# Instability of the Body-Centered Cubic Lattice within the Sticky Hard Sphere
and Lennard-Jones Model obtained from Exact Lattice Summations

Antony Burrows  Centre for Theoretical Chemistry and Physics, The New Zealand
Institute for Advanced Study, Massey University Auckland, Private Bag 102904,
0632 Auckland, New Zealand  Shaun Cooper  School of Natural and Computational
Sciences, Massey University Auckland, Private Bag 102904, 0632 Auckland, New
Zealand.  Peter Schwerdtfeger<EMAIL_ADDRESS>Centre for
Theoretical Chemistry and Physics, The New Zealand Institute for Advanced
Study, Massey University Auckland, Private Bag 102904, 0632 Auckland, New
Zealand

(August 29, 2024)

###### Abstract

A smooth path of rearrangement from the body-centered cubic (bcc) to the face-
centered cubic (fcc) lattice is obtained by introducing a single parameter to
cuboidal lattice vectors. As a result, we obtain analytical expressions in
terms of lattice sums for the cohesive energy. This is described by a Lennard-
Jones (LJ) interaction potential and the sticky hard sphere (SHS) model with
an $r^{-n}$ long-range attractive term. These lattice sums are evaluated to
computer precision by expansions in terms of a fast converging series of
Bessel functions. Applying the whole range of lattice parameters for the SHS
and LJ potentials demonstrates that the bcc phase is unstable (or at best
metastable) toward distortion into the fcc phase. Even if more accurate
potentials are used, such as the extended LJ potential for argon or chromium,
the bcc phase remains unstable. This strongly indicates that the appearance of
a low temperature bcc phase for several elements in the periodic table is due
to higher than two-body forces in atomic interactions.

##  I Introduction

The stability of different bulk phases and their possible connections through
distortions and rearrangements in phase transitions remain an open and
challenging field in solid-state physics Young (1991). Solid-to-solid phase
transitions are commonly modeled by computer intensive molecular dynamic or
Monte-Carlo simulations at finite temperatures and pressures Binder (1985);
Gomez  _et al._ (2019), or by various algorithms to find phase transition
paths on a Born-Oppenheimer hypersurface Caspersen and Carter (2005). For
example, the relative stability of the fcc versus the hexagonal close packing
(hcp) and possible transition mechanisms between these two phases for the rare
gas elements has been a matter of a long-standing controversy Mau and Huse
(1999); Stillinger (2001); Krainyukova  _et al._ (2012); Schwerdtfeger  _et
al._ (2016); Li  _et al._ (2017); Wiebe  _et al._ (2020); Schwerdtfeger  _et
al._ (2021). While fcc has a higher excess entropy compared to hcp by a rather
small difference (for the hard sphere model it is 0.00115$\pm$0.00004 $k_{B}$
per sphere Mau and Huse (1999)), the energetic stability of the fcc over the
hcp phase for the rare gas solid argon (at low temperatures and pressures) is
due to quantum effects (phonon dispersion) Schwerdtfeger  _et al._ (2016);
Wiebe  _et al._ (2020). Similarly, the transformation between the
bcc$\leftrightarrow$fcc phases has been the subject of many discussions as the
exact martensitic type of transformation path for a solid, such as in iron-
based materials, or in clusters, is still being debated Kraft  _et al._
(1993); Rollmann  _et al._ (2007); Cayron (2015).

It is commonly believed that strong repulsive forces favor close-packed
arrangements such as fcc or hcp, whereas soft repulsion favors less dense
packed structures such as bcc Agrawal and Kofke (1995); Prestipino  _et al._
(2005); Likos  _et al._ (2007); Bharadwaj and Singh (2017). Laird showed that
the bcc phase is unstable within the hard-sphere model Laird (1992), while
Hoover et al. and later Agrawi and Kofke showed that soft repulsive potentials
of the form $ar^{-n}$ with small $n$ values are required to stabilize the bcc
phase Hoover  _et al._ (1972); Agrawal and Kofke (1995). Very recently Ono and
Ito used phonon dispersion curves to show that soft Lennard-Jones forces are
required to turn the bcc phase into a minimum Ono and Ito (2021). However, as
minima can be very shallow on a energy hypersurface, one requires accurate
numerical or analytical methods to determine if the bcc phase represents a
(metastable) minimum for a two-body potential or not. Moreover, inverse power
law potentials such as the LJ potential have the advantage that properties
such as the cohesive energy can be evaluated analytically through lattice sums
Borwein  _et al._ (2013); Burrows  _et al._ (2020). If a single path through a
lattice parameter can be found Caspersen and Carter (2005) describing smoothly
the bcc$\leftrightarrow$fcc transition (not necessarily a minimum energy
path), one gains valuable insight into the stability of the bcc phase.

Conway and Sloane introduced the isodual mean-centered cuboidal lattice (mcc)
which can be seen as an average between the bcc and the fcc lattice Conway and
Sloane (1994). They introduced lattice vectors depending on two parameters
connecting the bcc, mcc and fcc lattices. Recently we were able to find fast
converging lattice sums for these cuboidal lattices derived from their
corresponding Gram matrices and quadratic forms Burrows  _et al._ (2021).
These lattice sums, which can be evaluated to computer precision, will be
introduced in the next section and applied to analyse the energy profile of
the bcc lattice distortion into the fcc densest packing using LJ and SHS
interaction potentials. For more realistic two-body forces we apply extended
Lennard-Jones potentials Schwerdtfeger  _et al._ (2006) for Ar2 Schwerdtfeger
_et al._ (2021) and Cr2.

##  II Method

Lattice vectors for the primitive cell depending on a single parameter $A$ are
defined by

$\displaystyle\boldsymbol{b}_{1}^{\top}(A)=\left(1,0,0\right)\quad,\quad\boldsymbol{b}_{2}^{\top}(A)=\left(\frac{A}{A+1},\frac{\sqrt{2A+1}}{A+1},0\right),$
(1)
$\displaystyle\boldsymbol{b}_{3}^{\top}(A)=\left(\frac{1}{A+1},\frac{1}{(A+1)\sqrt{2A+1}},\sqrt{\frac{4A}{(A+1)(2A+1)}}\right).$

The corresponding Gram matrix for the quadratic form is given by the scalar
product between these lattice vectors,

$G_{ij}(A)=\langle\boldsymbol{b}_{i}(A),\boldsymbol{b}_{j}(A)\rangle=\frac{1}{A+1}\begin{pmatrix}A+1&A&1\\\
A&A+1&1\\\ 1&1&2\end{pmatrix}.$ (2)

The cuboidal lattices are defined in the range $A\in[\frac{1}{3},1]$ Burrows
_et al._ (2021), and for the special values of $A=1/3$, $A=1/2$,
$A=1/\sqrt{2}$, and $A=1$ lattice vectors for the acc (axial centered cuboidal
Conway and Sloane (1994)), bcc, mcc and fcc lattices are obtained, with number
of nearest neighbors of 10, 8, 8 and 12 respectively. This sets the minimal
distance between two lattice points to 1 for the range $A\in[\frac{1}{3},1]$,
which ensures that the lattice deformation is compatible with the hard sphere
model. The volume spanned by these three vectors is $V(A)=\sqrt{{\rm
det}G(A)}=2A^{1/2}(A+1)^{-3/2}$ with a maximum volume at the bcc structure
($A=1/2$). The mcc lattice with the corresponding lattice vectors (1) is shown
in Figure 1.

Figure 1: The mcc ($A=1/\sqrt{2}$) lattice and corresponding non-orthogonal
basis vectors (red) according to (1). In green the atoms moving towards the
central atom (blue) becoming nearest neighbors in the fcc lattice are shown.

The choice of the basis vectors (1) has the advantage that only
$\boldsymbol{b}_{2}$ and $\boldsymbol{b}_{3}$ move in this 3D lattice
transformation. The length of $\boldsymbol{b}_{1}$ and $\boldsymbol{b}_{2}$ is
1 for all $A$ values considered, and the angle between $\boldsymbol{b}_{1}$
and $\boldsymbol{b}_{3}$ is the same as between $\boldsymbol{b}_{2}$ and
$\boldsymbol{b}_{3}$. From the Gram matrix one obtains the atomic packing
fraction or packing density Conway and Sloane (2013) for the cuboidal lattices
as $\rho(A)=\frac{\pi}{12}\sqrt{\frac{(A+1)^{3}}{A}}$ Burrows  _et al._
(2021). This yields the well known values for fcc ($\rho(1)=\pi\sqrt{2}/6$)
and bcc ($\rho(\frac{1}{2})=\pi\sqrt{3}/8$). In fact, from this formula we
deduce that bcc is the least packed arrangement of all the cuboidal lattices
considered here.

Using an $(a,b)$ LJ potential in its most general form Grüneisen (1912); Jones
and Ingham (1925)

$V_{\rm
LJ}(r,a,b)=\frac{ab}{a-b}\epsilon\left[\frac{1}{a}\left(\frac{r_{e}}{r}\right)^{a}-\frac{1}{b}\left(\frac{r_{e}}{r}\right)^{b}\right],$
(3)

where $r_{e}$ is the minimum (equilibrium) distance, $\epsilon>0$ is the
dissociation energy and $a>b>3$ are real numbers. We obtain an analytical
expression for the cohesive energy in terms of lattice sums $L(a,A)$ and the
nearest neighbor distance $R$ in the lattice Burrows  _et al._ (2020),

$E_{\rm
LJ}(R,a,b,A)=\frac{ab\epsilon}{2(a-b)}\left[\frac{1}{a}L(a,A)\left(\frac{r_{e}}{R}\right)^{a}-\frac{1}{b}L(b,A)\left(\frac{r_{e}}{R}\right)^{b}\right].$
(4)

Here, $b>3$ is required to avoid the singularity in $L(b,A)$ at $b=3$
Schwerdtfeger  _et al._ (2006) (although these lattice sums can be
analytically continued Borwein  _et al._ (2013, 1998); Burrows  _et al._
(2021)). The lattice sums $L(a,A)$ are defined through their corresponding
quadratic forms $\boldsymbol{i}^{\top}G\boldsymbol{i}\leavevmode\nobreak\
,\leavevmode\nobreak\ \boldsymbol{i}\in\mathbb{Z}^{3}$ by Conway and Sloane
(2013)

$L(a,A)={\sum_{\boldsymbol{i}\in\mathbb{Z}^{3}}}^{\prime}\left(\frac{1}{\boldsymbol{i}^{\top}G\boldsymbol{i}}\right)^{a/2}={\sum_{i,j,k}}^{\prime}\left(\frac{A+1}{A(i+j)^{2}+(j+k)^{2}+(i+k)^{2}}\right)^{a/2},$
(5)

where the prime symbol indicates that the term corresponding to
$\boldsymbol{i}^{\top}=(0,0,0)$ is omitted in the summation. For small values
of $a$, these triple sums are slowly convergent and one needs to find
expansions in terms of fast converging series to obtain computer precision
Borwein  _et al._ (2013). A number of methods to achieve this have recently
been introduced by our group Burrows  _et al._ (2020, 2021). A program to
evaluate these lattice sums including the cuboidal lattices considered here is
freely available from our website Schwerdtfeger and Burrows (2021). For this
work we use either the Terras decomposition of the Epstein zeta function
Terras (1973); Burrows  _et al._ (2020) or the decomposition in terms of
Jacobi $\theta$ functions and integral transforms to produce series expansions
in terms of Bessel functions Burrows  _et al._ (2020, 2021).

The SHS model can easily be obtained in the limit of $a\rightarrow\infty$ of
the LJ potential Trombach  _et al._ (2018), and the cohesive energy given by
the expression

$E_{\rm SHS}(R,b,A)=\lim_{a\rightarrow\infty}E_{\rm
LJ}(R,a,b,A)=-\frac{\epsilon}{2}L(b,A)\left(\frac{r_{e}}{R}\right)^{b},$ (6)

with $R\geq r_{e}$. This gives a direct relation between the SHS energy of the
solid and the corresponding lattice sum.

It is convenient to introduce dimensionless units ($R^{*}=R/r_{e}$ and
$E^{*}=E/\epsilon$). The minimum nearest neighbor distance for a cuboidal
lattice can be found from (4),

$R^{*}_{\rm min}(a,b,A)=\left[\frac{L(a,A)}{L(b,A)}\right]^{\tfrac{1}{a-b}}.$
(7)

For the SHS model this reduces to $R^{*}_{\rm min}=1$. The cohesive energy at
minimum becomes

$\displaystyle E^{*}$ $\displaystyle(R^{*}_{\rm
min},a,b,A)=-\frac{1}{2}\left[\frac{L(b,A)^{a}}{L(a,A)^{b}}\right]^{\tfrac{1}{a-b}},$
(8)

and for the SHS model we attain $E^{*}(R^{*}_{\rm min}=1;b,A)=-L(b,A)/2$.
Finally, a more realistic two-body potential is used, where lattice sum
techniques can still be applied. This requirement is fulfilled by the extended
Lennard-Jones (ELJ) potential, which is an inverse power series expansion in
terms of the distance $R$,

$E_{\rm
ELJ}(R,c_{n},A)=\frac{1}{2}\sum_{n=1}^{n_{\textrm{max}}}c_{n}L(a_{n},A)R^{-a_{n}},$
(9)

with $\sum_{n}c_{n}=-\epsilon$ and $a_{n}>3$ Schwerdtfeger  _et al._ (2006,
2021).

##  III Results

Starting with the discussion of the SHS model, the difference in cohesive
energies between the $A$-dependent cuboidal lattices and the fcc lattice
($A=1$) as a function of the two parameters $b$ and $A$, $\Delta
E^{*}(b,A)=\left[L(b,A=1)-L(b,A)\right]/2$ at $R^{*}_{\rm min}=1.0$, is shown
in Figure 2. It is evident that the SHS model predicts a maximum in energy at
the bcc structure. In fact, it was proved recently that $\partial
L(b,A)/\partial A=0$ and $\partial^{2}L(b,A)/\partial A^{2}>0$ at
$A=\frac{1}{2}$ (bcc) for all $b\in(3,\infty)$ Burrows  _et al._ (2021).
Despite the path being chosen along the $A$ parameter, this most likely does
not represent the true minimum energy path, it is clearly downhill
energetically towards the fcc structure. As a result, the bcc lattice is
unstable with respect to distortion to fcc within the SHS model. There is also
the opposite path towards the acc crystal ($A=1/3$), which has to our
knowledge not yet been observed in nature. Figure 2 shows that for low $b$
values, $\Delta E^{*}(b,A)$ starts to increase again (at lower exponents
$\Delta E^{*}(b,A)\rightarrow\infty$ for $b\rightarrow 3$). The most stable
bcc lattice is observed at $\Delta
E^{*}(b=5.49363406\dots,\frac{1}{2})=1.090510595\dots$, with a $b$ value close
to the exponent $b=6$ used for dispersive type of forces.

Figure 2: Difference in cohesive energies $\Delta
E^{*}(b,A)=\tfrac{1}{2}\left[L(b,A=1)-L(b,A)\right]$ between the cuboidal
lattices and fcc for various exponents $b$ and lattice parameter $A$ of the
SHS model. Contour interval chosen is 0.125. The vertical black line at
$b=5.493634$ shows the point of least instability for the bcc lattice.

As the SHS model clearly has its limitations, we turn to the more accurate
$(a,b)$ LJ potential, i.e., we introduce softer repulsive walls to the SHS
model. This will also remove the discontinuity in the $\Delta E^{*}(a,b,A)$
curve at the fcc point ($A=1$). Due to the attractive long-range lattice
forces, the minimum distance between two lattice points is $R^{*}_{\rm
min}(a,b,A)<1$ for finite $(a,b)$ exponents provided that $a>b>3$, see
eq.(23). $R^{*}_{\rm min}(a,b,A)$ does not vary much with respect to $A$ for a
fixed $(a,b)$ combination (see appendix). The minimum distance for the (12,6)
LJ potential for the bcc lattice is $R^{*}_{\rm
min}(12,6,\frac{1}{2})$=0.951864819 compared to $R^{*}_{\rm
min}(12,6,1.0)$=0.9712336910 for the fcc lattice. That is, lattice
interactions lead to a contraction in the nearest neighbor distance by about
4.8% for the bcc lattice. A deviation from this behavior is observed only for
large $a$ values, as eventually the minimum distance has to approach the SHS
limit of $R^{*}_{\rm min}=$1.0. Here $R^{*}_{\rm min}$ at bcc turns into a
very shallow maximum (see appendix). From eq.(23) and $\partial
L(b,A)/\partial A$=0 at $A=\frac{1}{2}$ Burrows  _et al._ (2021) it follows
that $\partial R^{*}_{\rm min}(a,b,A)/\partial A=0$ at $A=\frac{1}{2}$, and
the bcc point remains a critical point.

Shorter distances are usually associated with greater stability of the lattice
which could stabilize the bcc phase. This is however not the case as Figure 3
shows. In fact, the bcc lattice is not a stable lattice for any $(a,b)$ LJ
potential, i.e., it will continuously distort by lowering the energy along the
distortion parameter $A$ toward the most densely packed and stable fcc
lattice. From eq.(8) the energy difference between the bcc and the fcc lattice
for a $(a,b)$ LJ potential becomes $\Delta E^{*}_{\rm
bcc,fcc}(a,b)=E^{*}(R^{*}_{\rm min,bcc},a,b,\tfrac{1}{2})-E^{*}(R^{*}_{\rm
min,fcc},a,b,1)$. Figure 4 shows that $\Delta E^{*}_{\rm bcc,fcc}(a,b)>0$ for
all $a>b>3$, thus bcc remains energetically unstable for a general LJ
potential. For a very small $(a,b)$ range, however, the bcc phase becomes
metastable, i.e., the minimum at $A<\frac{1}{2}$ shifts toward the bcc
structure, see Figure 4. The $(a,b)$ phase transition line from the unstable
to the metastable bcc lattice is approximately described by the polynomial
$a_{\rm PT}=-6.3829845\times 10^{-4}b_{\rm PT}^{3}+3.8186745\times
10^{-2}b_{\rm PT}^{2}-1.3466248b_{\rm PT}+1.1373783\times 10^{1}$ with $a_{\rm
PT}>b_{\rm PT}\in(3,5.25673]$ (see appendix), and we see an almost linear
behavior as shown in Figure 4. This also explains why Ono and Ito obtained
imaginary phonon frequencies for some low $(a,b)$ combinations Ono and Ito
(2021) (their results have to be taken with some care as the $r^{-3}$
potential used leads to a singularity in the cohesive energy). In fact, the
bcc structure becomes metastable if and only if
$L(a,A)\partial^{2}L(a,A)/\partial A^{2}<L(b,A)\partial^{2}L(b,A)/\partial
A^{2}$ for $A=\frac{1}{2}$ and $a>b>3$ (see appendix). However, these minima
appear at energies $\Delta E^{*}_{\rm bcc,fcc}(a,b)>$0.2 ($a<7.660388$) for
rather unphysical potentials, with low $\Delta E^{*}$ values only if $a\approx
b$. As an example, for a (4,3.1) LJ potential the bcc structure is a minimum
at $\Delta E^{*}_{\rm bcc,fcc}=170.2$ with an activation barrier of ${\Delta
E^{*}}^{\\#}=12.2$ situated at $A=0.6$ on the path toward the distortion to
the fcc structure. The distortion along the $A$ parameter can also occur
towards a metastable lattice with $A<\frac{1}{2}$ and higher packing density.
Alternatively, with a (12,6) LJ potential the metastable minimum sits at a
lattice with $A=0.3962483\dots$ and packing density $\rho=0.6861655\dots$.
Finally, the mcc lattice is just a lattice along the energetic downward path
towards fcc as for the SHS model.

Figure 3: Cohesive energy differences $\Delta E^{*}(R^{*}_{\rm
min},a,b,A)=E^{*}(R^{*}_{\rm min},a,b,A)-E^{*}(R^{*}_{\rm min},a,b;A=1)$ for
the $(a,b)$ LJ potential dependent on the lattice parameter $A$, and for the
two ELJ potentials of argon and chromium (see appendix). Figure 4: Energy
difference $\Delta E^{*}_{\rm bcc,fcc}(a,b)=E^{*}(R^{*}_{\rm
min,bcc},a,b,\tfrac{1}{2})-E^{*}(R^{*}_{\rm min,fcc},a,b,1)$ between the bcc
and fcc lattice for the $(a,b)$ LJ potential. The yellow curve indicates the
phase transition line to a metastable bcc state.

As $\partial L(b,A)/\partial A=0$ at $A=\frac{1}{2}$ Burrows  _et al._ (2021)
we obtain $\partial E^{*}(R^{*}_{\rm min},a,b,A)/\partial A=0$ at
$A=\frac{1}{2}$ (see appendix), and the bcc structure remains a critical point
for all $(a,b)$ combinations. Moreover, if the exponent $a$ responsible for
the repulsive wall increases, we approach the limit of the SHS potential with
much higher energies compared to the LJ potential. Here we mention that by
applying an inverse power law potential for the repulsive wall (opposed to the
long-range part in the SHS model), Agrawal and Kofke showed from Monte Carlo
simulations that the bcc phase is unstable Agrawal and Kofke (1995).

The Einstein frequency $\omega_{E}$ of a single atom moving in the field of
all other atoms can be expressed analytically in terms of lattice sums
Schwerdtfeger  _et al._ (2021). $\omega_{E}(a,b,A)>0$ for all
$A\in[\frac{1}{3},1]$ and $a>b>3$. As a consequence, a single atom is locked
and more than one atom has to move simultaneously along the
bcc$\rightarrow$fcc path similar to a Zener martensitic transformation Rifkin
(1984); Cayron (2015).

The question remains as to why bcc lattices are observed in nature given their
instability, large volume and small bulk modulus within the cuboidal
structures. It is clear that two-body forces favor dense packings with the
largest kissing number for an atom, that is fcc or hcp. The answer therefore
lies in the failure of the two-body potential to correctly describe the
interactions in the crystal, i.e., neglecting important higher than two-body
interactions (and perhaps quantum effects for quantum solids such as helium).
It is well known that the many-body expansion is only slowly convergent for
metallic systems Kaplan  _et al._ (1996); Hermann  _et al._ (2007). To see if
the form of the LJ potential limits our conclusion, a more accurate ELJ two-
body potential is taken, derived from relativistic coupled cluster theory for
argon Cybulski and Toczyłowski (1999); Schwerdtfeger  _et al._ (2016). As in
the case for the (12,6) LJ potential, the ELJ potential has a minimum
$R^{*}_{\rm min}(A)$ value at the bcc structure (see appendix). More
importantly, the $E_{\rm ELJ}^{*}(A)$ curve does not change substantially in
shape and is only slightly shifted compared to the (12,6) LJ potential, as
shown in Figure 3. This is perhaps expected from the comparison between the
two potentials (see appendix), and from the fact that for the fcc structure
$E^{*}(R^{*}_{\rm min},1.0)=7.8532$ Schwerdtfeger  _et al._ (2021) for the ELJ
potential and close to $E^{*}(R^{*}_{\rm
min},12,6,1.0)=-L_{6}^{2}/(2L_{12})=8.6102$ for the (12,6) LJ potential (exp.
$E^{*}=6.4951$ using the data from Ref.Schwalbe  _et al._ (1977)).

Figure 5: Potential energy curves $V^{*}(r^{*})$ (in dimensionless units) for
a (12-6) LJ potential, and for Ar2, Li2 and Cr2 (see appendix).

To underscore our argument even further the unusual potential energy curve for
Cr2 is considered. Here we use experimental potential values of Casey and
Leopold Casey and Leopold (1993), but attenuated for the long range dispersion
using the $C_{6}$ coefficient of Roos and co-workers Pavlović  _et al._
(2004), and finally fitted to an extended Lennard Jones potential potential
(see appendix). This potential curve, shown in Figure 5, is extremely broad
and has a large dip in the medium distance range $r\in[1.3,1.7]r_{e}$, and
therefore deviates substantially from a typical potential energy curve such as
LJ or Morse Vancoillie  _et al._ (2016). As it turns out, this potential leads
to far too small distances and far too large cohesive energies for the solid
state (see appendix). However, the chromium $\Delta E^{*}(R^{*}_{\rm
min},a,b,A)$ curve in Figure 3 shows that bcc remains a transition state along
the distortion parameter $A$ in line with all the other two-body potentials.

We also looked at lithium, which adopts a bcc structure at normal conditions.
Lithium has an extremely broad potential energy curve (see Figure 5) even in
the repulsive region Barakat  _et al._ (1986), which leads to a collapse of
the crystal to a very small nearest neighbor distance (see appendix). It is
clear that $N$-body forces describing correctly the confinement of the atoms
in the solid state become very important here, i.e., the $N$-body expansion is
not converging smoothly with increasing $N$ for metals such as lithium or
chromium Hermann  _et al._ (2007). One may argue that a broad potential energy
curve such as for Li2 gives lower exponents for a LJ potential energy curves
typical for metallic systems. It should be pointed out however, that the long
range has to be correctly described and potential curves containing terms of
$r^{-n},n\leq 3$ in the interaction between atoms in the solid lead to
divergent series (if not analytically continued).

##  IV Conclusions

From exact lattice summations we were able to produce cohesive energies within
the SHS and LJ models to computer precision. Both potentials result in an
unstable bcc phase distorting toward the fcc phase or toward a phase in-
between acc and bcc. The metastable bcc phase for an $(a,b)$ LJ potential
occurs for unphysical potentials with very low $(a,b)$ values. The situation
does not change if accurate two-body potentials are used such as for argon or
chromium, the latter known to crystallize in the bcc phase. As a result, the
bcc phase (at low temperatures and pressures) is stabilized only by higher
than two-body forces, which have to be large enough to compete with the fcc
(or hcp) structure. The mcc lattice introduced by Conway and Sloane Conway and
Sloane (1994) is merely a point on a energetic bcc$\rightarrow$fcc downhill
path. How well effective two-body potentials Johnson (1988) will work for the
bcc problem remains to be seen. We are currently investigating the high-
pressure and temperature bcc phase stability of the bcc lattice for Lennard-
Jones type of potentials.

Acknowledgements. We acknowledge financial support by the Marsden Fund of the
Royal Society of New Zealand (MAU1409).

## Appendix: Lattice Sums and Their Derivatives

In this section we give more details about lattice sums and their derivatives,
critical point analyses for the bcc structure, and potential energy curves
using extended Lennard-Jones (LJ) potentials for argon, lithium and chromium.

The Gram matrix $G$ introduced in the main paper leads to the following
lattice sum,

$\displaystyle L(a,A)$
$\displaystyle=L(2s,A)=\mathcal{L}(s,A)={\sum_{\vec{i}\in\mathbb{Z}^{3}}}^{\prime}\left(\vec{i}^{\top}G\vec{i}\right)^{-s}$
(10)
$\displaystyle={\sum_{i,j,k\in\mathbb{Z}}}^{\prime}\left(\frac{A+1}{A(i+j)^{2}+(j+k)^{2}+(i+k)^{2}}\right)^{s},$

with the prime indicating that the term with $\vec{i}^{\top}=(0,0,0)$ is not
included, and $A\in[\frac{1}{3},1]$ for the cuboidal lattices considered here.
These sums are important for inverse power law potentials such as LJ Burrows
_et al._ (2021). Here the exponent $s$ is set to $s=\frac{a}{2}$ for
simplicity compared to the main paper. The lattice sums for the acc, bcc, mcc,
and fcc lattices are obtained for the values $A=1/3$, $A=1/2$, $A=1/\sqrt{2}$
and $A=1$, respectively. We split the lattice sum into two sums according to
Ref.Burrows  _et al._ (2021),

$\displaystyle\mathcal{L}(s,A)$
$\displaystyle=\frac{(A+1)^{s}}{2}\left[S_{1}(s,A)+S_{2}(s,A)\right]$ (11)
$\displaystyle\text{with}\quad S_{1}(s,A)=\
{\sum_{i,j,k\in\mathbb{Z}}}^{\prime}(Ai^{2}+j^{2}+k^{2})^{-s}$
$\displaystyle\text{and}\quad
S_{2}(s,A)={\sum_{i,j,k\in\mathbb{Z}}}^{\prime}(-1)^{i+j+k}(Ai^{2}+j^{2}+k^{2})^{-s}.$

For the special case of $A=1$, the sum $S_{1}(1,s)$ represents the lattice sum
for the simple cubic (sc) lattice, and the alternating sum $S_{2}(1,s)$ is
known as the Madelung constant when $s=\frac{1}{2}$ Madelung (1918). In the
following, we only consider $s>\frac{3}{2}$, keeping in mind that the lattice
sums are valid for all $s\in\mathbb{R}$ through analytical continuation and
that $S_{1}(s,A)$ (and therefore $\mathcal{L}(s,A)$) has a singularity at
$s=\frac{3}{2}$.

The two lattice sums have been expanded in terms of modified Bessel functions
of the second kind $K_{s}(x)$ in Ref.Burrows  _et al._ (2021),

$\displaystyle S_{1}(s,A)=a_{1}(s)+a_{2}(s)A^{1-s}$
$\displaystyle+a_{3}(s)A^{(1-s)/2}\sum_{i=1}^{\infty}\sum_{N=1}^{\infty}c_{iN}(s)K_{s-1}\left(d_{iN}(s)\sqrt{A}\right)$
(12) $\displaystyle
S_{2}(s,A)=b_{1}(s)+a_{3}(s)A^{(1-s)/2}\sum_{i=1}^{\infty}\sum_{N=0}^{\infty}p_{iN}(s)K_{s-1}\left(q_{iN}(s)\sqrt{A}\right),$
(13)

with the following coefficients

$\displaystyle a_{1}(s)=4\zeta(s)\beta(s)\quad,\quad
a_{2}(s)=\frac{2\pi}{(s-1)}\zeta(2s-2)$ $\displaystyle
a_{3}(s)=\frac{4\pi^{s}}{\Gamma(s)}\quad,\quad
b_{1}(s)=-4(1-2^{1-s})\zeta(s)\beta(s)$ $\displaystyle
c_{iN}(s)=r_{2}(N)\left(i^{-2}N\right)^{(s-1)/2}\quad,\quad d_{iN}(s)=2\pi
i\sqrt{N}$ (14) $\displaystyle
p_{iN}(s)=(-1)^{i}r_{2}(4N+1)\left(\frac{4N+1}{2i^{2}}\right)^{(s-1)/2}$
$\displaystyle q_{iN}(s)=\pi i\sqrt{8N+2}\,.$

$\zeta(s)$ is the Riemann zeta function, $\beta(s)$ the Dirichlet beta
function, and $r_{2}(N)$ the number of representations of number $N$ as a sum
of two squares.

We are interested in the first and second derivatives,
$\partial_{A}\mathcal{L}(s,A):=\partial\mathcal{L}(s,A)/\partial A$ and
$\partial^{2}_{A}\mathcal{L}(s,A):=\partial^{2}\mathcal{L}(s,A)/\partial
A^{2}$, of the lattice sums. It was already proven directly from (11) that
$\partial_{A}\mathcal{L}(s,A)|_{A=1/2}=0$ and
$\partial^{2}_{A}\mathcal{L}(s,A)|_{A=1/2}>0$ if $s>\frac{3}{2}$ Burrows  _et
al._ (2021). We therefore derive from eq.(11) the following expressions,

$\displaystyle\partial_{A}\mathcal{L}(s,A)=\frac{s}{A+1}\mathcal{L}(s,A)+\frac{(A+1)^{s}}{2}\left[\partial_{A}S_{1}(s,A)+\partial_{A}S_{2}(s,A)\right]$
(15)

and

$\displaystyle\partial^{2}_{A}\mathcal{L}(s,A)=-\frac{s(s+1)}{(A+1)^{2}}\mathcal{L}(s,A)+\frac{2s}{A+1}\partial_{A}\mathcal{L}(s,A)$
$\displaystyle+\frac{(A+1)^{s}}{2}\left[\partial^{2}_{A}S_{1}(s,A)+\partial^{2}_{A}S_{2}(s,A)\right]\,.$
(16)

The derivatives $\partial_{A}S_{1}(s,A)$, $\partial_{A}S_{2}(s,A)$,
$\partial^{2}_{A}S_{1}(s,A)$ and $\partial^{2}_{A}S_{2}(s,A)$ are evaluated
from the Bessel function expansions (12) and (13). For this, the following
relations are required,

$\displaystyle K_{s}(x)$ $\displaystyle=K_{s+2}(x)-\frac{2(s+1)}{x}K_{s+1}(x)$
(17) $\displaystyle\partial_{x}K_{s}(x)$
$\displaystyle=\frac{s}{x}K_{s}(x)-K_{s+1}=-\frac{s}{x}K_{s}(x)-K_{s-1}$
$\displaystyle=-\frac{1}{2}\left[K_{s-1}(x)+K_{s+1}(x)\right]\,.$ (18)

After some algebraic manipulations the following expressions are obtained

$\displaystyle\partial_{A}$ $\displaystyle S_{1}(s,A)=-(s-1)a_{2}(s)A^{-s}$
$\displaystyle-\frac{a_{3}(s)}{2}A^{-\frac{s}{2}}\sum_{i=1}^{\infty}\sum_{N=1}^{\infty}c_{iN}(s)d_{iN}(s)K_{s}\left(d_{iN}(s)\sqrt{A}\right)$
(19) $\displaystyle\partial_{A}$ $\displaystyle S_{2}(s,A)=$
$\displaystyle-\frac{a_{3}(s)}{2}A^{-\frac{s}{2}}\sum_{i=1}^{\infty}\sum_{N=0}^{\infty}p_{iN}(s)q_{iN}(s)K_{s}\left(q_{iN}(s)\sqrt{A}\right)$
(20) $\displaystyle\partial^{2}_{A}$ $\displaystyle
S_{1}(s,A)=s(s-1)a_{2}(s)A^{-s-1}$
$\displaystyle+\frac{a_{3}(s)}{4}A^{-\frac{s+1}{2}}\sum_{i=1}^{\infty}\sum_{N=1}^{\infty}c_{iN}(s)d^{2}_{iN}(s)K_{s+1}\left(d_{iN}(s)\sqrt{A}\right)$
(21) $\displaystyle\partial^{2}_{A}$ $\displaystyle S_{2}(s,A)=$
$\displaystyle\frac{a_{3}(s)}{4}A^{-\frac{s+1}{2}}\sum_{i=1}^{\infty}\sum_{N=0}^{\infty}p_{iN}(s)q^{2}_{iN}(s)K_{s+1}\left(q_{iN}(s)\sqrt{A}\right)\,.$
(22)

The Bessel function sums are fast converging, therefore making the evaluation
of lattice sums and their derivatives to computer precision attainable within
less than a second on a modern laptop computer Schwerdtfeger and Burrows
(2021).

### Critical Points for the bcc structure

For the following, we set $a$ to $2s$, making the lattice sum
$L(a,A)=L(2s,A)$, which is more convenient for the LJ potential. Figure 1
shows the lattice sums and their second derivative for $A=\frac{1}{2}$ (bcc
lattice) as a function of the exponent $a$. It is clear that
$\partial^{2}_{A}L(A,a)|_{A=\frac{1}{2}}$ has a peculiar form with a minimum
at $a=5.52534$ and a maximum at $a=12.57676$, this becomes important in the
discussion of the bcc stability for Lennard-Jones systems detailed below.
However, it is illustrative to evaluate the minimum distance derivatives
$\partial^{n}_{A}R^{*}_{\rm min}(a,b,A)$ for $A=\frac{1}{2}$ and $n=1,2$
(using dimensionless quantities as discussed in the main paper).

Figure 6: Lattice sums $L(a,\frac{1}{2})$ and
$\partial^{2}_{A}L(a,A)|_{A=\frac{1}{2}}$ (bcc lattice) as a function of the
exponent $a$. Note that $\partial_{A}L(a,A)|_{A=\frac{1}{2}}=0$ for all $a$
values.

The minimum distance for the cuboidal lattices is given by

$R^{*}_{\rm
min}(a,b,A)=\left(\frac{L(a,A)}{L(b,A)}\right)^{\tfrac{1}{a-b}}\,,$ (23)

and is shown in Figure 7 for various $(a,b)$ combinations.

Figure 7: Minimum distance $R^{*}_{\rm min}(a,b,A)$ for various $(a,b)$ LJ
potentials and for the ELJ potential (taken from Ref.Schwerdtfeger  _et al._
(2016)) dependent on the lattice parameter $A$.

The first derivative for $a>b>3$ is given by

$\partial_{A}R^{*}_{\rm min}(a,b,A)=\frac{R^{*}_{\rm
min}(a,b,A)}{a-b}\left(\frac{\partial_{A}L(a,A)}{L(a,A)}-\frac{\partial_{A}L(b,A)}{L(b,A)}\right)\,,$
(24)

which for the bcc lattice ($A=\frac{1}{2}$) is zero because
$\partial^{A}L(a,A)|_{A=\frac{1}{2}}=0$ identically for all values of $a>3$
Burrows  _et al._ (2021). The second derivative evaluated at $A=\frac{1}{2}$
is given by

$\displaystyle\partial^{2}_{A}R$ ${}^{*}_{\rm min}(a,b,A)|_{A=\frac{1}{2}}=$
(25) $\displaystyle\left[\frac{R^{*}_{\rm
min}(a,b,A)}{a-b}\left(\frac{\partial^{2}_{A}L(a,A)}{L(a,A)}-\frac{\partial^{2}_{A}L(b,A)}{L(b,A)}\right)\right]_{A=1/2}\,.$

Evaluating the expression in parentheses in (25) shows that $R^{*}_{\rm
min}(a,b,A)$ has a minimum at $A=\frac{1}{2}$ if $a<14.17598$. For values
$a>14.17598$ we have a certain range of $b$ values where $R^{*}_{\rm
min}(a,b,A)$ becomes a shallow maximum as is the case for the (30,6) LJ
potential shown in Figure 7.

In a similar way we evaluate the cohesive energy for an $(a,b)$ LJ potential
at $R^{*}_{\rm min}(a,b,A)$,

$\displaystyle E^{*}$ $\displaystyle(R^{*}_{\rm min},a,b,A)=$
$\displaystyle\frac{1}{2(a-b)}\left[bL(a,A)\left(\frac{L(b,A)}{L(a,A)}\right)^{\tfrac{a}{a-b}}-aL(b,A)\left(\frac{L(b,A)}{L(a,A)}\right)^{\tfrac{b}{a-b}}\right]$
$\displaystyle=-\frac{1}{2}\left[\frac{L(b,A)^{1/b}}{L(a,A)^{1/a}}\right]^{\frac{ab}{a-b}}\,.$
(26)

The first and second derivatives are evaluated as,

$\displaystyle\partial_{A}E^{*}(R^{*}_{\rm min},a,b,A)=$ (27) $\displaystyle
E^{*}(R^{*}_{\rm
min},a,b,A)\frac{ab}{(a-b)}\left[\frac{1}{b}\frac{\partial_{A}L(b,A)}{L(b,A)}-\frac{1}{a}\frac{\partial_{A}L(a,A)}{L(a,A)}\right]$
and $\displaystyle\partial_{A}^{2}E^{*}(R^{*}_{\rm
min},a,b,A)=\frac{\left\\{\partial_{A}E^{*}(R^{*}_{\rm
min},a,b,A)\right\\}^{2}}{E^{*}(R^{*}_{\rm min},a,b,A)}$ (28)
$\displaystyle+\frac{ab}{a-b}E^{*}(R^{*}_{\rm
min},a,b,A)\bigg{[}\frac{1}{b}\frac{\partial_{A}^{2}L(b,A)}{L(b,A)}-\frac{1}{a}\frac{\partial_{A}^{2}L(a,A)}{L(a,A)}$
$\displaystyle-\frac{1}{b}\frac{\left\\{\partial_{A}L(b,A)\right\\}^{2}}{L(b,A)^{2}}+\frac{1}{a}\frac{\left\\{\partial_{A}L(a,A)\right\\}^{2}}{L(a,A)^{2}}\bigg{]}\,.$

The first derivative is zero for the bcc lattice ($A=\frac{1}{2}$) because
$\partial^{A}L(a,A)|_{A=\frac{1}{2}}=0$ as mentioned above. This makes the bcc
point strictly an extremum along the $A$ coordinate for any $(a,b)$
combination of the LJ potential. The second derivative evaluated at
$A=\frac{1}{2}$ gives

$\displaystyle\partial_{A}^{2}$ $\displaystyle E^{*}(R^{*}_{\rm
min},a,b,A)|_{A=\frac{1}{2}}=$ (29)
$\displaystyle\frac{ab}{a-b}E^{*}(R^{*}_{\rm
min},a,b,\tfrac{1}{2})\left[\frac{1}{b}\frac{\partial_{A}^{2}L(b,A)|_{A=\frac{1}{2}}}{L(b,\frac{1}{2})}-\frac{1}{a}\frac{\partial_{A}^{2}L(a,A)|_{A=\frac{1}{2}}}{L(a,\frac{1}{2})}\right]\,.$

Hence, the bcc instability can be a maximum or a (metastable) minimum
depending on the sign of the expression in the square brackets. The transition
to a metastable phase occurs at

$\displaystyle\frac{\partial_{A}^{2}L(b,A)|_{A=\frac{1}{2}}}{bL(b,\frac{1}{2})}=\frac{\partial_{A}^{2}L(a,A)|_{A=\frac{1}{2}}}{aL(a,\frac{1}{2})}\,,$
(30)

with $b<a$. For the singularity at $a=3$ we get from computation,

$\displaystyle\lim_{a\rightarrow\
3}\frac{\partial_{A}^{2}L(a,A)|_{A=\frac{1}{2}}}{aL(a,\frac{1}{2})}=\frac{4}{9}\,,$
(31)

which is shown on Figure 8. This can be proven using a Laurent expansion
around the simple pole at $a=3$ Burrows  _et al._ (2021),

$\displaystyle
L(A;s)=\frac{2c_{-1}(A)}{a-3}+c_{0}(A)+\sum_{n=1}^{\infty}2^{-n}c_{n}(A)(a-3)^{n}$
(32)

with

$\displaystyle c_{-1}(A)=\pi\sqrt{\frac{(A+1)^{3}}{A}}\quad{\rm
and}\quad\partial^{2}_{A}c_{-1}(A)=\frac{3\pi}{4A^{2}\sqrt{A(A+1)}}\,.$ (33)

This gives

$\displaystyle\frac{\partial_{A}^{2}L(a,A)}{L(a,A)}=\frac{\partial^{2}_{A}c_{-1}(A)}{c_{-1}(A)}+\mathcal{O}(a-3)=\frac{3}{4A^{2}(A+1)^{2}}+\mathcal{O}(a-3)$
(34)

which results in (31) for $a=3$ and $A=\frac{1}{2}$.

From this limit it is clear that a metastable minimum can only exist if
$a<a_{\text{MS}}=7.66039$, but with a limited range of small $b$ values
evident from (30) and Figure 8. The maximum of the curve shown in Figure 8 is
at $a_{\text{max}}$=5.25673, for which all $b<a<a_{\text{max}}$ values result
in a metastable state. We note that the curve in Figure 8 is almost (but not
quite) symmetric around the maximum. This makes the phase transition line from
the unstable to the metastable bcc phase almost linear in the $(a,b)$ plane.

Figure 8: Lattice sum ratio
$\frac{\partial_{A}^{2}L(a,A)|_{A=\frac{1}{2}}}{aL(a,\frac{1}{2})}$ against
the exponent $a$ at lattice parameter $A=\frac{1}{2}$.

Next, we consider the Einstein frequency of a single atom of mass $M$ moving
in the field of other atoms (in atomic units) for an $(a,b)$ LJ potential
Schwerdtfeger  _et al._ (2021),

$\displaystyle\omega_{E}$
$\displaystyle(R,a,b,A)=\frac{1}{3r_{e}}\sqrt{\frac{3\epsilon}{M}}\sqrt{\frac{ab}{a-b}}\left(\frac{r_{e}}{R}\right)^{\frac{a}{2}+1}$
$\displaystyle\times\left[(a-1)L(a+2,A)-(b-1)L(b+2,A)\left(\frac{R}{r_{e}}\right)^{a-b}\right]^{\frac{1}{2}}.$
(35)

It is clear that $\omega_{E}(R)$ describes the instability of lattice by
moving a single atom as opposed to a collective movement of several atoms in
the lattice. However, at $R_{\text{min}}$ we always arrive at
$\omega_{E}(R_{\text{min}})>0$ for a finite mass $M$. To prove this we have to
show that the term in the square brackets stays positive at $R_{\text{min}}$
for a fixed $A$ value, that is

$\frac{L(a+2,A)}{L(b+2,A)}>\frac{(b-1)L(a,A)}{(a-1)L(b,A)}\,.$ (36)

As $a>b$ it suffices to show that

$\frac{L(a+2,A)}{L(b+2,A)}\geq\frac{L(a,A)}{L(b,A)}\,,$

or more generally

$\frac{L(b,A)}{L(b+h,A)}\geq\frac{L(a,A)}{L(a+h,A)}$ (37)

for any $h>0$ and $a>b>3$. The proof is given further below in the last
section.

### Extended Lennard-Jones Potentials for Li2, Ar2 and Cr2

The extended Lennard-Jones potential is defined by

$V_{\rm ELJ}(r,c_{n})=\sum_{n=1}^{n_{\textrm{max}}}c_{n}r^{-a_{n}}\quad{\rm
with}\quad\sum_{n=1}^{n_{\textrm{max}}}c_{n}r_{e}^{-a_{n}}=-D_{e}\,.$ (38)

It then follows that the cohesive energy for an extended Lennard-Jones
potential becomes,

$E_{\rm
ELJ}(R,c_{n},A)=\frac{1}{2}\sum_{n=1}^{n_{\textrm{max}}}c_{n}L(a_{n},A)R^{-a_{n}}$
(39)

with $R$ being the nearest neighbor distance in the solid. The corresponding
parameters for the potential energy curves in reduced units, $V^{*}(r^{*})$,
for Ar2, Li2 and Cr2 are listed in Table 1. For Ar2 the extended LJ potential
from Ref.Schwerdtfeger  _et al._ (2016) has been converted to dimensionless
units for this work ($r^{*}=r/r_{e}$, $V^{*}(r)=V^{*}(r)/D_{e}$, from which
follows that $r^{*}_{\text{min}}=1$ and $V^{*}_{\text{min}}=-1$).

Table 1: Potential parameters for the Ar, Li and Cr dimers obtained from a least-squares fit to the (a) analytical form of Cybulski and Toczyłowski for Ar2 Cybulski and Toczyłowski (1999); Schwerdtfeger  _et al._ (2016), (b) exp. determined potential of Barakat et al. Barakat  _et al._ (1986), and (c) exp. determined potential of Casey and Leopold Casey and Leopold (1993) as described in detail in the text. Dimensionless units are used. For Li2 and Cr2 the potential parameters are only valid for the region $V^{*}(r^{*})<0$. $n$ | $a_{n}$ | $c_{n}$ | $n$ | $a_{n}$ | $c_{n}$
---|---|---|---|---|---
Ar |  |  |  |  |
1 | 6 | -2.112319339 | 2 | 8 | 7.126409258
3 | 10 | -21.30053312 | 3 | 12 | 24.42390886
5 | 14 | -10.89025935 | 6 | 16 | 1.752793693
Li |  |  |  |  |
1 | 6 | -2.185099402 | 2 | 8 | 1588.743093
3 | 9 | -13096.66094 | 4 | 10 | 44937.24250
5 | 11 | -85547.67477 | 6 | 12 | 100055.5130
7 | 13 | -74450.14624 | 8 | 14 | 35150.12854
9 | 15 | -10264.39581 | 10 | 16 | 1744.182010
11 | 17 | 25.87885791 | 12 | 18 | -237.6273332
13 | 19 | 114.6392978 | 14 | 20 | -18.63705649
Cr |  |  |  |  |
1 | 6 | -15.20122639 | 2 | 8 | 13471.86476
3 | 9 | -124591.4050 | 4 | 10 | 464698.3696
5 | 11 | -888076.6787 | 6 | 12 | 854878.9650
7 | 13 | -190568.3900 | 8 | 14 | -441981.1016
9 | 15 | 487340.5171 | 10 | 16 | -209652.8384
11 | 17 | 34494.89857 | 12 | 18 | 0.000016589

For Cr2 we took the potential curve from experimental data of Casey and
Leopold, who obtained the potential energy curve $V(r)$ from vibrational data
through the RKR method Casey and Leopold (1993). This potential curve only
describes the medium range of the potential energy curve. We therefore
attenuated the long range by matching the last point $R_{\text{max}}=$3.35 Å
to a $-C_{6}/r^{-6}$ dispersion curve. Finally, the points are used to fit an
inverse power potential (extended Lennard-Jones) to the potential energy curve
fixing the Van der Waals coefficient to $C_{6}=$800 a.u. according to Roos and
co-workers Pavlović  _et al._ (2004). Because of the peculiar shape of the Cr2
potential energy curve the fit was rather difficult to achieve, but is
accurate enough ($R^{2}$=0.9984) for the discussion of the bcc instability.
The potential energy curve for Cr2 was then converted to dimensionless units.
For the ELJ form we obtain $E^{*}=$24.0 and 23.3 for the fcc and bcc
structures respectively. These values are unusually large, but perhaps not
surprising given the broad potential energy curve of Cr2. In fact, using the
original potential energy curve we obtain a nearest neighbor distance for bcc
chromium of $R_{\rm min}=1.479$ Å, just above the hard sphere radius of the
diatomic potential energy curve with $\sigma=1.467$ Å, and a cohesive energy
$E_{coh}=33.6$ eV. This is in stark disagreement with the experimental values
of $R_{\rm min}=2.52$ Å and $E_{\rm coh}=4.1$ eV Kittel  _et al._ (1996). It
clearly demonstrates that the direct use of potential curves from the free
unconfined diatomic is not useful to describe the solid state of metals as the
many-body expansion is not converging fast and smoothly.

We briefly discuss lithium. For Li2 we used the Rydberg-Klein-Rees (RKR)
potential curve of Barakat et al. Barakat  _et al._ (1986) and fixed the Van
der Waals coefficient $C_{6}=$1408 a.u. Gould and Bučko (2016). For the fit to
an extended LJ potential we obtained with an $R^{2}$ value of 0.99997, but
only by including terms up to $1/r^{20}$. However, the situation here is even
worse compared to chromium as the Li2 potential energy curve is so broad in
both the repulsive and attractive region that crystal optimizations entered
the repulsive wall well below the hard-sphere radius of $\sigma=1.822$ Å,
where our extended LJ potential is not accurate anymore. In general, a fit to
an extended LJ form works reasonably well for the whole distance region if it
deviates not too much from an ideal ($a,b$)-LJ potential, which is certainly
not the case for Li2. In fact, if we optimize the exponents $a,b$ for the LJ
potential we get $a\approx b<3$ left of the singularity at $b=3$ and therefore
an unphysical result. Using the far more accurate extended Morse potential by
LeRoy and co-workers Roy  _et al._ (2011), which correctly describes the
repulsive region, we obtain from crystal optimizations Schwerdtfeger (2020) a
nearest neighbor distance of $R_{\rm min}=0.21$ Å and a cohesive energy of
$E_{\rm coh}=9.2\times 10^{3}$ eV for bcc lithium. This can be best described
as a collapse of the crystal to small internuclear distances with large
overbinding, and clearly demonstrates that many-body forces in a confined bulk
system cannot be neglected.

### Proof of inequality eq.(37)

A function $g(x)$ is said to be logarithmically convex on an interval if
$g(x)>0$ and $\log g(x)$ is convex on the interval. It can be shown that the
sum of logarithmically convex functions is logarithmically convex, e.g., see
(Roberts and Varberg, 1973, p. 19). It follows that the lattice sum $L(x,A)$
is a logarithmically convex function of $x$ because it is a sum of terms of
the form $n^{-x}$, each of which is logarithmically convex.

Now suppose that $f(x)$ is a convex function, and $x_{1}$, $x_{2}>0$. By
applying the definition of convexity to the interval $[0,x_{1}+x_{2}]$ we have

$f(x_{1})\leq\frac{x_{2}}{x_{1}+x_{2}}f(0)+\frac{x_{1}}{x_{1}+x_{2}}f(x_{1}+x_{2}),$

while interchanging $x_{1}$ and $x_{2}$ gives

$f(x_{2})\leq\frac{x_{1}}{x_{1}+x_{2}}f(0)+\frac{x_{2}}{x_{1}+x_{2}}f(x_{1}+x_{2}).$

Adding the inequalities gives

$f(x_{1})+f(x_{2})\leq f(0)+f(x_{1}+x_{2}).$ (40)

Incidentally, it can be shown from this using mathematical induction that

$f(x_{1})+f(x_{2})+\cdots+f(x_{n})\leq(n-1)f(0)+f(x_{1}+x_{2}+\cdots+x_{n}),$

a result known as Petrović’s inequality, e.g., see (Mitrinović and Vasić,
1970, p. 22), Petrović (1932). We shall only require the case $n=2$ as given
by (40).

Suppose $a>b$, $h>0$, and $g(x)$ is a convex function for $x\geq b$. Let
$f(x)=g(x+b)$ and take $x_{1}=h$ and $x_{2}=a-b$. Then Petrović’s inequality
(40) gives

$f(h)+f(a-b)\leq f(0)+f(a-b+h).$

This implies

$g(b+h)+g(a)\leq g(a+h)+g(b)$

which is equivalent to

$g(b+h)-g(b)\leq g(a+h)-g(a).$

It follows that if $G(x)$ is logarithmically convex, then

$\log G(b+h)-\log G(b)\leq\log G(a+h)-\log G(a).$

This can be rearranged to give

$\frac{G(b+h)}{G(b)}\leq\frac{G(a+h)}{G(a)},$

which is exactly the inequality we seek for the lattice sums.

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# Thermal and Quantum Phase Transitions of the $\phi^{4}$ Model

István Gábor Márián  HUN-REN Atomki, P.O.Box 51, H-4001 Debrecen, Hungary
University of Debrecen, Institute of Physics, P.O.Box 105, H-4010 Debrecen,
Hungary  Andrea Trombettoni  Department of Physics, University of Trieste,
Strada Costiera 11, I-34151 Trieste, Italy  CNR-IOM DEMOCRITOS Simulation
Center, Via Bonomea 265, I-34136 Trieste, Italy  István Nándori  University of
Debrecen, Institute of Physics, P.O.Box 105, H-4010 Debrecen, Hungary  HUN-REN
Atomki, P.O.Box 51, H-4001 Debrecen, Hungary

###### Abstract

The critical behaviour of the classical (thermal) phase transition (CPT) of
the Ising model is identical to the quantum phase transition (QPT) of the
$\phi^{4}$ Quantum Field Theory (QFT). Our goal in this work is to map out the
QPT-CPT diagram of the $\phi^{4}$ QFT, and study its relation to the well-
known QPT-CPT phase diagram of the Ising model. To do this, we propose a
modification of the usual finite-temperature Renormalization Group (RG)
approach by relating the temperature parameter to the running RG scale,
$T\equiv k_{T}=\tau k$ where $k_{T}$ is the running cutoff for thermal, and
$k$ is for the quantum fluctuations. In this case, the temperature is related
to the dimensionless quantity $\tau$. We apply this new thermal RG approach
for the $\phi^{4}$ model in lower dimensions and construct its QPT-CPT phase
diagram. We formulate requirements for the QPT-CPT phase diagram of the
$\phi^{4}$ theory based on known properties of the QPT-CPT phase diagram of
the Ising model. Finally, we use these requirements to check the viability of
the thermal RG method proposed in this work.

##  I Introduction

A quantum phase transition (QPT) occurs at the quantum critical point at zero
temperature where thermal fluctuations are absent. The QPT is understood as a
transition between different quantum phases and driven by quantum fluctuations
arising from Heisenberg’s uncertainty principle. Contrary to a classical phase
transition (CTP) which is also called as thermal phase transition, the QPT is
accessed at absolute zero temperature by varying a physical parameter such as
magnetic field or pressure. In this work, we refer to this as the ’quantum’
parameter. At finite but very low temperatures, classical and quantum
fluctuations compete with each other. In the quantum critical region near the
quantum critical point, quantum fluctuations dominate the system’s behavior
and one can study properties of the QPT. Thus, a so called QPT-CPT phase
diagram can be drawn where the ordered and the disordered phases are separated
by each other by a critical line which connects the critical temperature and
the quantum critical point. Around the classical phase transition, in the
classical critical region, the system is governed by thermal fluctuations and
this region becomes narrower with decreasing temperatures and converges
towards the quantum critical point monotonically.

The Ising model in $d=2$ or $d=3$ dimensions represents an excellent
playground to test and to understand the interplay between quantum and
classical phase transitions. The quantum Ising model tranverse_ising  can be
realised by adding a transverse, external magnetic field to the usual
Hamiltonian of the classical Ising model. This is the so called transverse
field Ising model where the QPT is realised by varying the magnitude of the
external magnetic field tranverse_ising . The QPT-CPT phase diagram of the
Ising model is well-known. The classical Ising model with vanishing external
magnetic field undergoes a second order classical (thermal) phase transition
at a finite critical temperature $T_{c}$. The quantum Ising model at zero
temperature has a QPT at a critical magnetic field $h_{c}$.

It is also well-known that the classical Ising model (without any transverse
magnetic field) can be mapped onto the $\phi^{4}$ Quantum Field Theory (QFT)
which has a QPT where its quantum critical behaviour is found to be identical
to the classical critical behaviour of the Ising model. As a consequence, the
two models belong to the same universality class and their critical exponents
are identical. It is a natural question to ask: what is the QPT-CPT diagram of
the $\phi^{4}$ QFT and how it is related to that of the Ising model.

In order to map out the QPT-CPT diagram of the $\phi^{4}$ field theory one has
to perform its thermal RG analysis where both quantum and thermal fluctuations
are taken into account. This can be done by e.g., using the finite temperature
extension of the zero-temperature Functional Renormalization Group (FRG)
method. The modern formulation of the FRG method is the Wetterich equation
eea_rg  which is written in Euclidean spacetime in the framework of the zero-
temperature QFT. Its generalisation to finite-temperatures requires the
inclusion of the inverse temperature parameter $\beta=1/T$ as the upper bound
for the imaginary time integral, see e.g., Ref. thermal_rg_phi4 ;
thermal_rg_phi4_pressure ; thermal_rg_phi4_volume .

As a final step, one has to relate the temperature parameter to a momentum
scale by using a unit in which $c=\hbar=k_{B}=1$. In the usual perturbative
approach the standard choice is $\mu=2\pi T$ where $\mu$ is the perturbative
renormalization group (RG) scale which is related to the non-perturbative one,
i.e., $\mu\sim k$. However, it is known that the k-dependence of the effective
action of the Wetterich FRG equation is only introduced artificially to
implement the Wilsonian integration wilson  of fluctuating modes and the
quantised theory is obtained in the physical limit $k\to 0$. Thus, it makes no
sense to study the system at finite $k$, i.e., one has to take the limit $k\to
0$. So, in Refs. thermal_rg_phi4 ; thermal_rg_phi4_pressure ;
thermal_rg_phi4_volume  the running couplings are defined at an arbitrary but
fixed intermediate momentum scale $k_{\star}=2\pi T=2\pi\tau\Lambda$ where
$\Lambda$ is the UV initial value of the running momentum cutoff $k$. So, the
temperature parameter is linked to the UV cutoff, i.e., $T=\tau\Lambda$. Once
the couplings are defined, the zero-temperature FRG equation is integrated
from $k_{\star}$ up to $\Lambda$ and then starting from these bare parameters,
the FRG equation is solved down from $k=\Lambda$ to $k=0$ but this time with
the temperature $T$ turned on and the physical quantities are obtained at
$k=0$. However, this choice has a drawback, the explicit $k$-dependence cannot
be removed from the dimensionless RG flow equations. For example, the
dimensionless thermal FRG equation (3.9) of thermal_rg_phi4  which is given in
the so called local potential approximation, has an explicit k-dependence and
it blows up in the IR limit, i.e., for $k\to 0$ where the attractive fixed
points are always given, so it makes no room for non-trivial fixed points such
as the Wilson-Fisher one.

In this work we make an attempt to solve the above problems by suggesting the
identification $T\equiv k_{T}=\tau k$ where $k_{T}$ is the running cutoff for
thermal, and $k$ is for the quantum fluctuations. This choice can be supported
by the following argument: the Wilsonian approach requires the rescaling of
the upper bound of the imaginary time integral in every blocking step, so, it
has to be linked to the running momentum cutoff $k$ but in the FRG method one
has to take the limit $k\to 0$, therefore, the upper bound cannot be
considered as the (inverse) temperature but can be used as a running momentum
cutoff $T\equiv k_{T}$. Thus, the temperature is related to the dimensionless
quantity $\tau$ and not to $k_{T}$. In the Wilsonian approach fluctuations are
taken into account by the successive elimination (integration) of degrees of
freedom above these running cutoffs which are chosen to be different in order
to make difference between thermal and quantum fluctuations. In other words,
for $k_{T}=k$ (i.e., for $\tau=1$) one cannot distinguish between thermal and
quantum fluctuations.

Indeed, thermal field theory is a very well-developed framework for finite
temperature applications kapusta  and it is employed in many-body physics
negele ; kapusta  which relates a quantum system in $d-1$ (spatial) dimensions
at $T=0$ and the corresponding classic system in $d$ dimensions at finite
temperature sachdev . It generates the discretisation of the imaginary time-
integral in momentum space, i.e., summation on Matsubara frequencies negele ;
kapusta . At nonzero temperatures, classical fluctuations with an energy scale
of $k_{B}T$ compete with the quantum fluctuations of energy scale
$\hbar\omega$ where $\omega$ is the frequency of quantum oscillations. By
using natural units, one finds that $\omega$ must be compared to $T$ and in
the RG method one has to relate the frequency $\omega$ to the running cutoff
$k$ which is used to integrate quantum fluctuations systematically, so one
finds $\omega=k$. Thus, one can introduce a dimensionless quantity,
$\tau=T/k=T/\omega$ which is used to compare the strength of quantum and
thermal fluctuations. If $\tau<1$ then quantum fluctuations dominate, if
$\tau>1$ then thermal fluctuations dominate.

Our goal in this work is to explore the possibility of a thermal RG approach
in the framework of the FRG method using the identification $T\equiv
k_{T}=\tau k$ where the temperature is related to the dimensionless quantity
$\tau$. We apply this new thermal RG approach for the $\phi^{4}$ model in
lower dimensions. It is important to note that in Ref. thermal_rg_cosmo  we
have already applied this new thermal RG method for the 4-dimensional
$\phi^{4}$ theory and studied its consequences on some key issues of
Inflationary Cosmology. Our goal here is to map out the QPT-CPT phase diagram
of the $\phi^{4}$ theory in lower dimensions and compare it to the QPT-CPT
diagram of the Ising model. We discuss known properties of the thermal and
quantum phase transitions of the Ising model and based on these, we formulate
requirements for the QPT-CPT phase diagram of the $\phi^{4}$ theory. We
construct the QPT-CPT phase diagram of the $\phi^{4}$ theory (in lower
dimensions) by the thermal RG approach suggested in this work and compare it
to the formulated requirements in order to check the viability of the new
method.

##  II Modified thermal RG equation

The Wetterich FRG equation eea_rg  at zero-temperature is formulated in
Euclidean spacetime and reads as

$\displaystyle
k\partial_{k}\Gamma_{k}[\phi]=\frac{1}{2}\int\frac{d^{d}p}{(2\pi)^{d}}\,\frac{k\,\partial_{k}R_{k}(p)}{R_{k}(p)+\Gamma^{(2)}_{k}[\phi]},$
(1)

where $\Gamma_{k}[\phi]$ is the running effective action with its Hessian
$\Gamma^{(2)}_{k}[\phi]$ and $k$ is the running momentum cutoff (i.e., the RG
scale) and $R_{k}(p)$ is an appropriately chosen regulator function. In order
to handle the FRG equation one has to use approximations. In the so called
gradient expansion, the action is expanded in terms of the derivatives of the
field. The lowest order of the gradient expansion is the so called, local
potential approximation (LPA). In LPA when the couplings of the scaling
potential $V_{k}(\phi)$ carry RG scaling only, the Wetterich FRG equation (1)
is written as

$\displaystyle
k\partial_{k}{V_{k}(\phi)}=\frac{1}{2}\int_{-\infty}^{\infty}\frac{d^{d}p}{(2\pi)^{d}}\,\frac{k\partial_{k}R_{k}}{R_{k}+p^{2}+V^{\prime\prime}_{k}(\phi)}\,,$
(2)

with $V^{\prime\prime}_{k}=\partial^{2}_{\phi}V_{k}$.

The extension of the zero-temperature FRG method for finite temperatures
requires the modification of the imaginary time $\tilde{t}$ integral $\int
d^{d}x\to\int_{0}^{\beta}d\tilde{t}\int d^{d-1}x$ where $\beta=1/T$ and $T$ is
the temperature parameter. As a next step, the momentum integral with respect
to the imaginary time is modified $\int d^{d}p\to T\sum_{\omega_{n}}\int
d^{d-1}p$ by using Matsubara frequencies where the summation is performed over
these discrete frequencies; for bosonic degrees of freedom $\omega_{n}=2\pi
nT$.

###  II.1 Thermal RG with the Litim regulator

In order to proceed further one has to assume that the regulator function is
independent of the Matsubara frequency, but othevise the same as for the zero-
temperature case. This is the case of the ”frequency independent regulator”.
In addition, it is also important to note that once approximations are used,
the concrete form of the FRG equation start to depend on the particular choice
of the regulator function. There are two types of regulator functions where
the momentum integrals in (2) and its finite-temperature counterpart can be
performed analytically in LPA. The first one is the Litim regulator Litim2000
, which has the following form at zero-temperature,

$R_{k}(p)=(k^{2}-p^{2})\theta(k^{2}-p^{2})\,,$ (3)

where $\theta(y)$ is the Heaviside step function. The frequency-independent
form of the Litim regulator results in the following thermal RG equation in
LPA thermal_rg_phi4 ,

$k\partial_{k}V_{k}(\phi)=\frac{2\alpha_{d-1}}{d-1}\,k^{d-1}T\sum_{n=-\infty}^{\infty}\frac{k^{2}}{k^{2}+\omega_{n}^{2}+\partial^{2}_{\phi}V_{k}(\phi)}\,,$
(4)

with $\omega=2\pi nT$ and $\alpha_{d}=\Omega_{d}/(2(2\pi)^{d})$, where
$\Omega_{d}=2\pi^{d/2}/\Gamma(d/2)$ is the d-dimensional solid angle. The
summation can be performed thermal_rg_phi4  which results in an RG equation
(identical to Eq.(3.9) of thermal_rg_phi4  for $N=1$)

$\displaystyle
k\partial_{k}V_{k}(\phi)=\frac{2\alpha_{d-1}}{d-1}\,k^{d+1}\frac{2n(\omega_{k})+1}{2\omega_{k}}\,,$

where $n(\omega_{k})=(\exp{(\omega_{k}/T)}-1)^{-1}$ is the bosonic
distribution function with
$\omega_{k}=\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}$ which appears in the
so called ”thermal contribution” which vanishes in the limit $T\to 0$ and the
remaining term is the so called ”vacuum contribution”. This equation can be
rewritten as,

$\displaystyle
k\partial_{k}V_{k}(\phi)=\frac{\frac{2\alpha_{d-1}}{d-1}\,k^{d+1}}{2\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}\frac{\exp{\left(\frac{\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}{T}\right)}+1}{\exp{\left(\frac{\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}{T}\right)}-1}$

and can be further simplified by using the identity
$\coth(x/2)=\frac{\exp(x)+1}{\exp(x)-1}$ which results in,

$\displaystyle
k\partial_{k}V_{k}(\phi)=\frac{2\alpha_{d-1}}{d-1}\,k^{d+1}\frac{\coth{\left(\frac{\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}{2T}\right)}}{2\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}\,.$
(5)

Let us note, the above thermal RG equation is well-known. For example, in Ref.
thermal_rg_phi4  the authors performed a detailed analysis of the O(N) scalar
field theory by the thermal RG equation (5) which was also discussed in
thermal_rg_phi4_pressure  with the inclusion of pressure and in
thermal_rg_phi4_volume  by taking into account volume fluctuations too. Finite
temperature and volume effects were considered in finite_T_V_1 ; finite_T_V_2
; finite_T_V_3 ; finite_T_V_5  and also discussed in finite_T_V_6 ;
finite_T_V_7 ; finite_T_V_9 . In these works the temperature is linked to the
UV cutoff $\Lambda$, i.e., $T=\tau\Lambda$.

The schematic thermal RG flow with $T=\tau\Lambda$ was shown in Fig. 6 of
finite_T_V_6  and Fig. 4 of finite_T_V_7  (which are identical figures) and
given here on Fig. 1.

Figure 1:  Schematic thermal RG flow with $T=\tau\Lambda$. The figure is taken
from finite_T_V_6 ; finite_T_V_7 .

It is clear from Fig. 1 that the zero-temperature FRG equation is integrated
from $k=0$ up to $\Lambda$ while the couplings are defined at an arbitrary but
fixed intermediate momentum scale $k_{\star}=2\pi T=2\pi\tau\Lambda$. Then
starting from these bare parameters, the thermal ($T\neq 0$) FRG equation is
solved from $k=\Lambda$ to $k=0$. It is also clear, that RG flows at zero and
at finite temperatures do not differ from each other for $k>k_{\star}$ which
is evident from (5) since in the limit $x\to\infty$ one finds $\coth(x)\to 1$.
However, for $k<k_{\star}$ they start to deviate from each other, so, the
physical quantities can be obtained at $k=0$ of the thermal RG flow.
Nevertheless, this procedure has a drawback, the explicit $k$-dependence
cannot be removed from the dimensionless RG flow equations.

In this work we use the identification $T\equiv k_{T}=\tau k$ where ”$\tau$”
plays the role of the (dimensionless) temperature which distinguishes between
thermal quantum fluctuations. We suggest a modified thermal RG equation,

$k\partial_{k}V_{k}(\phi)=\frac{2\alpha_{d-1}}{d-1}\,k^{d+1}\frac{\coth{\left(\frac{\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}{2\tau
k}\right)}}{2\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}\,,$ (6)

which was given in Ref. thermal_rg_cosmo , too. The advantage of this modified
thermal RG equation is the absence of $k$-dependence in the dimensionless RG
flow equations.

###  II.2 Thermal RG with the sharp-cutoff regulator

As we argued, in order to perform the momentum integrals in (2) and its
finite-temperature counterpart analytically one has to choose a special type
of regulator. In the previous subsection we discussed the case of the Litim
regulator Litim2000 . In this subsection we choose the sharp-cutoff which has
the following form at zero temperature,

$R_{k}(p)=p^{2}\left(\frac{1}{\theta(p^{2}-k^{2})}-1\right)$ (7)

and it is considered as the the simplest regulator function. The frequency-
independent form of the sharp-cutoff regulator results in the following
thermal RG equation in LPA,

$\displaystyle k\partial_{k}V_{k}(\phi)$ $\displaystyle=$
$\displaystyle-\alpha_{d-1}\,k^{d-1}T$
$\displaystyle\sum_{n=-\infty}^{\infty}\ln\left(\frac{k^{2}+\omega_{n}^{2}+\partial^{2}_{\phi}V_{k}(\phi)}{k^{2}+\omega_{n}^{2}}\right)\,,$

where the denominator inside the logarithm can be chosen with some freedom but
it must have the dimension of momentum square and it has to be field-
independent. For example, it could be $k^{2}$ or $T^{2}$. Here we take the
choice when it becomes identical to the nominator in the absence of fields. In
this case, the summation can be performed which results in a thermal RG
equation

$\displaystyle k\partial_{k}V_{k}(\phi)$ $\displaystyle=$
$\displaystyle-\alpha_{d-1}\,k^{d-1}2T$
$\displaystyle\ln\left[\rm{csch}\left(\frac{k}{2T}\right)\sinh\left(\frac{\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}{2T}\right)\right]\,,$

and by using the identification $T\equiv k_{T}=\tau k$ the following modified
thermal RG equation can be obtained,

$\displaystyle k\partial_{k}V_{k}(\phi)$ $\displaystyle=$
$\displaystyle-\alpha_{d-1}\,k^{d}2\tau$
$\displaystyle\ln\left[\rm{csch}\left(\frac{1}{2\tau}\right)\sinh\left(\frac{\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}{2\tau
k}\right)\right]\,.$

##  III The zero-temperature limit

Before we start to apply the modified thermal RG equations (6) and (II.2) let
us discuss in this section an important issue, the zero-temperature limit. As
we argued, the finite-temperature approach requires the modification of the
imaginary time integral by introducing an upper bound which is the inverse
temperature parameter, $\beta=1/T$. This results in the replacement of the
continuous (imaginary time) integral by a sum over discrete Matsubara
frequencies in the momentum space in RG equations (1) and (2). Here, the zero-
temperature limit, i.e., $\beta\to\infty$ must recover the RG equations (1)
and (2).

However, if one would like to perform the momentum integrals (and the
summation) the use of the regulator function is unavoidable. The finite-
temperature formalism requires a frequency independent regulator which does
not regulate the Matsubara sum. This is not true for the zero-temperature case
where momentum integrals (including the imaginary time direction) are
performed by the regulator function (i.e., with a frequency-dependent
regulator). Thus, the zero-temperature limit of RG equations (5) and (II.2)
may differ from their counterparts at zero temperature.

Indeed, by choosing the frequency independent Litim regulator and taking the
limit of zero-temperature, the thermal RG equation (5) reduces to,

${\rm T\to 0:}\hskip
14.22636ptk\partial_{k}V_{k}(\phi)=\frac{2\alpha_{d-1}}{d-1}\frac{k^{d+1}}{2\sqrt{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}}\,,$
(11)

which is not identical to its zero-temperature counterpart

${\rm T=0:}\hskip
14.22636ptk\partial_{k}V_{k}(\phi)=\frac{2\alpha_{d}}{d}\,\frac{k^{d+2}}{k^{2}+\partial^{2}_{\phi}V_{k}(\phi)}\,,$
(12)

where the frequency-dependent Litim regulator is used to perform the momentum
integrals of (2). The situation is similar for the sharp cutoff case, where
the zero-temperature limit of the thermal RG equation (II.2) is not identical
to its zero-temperature counterpart, i.e., the Wegner-Houghton equation wh_rg
at LPA.

The essence of this problem is whether the frequency dependence is taken
separately or not which is leading to cylindrically (frequency-dependent
regulator) or spherically (frequency independent regulator) symmetric geometry
cylindrical_symm_1 ; cylindrical_symm_2 . The cylinder can have finite or
infinite size. An infinite size cylinder with the Litim regulator results in
an FRG equation (see Eq. (7) of cylindrical_symm_2 ) identical to (11).

It is clear that equations (11) and (12) have different forms, however, they
have the same singularity structure ($k^{2}+\partial^{2}_{\phi}V_{k}=0$) which
results in a convex dimensionful potential in the IR limit ($k\to 0$) in both
cases. The position and the critical behaviour of the Wilson-Fisher (WF) fixed
point obtained by equations (11) and (12) may differ from each other, i.e.,
the zero-temperature limit requires attention. Actually, fixed points are not
connected to physical observables, but the critical exponents which are used
to characterise the critical behaviour around the WF fixed point can be
measured directly, thus, they can be used to study the zero-temperature limit
problem.

One of our goals in this work is to use measurable quantities, i.e., critical
exponents to study the zero-temperature limit of the thermal RG approach by
using the modified RG equations (6) and (II.2)

##  IV Thermal RG study of the $\phi^{4}$ model in lower dimensions

As a first step we apply the thermal RG equation (6) to the $\phi^{4}$ (more
precisely, to the $\phi^{2n}$) model in low dimensions. For the sake of
simplicity we consider the following scalar potential,

$V_{k}(\phi)=\sum_{n=1}^{\rm NCUT}\frac{g_{2n,k}}{(2n)!}\phi^{2n}\hskip
0.0pt\to\hskip 0.0pt\tilde{V}_{k}(\tilde{\phi})=\sum_{n=1}^{\rm
NCUT}\frac{\tilde{g}_{2n,k}}{(2n)!}{\tilde{\phi}}^{2n}$ (13)

where the RG scale-dependence is encoded in the couplings and we introduced
dimensionless quantities denoted by the tilde superscript. The dimensionful RG
flow equations for two couplings, i.e., for NCUT=2 have the following forms,

$\displaystyle k\partial_{k}$ $\displaystyle
g_{2,k}=\frac{2\alpha_{d-1}}{d-1}\,k^{d+1}\left[-\frac{g_{4,k}\coth\left(\frac{\sqrt{k^{2}+g_{2,k}}}{2\tau
k}\right)}{4\left(k^{2}+g_{2,k}\right)^{3/2}}\right.$
$\displaystyle\left.-\frac{g_{4,k}\;\text{csch}^{2}\left(\frac{\sqrt{k^{2}+g_{2,k}}}{2\tau
k}\right)}{8\tau k\left(k^{2}+g_{2,k}\right)}\right],$ (14) $\displaystyle
k\partial_{k}$ $\displaystyle
g_{4,k}=\frac{2\alpha_{d-1}}{d-1}\,k^{d+1}\left[\frac{9g_{4,k}^{2}\coth\left(\frac{\sqrt{k^{2}+g_{2,k}}}{2\tau
k}\right)}{8(k^{2}+g_{2,k})^{5/2}}\right.$
$\displaystyle\left.+\frac{9g_{4,k}^{2}\text{csch}^{2}\left(\frac{\sqrt{k^{2}+g_{2,k}}}{2\tau
k}\right)}{16\tau k(k^{2}+g_{2,k})^{2}}\right.$
$\displaystyle\left.+\frac{3g_{4,k}^{2}\coth\left(\frac{\sqrt{k^{2}+g_{2,k}}}{2\tau
k}\right)\text{csch}^{2}\left(\frac{\sqrt{k^{2}+g_{2,k}}}{2\tau
k}\right)}{16\tau^{2}k^{2}(k^{2}+g_{2,k})^{3/2}}\right],$ (15)

where $g_{4,k}$ and $g_{2,k}$ are dimensionful couplings. One can switch from
dimensionful to dimensionless couplings by introducing
$g_{2,k}=\tilde{g}_{2,k}k^{2}$ and $g_{4,k}=\tilde{g}_{4,k}k^{(4-d)}$. The RG
flow equations given for dimensionless couplings $\tilde{g}_{2,k}$ and
$\tilde{g}_{4,k}$ have no explicit $k$-dependence. Thus, one can look for non-
trivial, WF fixed point solutions. Indeed, Fig. 2 shows the thermal RG flow
diagram of the $\phi^{4}$ model in $d=3$ dimensions.

Figure 2:  Thermal RG flow diagram of the $\phi^{4}$ model in $d=3$ dimensions
based on flow equations (IV) and (IV). Circles show how the position of the
Wilson-Fisher fixed point changes with $\tau$.

Empty circles show how the position of the WF fixed point changes with $\tau$.
The RG trajectory which runs from the Gaussian ($\tilde{g}_{2}=0$ and
$\tilde{g}_{4}=0$) fixed point to the WF one separates the phases of the
model. If one increases the value of $\tau$, the slope of the critical
trajectory and consequently the broken phase is decreased. For $\tau\to\infty$
it is not possible to find any starting point in the vicinity of the Gaussian
fixed point from which an RG trajectory can run into the broken phase. Since
$\tau$ measures how thermal fluctuations are important compared to quantum
fluctuations, one can say that by changing $\tau$ a thermal phase transition
occurs. For an arbitrary but fixed starting point taken from the vicinity of
the Gaussian fixed point, see the black cross on Fig. 3 one can always
determine a critical value $\tau_{c}$: if $\tau<\tau_{c}$ the RG trajectory
from that starting point runs into the broken (low-temperature) phase and for
$\tau>\tau_{c}$ the RG trajectory ends up in the symmetric (high-temperature)
phase.

Figure 3:  Thermal RG flow diagram of the $\phi^{4}$ model in $d=3$ dimensions
with $\tau=1$ based on the RG flow equations (IV) and (IV). Black cross
denotes an initial condition which lies on the separatrix. For $\tau<1$ the RG
trajectory from that starting point runs into the broken (low-temperature)
phase and for $\tau>1$ the RG trajectory ends up in the symmetric (high-
temperature) phase.

Let us consider the RG flow diagram shown in Fig. 3 with respect to the change
in the ’quantum’ parameter which results in a QPT. The ’quantum’ parameter is
the slope of RG trajectories taken at the vicinity of the Gaussian fixed
point. For a fixed temperature, i.e., for fixed $\tau$, one can determine a
critical value which is the slope of the separatrix, see dashed line of Fig.
3. If the slope of an RG trajectory is smaller or larger then this critical
value, the corresponding trajectory runs into the broken or the symmetric
phase respectively which signals the QPT. A good approximation for the
critical value of the ’quantum’ parameter is the ratio
$(\tilde{g}_{4}/|\tilde{g}_{2}|)_{\rm{WF}}$ where $\tilde{g}_{4}$ and
$\tilde{g}_{2}$ are the coordinates of the WF fixed point. Thus, the critical
line on the QPT-CPT diagram of the $\phi^{4}$ model can be given in terms of
$\tau_{c}$ and $(\tilde{g}_{4}/|\tilde{g}_{2}|)_{\rm{WF}}$ which are related
to each other.

##  V QPT-CPT diagram of the $\phi^{4}$ model

As a next step we discuss some key issues regarding the comparison of the QPT-
CPT diagrams of the Ising model (which is well-known) and the $\phi^{4}$
theory (which we would like to map out here). The schematic pictures of these
QPT-CPT diagrams are shown in Fig. 4.

Figure 4:  Schematic QPT-CPT diagram of the Ising model (to the top) and the
$\phi^{4}$ theory (to the bottom) which are ”dual to each other”, i.e.,
$(T/J)$ corresponds to ($\tilde{g}_{4}/\tilde{g}_{2}$) while $\tau$ is related
to $(h/J)$. Thus, the critical behaviour on blue lines known to be identical
and expected to be the same along the green critical lines but has a
discontinuity at their endpoints.

It is known that the classical Ising model is equivalent to the zero-
temperature $\phi^{4}$ QFT. Thus, the vertical axis of the he QPT-CPT diagram
and the critical temperature of the Ising model can be mapped onto the
horizontal axis of the he QPT-CPT diagram and the critical ’quantum’ parameter
of the (zero-temperature) $\phi^{4}$ QFT, see the blue lines of Fig. 4. In
addition, the thermal critical behavior of the classical Ising model must be
identical to the quantum critical behavior of the $\phi^{4}$ QFT.

It is clear from Fig. 4, that the ’quantum’ parameter of the Ising model is
$h/J$ which is the strength of the transverse-field in terms of $J$. The
quantum critical behaviour of the $\phi^{4}$ QFT is given in terms of the
irrelevant coupling around the WF saddle point, however, this irrelevant
coupling is related to the ratio $(\tilde{g}_{4}/|\tilde{g}_{2}|)_{\rm{WF}}$
and in the previous section we argued, that this ratio can serve as a good
’quantum’ parameter for $\phi^{4}$ QFT. Thus, if the temperature is zero, both
the quantum Ising model and the $\phi^{4}$ QFT are non-trivial, they undergo a
QPT with a finite $h_{c}$ and finite ratio
$(\tilde{g}_{4}/|\tilde{g}_{2}|)_{\rm{WF}}$.

For vanishing quantum parameter, i.e., $h/J=0$, the Ising model becomes the
well-known classical one, which is non-trivial, it has a classical (thermal)
phase transition with a finite transition temperature $T_{c}<\infty$. On
contrary, for a vanishing ’quantum’ parameter, i.e., for
$(\tilde{g}_{4}/|\tilde{g}_{2}|)_{\rm{WF}}=0$, the $\phi^{4}$ model becomes a
trivial massive free field theory (with thermal fluctuations), which has no
phase transition, so there is no room for any finite transition temperature.
On the one hand, the critical temperature of the $\phi^{4}$ model must be zero
if $(\tilde{g}_{4}/|\tilde{g}_{2}|)_{\rm{WF}}=0$. On the other hand, the
critical line must be a strictly monotonic function, so one expects
$T_{c}\to\infty$ for the $\phi^{4}$ in the limit
$(\tilde{g}_{4}/|\tilde{g}_{2}|)_{\rm{WF}}\to 0$. Therefore, one expects a
discontinuity in the classical limit of the $\phi^{4}$ theory; this is why we
use an empty circle at $\tau_{c}=\infty$ on the lower panel of Fig. 4 and in
its inset. In addition, one can conclude that the Ising model has a thermal
(classical) and a quantum critical behavior, too but the $\phi^{4}$ model has
a quantum critical behavior, only.

Let us pay the attention of the reader that the terminology ’classical’ has to
be used with great care. In this work when we use ’classical Ising model’ we
refer to the Ising model with thermal fluctuations, however, when we use
’classical’ $\phi^{4}$ theory it is understood as a model with no quantum and
thermal fluctuations. The change in the sign of the mass parameter of the
classical $\phi^{4}$ theory results in two different (single and double well)
form for the potential but actually it is not considered as a phase
transition. As we argued, the ’quantum’ parameter of the $\phi^{4}$ QFT is the
irrelevant direction around the WF fixed point which is a combination of the
quartic and quadratic couplings. Thus, if one takes the limit of vanishing
’quantum’ parameter (in some sense the classical limit) one arrives at a free
field theory (with no quartic term) which is not equivalent to the ’classical’
$\phi^{4}$ model.

It is also known that the critical exponents of the finite temperature quantum
Ising model are equal to the critical exponents of the classical Ising model,
see the inset of the upper panel of Fig. 4. Since the Ising model and the
$\phi^{4}$ theory are ”dual to each other”, i.e., $(T/J)$ corresponds to
($\tilde{g}_{4}/\tilde{g}_{2}$) while $\tau$ is related to $(h/J)$, the
critical behaviour is expected to be the same along the green critical lines
of Fig. 4 and the critical exponents of the finite temperature $\phi^{4}$ QFT
must be equal to their zero-temperature values, see the inset on the lower
panel of Fig. 4. Thus, for finite temperatures and finite ’quantum’
parameters, the Ising and the $\phi^{4}$ models are expected to belong to the
same universality class.

However, it is well-known that the critical behavior of the Ising model has a
discontinuity in the limit $T\to 0$, see the empty circle at $T=0$ (i.e., at
$h_{c}$) on the upper panel of Fig. 4. So, critical lines (green lines on the
upper and lower panels of Fig. 4) have a discontinuity at their endpoints: for
the Ising it happens at $h/J=(h/J)_{c}$ and for the $\phi^{4}$ theory it
situates at $\tau=\tau_{c}=\infty$.

These conclusions can be drawn without performing any explicit calculations on
the QPT-CPT diagram of the $\phi^{4}$ model. In other words, one can assume
the above statements which can be used to test the thermal RG equations (6)
and (II.2) suggested by us.

The QPT-CPT diagram of the three-dimensional $\phi^{4}$ model is constructed
by the numerical solution of the thermal RG equations (6) and (II.2), see the
solid (Litim cutoff) and the dashed (shapr-cutoff) lines of Fig. 5.

Figure 5:  QPT-CPT diagram of the $\phi^{4}$ model in $d=3$ dimensions based
on the thermal RG equations (6) and (II.2). The solid (dashed) line stands for
results obtained by the Litim (sharp-cutoff) regulator. The inset shows the
critical exponent $\nu$ as a function of $\tau$ which is related to the
temperature. The shaded areas stand for $\tau<1$.

The QPT-CPT diagram based on thermal RG equations with the Litim and the
sharp-cutoff regulators are similar but not identical: between $\tau=1$ and
$\tau=\infty$ they coincide but differ from each other between $\tau=0$ and
$\tau=1$.

The critical exponent $\nu$ of the $\phi^{4}$ model is expected to show a
small dependence on the parameter $\tau$, practically they must be constant
along the critical lines which are see the solid (Litim regulator) and dashed
(sharp cutoff regulator) lines of Fig. 5. Indeed, the solid and the dashed
lines of the inset on Fig. 5 confirms this. Between $\tau=1$ and $\tau=\infty$
the solid line is practically constant and it has a relatively small
dependence between $\tau=0$ and $\tau=1$. The dashed line on the inset of Fig.
5 shows that by using the thermal RG equation with the sharp-cutoff regulator,
the critical exponent $\nu$ has a larger dependence on the parameter $\tau$,
indeed $\nu$ varies with $\tau$ even in the range $\tau>1$ which was not true
for the Litim case (solid line). Nevertheless, one can argue that the Limit
regulator always provides us a better convergence and as a consequence, more
reliable results in the zero-temperature limit, i.e., for $\tau\to 0$.
However, this requires further investigations by going beyond LPA and by the
study of different models, such as the two-dimensional sine-Gordon theory. In
both cases, i.e., for the Litim and the sharp-cutoff regulators the critical
exponent $\nu$ tends to a constant value in the limit $\tau\to\infty$ which is
identical to that of obtained by the zero-temperature FRG equation. Of course,
this constant value depends on the choice of the regulator. Let us draw the
attention of the reader that the Litim regulator is known to give the closest
result among all regulators to ”exact” value ($\nu=0.629971$).

It is also expected that the critical line of the $\phi^{4}$ model must tend
to infinity in the limit of vanishing quantum parameter which is clearly
supported by the thermal RG studies, see the solid and dashed lines of Fig. 5.
Our results are tested for larger values of NCUT and we have found the same
qualitative behaviour.

##  VI Conclusions

In this work, we proposed a modified finite-temperature FRG approach by
relating the temperature parameter to the running RG scale, $T\equiv
k_{T}=\tau k$ where $k_{T}$ is the running cutoff for thermal, and $k$ is for
the quantum fluctuations. In this case, the temperature must be related to the
dimensionless quantity $\tau$. This choice is supported by the idea of the
Wilsonian approach where fluctuations are taken into account by the
integration of degrees of freedom above these running cutoffs which are chosen
to be different in order to make difference between thermal and quantum
fluctuations. For $k_{T}=k$ (i.e., for $\tau=1$) one cannot distinguish
between thermal and quantum fluctuations. We applied this new thermal RG
approach for the $\phi^{4}$ model in lower dimensions and constructed its QPT-
CPT phase diagram which was used to test the new thermal RG approach. Indeed,
we formulated requirements for the QPT-CPT phase diagram of the $\phi^{4}$
theory based on known properties of the QPT-CPT phase diagram of the Ising
model and we used these requirements to prove the viability of the thermal RG
method proposed in this work.

It is important to note that previous studies based on the finite-temperature
FRG method had to discuss findings with special emphasis on the interplay of
the temperature $T$, and the running RG scale $k$. Above Eq. (58) of
thermal_rg_phi4_volume  one finds for example, the RG flow equation of the
infinite volume Stefan–Boltzmann pressure
$\partial_{k}p(T)=-k^{4}/12\pi^{2}[\coth(k/2T)-1]$. To draw conclusions on,
one has to discuss how the running RG scale $k$ is related to the temperature
which is fixed to the UV cutoff, i.e., $T=\tau\Lambda$. For large cutoff
scales, $k>>T$ the flow decays exponentially with $\exp(-k/T)$ which is given
by Eq. (58) of thermal_rg_phi4_volume . By using our suggestions where the
temperature parameter is related to the running scale $T=\tau k$, and the
dimensionless parameter $\tau$ plays the role of the temperature, this flow
equation is simplified as
$\partial_{k}p(\tau)=-k^{4}/12\pi^{2}[\coth(1/2\tau)-1]$ and conclusions can
be drawn based on the change of the dimensionless parameter $\tau$ only. Thus,
the modified thermal RG method of this work finds application in a natural way
in any previous thermal RG studies and simplifies the discussion.

## Acknowledgements

The support for the CNR/MTA Italy-Hungary 2023-2025 Joint Project ”Effects of
strong correlations in interacting many-body systems and quantum circuits” are
gratefully acknowledged.

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