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Ann. Henri Poincaré 4, Suppl. 2 (2003) S683 – S692
c Birkhäuser Verlag, Basel, 2003
1424-0637/03/02S683-10
DOI 10.1007/s00023-003-0954-6
Annales Henri Poincaré
Sliding Phases: From DNA-lipid Complexes
to Smectic Metals
T.C. Lubensky
Abstract. Materials consisting of a stack of “two-dimensional” layers that exhibit
2D power-law order in the absence of coupling between layers include systems as
diverse as layered magnets, DNA-lipid complexes, and coupled Luttinger liquids.
It is common belief that these systems exhibit ”three-dimensional” order whenever
there is a nonvanishing interlayer coupling. However, with appropriately chosen
interlayer gradient couplings between layers, they can exhibit unusual equilibrium
“Sliding Phases” characterized by 2D-like power-law correlations. After reviewing
the various types of sliding phases that can occur, this paper will discuss in detail the
properties of the simplest sliding phases, the sliding xy phases in three-dimensional
stacks of planar xy models.
There are many materials in nature consisting of a stack of two-dimensional planes,
within which order that breaks a continuous xy-like symmetry can develop. Examples include layered magnets with in-plane spin order [Fig. 1(a)], layered superconductors [1], DNA-lipid complexes in which strands of DNA intercalated between
lipid bilayers form a periodic stripe or smectic pattern [2, 3, 4] [Fig. 1(b)], and
lyotropic lamellar phases consisting of stacks of bilayer membranes punctured by
transmembrane proteins that can form hexagonal lattices [5] [Fig. 1(c)]. If there
were no coupling between layers in these stacks, then each layer would be an isolated two-dimensional system with power-law destruction of long-range order in the
low-temperature phase and a Kosterlitz-Thouless transition to a high-temperature
disordered phase. If the coupling is strong, then these systems are true threedimensional systems with true long-range order in the low-temperature phase and
transitions to a high-temperature disordered phase in a universality class determined by the symmetry of the 3D order parameter. What happens when there
is weak coupling between layers? Does arbitrarily weak coupling drive the system
immediately to three-dimensional behavior at the longest length scales? Or is it
possible for there to be a regime of non-zero couplings in which the system continues to behave like a two-dimensional system in spite of interlayer couplings. In this
talk I will review the case for the existence, at least in theory, of phases, called
sliding phases [6, 7], in three-dimensional stacks of two-dimensional layers with
2D-like power-law inplane order in spite of the presence of appropriate interlayer
coupling.
Similar questions arise in arrays of one-dimensional wires. Coulomb interactions in isolated one-dimensional wires lead to a destruction of the Fermi surface
and to Luttinger liquids [8] that can be described by quantum xy-models in two-
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T.C. Lubensky
y
DNA
Ann. Henri Poincaré
d
x
z
a
(a)
(b)
(c)
Figure 1: more to come
dimensional space time. Examples of systems that can be described as coupled
Luttinger liquids include the proposed stripe phases in high-Tc superconductors
[9, 10] and in Quantum Hall fluids [11, 12]. What happens when tunneling interactions are turned on between neighboring wires in a two-dimensional array
of parallel wires? Does this system become a two-dimensional metal as soon as
interwire interactions are turned on? Or is it possible that the unusual behavior of
one-dimensional wires (with such exotic properties as spin-charge separation) survive over some range of couplings? Again, within the context of theoretical models,
sliding phases of arrays of coupled arrays Luttinger liquids with power-law correlations characteristic of one-dimensional wires, can exist with appropriate coupling
between wires [13, 14].
The traditional wisdom is that three-dimensional stacks of two-dimensional
classical systems or two-dimensional arrays of one-dimensional quantum systems,
respectively, become true three- and two-dimensional systems as soon as interactions are turned on. The basic argument for this behavior is best seen in the
context of a model system consisting of a stack of xy-models whose dynamical
variables are angles θn (r) at the two-dimensional coordinate r in layer n. Isolated
layers are characterized by power-law decay of spin correlation functions at low
temperature and a Kosterlitz Thouless transition to a disordered phase at temperneighboring layers are described by a Josephson
ature TKT . Interactions between
coupling of the form −V d2 r cos[θn (r) − θn+1 (r)]. To lowest order in V , this coupling evaluated in the low-temperature phase of the uncoupled planes scales to
zero with system size and is irrelevant for temperatures greater than a decoupling
temperature Td ; for temperatures less Td , it is relevant, and the system flows to
true three-dimensional behavior. Thus, for Td less than TKT , coupling between
layers is irrelevant, and in-plane power-law order persists when Td < T < TKT .
On the other hand, for Td > TKT , the individual planes melt before the Josephson
coupling becomes irrelevant, and the system is either ordered three-dimensionally
at low temperature, or it is disordered at high temperature. If the only coupling
Vol. 4, 2003
Sliding Phases: From DNA-lipid Complexes to Smectic Metals
S685
between layers is a Josephson coupling, then Td is always greater than TKT and
the system behaves like a true three-dimensional system in spite of the strong
anisotropy imposed by the existence of layers. It is only when additional nonJosephson couplings between layers of the form n,m Knm d2 r∇θn (r) · ∇θm (r)
are added that the existence of sliding phases becomes possible. Figure 2 contrasts
phase sequences in typical three dimensional systems with those of systems that
can exhibit sliding phases.
The Hamiltonian for such a system consists of (1) a sum of independent
2D-xy, Hamiltonia
K
2
(1)
d2 r [∇⊥ θn (r)] ,
H0 =
2 n
and interlayer couplings
HJ [sn ] = −VJ [sn ]
d2 r cos
sp θn+p (r) ,
(2)
p
where sn is an integer-valued function of layer number n satisfying n sn = 0 if
there are no external fields inducing long-range order. If all VJ [sn ] are zero, then
in the low-temperature phase θn2 (r)0 = η log(L/b) and cos[θn (r) − θn (0)] ∼ r−η ,
where
T
,
(3)
η=
2πK
L is the sample width, b is a short-distance cutoff in the x-y plane, and .0 refers to
an average with respect to H0 . The averages of the Josephson
Hamiltonians with
respect to H0 scale as HJ [sn ] ∼ L2−η[sn ] where η[sn ] = η2 p s2p . Clearly, the
most relevant Josephson coupling is the one with the smallest value
of η[sn ], which
results when sn is non-zero on the smallest number of planes. Since n sn = 0, the
smallest value of η[sn ] is obtained for couplings between two layers separated by
p layers with sn = spn = δn,0 − δn,p . For these two-layer couplings, η p = η[spn ] = η
for all values of p. Thus, the decoupling temperature above which all Josephson
couplings are irrelevant is determined by η = 2 or Td = 4πK. The 2D KT transition
for decoupled layers is TKT = πK/2; this implies TKT /Td = 1/8 < 1, and there is
no decoupled phase with power-law correlations.
Josephson couplings are not, however, the only ones permitted by symmetry.
Gradients of θn in different layers may also be coupled. The Hamiltonian for the
ideal sliding phase is HS = H0 + Hg , where
Hg =
1
2 n,m
d2 r
Um
2
[∇⊥ (θn+m (r) − θn (r))] .
2
(4)
This Hamiltonian is invariant with respect to θn (r) → θn (r) + ψn for any angle ψn
depending on n but not r, i.e., the energy is unchanged when angles in different
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T.C. Lubensky
3D xy
xy model
Crystal
3D
crystal
Disordered
Hexatic
T
SmA
T
2D
Lamellar
SmA
Columnar columnar Nematic
(1+1) D
(1+1) D
Spin Gap Superconductor
CDW Insulator
LL
(a)
Crystal
g
3D
crystal
2D
Columnar col.
Spin Gap
LL
T
Sliding xy Disordered
3D xy
Sliding
Sliding
crystal Hexatic hexatic SmA
xy model
Sliding Lamellar Sliding
col. nematic nematic
(1+1) D
SC
Sliding
LL
Ann. Henri Poincaré
T
T
SmA
T
(1+1) D
CDW Ins
g
(b)
Figure 2: (a) Phase sequences in three-dimensional systems with low-temperature
ordered and high-temperature disordered phases. In the xy model, there is a single
transition with a divergent specific heat from a low-temperature xy ferromagnet
to a high-temperature paramagnet. In a hexatic smectic liquid crystal, the lowtemperature phase is a true crystalline solid. At intermediate temperatures, there
is a hexatic phase [15] with long-range hexatic order. Finally at high temperatures,
there is a simple smectic phase with no inplane order. In lamellar columnar systems
such the DNA-lipid complexes, the low-temperature phase is a columnar crystal
with the DNA stands forming a true two-dimensional lattice in the plane perpendicular to the layers. The columnar crystal can melt to a lamellar nematic in which
the DNA columns lose their long-range positional order but retain their directional
order. Finally at the highest temperature, the DNA strands loose all order, leaving
only the lamellar smectic-A phase. The final figure shows the phase sequence of
coupled spin-gap Luttinger liquids at zero temperature from a superconducting
to a CDW phase. (b) Phase sequences in layered systems with sliding phases. In
the xy model, a sliding phase with power-law spin ordering intervenes between
the ferromagnetic phase and the paramagnetic phase. The transitions from both
the ferromagnetic phase to the sliding phase and from the sliding phase to the
paramagnetic phase are of the Kosterlitz-Thouless type with no visible anomalies
in the specific heat. In the hexatic system a sliding crystal with power-law positional order and a sliding hexatic phase with power-law orientational order can
intervene, respectively, between the crystal and hexatic phases and between the
hexatic and smectic-A phases. Similarly in the columnar phase, sliding columnar
and sliding nematic phases appear; and in the Luttinger liquid system, a sliding
Luttinger liquid or smectic metal phase intervenes between the superconducting
and CDW phases.
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Sliding Phases: From DNA-lipid Complexes to Smectic Metals
S687
layers slide relative to one another by arbitrary amounts. The sliding Hamiltonian
can be written as
1
(5)
d2 r Knn ∇⊥ θn (r) · ∇⊥ θn (r),
HS =
2 nn
where Knn ≡ Kn−n , with K0 = K + 2 m Um and Kn = −Un for n = 0 . Spin
correlation functions with respect to HS are easily calculated:
η(p)
(L/b)−
p = 0
G(r, p) = cos[θn+p (r) − θn (r)] =
(6)
−η
S (0)
p = 0,
(r/b )
where
T
(K −1 − Kp−1 )
2π 0
T −1
K
2π p
η(p) =
η(p)
=
(7)
with
Kp−1
π
=
0
K(k) =
dk cos kp
π K(k)
eipk Kp .
(8)
p
Thus, we see that if system is described by the sliding phase Hamiltonian HC with
no Josephson couplings, it has the following properties: (1) Correlations within
the same layer die off with a power-law with separation as they do in a true twodimensional system, (2) Correlations between layers die to zero with a power of
the systems size, and (3) the exponents controlling the power-laws are determined
by the full functional form of Kp .
To determine is the sliding phase can be stable, we must (1) determine at
what temperature Td the Josephson couplings first become relevant and (2) calculate the KT transition temperature of the sliding phase, which now also depends
in general on the interlayer couplings Kp for p = 0. Calculation of the decoupling temperatures is straightforward. The expectation of the general Josephson
coupling [Eq. (2)] relative to the sliding phase is
η [sn ]
,
HJ [sn ] ∼ L2−
where
η[sn ] =
T −1
sn sm Kn−m
.
2π n,m
(9)
(10)
S688
T.C. Lubensky
Ann. Henri Poincaré
The decoupling temperature for the particularcase of simple Josephson couplings
between layers n and p of the form Hp = −Jp d2 r cos[θn+p (r) − θn (r)] is
Td (p) =
4π
.
K0−1 − Kp−1
(11)
If we restrict ourselves to couplings of the form of Hp , the decoupling temperature
is then,
(12)
Td = max[Td (p)].
p
Recall that the sliding phase will be stable if Td < TKT . The value of Td can be
adjusted by changing the gradient interactions Kp between layers. Thus, the goal
will be to find a set of couplings that decreases Td while not significantly altering
TKT .
As in the true two-dimensional power-law xy phase, vortices proliferate and
eventually destroy the sliding phase. Vortices in sliding phases are more complex
than those of pure 2D systems. There can be composite vortices that extend across
several layers. For example, two vortices in adjacent layers could form a bound state
with energy less than the sum of the energies of two independent vortices in the
two layers. The entropy of bound vortices like that of single-layer vortices, grows
as the logarithm of the system size. The transition to the disordered phase occurs
when the energy of the lowest-energy vortex becomes less that T times the entropy
of the vortex. Thus, we need to calculate the energy of all composite vortices.
Let
knl be the strength of a vortex at position rnl is layer n, and let σn = l knl be
the total vortex strength in layer l. Then standard procedures yield
Kn−m σn σm ln(L/b) − π
Kn−m knl km,l F (|rnl − rml |) (13)
EV = π
n,m,l,l
n,m
for the energy of all vortices, where F (r) is function that tends to zero as r → 0
and is proportional to ln r as r → ∞.
The vortex energy EV and Boltzmann statistics imply that the number of
times a given configuration of vortices occurs in the system scales with system size
as L2−ηKT [σn ] , where
π ηKT [σn ] =
Kn−n σn σn ,
(14)
T
n,n
2
and the factor of L counts the number of places in the 2D plane the configuration
can be placed. Clearly, if η[σn ] < 2, the particular vortex configuration {σn } will
proliferate. The “Kosterlitz-Thouless unbinding” for {σn } therefore occurs at a
temperature
π
Kn−n σn σn .
(15)
TKT [σn ] =
2
n,n
If there is only one vortex in layer 0, then σn ≡ σn0 = δn,0 and TKT [σn0 ] = πK0 /2. If
there is a +1 vortex in layer zero and a ±1 vortex in layer p, σn ≡ σnp± = δn,0 ±δn,p
Vol. 4, 2003
Sliding Phases: From DNA-lipid Complexes to Smectic Metals
S689
and TKT [σnp± ] = π(K0 ± Kp ). Note that when Kp is nonzero TKT [σnp± ] is not twice
TKT [σn0 ]. In fact it is possible for TKT [σn ] to be less than TKT [σn0 ] for one or more
configurations {σn }. The interactions between layers lead to composite multi-layer
vortices that cost less energy to create than a single vortex in an individual layer.
Unbinding of bound pairs of any set of individual-layer or composite vortices will
destroy the rigidity within those layers. Thus, the transition temperature to the
disordered state is
(16)
TKT = min TKT [σn ].
{σn }
For the models, we consider it turns out that the minimum TKT [σn ] occurs for
composite vortices formed by binding two +1 vortices in neighboring layers together for which σn = δn,p + δn,p+1 for some p.
As discussed earlier, the sliding phase exists provided all Josephson interactions between layers become irrelevant before the layers Kosterlitz-Thouless melt,
i.e., provided
minσn TKT [σn ]
TKT
> 1.
(17)
=
β=
Td
maxp Td (p)
The goal then is to find a set of potential Kn that minimize Td while not decreasing TKT too much. From Eq. (11) for Td (p), it is clear that to minimize Td , we
need to maximize K0−1 − Kp−1 , which in turn implies choosing potentials Kn that
produce minima in the Fourier transform K(k) at appropriate wavenumbers k.
For simplicity, we consider a model with only nearest-neighbor and next-nearest
neighbor couplings for which K(k) can be written as
K(k) = K [1 + γ1 (1 − cos k) + γ2 (1 − cos 2k)] ,
(18)
and we choose γ1 and γ2 such that K(k) has a minimum with value K∆ at k = k0 .
Minima in 1/Td(p) occur when minima in K(k) occur at k0 = 2π(p/l) for integers
p and l. The value of Td−1 varies with k0 ; it has minima at k0 = 2π(p/l) and grows
as [k0 − 2π(p/l)]2 away from these minima. Figure 3 plots β as a function of k0 for
∆ = 10−5 and clearly shows that there are regions in which β > 1, and the sliding
phase is stable with respect to Josephson couplings between any two layers.
The analysis just presented for sliding xy-models applies with minor modification to sliding columnar phases in DNA lipid complexes arrays of coupled Luttinger liquids. In sliding columnar phase, each layer is treated as a two-dimensional
smectic with a mass-density-wave phase variable un (r) replacing the angle variable
θn (r). The fact that a smectic breaks both translational and rotational symmetry
leads to some complications not encountered in the xy model, but the basic conclusions about the existence of the sliding phase and the existence of power-law
correlations remain.
Luttinger liquids are conveniently described in terms of a bosonized action
with dual phase variables θn (x, t) and φn (x, t) on wire n describing, respectively,
charge and superconducting fluctuations. The two-dimensional variable r is replaced by the two-dimensional space-time variable (x, t) of a one-dimensional wire.
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T.C. Lubensky
4
Ann. Henri Poincaré
(3,23)
3.5
(3,22)
3
(3,20)
(3,19)
2.5
β
(3,17)
(3,16)
(4,23)
(4,21)
2
(2,15)
1.5
1
(2,13)
(1,8)
0
(2,11)
(1,7)
0.5
(1,6)
0.26
0.3
0.34
(1,5)
0.38
k0 /π
Figure 3: β = TKT /Td is plotted versus k0 /π. Local minima near k0 /π = 2l/p
are labeled by (l, p). Other possible integer pairs either do not fall in the range
0.24 < k0 /π < 0.40 or yield larger values of β than those shown above [7]
The analog of the sliding phase of the stack classical xy-models in a 2D array of
coupled Luttinger liquids in the spin-gap phase is a smectic-metal phase [10] characterize by superconductivity with vanishing resistivity ρxx along the wires and
vanishing conductivity σyy perpendicular to the wires [13]. At finite temperature,
these quantities vanish as a power-law in T :
σxx ∼ T ∆CDW −2 ,
ρyy ∼ T 2∆SW −3 ,
(19)
where ∆CDW > 2 and 2∆SW > 3. Figure 4 shows a phase diagram for a 2D array
of parallel Luttinger wires as a function of the wavenumber k of the analog of the
coupling K(k) of the xy stripe phases, and a coupling constant K along the wires.
Crossed arrays of Luttinger liquids can also have sliding phases with an
isotropic conductivity that diverges as a power law with temperature [16]. In addition, arrays of Luttinger liquids in an external field provide an interesting approach
to the hierarchy of fractional quantum Hall states [14].
In this talk I have reviewed the properties of sliding phases that can in principal exist in systems as diverse as DNA-lipid complexes and quantum Hall fluids.
Classical systems that can exhibit sliding phases consist of stacks of coupled twodimensional layers with xy-like order; they exhibit power-law correlations characteristic of two-dimensional systems even though layers are coupled in threedimensions. Quantum systems that can exhibit sliding phases consist of arrays
Sliding Phases: From DNA-lipid Complexes to Smectic Metals
S691
k /F
Vol. 4, 2003
SM
0.69
SC
0.67
FL
0.65
0
2
4
K
CDW
6
8
Figure 4: Phase diagram, showing superconducting (SC), Fermi Liquid (FL),
charge-density-wave (CDW), and Smectic metal or striped phases, for an array
of parallel Luttinger-liquid wires as a function of coupling constant K along the
wires, and the wavenumber q0 of the interwire coupling [13]
of coupled one-dimensional wires; they exhibit power-law behavior in space and
time characteristic of one-dimensional systems. I showed that sliding phases can
be stable with respect to a restricted class of interlayer (or interwire) couplings
that couple only two layers. The region of stability of the sliding phase is is reduced when multi-layer couplings are included. The stability of sliding phases with
respect to all multi-layer couplings remains to be investigated.
This work was supported in part by grants from the US National Science
Foundation under grant Nos. DMR 97-30405 and DMR00-96531.
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Ann. Henri Poincaré
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T.C. Lubensky
Department of Physics and Astronomy
University of Pennsylvania
Philadelphia, PA 19104
USA