First-order transition into a charge-density wave phase
A. V. Rozhkov





Institute for Theoretical and Applied Electrodynamics, Russian
Academy of Sciences, 125412 Moscow, Russia









§ ABSTRACT 
The transition into a charge-density wave phase is analyzed theoretically.
We argue that, for commensurate or nearly-commensurate charge-density wave,
the transition may become first order. Such a remarkable departure from the
more common second-order transition scenario is associated with additional
`umklapp' terms one can include in the Landau free energy expansion in the
commensurate or nearly-commensurate cases. The proposed mechanisms may be
relevant for certain alloys demonstrating first-order transition into
charge-density wave state. It may be generalized for spin-density wave
phases as well.






First-order transition into a charge-density wave phase
    A.V. Rozhkov
    September 9, 2024
=======================================================






§ INTRODUCTION





Canonical theory of the charge-density wave (CDW) thermodynamic phase
(e.g., 
Ref. gruner_book)
concludes that the transition from a disordered phase into a CDW state is
continuous (second-order). This basic theoretical expectation is indeed
confirmed by numerous experiments. For example, 
Fig. 5
in review
paper gruner_review_dynamics1988
demonstrates continuous decay of the CDW order parameter to zero as a
function of increasing temperature, for three different CDW-hosting alloys
[NbSe_3, (TaSe_4)_2I, and K_0.3MoO_3].
As a more recent demonstration of the same behavior, we can mention
Fig. 3 in
Ref. cdw2017cont,
where order-parameter-versus-temperature data for
TiSe_2
are presented.
Reference cdw_fluct2008exper
examined the transition type for
TbTe_3.
Continuous decay of the order parameter, absence of hysteresis, and
critical fluctuations all point to the second-order transition in the
latter compound.

Yet for many crystals a CDW phase is separated from a disordered state by a
discontinuous (first-order) transition. Namely, in
IrTe_2  
the formation of a commensurate CDW (CCDW) is accompanied by pronounced
hysteresis inside heating-cooling
cycle <cit.>,
the signature of a first-order transition. Another alloy demonstrating 
the first-order transition between a CCDW and a disordered state is
Lu_5Ir_4Si_10,
see
Ref. Lu5Ir4Si10first_orderCDW1999exper.

Compound
Er_2Ir_3Si_5
presents a similar case but for a nearly-commensurate CDW (NC-CDW)
phase <cit.>. 

Other materials for which this phenomenon was reported are
Lu_2Ir_3Si_5,
see
Refs. Lu2Ir3Si5_first_orderCDW2015exper,
Lu2Ir3Si5_cdw2006hysteresis_exper;
EuTe_4,
see
Refs. EuTe4hysteresis2022exper, EuTe4hyster2019exper.

A first-order transition into a CDW state may be explained by incorporating
the so-called `umklapp' contributions into a Landau-type model, as it was
done in
Ref. TaSe2McMillan1975theory_umklapp
in the context of the CDW state in
TaSe_2.
However, as a CDW-hosting material,
TaSe_2
is quite peculiar for it can host three non-identical order parameters
connected by C_3 rotations. As one can expect, this symmetry feature is
of crucial importance for the structure  of the Landau functional. At the
same time, it makes the corresponding formalism too specialized to be
directly applicable in many relevant situations. 

In this paper, we aim to expand the ideas of 
Ref. TaSe2McMillan1975theory_umklapp
to a broader context. Namely, using the Landau free energy framework, we
study phase transitions into commensurate and nearly-commensurate CDW
states.  We argue that, in various rather general situations, the
transition into these states may be discontinuous. Moreover, for some
conditions our theory predicts that a material can demonstrate
two-transition sequence: a continuous normal-to-CCDW transition is followed
by a first-order CCDW-CCDW transition, the latter connecting the phases
that differ only by order parameter magnitudes. 




As for NC-CDW, we show that, if we incorporate lattice distortions into the
model, suitably constructed umklapp terms become symmetry-allowed. When the
lattice distortions fields are eliminated from the free energy, the
resultant effective model is equivalent to CCDW Landau free energy, and a
first-order
transition can be recovered.

Our paper is organized as follows.
Section <ref> 
is dedicated to formulation of a Landau free energy function valid for
unidirectional incommensurate CDW. Various models of CCDW are introduced
and analyzed in
Sec. <ref>.
The case of NC-CDW is presented in
Sec. <ref>.
Our results are discussed in
Sec. <ref>.



§ GENERAL CONSIDERATIONS


 


It is common to describe transition into a CDW state within the framework
of the Landau free energy 

    F_0 (ρ_ cdw)
    =
    a/2 |ρ_ cdw|^2 + b/4 |ρ_ cdw|^4,

where the coefficients a and b satisfy the well-known conditions
b>0
and
a (T) = α (T - T_ CDW).
Here T is temperature, 
T_ CDW
is the CDW transition temperature, and coefficient α is positive.

As for the complex order parameter
ρ_ cdw
=
|ρ_ cdw| e^i φ,
it represents charge density modulation. In many situations it is
conveniently approximated by a single harmonic term

    ρ(  R )
    ≈ρ_ cdw e^i  k· R
    + 
     c.c.
    =
    2 |ρ_ cdw| cos (  k· R + φ) ,
 
where the wave vector
k
characterizes CDW spatial periodicity. Following the standard prescription,
one minimizes F over
ρ_ cdw
to derive

    |ρ_ cdw|
    = θ( T_ CDW - T) √(α ( T_ CDW - T) /b),
 
where
θ(x)
is the Heaviside step-function. This formula explicitly demonstrates that
the order parameter strength
|ρ_ cdw|
is a continuous function of T, a hallmark of the second-order transition. 

Unlike the absolute value
|ρ_ cdw|,
the order parameter phase φ remains undetermined, which is a
manifestation of the U(1)-symmetry of F: the change

    φ→φ + δφ
 
keeps F the same. Physically, this can be viewed as a invariance of the
CDW state under arbitrary uniform translation 

    R→ R +  t,
       t∈ℝ^3.
 
It is easy to check that, for a given
t,
the phase change is
δφ = ( k· t),
or, equivalently the order parameter transforms according to

    ρ_ cdw→ρ_ cdw e^i  ( k· t)
 
under the translation
t.




§ COMMENSURATE CDW







 §.§ Landau free energy with `umklapp' contribution


The invariance of the CDW Landau free energy relative to arbitrary
translations (<ref>)
is, by itself, a very excessive constraint on the model: in any crystal
the translation group must be limited to lattice translations only, that
is, instead of
t∈ℝ^3,
the allowed 
t's
are

    t = m_1  a_1 + m_2  a_2 + m_3  a_3,
 
where 
m_i
are integers, and
a_i
are elementary lattice vectors.

The reduction of the invariance group implies that additional terms may be
introduced into the Landau free energy. Below we explicitly construct these
terms for commensurate CDW order. 

By definition, a commensurate CDW satisfies the following conditions

    ( k· a_i ) = 2 π p_i/q_i,
      
    i = 1,2,3,

where integer 
p_i
is co-prime with
q_i
for all i. Formally, of course, any measured 
k
can be described in this manner, with arbitrary large
q_i's.
However, for practical matters, a wave vector is considered to be
commensurate only when all three
q_i's
are not too large.

For these three 
q_i
we introduce their least common multiple
n =  lcm (q_1, q_2, q_3),
referred below to as commensuration degree. Then the monomial
ρ_ cdw^n
is invariant under arbitrary lattice translations. To prove this claim, we
start with
Eq. (<ref>)
and write




ρ_ cdw^n
→ρ_ cdw^n e^i  ( k· t) n,
where

    ( k· t) n
    =
    2π∑_i m_i p_i n/q_i.
 
Since n is a multiple of a
q_i
for any i, one establishes that
( k· t) n = 2π N,
where N is an integer. Thus, 
ρ_ cdw^n
is invariant for any 
t
described by
Eq. (<ref>)

Note that, while n is introduced as least common multiple of three
denominators
q_i's,
in many realistic situations, however, no significant number-theoretical
calculations are required, as the commensuration degree is quite obvious
from the data. For example, if 
(p_1/q_1, p_2/q_2, p_3/q_3) = (1/5, 0, 1/5),
as in
Ref. Ivashko2017,
then
n=5.

Since
ρ_ cdw^n
is invariant, we conclude that, for any complex number
c_n = |c_n| e^i γ,
the nth degree `umklapp' contribution

    F_ u^(n) = - c_n/2nρ_ cdw^n +  c.c.
    =
    -|c_n|/n |ρ_ cdw|^n cos ( n φ + γ)
 
is explicitly real and invariant under lattice translations. Consequently,
the free energy

    F^(n) = F_0 + F_ u^(n).

can be used as a model for a commensurate CDW state.

Inclusion of
F_ u^(n)
into the free energy shrinks the symmetry group of the Landau energy from
U(1) to
Z_n:
function 
F^(n)
is no longer invariant under an arbitrary phase shift, only discrete
shifts

    φ→φ + 2π m/n,    m ∈ℤ,
 
do not change the free energy. 

Let us now search for minima of
F^(n).
Minimization with respect to φ is very simple. As this variable
enters
F_ u^(n)
term only, it is easy to demonstrate that the Landau energy is the lowest
when
φ = φ^*_m,
where

    φ^*_m = 2π m/n - γ/n,
      
    m = 0, 1, …, n-1.
 
We see n minima evenly distributed over a unit circle. This arrangement
resembles a clock dial, thus a common name for a model of this kind is the
n-state clock model. Another frequently used designation is the
Z_n
model, a reference to the invariance group of
F^(n).

Every 
φ^*_m
in
Eq. (<ref>)
represents a particular localization of CDW distortions relative to the
underlying lattice, see
Fig. <ref>.
There are n such localizations, all of them are degenerate. Speaking
heuristically, one can say that a CDW with nth degree commensuration is
always pinned by the lattice to one of n possible minima, as illustrated
by
Fig. <ref>.
This pinning decreases the symmetry of the Landau free energy from U(1) to
Z_n.



At any of these minima
the cosine in
Eq. (<ref>)
is equal to unity. Thus, the Landau free energy can be re-written as the
following function of a single non-negative variable
|ρ_ cdw|

    F^(n)
    =
    a/2 |ρ_ cdw|^2 + b/4 |ρ_ cdw|^4
    - |c_n|/n |ρ_ cdw|^n
    + …,
 
where ellipses stand for higher-order terms that might be necessary to
include in order to maintain stability of the free energy, and the tilde
over F implies that this free energy does not depend on φ.

For
n=2
the contribution proportional to
|c_2|
acts to renormalize a, effectively increasing the transition temperature.
The transition remains continuous for all 
|c_2|.
If 
n>2,
the contribution coming from the “umklapp" term
F^(n)_ u
may qualitatively alter the behavior of the system near the transition
point, as discussed below.



 §.§ Z_3 model of CDW





The value
n=3
represents the CCDW phase whose unit cell is three times larger
than the unit cell of the underlying lattice. (This type of order is
schematically shown in
Fig. <ref>.)
Specializing 
Eq. (<ref>)
for
n=3,
one can express the free energy 
F^(3)
as

    F^(3) (|ρ_ cdw|)
    =
    a/2 |ρ_ cdw|^2 - |c_3|/3 |ρ_ cdw|^3
    + b/4 |ρ_ cdw|^4.
 
We see that this free energy is stable in the sense that, for large 
|ρ_ cdw|,
function
F^(3)
grows, which guarantees that an equilibrium value of the order parameter
is bounded.



For
a>0
this free energy has a
|ρ_ cdw|=0
minimum that represents (meta)stable disordered state. Additionally, for
a < |c_3|^2/(4b)
there is a minimum of
F^(3)
at

    |ρ_ cdw|
    = 
    1/2b( |c_3| + √(|c_3|^2 - 4ab)),
 
see
Fig. <ref>.
It is easy to check that 
Eq. (<ref>)
describes the global minimum of 
F^(3)
when
a < 2|c_3|^2/(9b).

By exploiting the commonly assumed linearization

    a = a(T) ≈α (T-T_*),
      α > 0,
 
where 
T_*
is the temperature for which
a(T)
passes through zero, the CDW transition temperature can be expressed as

    T_ CDW = T_* + 2 |c_3|^2/9 α a b > T_*.
 
We see that
T_*
by itself does not have any special meaning. However, in the limit
|c_3| → 0
the transition temperature
T_ CDW
approaches
T_*.

At the transition, the coefficient a is not zero, but rather 
a = 2|c_3|^2/(9b).
Substituting this value in
Eq. (<ref>),
one finds that
|ρ_ cdw|
jumps from 0 to
2|c_3|/(3b),
see also
Fig. <ref>.
Thus, we conclude that, at finite
|c_3|,
the transition is discontinuous. On the other hand, one must remember that
at small
|c_3|
the transition is formally indeed first-order, yet, in this regime, the
discontinuity of order parameter becomes weak, and
difficult to detect. This observation remains relevant for other signatures
of first-order transition.



 §.§ Z_4 model of CDW





The
n=4
phase diagram differs qualitatively from the
n=3
situation. The 
n=4
Landau free energy reads

    F^(4)
    =
    a/2 |ρ_ cdw|^2 +  b/4 |ρ_ cdw|^4
    + d/6 |ρ_ cdw|^6,
 
where
b = b - |c_4|.
In other words,
|c_4|
effectively renormalizes b. Since 
b
can be either positive, or negative, depending on the relation between b
and
|c_4|,
we retained here the sixth-order term to prevent uncontrollable growth of
the order parameter at 
b < 0.

When
|c_4| < b,
the free energy 
F^(4)
describes second-order transition that occurs at
a=0.
If
linearization (<ref>)
is assumed, then the transition temperature coincides with
T_*.

At negative
b,
the transition into the CDW phase becomes first-order. (Qualitatively, the
behavior of
F̃^(4)
in this regime is very similar to the graphs of
F̃^(3)
in
Fig. <ref>.)
For
0<4ad <  b^2
the free energy has three extrema: one at zero, and two more at

    |ρ_ cdw|
    =
    √(1/2d( | b| ±√( b^2 - 4ad))
    ).
 
The minimum (maximum) corresponds to the plus (minus) sign in this formula.
The transition into the ordered state takes place when the free energy at
the non-trivial minimum becomes equal to zero, which is the free energy at
the trivial minimum
|ρ_ cdw| = 0.
This occurs at
a = 3  b^2/(16 d)
if
b < 0.

At arbitrary sign of
b
the transition temperature can be compactly expressed as

    T_ CDW = T_* +  3 (b-|c_4|)^2/16 α dθ(|c_4| - b).
 
This shows that, unlike the
n=3
case, 
arbitrary weak 
n=4
“umklapp" term cannot change the continuous type of the transition. Only
when
|c_4|
exceeds b, the transition becomes discontinuous. The point
a=0,
|c_4| = b
is a tricritical point on the phase diagram.



 §.§ Z_5 and Z_6 models


For
n=5
and
n=6,
the phase diagram acquires additional complexity. We start our analysis by
writing the 
n=5
Landau free energy as

    F^(5) = a/2 |ρ_ cdw|^2 +b/4 |ρ_ cdw|^4
    -|c_5|/5 |ρ_ cdw|^5 +d/6 |ρ_ cdw|^6,
 
where, as before, we included the
O(|ρ_ cdw|^6)
term to provide proper growth of 
F̃^(5)
at 
|ρ_ cdw| → +∞.
Due to relative complexity of the
n=5
and
n=6
cases, it is convenient to introduce normalized quantities. Namely, the
dimensionless form of 
F^(5)
reads

    F^(5)/ F_0 
    =
    A/2 y^2 + 1/4 y^4
    -
    C/5 y^5 + 1/6 y^6.
 
The coefficient A in this formula is

    A = a d/b^2 = α d /b^2(T - T_*).
 
Other quantities are

    F_0 = b^3/d^2,
      
    y = √(d/b)|ρ_ cdw|,
      
    C = |c_5|/√(bd).
 
Here energy
F_0
sets the overall scale for
F^(5),
and y is the dimensionless order parameter.







For 
A>0,
the disordered state
y=0
is absolutely unstable. It is at least metastable when
A<0.
As for ordered states (stable, metastable, or unstable), they are
represented by roots of the equation

    P^(5)_C (y) = - A,
 
where
P^(5)_C (y)
is a family of polynomials of variable y 

    P^(5)_C (y) = y^2(1 - Cy + y^2),
 
parameterized by
C>0.

For small C and positive y, the polynomials are positive increasing
functions, see
Fig. <ref>.
Thus,
Eq. (<ref>)
has one solution for negative A. No solution exists when
A>0.
If C is fixed, this describes an order-disorder continuous phase
transition at
A=0,
or, equivalently, at
T=T_*.

This simple picture is not applicable for
C>C_ hi = 4√(2)/3.
Indeed, at
C=C_ hi
a horizontal inflection at
y = 1/√(2)
is formed (hence, the subscript `hi'). For
C > C_ hi
the polynomial
P^(5)_C (y)
is no longer monotonic as a function of y, and more than one  solution
become possible for appropriate (negative) values of A. Since
P_ C_ hi (1/√(2)) = 1/12,
these roots are realized for
A>-1/12.
Multiple non-trivial roots of
Eq. (<ref>)
implies that first order transitions between CDW states emerges. Note that
the states separated by this transition have identical symmetries. The only
difference is the magnitude of
|ρ_ cdw|.

As one can see from
Fig. <ref>,
for sufficiently large C there are finite intervals of y in which the
value
P^(5)_C (y)
is negative. For such C,
Eq. (<ref>)
has two roots even for positive
A>0.
When A grows, the roots approach each other, merge, and ultimately
disappear, signaling a first-order transition into the disordered phase.

The resultant phase diagram is shown in
Fig. <ref> (left).
It features a second-order transition line reaching the first-order
transition curve. The latter terminates at a critical point inside the CDW
phase. This point corresponds to the horizontal inflection point for
P^(5)_C.
The tricritical point `T', where two transition lines meet, can be found by
solving
Eq. (<ref>)
simultaneously with
F̃^(5)(y) = 0,
both in the limit
A=0.
These two equations can be satisfied when
C=5/√(6),
which is the horizontal coordinate of `T'.

Depending on the value of C, the behavior
ρ_ cdw = ρ_ cdw (T)
may vary significantly, see
Fig. <ref>.
If
C > 5/√(6),
which corresponds to the area to the right of the `T' on the phase diagram,
the order-disorder transition is discontinuous. To the left of point `Cp'
(C<4√(2)/3),
the transition is continuous, at
A=0.
In the interval
4√(2)/3 < C < 5/√(6)
the model exhibits a cascade of two transitions (a first-order CDW-CDW
transition followed by a second-order CDW-disorder transition). When
coefficient C is fine-tuned to be
C=4√(2)/3,
the order parameter discontinuity shrinks to zero and becomes a continuous
singularity, as shown in
Fig. <ref>.

For
n=6
commensuration, the Landau free energy can be expressed as

    F^(6) = a/2 |ρ_ cdw|^2 +b/4 |ρ_ cdw|^4
    -d̃/6 |ρ_ cdw|^6
    +e/8 |ρ_ cdw|^8,

where
d̃ = |c_6| - d,
and
d, e > 0.
Normalized form of this free energy is easy to establish

    F^(6)/ F_0 
    =
    A/2 y^2 + 1/4 y^4
    -
    D/6 y^6 + 1/8 y^8.
 
Here, under assumption
b>0,
we introduced the following set of parameters

    y = (e/b)^1/4|ρ_ cdw|,
       F_0 = b^2/e,
    
    
    A = a √(e/b^3),
      
    D = d̃/√(b e).
 
Similar to 
Eq. (<ref>),
ordered phases of the
n=6
model are represented by roots of equation

    P^(6)_D (y) = -A,
      where   P^(6)_D = y^2 ( 1 - D y^2 + y^4).
 
Analysis of
Eq. (<ref>)
can be adopted for the latter equation, and an
n=6
phase diagram can be constructed, see
Fig. <ref> (right).
It is clear that both diagrams in
Fig. <ref>
are qualitatively similar. 




§ NEAR-COMMENSURATE CDW





We demonstrated in the previous section that symmetry-allowed “umklapp"
terms enhance complexity of the model's phase diagram. In particular, a
first-order transition line emerges. Let us now generalize our approach to
the case of NC-CDW. 

For an NC-CDW, vector
n  k
does not belong to the reciprocal lattice of a host crystal for any
n ∈ℕ,
however, one can find a (small) integer m and a reciprocal lattice vector 
b 0
such that a vector

    δ q =  b - m  k
 
is small in the sense that
|( δ q· a_i )| ≪ 1
for all
i=1,2,3.

Since
k
is not commensurate,
ρ_ cdw^n
is not compatible with the lattice translation group for any
n ∈ℕ.
Yet, an umklapp contribution associated with the NC-CDW order can emerge
through the following mechanism. Note that a monomial
ρ_ cdw^m,
where m is defined in
Eq. (<ref>),
transforms according to the rule
ρ_ cdw^m → e^- i (δ q· a_i)ρ_ cdw^m
upon a translation on the elementary lattice vector
a_i.
Although “the elementary defects"
e^- i (δ q· a_i)
are close to unity, the exponent oscillates for longer translations,
indicating that the contribution
∝ρ_ cdw^m
averages to zero upon summation over the whole sample. 

Fortunately, since
|δ q|
is small, the lattice can adjust its structure to allow the umklapp term.
Imagine that the lattice, in response to the CDW presence, experiences an
additional periodic distortion with the wave vector 
δ q.
Representing such a distortion by a complex quantity u, we can devise a
contribution
u ρ_ cdw^n
that is invariant under the lattice translations. (Conceptualizing u, one
can think of it as a “frozen", or “condensed" phonon mode, whose wave
vector is
δ q.)

To check the invariance, observe that a translation on 
a_i
transforms u according to the rule
u → e^i (δ q· a_i) u.
This makes the products
u ρ_ cdw^m
and
(u ρ_ cdw^m)^*
translation-invariant, and permissible to enter the Landau free energy.

With this in mind, we write the following Landau-type model

    F_ NC (ρ_ cdw, u)
       =   
    F_0 (ρ_ cdw)
    
       +   ε_δ q |u|^2
    +
    1/√(4m)( g_ nc u ρ_ cdw^m +  c.c.).
 
Here
ε_δ q |u|^2 ≥ 0
is the elastic energy associated with the distortion u, complex
coefficient 
g_ nc
is a coupling constant, and factor
(4m)^-1/2
is introduced into this formula to make expression below consistent with
previous definitions.

Minimizing this energy over u, we obtain

    u
    =
    - g_ nc^*/√(4m)ε_δ q
     (ρ_ cdw^*)^m.

This relation demonstrates that in the NC-CDW phase the distortion u is
always present. This relation can be checked in experiment. 

Substituting the equality for u into
Eq. (<ref>)
one derives the reduced free energy that depends on
|ρ_ cdw|
only

    F_ NC
    =
    F_0 
    -
    |c_2m|/4m |ρ_ cdw|^2m,
     where 
    |c_2m| = |g_ nc|^2/ε_δ q.
 
Note that this reduced Landau free energy is identical to
F^(n) (|ρ_ cdw|)
for 
n=2m.
As a result, it can describe a first-order transition between the
disordered and ordered phases, as we have seen above for
Z_4
and
Z_6
models.





§ DISCUSSION





We argued above that the transition into a commensurate or
nearly-commensurate CDW phase can be discontinuous. The mechanism that
turns an anticipated second-order transition into a first-order one relies
on the umklapp contributions to the CDW Landau free energy. The proposed
model may be viewed as an expansion of the ideas previously formulated for
TaSe_2
in
Ref. TaSe2McMillan1975theory_umklapp
to CDW phases with different types of order parameter symmetries. 

The effects of the umklapp contribution on the model phase diagram depend
on the commensuration degree n: the larger n the richer the model's
phase diagram. Indeed, for 
n=2
the umklapp contribution does nothing but corrects the transition point,
when n is as large as 5 or 6, the phase diagram displays such elements as
tricritical point, critical point, continuous and discontinuous transitions
lines, see
Fig. <ref>.

Moreover, the
Z_5,6
models allow for a possibility that the destruction of the order may occur
through a two-step process: lower-temperature CDW-CDW discontinuous
transition followed by higher-temperature CDW-disorder continuous
transition, as 
Fig. <ref>
illustrates for
Z_5
model. Superficially, one may argue that such a cascade was already
discussed quite some time ago (see, for instance,
Fig. 1 
in
Ref. TaSe2McMillan1977microscopic_theory,
or
Fig. 2 
in
Ref. nakanishi1977domain).
However, there is an important difference. Indeed, in our case, the
first-order CDW-CDW transition occurs within the same commensurate phase.
This is very much unlike commensurate-incommensurate CDW lock-in
transitions of
Refs. TaSe2McMillan1975theory_umklapp,
TaSe2McMillan1977microscopic_theory,
as well as
other <cit.>
first-order transitions associated with discontinuous change of CDW wave
vector. 

We did not extend our analysis beyond
n=6
power. Unfortunately, we were unable to identify any general principle
restricting structure of the phase diagram of large-n models. In such a
situation any investigation of large-n model becomes problematic due to
ever increasing number of parameters one must keep in the Landau free
energy expansion to guarantee model stability. Yet, apart these purely
technical issues, the formulated analytical framework is perfectly
applicable for 
n>6
models.

Our argumentation can be extended to NC-CDW order parameters as well. This
is not ultimately that surprising: in a situation of small deviation
from commensurability
|δ q|
at not-too-large n a sufficiently soft hosting crystal lattice
reorganizes itself to lock-in with the CDW. This is the heuristic
understanding behind
Eq. (<ref>).
Such a model offers an alternative to a well-known lock-in
scenario <cit.>
in which the CDW adjusts its wave vector while the (infinitely rigid)
lattice does not participate at all.

At the same time, our approach to a NC-CDW phase bears clear similarity
with that of
Ref. nakanishi1977domain. 
To satisfy the commensurability
condition (<ref>),
we postulated that the lattice generates a deformation with small
|δ q|,
while in
Ref. nakanishi1977domain 
a commensurability condition is fulfilled by introducing additional smooth
modulations of the order parameter.

In our discussion we assumed that the system always chooses the global
minimum of the Landau free energy. Yet our models for
n>2
allows for metastable states. For example, metastable minima, both ordered
and disordered, are clearly visible in
Fig. <ref>,
which is plotted for
n=3.
Switching between two minima reveals itself as hysteresis, a common fixture
of experimental presentation of a first-order transition. Using the above
phase diagrams for experimental data analysis one must remember that our
calculations do not take  hysteresis into account. In principle, hysteresis
can be captured in the framework of the Landau theory of phase transitions.
However, the description of this kind oversimplifies the physics
significantly as it ignores various non-universal mechanisms affecting
hysteretic behavior in real materials.

Additionally, we can adapt the discussed ideas to the spin-density wave
(SDW) case. The SDW order parameter is a complex vector
S,
and
( S· S)
is a true complex scalar. Thus, for even 
n=2k
one can construct an umklapp term of the form
c_2k ( S· S)^k +  c.c.
that is consistent with the discrete translations, parity, and time
inversion symmetries.

Finally, let us make the following observation. A number of alloys
demonstrate the first-order transition between disordered and CDW phases.
Several papers reporting this also
commented <cit.>
that such an unusual transition type must be a consequence of “strong
coupling". Within the context of the described formalism the expectation of
strong coupling regime is quite natural: the umklapp coefficients 
|c_n|
are likely to be small for larger n unless the displacements associated
with the order parameter are significant.

To conclude, in this paper, within the Landau free energy framework, we
explored effects of the order parameter commensuration on the CDW
transition properties. We demonstrated that in the case of commensurate and
nearly-commensurate CDW the anticipated second-order transition may be
replaced by the first-order transition, as indeed  observed experimentally.
Under certain circumstances our model predicts a cascade of two transitions
(low-temperature CDW-CDW first-order transition is followed by
higher-temperature order-disorder second-order transition). These ideas may
be applicable to SDW phases as well.



§ ACKNOWLEDGMENTS


Author is thankful to B.V. Fine, B.Q. Lv, and Alfred Zong for illuminating
discussions.






18
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