Master M2 Sciences de la Matiere – ENS de Lyon – 2015-2016

Phase Transitions and Critical Phenomena

Phase transitions on a 2D Coulomb lattice

Sergey Vilov

January 14, 2016

Abstract

The present paper is intended to examine phase transitions on a 2D Coulomblattice where the electric field exists beyond the plane. The research is based ona comparison with critical phenomena discovered for a 2D Coulomb gas where thepotential has a logarithmic dependance on a distance. The study sets out to establishwhether the same transitions take place in both cases and to explain the differencesthat emerge.To this end, a series of Grand-Canonical Monte-Carlo simulations andtheoretical analysis are performed. It is shown that similar phase transitions happenin both cases. However, the possibility of a metal-insulator phase transition in thecase of 1/r potential in a bulk matter is questioned theoretically.

1 Introduction

Phase transitions in the 2D Coulomb gas model have been studied extensively formore than 20 years. In this model the interaction potential varies logarithmicallywith the distance between charges and the electric field is taken as confined withinthe plane. Three types of phase phenomena have been discovered: a first-order phasetransition separating insulting gas and insulting solid, a second-order Ising transitionbetween conducting solid and conducting liquid, two Kosterlitz-Thouless transitionsfrom insulting gas to conducting liquid and from insulting solid to conducting solid.However, in the real world the interaction potential should be a solution to the 3DLaplace’s equation even if electric charges are placed on a single plane. In this regardit is tempting to inspect differences between the 2D Coulomb gas model and one withthe interaction potential being a solution to the 3D Laplace’s equation. This studyis aimed to give a qualitative analysis of phase transitions in the latter model basedon results of a number of Grand-Canonical Monte-Carlo simulations. Notably, thetwo systems described above share the same number of order parameter components,they can be viewed as having the same number of dimensions and almost the sameinteraction range. It also turns out that the two Hamiltonians are similar and con-sequently may conserve the same symmetry features under the RG transformation.Thus, the universality hypothesis allows simplifying the task by providing basis forthe targeted search. In this sense, the results delivered in the pioneering paper byJ.-R. Lee and S. Teitel on phase transitions in classical 2D lattice Coulomb gases [1]are used as an inception point for the present study. The paper is organised as follows.Sec. II provides the definition of the model. Sec. III is composed of the theoreticalbasis of this research, including the finite-size-scaling methods used to determine thecritical behaviour of the system. Sec. IV contains the simulation details, results anddiscussion. The conclusion comprises Sec. V.

2 The 2D Coulomb lattice

The classical Coulomb gas Hamiltonian reads

HCG =1

2

∑i,j

qiV (ri − rj)qj −µ

2

∑i

q2i , (1)

where the sum is computed over all pairs of sites of a 2D periodic L× L = N lattice,µ is the chemical potential per a pair of charges. All of the charges are situated on thesame XY plane. The space is made discrete for all the dimensions to hold the samesymmetry. However, the Z dimension is not periodic. qi is the charge at site i andV (r) is the Coulomb potential which solves the equation

4f(r) = −4πδ(r). (2)

Here 4 is the discrete Laplase’s operator

4f(r) = 4xyf(r) +4zf(r) =1

a2

∑µ=±x0µ=±y0

[f(r + µ)− 2f(r) + f(r− µ)]+

1

a2

∑µ=±z0

[f(r + µ)− 2f(r) + f(r− µ)],

(3)

1

where |x0| = |y0| = |z0| = a, a is the lattice constant.Substituting the Fourier transforms on the plane

V (r) =1

Na2

∑kxy

Vkxy (z)eikxy·r, δ(r) =1

Na2

∑kxy

eikxy·rδ(z) (4)

into Eq. 2, one gets

Vkxy (z)∑µ

(eikxy·µ + e−ikxy·µ − 2) +4zVkxy (z) = −4πa2δ(z). (5)

To mimic an infinite plane periodic boundary conditions are imposed, namely kxy =(ni/L)x0 + (nj/L)y0, where ni, nj = 0, L− 1.Fourier transforms along the Z axis

Vkxy (z) =1

2π

∫ π/a

−π/aV (kxyz)eikzzdkz, δ(z) =

1

2π

∫ π/a

−π/aeikzzdkz (6)

lead to a simple equation

V (kxyz)(12− 4[cos (kxyz · x0) + cos (kxyz · y0) + coskxyz · z0]) = 4πa2, (7)

Taking a reverse Fourier transform, it is easy to show that

Vkxy (z = 0) =aπ√

[3− cos (kxy · x0)− cos (kxy · y0)]2 − 1. (8)

In order to treat the divergence in Eq. 8 at k → 0 the potential V (r) can be dividedinto two parts V (r) = V ′(r) + V (r = 0).Plugging this into Eq. 1, the Hamiltonian transforms into

HCG =1

2

∑i,j

qiV′(ri − rj)qj + V (0)[

∑i

qi]2 − µ

2

∑i

q2i . (9)

It can be clearly seen that only the nonsingular contribution V ′(r) can be used if thesystem satisfies the electroneutrality condition

∑i qi = 0.

The Hamiltonian of such a planar system will be

H =1

2

∑i,j

qiqj1

Na

∑kx,kyk 6=0

π√[3− cos (kxy · x0)− cos (kxy · y0)]2 − 1

(eik(ri−rj) − 1)

−µ2

∑i

q2i .

(10)In this research the absolute value of all the charges is equal to q0 and integer, whilethe electroneutrality condition is imposed.

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3 Phase transitions in the system

Since the universality postulate allows expecting the same phase transitions as in the2D Coulomb gas model, the following analysis concerns a second-order Ising transitionand a metal-insulator transition.Any second-order phase transition is associated with an order parameter which al-ters in the course of the transition. In this system it is convenient to take M =∑i qi(−1)xi+yi as the order parameter.

In fact, the phase dynamics does not depend on the Z direction as all the chargesreside on a single plane. Therefore, it is possible to take a = 1 and d = 2 without lossof generality while describing the second-order phase transition.The singular part of the free-energy density f ≡ L−dlnZ in the vicinity of the criticalpoint can be written as [2],[3]

fs(t, h, L) = b−dfs(tbyt , hbyh , L/b). (11)

In Eq. 11 t = (T−Tc)/T is the reduced temperature, h is the magnetic field conjugateto M, yt and yh are the eigenvalues of a renormalizaton group transformation,b is therescaling factor,d = 2 is the number of dimensions.Neglecting the contribution of the non-singular part of the free energy close to thecritical point, one can write for the heat capacity

C(T, L) = − 1

T

∂2f

∂(1/T )2= −L2yt−d

(TcT

)2

ftt(tLyt , 0, 1) (12)

and for the order-parameter susceptibility

χM =∂M

∂h

∣∣∣∣h=0

=∂2f

∂h2

∣∣∣∣h=0

= L2yh−dfhh(tLyt , 0, 1). (13)

In Eq. 12 and Eq. 13 ftt and fhh are the second derivatives of fs with respect to tand h correspondingly.The heat capacity can also be computed through its definition

C =1

Ld∂H

∂T=

1

LdT 2(< H2

CG > − < HCG >2), (14)

whereas the magnetic susceptibility can be derived through the fluctuation-dissipationtheorem

χM =Ld

T(< M2 > − < M >2). (15)

Combining Eq. 13 and Eq. 15 and considering < M >= 0 in the absence of anexternal field, one obtains

M2 = L2(yh−d)Tfhh(tLyt , 0, 1) (16)

It is well-known that the behaviour of the heat capacity C, order parameter M ,susceptibility χM and correlation length ξ in the proximity of the critical point isdetermined by the critical exponents α, β γ, ν.If λ is the critical exponent of a generic function f(x) then [3]

λ ≡ limx→0+

ln f(x)

lnx. (17)

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In particular,M ∝ |t|β , χM ∝ |t|−γ , C ∝ |t|−α, ξ ∝ |t|−ν . (18)

On the other hand, ξ can not exceed the system’s size in the finite-size system. Con-sequently, when ξ ≈ L the system is already ordered.There appears to be a pseudocritical point at

[Tc(Ld)− Tc(∞)]−ν ∝ L. (19)

Replacing |t| with L−1/ν in (18) and comparing (18) with Eq. 12, Eq. 13 and Eq. 16,one gets

α/ν = 2yt − d,γ/ν = 2yh − d,β/ν = d− yh.

(20)

To obtain a complete set of equations linking the critical exponents with the eigen-values of the renormalization group transformation, hyperscaling relations can beutilised[3]

α = 2− dν,α+ 2β + γ = 2.

(21)

Using Eq. 20 and Eq. 21 and expanding the singular part of the free energy near thecritical point in Eq. 16,it is possible to get

M2(T, L) = L−β/ν [A0 +A1L1/ν(T − Tc) +O(L2/ν(T − Tc)2)]. (22)

In a series of MC simulations in this study, critical exponents β and ν are computedthrough Eq. 22. To gain this, a number of points are generated and then processedwith a fitting procedure. The least-squared method [4] is employed for this fitting. Itshould be noted that the farther the point observed from the critical region, the moreterms must be included into expansion (22). Specifically, it follows from Eq. 19 thatwhen L is small and the points taken reside in the vicinity of the pseudocritical point,inclusion of few terms in Eq. 22 is likely to produce a considerable error.The critical exponent α is determined in the simulations from studying the depen-dency Cmax(L). According to Eq. 18 and Eq. 19 Cmax(L) ∝ Lα/ν . If α = 0 thenthere are two possibilities [3]: a logarithmic divergence Cmax(L) = A| lnL| + B or acusp singularity Cmax(L) = A−BLz. Both of them satisfy definition (17).

Another phase phenomenon taking place in the 2D Coulomb gas model is a so-called”metal-insulator” transition associated with the unbinding of neutral charge pairs andappearance of polarization in the system. As the interaction potential in the systemexamined is different one has to reestablish conditions which may lead to similar be-haviour.To achieve this, the simplest system consisting of two charges on the infinite XY planecan be considered. If there is an external field D along the X axis, the average dipolemoment component in this direction can be written as

< px >= limrmax→∞

∫ rmax

a

∫ 2π

0pxexp[−β(−pxD)− βH]rdφdr∫ rmax

a

∫ 2π

0exp[−β(−pxD)− βH]rdφdr

, (23)

where H is the interaction potential between two opposite charges, px = q0r cosφ isthe dipole moment projection on the X direction, β = 1/kBT , a is the lattice interval.

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Integrating Eq. 23 over φ, one obtains

< px >= limrmax→∞

∫ rmax

ar2exp[−βH]J1(q0Dβr)dr∫ rmax

arexp[−βH]J0(q0Dβr)dr

, (24)

where J0(Dβr) and J1(Dβr) are modified Bessel functions of the zeroth and the firstkind respectively. Since one is interested in the linear response to the external fieldD, it is worth studying the limit of Eq. 24 when D → 0. Expanding the functionsJ0(Dβr) and J1(Dβr) around D = 0, it follows that

< px >= limrmax→∞

q0Dβ∫ rmax

ar3exp[−βH]dr∫ rmax

arexp[−βH]J0(0)dr

(25)

For the KT transition βH = 2πKln(r/a). Therefore, the integral in the numeratorconverges only if 3 − 2πK < −1 which corresponds to the threshold of the KT tran-sition [5]. The integral in the denominator does not influence the result as even whenit is divergent it diverges more slowly than the other one. Now it becomes apparentthat for the 1/r potential exp[−βH] = exp[βq0/r] goes to zero for large r and < px >in Eq. 24-25 always explodes in the bulk matter regardless of the temperature. Inother words, there is no KT-like transition in the infinite system.A collapse of the inverse dielectric constant ε−1 is usually used as an indicator of a”metal-insulator” transition. For example, in 3D homogeneous substance an electro-dynamic relation [8] asserts

P =ε− 1

4πεD, (26)

where P =∑qiriV is the polarization vector. If Px = κxxDx (compare to Eq. 25) then

1εxx

= 14π − κxx diminishes to zero when κxx increases.

In order to derive ε−1 from a MC simulation, an equation connecting this quantitywith microscopic properties of the system can be used [6],[7]:

ε−1(T, L) = limk→0

{1− 2π

k2TNa< qkq−k >

}, (27)

where qk ≡∑i e−ikri is the Fourier transform of the charge density. However, in the

finite lattice case the limit k→ 0 can not be reached accurately.A more accurate equation can be derived as follows. The Hamiltonian of the systemunder an external field D reads

HCG(D) = HCG(D = 0) +V

8πE ·D = H(0) +

V

8πD2 − V

2P ·D, (28)

where V = a · Ld = aN is the volume of the system.The average polarization is

< P >=

∑e−βµNp

∑{q1...q2Np}

P e−β[H(0)−V2 P ·D]∑

e−βµNp∑{q1...q2Np}

e−β[H(0)−V2 P ·D]

=

1Z0

∑e−βµNp

∑{q1...q2Np}

P e−β[H(0)−V2 P ·D]

1Z0

∑e−βµNp

∑{q1...q2Np}

e−β[H(0)−V2 P ·D]

≈1Z0

∑e−βµNp

∑{q1...q2Np}

P e−βH(0)(1 + βV2 P ·D)

1Z0

∑e−βµNp

∑{q1...q2Np}

e−βH(0)(1 + βV2 P ·D)

≈ βV

2D < P 2 >,

(29)

5

where β = 1/kBT ,Np is the number of charge pairs in the system, Z0 is the Grand-Canonical partition function without the external field,

∑{q1...q2Np}

denotes all the

possible combinations of charges in the system.Redefining Pnew = V · Pold and substituting Eq. 29 into Eq. 26, one finds

1

ε= 1− 2π

TaN< P 2 > . (30)

In the simulations the inverse dielectric constant ε−1 is computed by means of Eq. 30.The squared polarization < P 2 > is averaged over the X and Y directions.

4 Results and discussion

In order to explore phase transitions in the system, a series of Grand-Canonical Monte-Carlo simulations was conducted. At each step a pair of opposite charges takenrandomly was forced at equal probability either to leave or to enter the system ife−βδE > r, where r is a random number uniformly distributed on the interval [0, 1)and δE = δEnew − δEold. Each MC pass consisted of L × L = N such steps. In allthe calculations a set of reduced variables was used

T ∗ = kBT/E0;µ∗ = µ/E0;H∗ = H/E0;

M∗ = M/q0;C∗ = Ca3/kB ; ε∗ = ε,(31)

where E0 = q20/a.As is mentioned in the introduction, the search of phase transitions in the system wasbased on the phase diagram discovered for the 2D Coulomb gas model. This diagramis shown in Fig. 1.

Figure 1: Phase diagram of 2D Coulomb gas, adapted from [1].

A number of scans performed at different values of the temperature T ∗ and the chem-ical potential µ∗ led to a conclusion that the system studied and the 2D Coulomb gasdo indeed have a lot in common.At low temperatures and low values of the chemical potential, the medium lacks chargepairs as the energy cost needed to accept a pair is much larger than the energy gaincarried by the pair. As soon as the Coulomb energy between the charges comprisingthe pair becomes equal to the chemical potential, this pair is allowed to enter thesystem. Each new pair leads to even a larger energy gain than the previous one re-sulting in an abrupt growth of the density.In order to distinguish a first-order phase

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transition, the order parameter distribution is typically observed. If a first-order phasetransition takes place, this distribution has a double-peak structure at the coexistenceline. This is demonstrated in Fig. 2 for µ∗ = 1.58367 and T ∗ = 0.115. Apparently,this temperature corresponds to the region close to the tricritical point in Fig. 1 asthe coexistence value of the chemical potential tends to increase with the temperature.However, there is no double-peak distributions in a series of plots for T ∗ = 0.15 inFig. 3. Therefore, this temperature might well be above the tricritical value.

Figure 2: Density distributions at µ∗ = 1.58367. Each point is averaged over 10 programruns, each consisting of 100000 MC passes. Errors are smaller than symbol’s size.L=12.

Figure 3: Density distributions at T ∗ = 0.15. Each point is averaged over 10 programruns, each consisting of 100000 MC passes. Errors are smaller than symbol’s size.L=12.

Figures 4-7 show how ρ∗,√< M∗2 >,C∗ and ε∗−1 vary at a fixed chemical poten-

tial. It appears that the order in which the transitions follow conforms to the phasediagram in Fig 1.

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Figure 4: Density ρ vs chemical potential µ∗ at different T ∗. At each point 50000 MCpasses were done.L=12.

Figure 5: Order parameter M∗2 vs chemical potential µ∗ at different T ∗. At each point50000 MC passes were done.L=12.

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Figure 6: Specific heat C∗ vs chemical potential µ∗ at different T ∗. At each point 50000MC passes were done.L=12.

Figure 7: Inverse dielectric constant ε∗−1 vs chemical potential µ∗ at different T ∗. At eachpoint 50000 MC passes were done.L=12.

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Figure 8: Inverse dielectric constant ε∗−1 vs temperature T ∗ at different µ∗. At each point50000 MC passes were done.L=12.

There are two clear peaks of the specific heat in Fig. 6. The first one might beattributed to the ”metal-insulator” transition while the other one may be a sign ofan Ising-like transition in the system. The third peak related to the KT transitionbetween conducting solid and insulting solid in the case of the 2D Coulomb gas turnsout to be absent here. However, the inverse dielectric constant in Fig. 7 grows backafter plunging. Probably, the rebinding of the charge pairs is so strongly coupled withthe Ising-like transition that both effects take place at the same point.At first sight, a finite value of the dielectric susceptibility in Fig. 7 and Fig. 8 con-fronts Eq. 25. Nevertheless, Eq. 25 permits non-vanishing ε−1 for a finite-size system.When a second-order phase transition happens, the order parameter changes. Thechange of the order parameter M∗ was observed at different values of the chemicalpotential for various temperatures (see Fig. 5). In order to determine critical ex-ponents, the point T ∗ = 0.25 was taken as it is quite far from the tricritical point.Fig. 9 illustrates the linear scaling of C∗max vs lnL, which is in agreement with Isingbehaviour(α = 0). The scaling behaviour of the order parameter M∗2 is shown in Fig.10. The fitting was done following expansion (22) up to the 5th order. The criticalexponents found are β = 0.099±0.004 and ν = 0.92±0.02. These results appear quiteclose to the exact exponents for the 2D Ising model, namely β = 1/8 and ν = 1.0, butthey are not the same as in the model under consideration the long-rage behaviourdominates contrary to nearest-neighbour interactions in the 2D Ising model.

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Figure 9: Maximum value of specific heat C∗max vs lnL at T ∗ = 0.25. Each point is

averaged over 10 program runs, each consisting of 50000 MC passes.L=10,12,14,16.

Figure 10: Finite-size-scaling behaviour of the order parameter M∗2 around T ∗ = 0.25.Each point is averaged over 10 program runs, each consisting of 50000 MC passes. Errorsare smaller than symbol’s size.L=8,10,12,14,16.

5 Conclusion

A new lattice model of an electrically neutral plane carrying positive and negativecharges interacting with the potential being a solution for the 3D Laplace’s equationwas described from the point of view of thermodynamics, in that the Hamiltonian (Eq.10) was obtained. An initial guess relying on the universality hypothesis that phasetransformations in this system and in the 2D Coulomb gas model should be alike wastested through a series of Monte-Carlo simulations. These simulations proved thatthe conjecture was correct, although the possibility of a ”metal-insulator” transitionin bulk matter was doubted on a theoretical basis (Eq.25). It is worth mentioningthat Eq. 25 can predict ”metal-insulator” transitions in a planar system of charges

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interacting with any potential. In addition, Eq. 30, derived in this study, seems tobe able to deliver more accurate values of the inverse dielectric constant in the finitelattice case than traditionally-used Eq. 27. The critical exponent ν computed for thesecond-order transition turned out to be slightly different from the exact value forthe 2D Ising model contrary to the critical exponents α and β which proved to beconsistent with their exact values for the 2D Ising model within the estimated errors.This difference in ν may arise from long-range interactions dominating in the systemexamined.Although this study gives an overwhelming qualitative description of phase transitionsin the system, it lacks quantitative assessment. For instance, the tricritical point needsto be described and then located accurately. In addition,one can perform a numberof extra Monte-Carlo simulations comparing systems of different sizes so as to checkwhether a ”metal-insulator” transition does not exist in an infinite medium.

References

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[2] M. P. Nightingale, Finite-size scaling and phenomenological renormalization,J.Appl. Phys. 53, 7927 (1982).

[3] H. E. Stanley, Introduction to Phase Transitions and Critical Phenomena, OxfordUniversity Press, Oxford, (1971).

[4] W. H. Press, B. Flannery, S. A. Teukolsky, and W. T. Vetter- ling, NumericalRecipes, Cambridge University Press, Cambridge, (1986).

[5] J. M. Kosterlitz and D. Thouless, Ordering,metastability and phase transitions intwo-dimentional systems, J. Phys. C. 6, 1181 (1973); J.M. Kosterlitz, The criticalproperties of the two-dimensional xy model, ibid., 7, 1046 (1974); A. P. Young,On the theory of the phase transition in the two-dimensional planar spin model,ibid., 11, L453, (1978).

[6] P. Minnhagen and G. G. Warren, Superfluid density of a two-dimensional fluid,Phys. Rev. B. 24, 2526 (1981).

[7] G. S.Grest, Critical behavior of the two-dimensional uniformly frustrated chargedCoulomb gas, Phys. Rev. B. 39, 9267 (1989).

[8] J.D. Jackson, Classical electrodynamics, Wiley, New York, (1962).

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