectations. We do this by generalizing the Ginzburg
criteria to understand when saddle point calculations accurately predict
critical exponents in the presence of fluctuations.

In order that the saddle point prediction is the dominant contribution, we
demand that the saddle point discontinuity $\Delta C_{r}\equiv
C_{r}^{+}-C_{r}^{-}$ is larger as compared to the contributions from
fluctuations. Only in such situations, we can trust the saddle point
computation. Using (31) in (37), we see that the the condition for saddle
point analysis to dominate over fluctuations is given by:

$\begin{split}\Delta C_{r}\gg\Delta
C_{r}^{\text{fluc.}}&=A_{3}\left(\left(\xi^{+}_{\parallel}\right)^{2r-d}+(n-1)\left(\xi^{+}_{\bot}\right)^{2r-d}\right)\delta_{r,2}-\left(A_{4}\left(\xi^{-}_{\parallel}\right)^{2r-d+\frac{2r(2-r)}{r-1}}\right)\\\
&=A^{\prime}\,\delta_{r,2}\,t^{-\left(\frac{2r-d}{2}\right)}+A^{\prime\prime}t^{\left(\frac{d}{2}-r\right)(r-1)+r(r-2)}\end{split}$
(38)

where $A^{\prime}$ and $A^{\prime\prime}$ are linear combinations of
proportionality constants $A_{3}$ and $A_{4}$. Notice that the term
$A^{\prime}$ drops out for higher order phase transitions ($r>2$), whereas for
$r=2$ both the terms have the same order of magnitude. Thus the requirement
for saddle point singularities to dominate over the fluctuations is given by

$\absolutevalue{t}\gg t_{G}\sim\left(\frac{A^{\prime\prime}}{\Delta
C_{r}}\right)^{-\frac{2}{\left(d-2r\right)(r-1)+2r(r-2)}}.$ (39)

For the case of $r=2$, we recover the standard Ginzburg criteria from equation
(38) which is given by

$\absolutevalue{t}\gg
t_{G}\sim\left(-\frac{A^{\prime}+A^{\prime\prime}}{\Delta
C_{V}}\right)^{\frac{2}{4-d}}.$ (40)

where we have used the fact that $C_{2}=-C_{V}$. The interpretation of this
temperature scale is that it is possible to recover the saddle point critical
exponents if the system does not go within a distance $t_{G}$ near the
critical temperature $T_{C}$.

##  4 Scaling relations among critical exponents

In the previous section, we saw that critical exponents obtain corrections due
to fluctuations. We now address the following question: Given that
fluctuations modify the saddle-point calculated values of the critical
exponents, can we still find some robust relations between the critical
exponents that are valid beyond the saddle point? As we already know, there
exist scaling relations between the exponents for second-order phase
transitions. Can we generalize them for $r$th order phase transitions?

Let us begin by looking at the free energy obtained using saddle point
description. It is given by

$f(t,h)\propto\begin{cases}\dfrac{t^{r}}{u^{r-1}},\quad&h=0;t<0\\\
\dfrac{h^{\frac{2r}{3}}}{u^{\frac{2r}{3}-1}}\quad&h\neq 0;t=0.\end{cases}$
(41)

Thus we can now write the above free energy (at saddle point) as a homogenous
function of $t$ and $h$.

$\begin{split}f(t,h)&=\absolutevalue{t}^{r}g\left(\frac{h}{t^{\Delta}}\right)=\absolutevalue{t}^{r}g(x),\quad\text{where}\quad
x\equiv\frac{h}{\absolutevalue{t}^{\Delta}}\end{split}$ (42)

Here $\Delta$ is the generalization of the familiar gap exponent which appears
for the case of second order phase transitions. The generalization of gap
exponent for higher order phase transition was also discussed in JANKE2006319
. Let us now look at the $h\to 0$ and the $t\to 0$ limits of the function
$g(x)$ such that the power of $h$ is kept intact. These go as

$\lim_{x\to
0}g(x)\sim\frac{1}{u^{r-1}},\quad\text{and}\quad\lim_{x\to\infty}g(x)\sim\frac{x^{\frac{2r}{3}}}{u^{\frac{2r}{3}-1}}.$
(43)

Equation (43) implies that the free energy takes the following form near
$h\neq 0$ and $t=0$:

$f\sim\absolutevalue{t}^{r}\frac{h^{\frac{2r}{3}}}{t^{\frac{2r\Delta}{3}}\left(u^{\frac{2r\Delta}{3}-1}\right)}.$
(44)

Since $f$ remains finite at $t=0$, the gap exponent is given by setting the
coefficient of $\absolutevalue{t}$ to 0, and we obtain $\Delta=\frac{3}{2}$.

###  4.1 Scaling form of observables from RG

In this subsection, we will argue that beyond the saddle point, the singular
part of the free energy has the following homogenous form:

$f_{\text{sing}}(t,h)=\absolutevalue{t}^{r-\alpha}g\left(\frac{h}{t^{\Delta}}\right)$
(45)

This form can be understood using Wilsonian Renormalization Group (RG) flow.
Under the RG transformations, the partition function remains unchanged and
hence the corresponding free energies are related by:

$Vf(t,h)=V^{\prime}f(t_{b},h_{b})$ (46)

where $t_{b}$ and $h_{b}$ are the rescaled couplings obtained by rescaling
$x\rightarrow x/b$, where $b>1$. Notice that we have suppressed the subscript
$r$ in $t_{r}$, which we will keep doing for the rest of this section as well.
Hence, in $d$\- dimensions, the rescaled volume gets smaller by a factor of
$b^{d}$ and we obtain:

$f(t,h)=b^{-d}f(b^{y_{t}}t,b^{y_{h}}h)$ (47)

where $y_{t}$ and $y_{h}$ are the mass dimensions of the couplings $t_{r}$ and
$h$ as given in (5). In case the coupling $t$ is relevant i.e. $y_{t}>0$, we
can choose $b=t^{-1/y_{t}}$, such that $b>1$. Notice that if the Lagrangian
does not contain any relevant coupling, this choice of $b$ is not possible and
hence we cannot write down the scaling relations derived below. With above
choice of $b$, the free energy takes the following form:

$f(t,h)=t^{d/y_{t}}f\left(1,\frac{h}{t^{y_{h}/y_{t}}}\right)\equiv
t^{d/y_{t}}g_{f}\left(\frac{h}{t^{y_{h}/y_{t}}}\right)$ (48)

Thus we obtain the same form as (45) with $r-\alpha=d/y_{t}$ and
$\Delta=y_{h}/y_{t}$. Notice that the above scaling is valid only when the
coupling $t$ is relevant.

Within the context of third and fourth order phase transition, a similar
scaling relation for the free energy was discussed in
doi:10.1080/13642810208223158 ; JANKE2006319 . We derive the above scaling
using renormalization group and as described previously, the expression is
valid only if the coupling $t_{r}$ in (3) is a relevant coupling within
Wilsonian RG.

We can obtain similar homogeneous scaling forms for the specific heat and
$C_{r}$ by differentiating $f$ w.r.t. $t$. We have:

$\begin{split}\frac{df}{dt}&=(r-\alpha)\absolutevalue{t}^{r-\alpha-1}g\left(\frac{h}{t^{\Delta}}\right)-\Delta
h\absolutevalue{t}^{r-\alpha-\Delta-1}g\left(\frac{h}{t^{\Delta}}\right)\\\
&=\absolutevalue{t}^{r-\alpha-1}\left[(r-\alpha)g\left(\frac{h}{t^{\Delta}}\right)-\Delta\frac{h}{t^{\Delta}}g\left(\frac{h}{t^{\Delta}}\right)\right]\\\
&=\absolutevalue{t}^{r-\alpha-1}\tilde{g}\left(\frac{h}{\absolutevalue{t}^{\Delta}}\right)\end{split}$
(49)

where
$\tilde{g}\left(\frac{h}{\absolutevalue{t}^{\Delta}}\right)=(r-\alpha)g\left(\frac{h}{t^{\Delta}}\right)-\Delta\frac{h}{t^{\Delta}}g\left(\frac{h}{t^{\Delta}}\right)$.
Hence the specific heat takes the form

$C^{\text{sing}}_{V}\sim-\frac{\partial^{2}f}{\partial
t^{2}}\sim\absolutevalue{t}^{r-\alpha-2}\widehat{g}\left(\frac{h}{t^{\Delta}}\right),$
(50)

while the $r$th derivative of the free energy takes the form

$C^{\text{sing}}_{r}\sim-\frac{\partial^{r}f}{\partial
t^{r}}\sim\absolutevalue{t}^{-\alpha}\bar{g}\left(\frac{h}{t^{\Delta}}\right).$
(51)

###  4.2 Identities satisfied by critical exponents

In this subsection, we write down similar homogeneous forms for other physical
observables near the critical point. We will use these scaling forms to obtain
relations between various critical exponents.

  1. 1.

Using the scaling of the free energy from eqn (45) and the relation between
the order parameter and the free energy, we conclude that $\phi$ scales as

$\phi^{2r-3}\sim|t|^{r-\alpha-\Delta}g_{\phi}\left(\frac{h}{t^{\Delta}}\right),$
(52)

using which we can conclude from equation (7) that $\beta$ is given by:

$\beta=\frac{r-\alpha-\Delta}{2r-3}.$ (53)

  2. 2.

When $x\to\infty$, the dominant contribution in $g_{\phi}$ scales as
$g_{\phi}\sim x^{p}$ where $p$ is the leading power in the expansion, and
therefore
$\phi^{2r-3}\sim\absolutevalue{t}^{r-\alpha-\Delta}\left(\frac{h}{t^{\Delta}}\right)^{p}$.
As this limit is $t$-independent, we have the following relation between the
exponents

$\Delta p=r-\alpha-\Delta,\quad\implies\quad\phi^{2r-3}(t=0,h)\sim
h^{\frac{r-\alpha-\Delta}{\Delta}}.$ (54)

Using equation (8) the above identity implies the following relation

$\delta=\frac{(2r-3)\Delta}{r-\alpha-\Delta}=\frac{\Delta}{\beta}.$ (55)

  3. 3.

Using the definition of generalized susceptibility as
$\chi=\frac{d\braket{\Phi}}{dh}$, and using the scaling of $\phi$ from eqn
(52), we see that $\chi$ scales as

$\chi(t,h)=\frac{d\braket{\Phi}}{dh}\sim\absolutevalue{t}^{\frac{r-\alpha-\Delta}{2r-3}-\Delta}g_{\chi}\left(\frac{h}{t^{\Delta}}\right).$
(56)

Using equation (11) we obtain:

$\gamma=\Delta-\left(\frac{r-\alpha-\Delta}{2r-3}\right).$ (57)

Thus we see that all the critical exponents
$\left(\alpha,\beta,\gamma,\delta,\Delta\right)$ can be derived from two
independent critical exponents (say from $\Delta$ and $\alpha$ in the case of
second order phase transition). Using equations (53), (55) and (57), we obtain
the following higher-order generalizations of the Rushbrooke identity
doi:10.1063/1.1734338  (the saturation of the Rushbrooke inequality), and the
Widom scaling law doi:10.1063/1.1696618  for higher order phase transitions:

$\alpha+2(r-1)\beta+\gamma=r\qquad\&\qquad\delta-1=\frac{\gamma}{\beta}$ (58)

Notice that the above identities are different from the identities derived in
doi:10.1080/13642810208223158 ; JANKE2006319 . This is because
doi:10.1080/13642810208223158  obtained their scaling relations by considering
a different magnetic field coupling as compared to our coupling in the
Hamiltonian given in (3). As a consistency check for our above relations, we
can set $r=2$ to obtain

$\alpha+2\beta+\gamma=2\qquad\&\qquad\delta-1=\frac{\gamma}{\beta}$ (59)

which are the Rushbrooke’s identity doi:10.1063/1.1734338  and Widom’s scaling
law doi:10.1063/1.1696618  respectively for the second order phase
transitions.

###  4.3 Scaling relation from the divergence of correlation length

In this subsection, we will use the two-point correlator given in equation
(15) to understand it’s divergence near the critical point. We hereby derive
the generalized Josephson identity for $r$th order phase transitions.
Following the main argument from our previous section, the correlation length
$\zeta(t,h)$ has the following form:

$\zeta(t,h)\sim\absolutevalue{t}^{-\nu}g_{\zeta}\left(\frac{h}{\absolutevalue{t}^{\Delta}}\right)$
(60)

In the vicinity of the critical point, the most important length scale is
$\zeta$, and the singular contribution to observables arise solely due to
$\zeta$’s singularity. A similar singular behaviour was demonstrated in the
case of quadratic fluctuations in §3, where the correlation length $\xi$
dictated the singular contributions in $d<2r$, with the only significant
difference being that the corrections led to non-analytic behaviour. Hence the
singular part of the free energy in terms of the correlation length $\zeta$ is
given by:

$V_{d}f(t,h)\sim\ln
Z=B_{1}\left(\frac{L}{\zeta}\right)^{d}+B_{2}\left(\frac{L}{a}\right)^{d},$
(61)

where $B_{1}$ and $B_{2}$ denote non-singular quantities, $L$ denotes the
system size, and $\Lambda=\frac{1}{a}$ is the UV cutoff. Since the singular
part of the free energy arises from the first term in the above equation, we
have

$f(t,h)\sim\frac{\ln
Z}{L^{d}}\sim\zeta^{-d}\sim\absolutevalue{t}^{d\nu}g\left(\frac{h}{t^{\Delta}}\right)$
(62)

This is the same scaling form as given in (48) with

$\nu=\frac{1}{y_{t}}\,,\qquad\qquad\Delta=\frac{y_{h}}{y_{t}}.$ (63)

The scaling form in (62) has the following important properties:

  1. 1.

From (61), we observe that the free energy grows as
$\left(\frac{L}{\zeta}\right)^{d}$ near the critical point. It can be
interpreted as a measure of the system size divided by the correlation length
$\zeta$, such that the system is divided into many small blocks. Each small
block can be thought of as an independent degree of freedom which contributes
a constant factor to the overall free energy. We also naturally obtain the
homogeneity of the singular part of the free energy.

  2. 2.

We arrive at the generalized Josephson’s identity by comparing (62) with the
homogenous form of free energy given in (45). The identity is given by

$r-\alpha=d\nu.$ (64)

Josephson’s identity is basically a hyperscaling relation that holds below and
at the upper critical dimension, but it does not hold above the upper critical
dimension. This is because the scaling form for $\zeta$ as given in equations
(60) and (61) does not hold above the upper critical dimension. Above the
upper critical dimension $g_{\zeta}(x)$ is singular in $x$ 363515  and can be
shown to grow as $g_{\zeta}(x)\sim\frac{1}{x}$ using mean-field theory, which
is singular when $x\to 0$.

We have summarized the four independent relations between the critical
exponents $\alpha,\beta,\gamma,\delta,\Delta$ and $\nu$ in Table 2. Thus all
the critical exponents can be written in terms of two independent ones, say in
terms of $\gamma$ and $\Delta$.

Serial No. | Exponents | Relation
---|---|---
1. | $\beta,\alpha,\Delta$ | $\beta=\dfrac{r-\alpha-\Delta}{2r-3}$
2. | $\delta,\alpha,\Delta$ | $\delta=\dfrac{(2r-3)\Delta}{r-\alpha-\Delta}$
3. | $\gamma,\alpha,\Delta$ | $\gamma=\Delta-\left(\dfrac{r-\alpha-\Delta}{2r-3}\right)$
4. | $\alpha,\nu$ | $r-\alpha=d\nu$
Table 2: Independent relations between different critical exponents

##  5 Critical exponents about fixed points of RG flow

In the previous sections, we calculated the critical exponents using the
saddle point approximation and derived relations between critical exponents
away from the saddle point. In this section, we will use perturbative RG flow
to find fixed points of the theory and use them to calculate the critical
exponents. Specifically, we will be looking at the case when fluctuations
dominate over saddle point contributions as quantified by the generalized
Ginzburg criteria in $d\leq 2r$. In that case, the critical exponents are
captured by relevant couplings about fixed points. We will perform linear
stability analysis near these fixed points to calculate the critical exponents
in their vicinity.

Let us denote the full coupling space of our theory by $S(t_{r},u_{r},\dots)$.
A particular Hamiltonian ($H$) is described by couplings at a point in this
space. RG flow involves rescaling the couplings and renormalizing the order
parameter, thereby taking us from one point in this parameter space to
another, i.e., $S_{b}\rightarrow R_{b}S$. Here we have denoted the action of
RG flow by $R_{b}$, with $b$ being the rescaling parameter defined in §4.1.
The fixed points of these flows are defined as $R_{b}S^{*}=S^{*}$. Since we
rescale the couplings in the RG procedure, the correlation length $\zeta^{*}$
should either go to zero or infinity at the fixed point. When $\zeta^{*}=0$,
the system is in a completely disordered phase, and when
$\zeta^{*}\rightarrow\infty$, the system is in the ordered phase.

We can now study the stability of the fixed points by linearizing the beta
function near them. The subspace of irrelevant couplings is known as the basin
of attraction since the flow in this subspace converges towards the fixed
point. Since the correlation length diverges at the fixed point, it diverges
at each point in the basin of attraction. This basin of attraction is
basically the critical surface at which the phase transition takes place. The
behavior of critical exponents near the phase transition is hence determined
only by the set of relevant couplings. We will use them to compute the value
of critical exponents beyond the saddle point approximation.

###  5.1 One-loop beta function

In this subsection, we compute the one-loop beta functions for the couplings
appearing in the Hamiltonian (3). We follow the procedure described in skinner
; Hollowood:2009eh . The Euclidean action for $r$th order phase transition at
zero external magnetic field is given by:

$S_{\Lambda}[\phi_{i},h=0]=\int
d^{d}x\,\left[\frac{1}{2}(\partial_{\mu}\phi^{i}\partial^{\mu}\phi_{i})+V(\phi_{i})\right]$
(65)

where $\Lambda$ is the UV cutoff and the potential is given by:

$V(\phi_{i})=t_{r}(\phi^{2})^{r-1}+u_{r}(\phi^{2})^{r}$ (66)

with $\phi^{2}=\sum_{i=1}^{N}\phi_{i}\phi^{i}$. We work with dimensionless
couplings which are defined as follows:

$t_{r}=\frac{g_{2(r-1)}}{(2r-2)!}\Lambda^{d-(r-1)(d-2)}\,,\qquad\qquad
u_{r}=\frac{g_{2r}}{2r!}\Lambda^{d-r(d-2)}\,.$ (67)

In terms of these dimensionless couplings, the potential is given by:

$V(\phi_{i})=\frac{g_{2(r-1)}}{(2r-2)!}\Lambda^{d-(r-1)(d-2)}(\phi^{2})^{r-1}+\frac{g_{2r}}{2r!}\Lambda^{d-r(d-2)}(\phi^{2})^{r}$
(68)

We will now use this potential to study RG flows using the Wilsonian
formalism. This involves splitting the scalar field in terms of high energy
and low energy modes and the subsequent procedure of systematically
integrating out the high energy modes. Therefore we split the field $\phi^{i}$
into the low energy modes $\varphi^{i}$ and the high energy modes $\chi^{i}$
i.e. $\phi^{i}=\varphi^{i}+\chi^{i}$. Hence the action can be expanded as:

$S[\phi^{i}]=S[\varphi^{i}]+\int
d^{d}x\,\left[\frac{1}{2}(\partial_{\mu}\chi^{i}\partial^{\mu}\chi_{i})+\frac{1}{2}\chi^{i}\frac{\partial^{2}V(\varphi^{k})}{\partial\varphi^{i}\partial\varphi^{j}}\chi^{j}+..\right]$
(69)

where we have chosen $\varphi^{i}$ such that it minimises the potential i.e.
$V^{\prime}(\varphi^{i})=0$. We will now integrate out the high energy modes
$\chi^{i}$ in the partition function, by lowering the scale infinitesimally
i.e. setting $\Lambda^{\prime}$ = $\Lambda-\delta\Lambda$. We will focus on
the first order terms in $\delta\Lambda$, which is similar to computing one-
loop effects only. It can be argued that at this order, only the quadratic
terms in $\chi^{i}$ contribute to the effective action skinner . Upon
integrating out the high energy modes, the one-loop correction to the
effective action is given by:

$\delta_{\Lambda}S[\phi]=\frac{\Lambda^{d-1}\delta\Lambda}{(4\pi)^{d/2}\Gamma(d/2)}\int
d^{d}x\,\textrm{ln}\bigg{[}\textrm{det}\left(\Lambda^{2}\delta_{ij}+\frac{\partial^{2}V(\varphi^{i})}{\partial\varphi^{i}\partial\varphi^{j}}\right)\bigg{]}$
(70)

This process of integrating out the high energy modes changes the effective
couplings depending on the UV scale at hand. The change in the $k$th coupling
is captured by the beta function $\left(\beta_{k}=dg_{k}/d\log\Lambda\right)$.
We can use (70) to compute the beta functions of the couplings $g_{2(r-1)}$
and $g_{2r}$ appearing in the potential (68), which are given by:

$\displaystyle\Lambda\frac{dg_{2(r-1)}}{d\Lambda}$ $\displaystyle=$
$\displaystyle[(r-1)(d-2)-d]g_{2(r-1)}-a\Lambda^{(r-1)(d-2)}\frac{\partial^{2(r-1)}}{\partial\varphi_{i}^{2(r-1)}}\bigg{[}\textrm{ln}\bigg{[}\textrm{det}\left(\Lambda^{2}\delta_{ij}+\frac{\partial^{2}V(\varphi^{i})}{\partial\varphi^{i}\partial\varphi^{j}}\right)\bigg{]}\bigg{]}\bigg{|}_{\varphi^{i}=0}$
$\displaystyle\Lambda\frac{dg_{2r}}{d\Lambda}$ $\displaystyle=$
$\displaystyle[r(d-2)-d]g_{2r}-a\Lambda^{r(d-2)}\frac{\partial^{2r}}{\partial\varphi_{i}^{2r}}\bigg{[}\textrm{ln}\bigg{[}\textrm{det}\left(\Lambda^{2}\delta_{ij}+\frac{\partial^{2}V(\varphi^{i})}{\partial\varphi^{i}\partial\varphi^{j}}\right)\bigg{]}\bigg{]}\bigg{|}_{\varphi^{i}=0}$
(71)

where we have defined $a$ as follows

$a=\frac{1}{(4\pi)^{d/2}\Gamma(d/2)}.$ (72)

Using (68) we compute the first derivative of potential which is given by:

$\displaystyle\frac{\partial V}{\partial\varphi^{i}}$ $\displaystyle=$
$\displaystyle\frac{g_{2(r-1)}}{(2r-3)!}\Lambda^{d-(r-1)(d-2)}\left(\sum_{m}\varphi_{m}^{2}\right)^{r-2}\varphi_{k}\delta^{ik}+\frac{g_{2r}}{(2r-1)!}\Lambda^{d-r(d-2)}\left(\sum_{m}\varphi_{m}^{2}\right)^{r-1}\varphi_{k}\delta^{ik},$

while the second derivative is given by:

$\displaystyle\frac{\partial^{2}V}{\partial\varphi^{i}\varphi^{j}}$
$\displaystyle=$
$\displaystyle\frac{g_{2(r-1)}}{(2r-3)!}\Lambda^{d-(r-1)(d-2)}\left(\sum_{m}\varphi_{m}^{2}\right)^{r-2}\delta^{ij}+\frac{g_{2r}}{(2r-1)!}\Lambda^{d-r(d-2)}\left(\sum_{m}\varphi_{m}^{2}\right)^{r-1}\delta^{ij}$
$\displaystyle+2(r-1)\frac{g_{2r}}{(2r-1)!}\Lambda^{d-r(d-2)}\left(\sum_{m}\varphi_{m}^{2}\right)^{r-2}\varphi_{k}\varphi_{l}\delta^{ik}\delta^{jl}$
$\displaystyle+2(r-2)\frac{g_{2(r-1)}}{(2r-3)!}\Lambda^{d-(r-1)(d-2)}\left(\sum_{m}\varphi_{m}^{2}\right)^{r-3}\varphi_{k}\varphi_{l}\delta^{ik}\delta^{jl}.$

We can now compute the determinant appearing in the expression for beta
function (5.1) which takes the following form:

$\displaystyle\textrm{det}\left(\Lambda^{2}\delta_{ij}+\frac{\partial^{2}V(\varphi^{i})}{\partial\varphi^{i}\partial\varphi^{j}}\right)$
$\displaystyle=$
$\displaystyle\left[\Lambda^{2}+\left(\sum_{m}\varphi_{m}^{2}\right)^{r-2}\left(\frac{g_{2(r-1)}\Lambda^{d-(r-1)(d-2)}}{(2r-3)!}+\frac{g_{2r}\Lambda^{d-r(d-2)}}{(2r-1)!}\sum_{m}\varphi_{m}^{2}\right)\right]^{N-1}$
$\displaystyle\times$
$\displaystyle\left[\Lambda^{2}+\left(\sum_{m}\varphi_{m}^{2}\right)^{r-2}\left(\frac{g_{2(r-1)}\Lambda^{d-(r-1)(d-2)}}{(2r-4)!}+\frac{g_{2r}\Lambda^{d-r(d-2)}}{(2r-2)!}\sum_{m}\varphi_{m}^{2}\right)\right]$

Substituting the expression for the determinant given by (5.1) in the beta
function (5.1), we can calculate the one-loop beta functions for various $r$.
In the rest of this section, we compute the critical exponents for phase
transitions of different orders.

###  5.2 Critical exponents for second order phase transitions $(r=2)$

In this subsection, we focus on second order phase transitions ($r=2$). We can
see that for $r=2$ the couplings $g_{2}$ and $g_{4}$ appear in the Hamiltonian
(3). We use (5.1) to compute the one-loop beta-functions for these couplings
and then evaluate the critical exponents for this case. The results in this
subsection are a review of previously known results for second order phase
transitions. Using (5.1) and (5.1), we find the following one-loop beta
functions for the couplings $g_{2}$ and $g_{4}$:

$\displaystyle\Lambda\frac{dg_{2}}{d\Lambda}$ $\displaystyle=$
$\displaystyle-2g_{2}-\frac{(N+2)}{(4\pi)^{d/2}\Gamma(d/2)}\frac{g_{4}}{3(1+g_{2})}$
(74) $\displaystyle\Lambda\frac{dg_{4}}{d\Lambda}$ $\displaystyle=$
$\displaystyle(d-4)g_{4}+\frac{(N+8)}{3(4\pi)^{d/2}\Gamma(d/2)}\frac{g_{4}^{2}}{(1+g_{2})^{2}}$
(75)

The fixed points of the RG flow can be found by setting $\beta(g_{2})$ and
$\beta(g_{4})$ to zero. There are two fixed points. The first one is a trivial
fixed point known as the Gaussian fixed point and it is given by:

$g_{2}^{*}=0\,,\qquad\qquad g_{4}^{*}=0$ (76)

Near the Gaussian fixed point, the coupling $g_{2}$ is relevant and $g_{4}$ is
marginal and their mass dimensions are given in (5). There also exists a non-
trivial Wilson-Fisher (WF) fixed point wf  given by:

$g_{2}^{*}=-\frac{(d-4)(N+2)}{d(N+2)-6(N+4)}\,,\qquad\qquad
g_{4}^{*}=-\frac{12(d-4)(N+8)}{a(d(N+2)-6(N+4))^{2}}$ (77)

where $a$ is defined in (72). We will now study flows about this fixed point,
which appears "just below" four dimensions. Defining $\epsilon\equiv 4-d$, the
Wilson-Fisher fixed point is given by:

$g_{2}^{*}=-\frac{\epsilon(N+2)}{2(N+8)}+O(\epsilon^{2})\,,\qquad\qquad
g_{4}^{*}=\frac{3\epsilon}{a(N+8)}+O(\epsilon^{2})$ (78)

We can now perform $\epsilon$-expansion of the beta functions near these fixed
points:

$\displaystyle\beta(g_{2}^{*}+\delta g_{2})=\beta(\delta g_{2})$
$\displaystyle=$ $\displaystyle-2(g_{2}^{*}+\delta
g_{2})-\frac{(N+2)}{3a}\frac{(g_{4}^{*}+\delta g_{4})}{(1+g_{2}^{*}+\delta
g_{2})}$ $\displaystyle=$ $\displaystyle-2\delta
g_{2}+\frac{\epsilon(N+2)}{(N+8)}\delta g_{2}-\frac{a(N+2)}{3}\delta
g_{4}\left(1+\frac{\epsilon(N+2)}{2(N+8)}\right)+O(\epsilon^{2})$
$\displaystyle\beta(g_{4}^{*}+\delta g_{4})=\beta(\delta g_{4})$
$\displaystyle=$ $\displaystyle-\epsilon(g_{4}^{*}+\delta
g_{4})+\frac{a(N+8)}{3}\frac{(g_{4}^{*}+\delta g_{4})^{2}}{(1+g_{2}^{*}+\delta
g_{2})^{2}}$ $\displaystyle=$ $\displaystyle\epsilon\delta
g_{4}+O(\epsilon^{2})$

Hence we obtain the following matrix of the linearized beta functions about
the Wilson-Fisher fixed point:

$\begin{pmatrix}\beta(\delta g_{2})\\\ \beta(\delta
g_{4})\end{pmatrix}=\begin{pmatrix}-2+\frac{\epsilon(N+2)}{(N+8)}\qquad&-\frac{a(N+2)}{3}\left(1+\frac{\epsilon(N+2)}{2(N+8)}\right)\\\
0\qquad&\epsilon\end{pmatrix}\begin{pmatrix}\delta g_{2}\\\ \delta
g_{4}\end{pmatrix}$ (79)

The eigenvalues333Note that our beta functions are negative as compared to
kardar_2007 ; kardarcourse , who perform renormalization by studying the
variation of couplings with respect to length scale, as opposed to momentum
scale in our case. are $-2+\frac{\epsilon(N+2)}{(N+8)}$ and $\epsilon$. The
first eigenvalue is negative and hence the coupling $g_{2}$ is relevant around
the WF fixed point and the scaling dimension $y_{t}$ is given by:

$y_{t}=2-\frac{\epsilon(N+2)}{(N+8)}$ (80)

On the other hand, the eigenvalue corresponding to the coupling $g_{4}$ is
positive for $d<4$. Hence the coupling $g_{4}$ is irrelevant (making the WF
fixed point stable along the corresponding eigenvector) for $d<4$. We can use
$y_{t}$ to calculate the critical exponent $\nu$ using the relation (63) and
it is given by:

$\nu=\frac{1}{y_{t}}=\left(2\left(1-\frac{\epsilon(N+2)}{2(N+8)}\right)\right)^{-1}=\frac{1}{2}+\frac{\epsilon}{4}\frac{(N+2)}{(N+8)}+O(\epsilon^{2})$
(81)

For $r=2$, the scaling dimension of the coupling $h_{i}$ is still given by
(5). This is because in the Lagrangian $h_{i}$ couples to the zero momentum
mode $\int d^{d}x\,\phi_{i}(x)=\tilde{\phi}_{i}(k=0)$, and hence integrating
out high energy modes using Wilsonian RG leaves $h_{i}$ unchanged. Therefore
for the special case of second order phase transition ($r=2$) the scaling
dimension of $h_{i}$ just below four dimensions is given by:

$y_{h}=3-\frac{\epsilon}{2}+O(\epsilon^{2})$ (82)

We can now calculate the gap exponent $\Delta$ from (63) which is given by:

$\Delta=\frac{y_{h}}{y_{t}}=\frac{3}{2}+\frac{\epsilon}{4}\left(\frac{3(N+2)}{N+8}-1\right)+O(\epsilon^{2})$
(83)

We can use the exponents $\nu$ and $\Delta$ to compute all other critical
exponents using relations derived in §4.2. These exponents are given by:

$\begin{split}\alpha&=\epsilon\left(\frac{1}{2}-\frac{N+2}{N+8}\right)+O(\epsilon^{2})\\\
\beta&=\frac{1}{2}-\frac{\epsilon}{4}\left(1-\frac{N+2}{N+8}\right)+O(\epsilon^{2})\\\
\gamma&=1+\frac{\epsilon}{2}\left(\frac{N+2}{N+8}\right)+O(\epsilon^{2})\\\
\delta&=3+\epsilon\left(\frac{N+2}{N+8}\right)+O(\epsilon^{2})\end{split}$
(84)

The above results match with the standard results for second order phase
transitions as given in kardar_2007 ; kardarcourse ; Peskin:1995ev .

###  5.3 Critical exponents for third order phase transitions $(r=3)$

In this subsection, we calculate the critical exponents for $r=3$. The
calculation of exponents for $r>2$ is exactly the same as for $r=2$ with an
important distinction. For $r=2$, integrating out high energy modes does not
renormalize the coupling $h_{i}$ because the term $h_{i}\phi_{i}$ is a zero
momentum mode. Hence the mass dimension of the coupling $h_{i}$ does not
receive any quantum corrections under Wilsonian renormalization. However we
can see from going to the Fourier space that $h_{i}\phi_{i}^{2r-3}$ no longer
has only zero-momentum contribution. This term can potentially become relevant
in the IR and give large contributions.

In order to simplify the detailed calculation and qualitatively understand the
underlying physics of the problem, we work in the limit when
$h_{i}\phi_{i}^{2r-3}$ is extremely small as compared to the other terms. In
other words, even though the coupling may be relevant, we will restrict
ourselves to integrating out modes with an IR cutoff such that quantum
corrections to $h_{i}$ do not blow out of proportion and consequently change
the nature of the fixed points. In the first order approximation, the scaling
dimension of $h_{i}$ can thus be taken to be its classical mass dimension.

Such a restriction of the coupling $h_{i}$ to simplify our calculation
essentially means that we are looking at the phase transitions at a tunable
mesoscopic limit, rather than in the exact thermodynamic limit (i.e. the IR
cutoff $(l^{*})$ can be taken large enough provided we appropriately fine-tune
the contribution $h_{i}\phi_{i}^{2r-3}$). This assumption is reasonable
because we want to capture details of higher order transitions very close to
the critical point, i.e., $h_{i}\to 0$, and also, the physics of different
phase transitions really depends on moving in the $t$ direction. Note that in
the special case where $h_{i}=0$, the effective Hamiltonian in (3) for $r=3$
describes third order phase transitions in the exact thermodynamic limit,
since $l^{*}$ can now go all the way up to $l^{*}\to\infty$, with the only
exception being that critical exponents involving $h_{i}$ are not defined.

Here we will compute the critical exponents solely below four dimensions even
though the upper critical dimension for $r=3$ is $d_{u}=6$. This is because
the couplings $g_{4}$ and $g_{6}$ appearing in the phenomenological
Hamiltonian for $r=3$ become irrelevant for $d>4$. Hence we cannot obtain
critical exponents using linear stability analysis as done previously in §5.2
for $d>4$.

With these subtleties in mind, the stage is clear to calculate the critical
exponents of $r=3$. We start with writing the beta functions of the couplings
$g_{4}$ and $g_{6}$ which appear in (3) for $r=3$. The beta functions are
given by:

$\displaystyle\Lambda\frac{dg_{4}}{d\Lambda}$ $\displaystyle=$
$\displaystyle[d-4]g_{4}+\frac{a(N+8)}{3}g_{4}^{2}$ (85)
$\displaystyle\Lambda\frac{dg_{6}}{d\Lambda}$ $\displaystyle=$
$\displaystyle[2d-6]g_{6}+a(N+14)g_{4}g_{6}-\frac{10a(N+26)}{9}g_{4}^{3}$ (86)

where $a$ is given in (72). As in the case of $r=2$, we obtain two fixed
points in this case as well. The trivial Gaussian fixed point is given by:

$g_{4}^{*}=0\,,\qquad\qquad g_{6}^{*}=0$ (87)

We now perform linear stability analysis about this fixed point by expanding
the beta function near the same. We again look at RG flows in the vicinity of
this fixed point just below four dimensions, where we have defined
$\epsilon=4-d$. We obtain the following linearized beta functions:

$\begin{pmatrix}\beta(\delta g_{4})\\\ \beta(\delta
g_{6})\end{pmatrix}=\begin{pmatrix}-\epsilon\qquad&0\\\
0\qquad&2(1-\epsilon)\end{pmatrix}\begin{pmatrix}\delta g_{4}\\\ \delta
g_{6}\end{pmatrix}$ (88)

Since $0<\epsilon\ll 1$, the coupling $g_{4}$ is relevant while $g_{6}$ is
irrelevant near this fixed point. We use the relevant coupling to compute the
critical exponents which are given below:

$\begin{split}\nu=\frac{1}{\epsilon}\,&,\qquad\qquad\Delta=\frac{1}{2}+\frac{1}{\epsilon}\\\
\alpha=4\left(1-\frac{1}{\epsilon}\right)\,&,\qquad\qquad\beta=-\frac{1}{2}+\frac{1}{\epsilon}\\\
\gamma=1\,&,\qquad\qquad\delta=\frac{2+\epsilon}{2-\epsilon}\end{split}$ (89)

Notice that for $d=3$ i.e. $\epsilon=1$, the above values of critical
exponents match the ones computed via saddle point approximation in Table 1.
This is because the one-loop corrections near the Gaussian fixed point do not
change the scaling dimensions of the couplings. Hence we do not get any
corrections to the saddle point result at this order.

Notice that there is no $N$-dependence in the critical exponents near the
Gaussian fixed point. This is because about the Gaussian fixed point, the
terms in (85) and (86) proportional to $N$ are quadratic or higher order in
$\delta g_{4}$ and $\delta g_{6}$. However since we are only considering
linear terms in our analysis, these terms do not contribute and hence we do
not get any $N$ dependence.

For $d<4$, there exists another non-trivial fixed point of the beta functions
in (85), which is given by:

$g_{4}^{*}=-\frac{3(d-4)}{a(N+8)}\,,\qquad\qquad
g_{6}^{*}=-\frac{30(d-4)^{3}(N+26)}{a^{2}(N+8)^{2}(6(N+20)-d(N+26))}$ (90)

The fixed point takes the following form in $d=4-\epsilon$ dimensions:

$g_{4}^{*}=\frac{3\epsilon}{a(N+8)}\,,\qquad\qquad
g_{6}^{*}=\frac{15\epsilon^{3}(N+26)}{a^{2}(N+8)^{3}}$ (91)

Linearizing the beta functions about this fixed point, we obtain:

$\begin{pmatrix}\beta(\delta g_{4})\\\ \beta(\delta
g_{6})\end{pmatrix}=\begin{pmatrix}\epsilon\qquad&0\\\
O(\epsilon^{2})\qquad&2+\epsilon\frac{N+26}{N+8}\end{pmatrix}\begin{pmatrix}\delta
g_{4}\\\ \delta g_{6}\end{pmatrix}$ (92)

Notice that near this fixed point both the couplings are irrelevant. Hence the
whole $g_{4}-g_{6}$ plane is a basin of attraction. As we change the
temperature i.e. vary the coupling $g_{4}$, the theory always flows to the
above non-trivial fixed point. As stressed in §4.1, we need at least one
relevant coupling to write down the scaling relations, hence we cannot use the
RG flow to obtain critical exponents near this fixed point. Hence the critical
exponents for $r=3$ are determined by RG flows near the Gaussian fixed point.

###  5.4 Critical exponents for fourth order phase transitions $(r=4)$

We will now compute the critical exponents for fourth order phase transitions.
The beta functions for the couplings $g_{6}$ and $g_{8}$ appearing in (3) for
$r=4$ are given by:

$\displaystyle\Lambda\frac{dg_{6}}{d\Lambda}$ $\displaystyle=$
$\displaystyle[2d-6]g_{6}-\frac{a(N+6)}{7}g_{8}$ (93)
$\displaystyle\Lambda\frac{dg_{8}}{d\Lambda}$ $\displaystyle=$
$\displaystyle[3d-8]g_{8}+\frac{7a}{5}(N+24)g_{6}^{2}$ (94)

Again we have a trivial Gaussian fixed point, i.e.

$g_{6}^{*}=0\,,\qquad\qquad g_{8}^{*}=0\,.$ (95)

We perform the linear stability analysis about the Gaussian fixed point just
below three dimensions, since by naive power-counting we know that these
couplings are irrelevant for $d\geq 4$. Defining $\lambda\equiv 3-d$, the
linearized expansion of beta functions near the Gaussian fixed point is given
by:

$\begin{pmatrix}\beta(\delta g_{6})\\\ \beta(\delta
g_{8})\end{pmatrix}=\begin{pmatrix}-2\lambda\qquad&-\frac{a}{7}(N+6)\\\
0\qquad&1-3\lambda\end{pmatrix}\begin{pmatrix}\delta g_{6}\\\ \delta
g_{8}\end{pmatrix}$ (96)

We see that the coupling $g_{6}$ is relevant while the coupling $g_{8}$ can
either be relevant, irrelevant or marginal depending on $\lambda$. For small
$\lambda$, only $g_{6}$ is relevant, which we use to determine the critical
exponents:

$\begin{split}\nu=\frac{1}{2\lambda}\,&,\qquad\qquad\Delta=\frac{1+3\lambda}{4\lambda}\\\
\alpha=\frac{3}{2}\left(3-\frac{1}{\lambda}\right)\,&,\qquad\qquad\beta=\frac{1-\lambda}{4\lambda}\\\
\gamma=1\,&,\qquad\qquad\delta=\frac{1+3\lambda}{1-\lambda}\end{split}$ (97)

The beta functions given by (93) also admit a non-trivial fixed point which is
given by:

$g_{6}^{*}=-\frac{10\left(3d^{2}-17d+24\right)}{a^{2}\left(N^{2}+30N+144\right)}\,,\qquad\qquad
g_{8}^{*}=-\frac{140(d-3)^{2}(3d-8)}{a^{3}(N+6)^{2}(N+24)},$ (98)

Notice that this fixed point exists only for $d<3$. For $d>3$, $g^{*}_{8}$
becomes negative, thereby making the Hamiltonian unbounded. Just below three
dimensions, the fixed point is given by:

$g_{6}^{*}=\frac{10(1-3\lambda)\lambda}{a^{2}\left(N^{2}+30N+144\right)}\qquad\qquad
g_{8}^{*}=\frac{140\lambda^{2}(3\lambda-1)}{a^{3}(N+6)^{2}(N+24)}$ (99)

Expanding the beta-function near the fixed point in $3-\lambda$ dimensions, we
obtain the following linearized form:

$\begin{pmatrix}\beta(\delta g_{6})\\\ \beta(\delta
g_{8})\end{pmatrix}=\begin{pmatrix}-2\lambda\qquad&-\frac{a}{7}(N+6)\\\
\frac{28\lambda}{a(N+6)}\qquad&1-3\lambda\end{pmatrix}\begin{pmatrix}\delta
g_{6}\\\ \delta g_{8}\end{pmatrix}$ (100)

The eigenvalues of above matrix are given by:

$\frac{1}{2}\left(1-5\lambda-\sqrt{\lambda^{2}-18\lambda+1)}\right)\,\qquad\text{and}\qquad\frac{1}{2}\left(1-5\lambda+\sqrt{\lambda^{2}-18\lambda+1}\right).$

Since $\lambda$ is small, we can linearly expand the above eigenvalues while
ignoring higher order terms in $\lambda$. At the first order in $\lambda$, the
eigenvalues are given by:

$2\lambda+O(\lambda^{2})\,,\qquad\qquad 1-2\lambda+O(\lambda^{2})$

We find that both the eigenvalues are positive i.e. both the couplings $g_{6}$
and $g_{8}$ are irrelevant near the non-trivial fixed point. Hence the
critical exponents are given by the values obtained near the Gaussian fixed
point, similar to our discussion for $r=3$. Notice that just like the case for
$r=3$, the critical exponents are independent of $N$ here as well.

For $r>4$, the one-loop corrections to the beta functions of the corresponding
couplings vanish. Hence we need to look at higher loop corrections to the beta
function in order to obtain the values of critical exponents beyond saddle
point. The main results of this section are summarized in Table 3.

r | $\alpha$ | $\beta$ | $\gamma$ | $\delta$ | $\nu$ | $\Delta$
---|---|---|---|---|---|---
2 | $\epsilon\left(\frac{1}{2}-\frac{N+2}{N+8}\right)$ | $\frac{1}{2}-\frac{\epsilon}{4}\left(1-\frac{N+2}{N+8}\right)$ | $1+\frac{\epsilon}{2}\left(\frac{N+2}{N+8}\right)$ | $3+\epsilon\left(\frac{N+2}{N+8}\right)$ | $\frac{1}{2}+\frac{\epsilon}{4}\frac{(N+2)}{(N+8)}$ | $\frac{3}{2}+\frac{\epsilon}{4}\left(\frac{3(N+2)}{N+8}-1\right)$
3 | $4\left(1-\frac{1}{\epsilon}\right)$ | $-\frac{1}{2}+\frac{1}{\epsilon}$ | 1 | $\frac{2+\epsilon}{2-\epsilon}$ | $\frac{1}{\epsilon}$ | $\frac{1}{2}+\frac{1}{\epsilon}$
4 | $\frac{3}{2}\left(3-\frac{1}{\lambda}\right)$ | $\frac{1-\lambda}{4\lambda}$ | 1 | $\frac{1+3\lambda}{1-\lambda}$ | $\frac{1}{2\lambda}$ | $\frac{1+3\lambda}{4\lambda}$
Table 3: One-loop corrections to the critical exponents. Here $\epsilon=4-d$
and $\lambda=3-d$. Note that the critical exponents for $r=2$ are calculated
about the Wilson-Fisher fixed point, whereas the critical exponents for $r=3$
and $r=4$ are calculated about the Gaussian fixed point. This is due to the
fact that for phase transitions of order $r=3,4$, there exists a relevant
coupling only about the Gaussian fixed point.

###  5.5 Discussion of results for $r=3,4$

We now give a summary of results regarding critical exponents in various
dimensions444We thank anonymous referee 2 for their insightful questions,
which have been instrumental for the analysis in this subsection.. As we
pointed out in §3, saddle point calculations dominate over fluctuation
contributions above $d=2r$. Below this upper critical dimension, saddle point
dominates over fluctuations only if the generalized Ginzburg criteria is
satisfied. If this is not the case, we need to take fluctuation contributions
into account by venturing away from the saddle point, which gives rise to
various scaling relations as derived in §4. In order to calculate all possible
critical exponents using these scaling relations, we need to estimate the
values of two of them. In this section, we have evaluated the values of $\nu$
and $\Delta$, and consequently evaluate all of the critical exponents.

In order to calculate all critical exponents using the scaling relations, we
need to work within the regime where the coupling $t_{r}$ is relevant, which
is always true for $r=2$, and conditionally true for $r=3$ and $4$ provided we
work in $d\leq 4$ and $d\leq 3$ respectively. In this section, we have
calculated $\nu$ and $\Delta$ using the one-loop beta functions for certain
higher order phase transitions, i.e., for $r=2,3$ and $4$.

We find that there exist two fixed points: one of them being the trivial
Gaussian fixed point, i.e., the free theory limit, while the other point for
$r=3,4$ below $d=4,3$ respectively is analogous to the Wilson-Fisher (WF)
fixed point for $r=2$. We observe that within the $\epsilon$-expansion, there
exist no relevant couplings about the WF fixed point for $r=3,4$ cases, in
contrast to the $r=2$ case. Since there are no relevant directions, the
scalings and scaling relations of physical observables derived in Section 4
are not valid near WF fixed point. Consequently we perform our analysis about
the Gaussian fixed point for $r=3,4$.

Note that $\epsilon$ is not the parameter characterizing our perturbative
expansion for the $r=3,4$ cases as compared to the $\epsilon$-expansion
performed for $r=2$. This is because the values of couplings at Gaussian fixed
point are independent of $\epsilon$ (i.e., the couplings are zero), in
contrast to the WF fixed point where the couplings depend on $\epsilon$.
Consequently the critical exponents computed about the Gaussian fixed points
are exact in $\epsilon$, in contrast to the computation of critical exponents
in the expansion about WF point. Hence perturbation theory remains valid even
when we take $\epsilon$ large enough, since expanding about the Gaussian fixed
point is a perturbative expansion in the couplings, and not in terms of
$\epsilon$ since the couplings are independent of $\epsilon$.

Our analysis about the Gaussian point for $r=3,4$ shows that some critical
exponents diverge in the $\epsilon\,(\text{or}\,\lambda)\to 0$ limit, as given
in Table 3. This is because the couplings become marginal when this limit is
imposed, and consequently our scaling relations in §4 become ill-defined,
since they are valid only under the assumption that the coupling $t_{r}$ is
relevant. This suggests us to go to finite non-zero values of
$\epsilon(\lambda)$ to ensure the relevance of couplings and the subsequent
validity of scaling relations in §4. The results given in Table 3 therefore
make sense when $\epsilon(\lambda)$ is finitely non-zero, and not when
$\epsilon(\lambda)\to 0$. Coincidentally, for $\epsilon=1$, the one-loop
results for $r=3$ match the expected saddle point results for third order
phase transitions in $d=3$, and similarly for $\lambda=\frac{1}{3}$ for $r=4$.
In short, the "divergence" of the critical exponents neither invalidates
perturbation theory and nor the $\frac{1}{\epsilon}$ factor appearing in Table
3 suggests anything dramatic, but are essential features of the phase
transition. Since perturbation theory remains valid, we strongly expect other
perturbative methods, such as dimensional regularization to give us same
results about the Gaussian fixed point. We also note that our results in this
subsection are not entirely unexpected, given the classical mass dimensions of
the couplings obtained in (5).

Further, the absence of $N$-dependence in Table 3 is a standard feature of
expansion about the Gaussian fixed point. It should be noted that the
expansion about the Gaussian fixed point for $r=2$ does not lead to any
$N$-dependence in the critical exponents as well.

We also notice from Table 3 that if $\epsilon(\lambda)$ is close enough to
zero, the critical exponent $\alpha$ can become drastically different from the
saddle point expectation $\alpha=0$. In particular, an interesting case arises
when $\absolutevalue{\alpha}\geq 1$, which basically implies that the order of
the phase transition can potentially get changed under RG flows. Note that the
changing of order of phase transition in this fashion is different from
explicitly introducing lower order terms into (3).

##  6 Summary and Discussion

We briefly summarize our work and discuss our conclusions here. In our work,
we look at higher order thermodynamic phase transitions. For phase transitions
involving a local order parameter, we write a generalized phenomenological
Landau Hamiltonian, which describes $r$th order phase transitions. Near
criticality, these phase transitions are characterized by divergences in the
physical observables. We capture such divergences by introducing critical
exponents. As a first step, we calculate these critical exponents by using the
saddle point approximation.

Next, we investigate the role of fluctuations. We consider fluctuations giving
rise to polynomials in the local field whose order is smaller than the order
of terms in the Lagrangian. We show that such fluctuations lead to non-
analyticity in the critical exponents. We also show that there is a
generalization of the notion of upper and lower critical dimensions, which is
a straightforward extension of the $r=2$ case. Above the upper critical
dimension $d=2r$, the saddle point calculation is valid since fluctuations do
not introduce new singularities. Below the upper critical dimension, we
generalize the Ginzburg criteria to precisely quantify when fluctuations
dominate over the saddle point calculation.

Next, we introduce scaling forms for physical observables derived from the
partition function. These scaling forms can be conveniently derived using
Wilsonian renormalization. We use these forms to obtain scaling relations
between the critical exponents, which continue to hold beyond the saddle point
approximation. We show that given relevant couplings $t$ and $h$, one can use
their mass dimensions and these scaling relations to determine all the
critical exponents for these transitions.

We further use the renormalization group to compute corrections to the
critical exponents by calculating the one-loop beta functions. The one-loop
beta functions are used to identify fixed points of the RG flow. We calculate
the scaling dimensions of the couplings about these fixed points and use them
to write the corrections to the critical exponents beyond the saddle point.
For $r=3$ and $r=4$, we determine the corrected critical exponents of the
couplings near the Gaussian fixed point. We also find a non-trivial fixed
point in both cases. However, there are no relevant couplings in the vicinity
of the non-trivial fixed point, and consequently, the sole corrections arise
due to flows near the Gaussian fixed point. For $r\geq 5$, the one-loop beta
function vanishes, and hence we need to go beyond one-loop calculations to
find corrections to the critical exponents.

We now briefly discuss why we rarely observe higher order local phase
transitions in physical systems. Higher order phase transitions with a local
order parameter require a delicate fine-tuning which sets lower order terms in
the Hamiltonian to zero. Such a fine-tuning is uncommon in most systems of
physical interest, where such terms can arise due to fluctuations. These lower
order terms can be avoided due to the existence of a symmetry, however we
presently do not know any symmetry argument which explicitly rules out such
terms.

A conceptual feature we wish to convey through our work is that the one-loop
method used in our of calculation of critical exponents in §5 is possibly more
convenient than using Feynman diagrammatics to do the same, as performed for
$r=2$ in Peskin:1995ev ; kardar_2007 . One important question which remains to
be answered is to estimate the contribution of higher loop corrections to the
critical exponents.

We will list out possible avenues that branch out from our work. An immediate
direction is to construct a guiding principle that classifies higher order
non-local phase transitions. Using such a principle, we hope to write a
phenomenological Hamiltonian describing higher-order non-local phase
transitions. An important class of non-local third order phase transitions
arise in Gross-Witten-Wadia type models. In a forthcoming work, we will
investigate such phase transitions and calculate critical exponents for the
same. In general, one can investigate such phase transitions in the context of
finite-range Coulomb gas models. We visualize our present work to be a
starting basis for looking at these problems and hope to uncover some of the
above mysteries in future works.

## Acknowledgements

We thank Abhishek Dhar and Akhil Sivakumar, and especially anonymous referee
for their comments on the draft. We are also grateful to Spenta Wadia for
going through the earlier version of the draft meticulously and for suggesting
corrections. The authors acknowledge gratitude to the people of India for
their steady and generous support to research in basic sciences.

##  Appendix A Why critical exponents are single valued in the $(h,t)$ plane?

In this appendix, we will see that single valuedness of critical exponents in
the $(h,t)$ plane follows from the analyticity assumption combined with our
scaling conjecture. Let us consider a more general scaling form such that
$C^{\pm}_{V},\alpha_{\pm},g_{\pm},\Delta_{\pm}$ are different for $t>0$ and
$t<0$ respectively. As an example, we will consider that the singular part of
specific heat is given by the following function

$C^{\pm}_{V}\sim\absolutevalue{t}^{r-\alpha_{\pm}-2}\widehat{g}_{\pm}\left(\frac{h}{t^{\Delta_{\pm}}}\right).$
(101)

However we argue that this is ruled out by free energy being analytic
everywhere apart from $h=0$ and $t<0$. To understand this, consider a point in
the $(h,t)$ plane at $t=0$ and finite $h$. Since the function $C$ is analytic
in the vicinity of this point, it can be expanded in terms of a Taylor series

$C_{V}(t\ll h^{\Delta})=A(h)+tB(h)+\text{O}(t^{2})+\dots$ (102)

Since we already have a power series description, $C$ can be obtained from the
two power series expansions given by

$C_{V}^{\pm}=\absolutevalue{t}^{r-\alpha_{\pm}-2}\left[A_{\pm}\left(\frac{h}{t^{\Delta_{\pm}}}\right)^{p_{\pm}}+B_{\pm}\left(\frac{h}{t^{\Delta_{\pm}}}\right)^{q_{\pm}}+\dots\right],$
(103)

where $\left(p_{\pm},q_{\pm}\right)$ are powers of the largest terms of
$\widehat{g}_{\pm}$. We will now match the expressions for $C_{V}^{\pm}$ with
the power series given in (102), which leads to the identification

$r-\alpha_{\pm}-2=-p_{\pm}\Delta_{\pm}\quad\&\quad
r-\alpha_{\pm}-1=-q_{\pm}\Delta_{\pm}$ (104)

Thus the series expansions can be written as

$C_{V}^{\pm}(t\ll
h^{\Delta})=A_{\pm}\,h^{\frac{-(r-\alpha_{\pm}-2)}{\Delta_{\pm}}}+B_{\pm}\,\absolutevalue{t}\,h^{\frac{-(r-\alpha_{\pm}-1)}{\Delta_{\pm}}}+\dots$
(105)

As a consequence of continuity at $t=0$, we can use the equation above to
obtain the following relations between the critical exponents

$\alpha_{+}=\alpha_{-}\equiv\alpha,\quad\Delta_{+}=\Delta_{-}\equiv\Delta,$
(106)

and so on for other critical exponents as well since these arguments
generalize for any scaling function $X(t,h)$ which possess analyticity
everywhere apart from the line $h=0$ and $t<0$.

##  Appendix B Kinetic fluctuations in $d\leq 2$ and loss of order

Consider the Hamiltonian (3) where the order parameter given by
$\vec{\phi}=\phi\hat{\phi}$ is $N$-dimensional, i.e. the target space is
$N$-dimensional. Taking the saddle point of the partition function fixes the
magnitude of the order parameter, thereby spontaneously breaking the O($N$)
symmetry to O$(N-1)$, with the $N-1$ angles remain unconstrained. Denoting
these angles by $\Theta^{\alpha}$ and the metric on the target space as
$g_{\alpha\beta}$, we can rewrite the kinetic part of the Hamiltonian, which
describes the Goldstone modes parametrized by $\Theta^{\alpha}$ as

$\beta H=\int
d^{d}x\left[\frac{K}{2}\phi^{2}g_{\alpha\beta}\,\frac{\partial\Theta^{\alpha}}{\partial
x^{i}}\frac{\partial\Theta^{\beta}}{\partial
x^{i}}+t_{r}\,\phi^{2(r-1)}(x)+u_{r}\,\phi^{2r}(x)-(h.\phi)\,\phi^{2(r-2)}\right].$
(107)

Our analysis can be made easier by considering physical target spaces where
$g_{\alpha\beta}$ has no off-diagonal terms. In this case, the two point
correlator can be found out by solving the Green function equation and is
given by

$\braket{\Theta^{\alpha}(x)\Theta^{\beta}(x^{\prime})}=-\frac{2C(x-x^{\prime})}{K\phi^{2}\,g_{\alpha\beta}},$
(108)

where $C(x)$ in the long distance limit is given by

$\lim_{x\to\infty}C(x)=\begin{cases}0\quad&d>2\\\
\dfrac{x^{2-d}}{(2-d)S_{d}}\quad&d<2\\\ \dfrac{\ln{x}}{2\pi}\quad&d=2\\\
\end{cases}$ (109)

Thus we see that kinetic fluctuations or Goldstone modes destroy long range
order in $d\leq 2$, as correlators in these dimensions grow with distance.
This leads to a destruction of order in $d\leq 2$ for systems possessing
continuous symmetry, which defines the lower critical dimension. The analysis
in this subsection is essentially the content of the Mermin Wagner theorem
PhysRevLett.17.1133 ; Coleman:1973ci , since Goldstone modes in the system
would have infrared divergent two-point correlator as given by (109). For
discrete systems, the lower critical dimension is $d_{l}=1$, the corresponding
argument remains unchanged as well for the case of higher order phase
transitions.

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