Classification of Phase Transitions
The description and analysis of phase transitions requires the use of thermodynamics and statistical physics, and so we will now summarize the thermodynamics of a many-body system [1] In
In thermodynamics, each state of a system is characterized by specific energy levels The free energy of a system is determined by its temperature (T) and either its pressure (P) or volume (V), defining the system's overall state.
The energy of a system at absolute zero is represented as E, while the energy that varies with temperature and entropy is represented by the Gibbs free energy (G) when considering temperature and pressure, defined as G = E − TS + PV Alternatively, when temperature and volume are the independent variables, the Helmholtz free energy (F) is used, expressed as F = E − TS.
The differentials of these free energies for a simple system are dG=−SdT +V dP, dF =−SdT −P dV (1.1)
In systems with a magnetic moment, an additional term, M dH, is included in the expressions, while a variable number of particles necessitates the incorporation of the term àdN, where à represents the chemical potential Consequently, the first derivatives of the free energy reveal critical physical properties of the system, including specific volume.
(V /N = [1/N]∂G/∂P), entropy (S = −∂G/∂T) and magnetic moment (M = −∂G/∂H), while its second partial derivatives give properties such as the specific heat (C p =T ∂S/∂T =−T ∂ 2 G/∂T 2 ), the compressibility and the magnetic susceptibility of the system.
The effect of changing an external parameter, such as temperature, on the free energy \( G \) of a system is governed by the conservation of energy, preventing sudden energy changes The free energy per unit volume \( g \) relates to \( G \) as \( G = gV \), leading to two scenarios for changes in \( G \): either through a change in free energy density \( g \) (expressed as \( \delta G = V \delta g \)) or through a change in volume \( V \) (expressed as \( \delta G = g \delta V \)) During a phase transition, properties can change uniformly across the system or initially only in specific regions If the new phase emerges through \( \delta G = g \delta V \) in localized areas, it necessitates the formation of stable nuclei, which are sufficiently large regions of the new phase that can grow rather than diminish.
In the context of phase transitions, the change in free energy (δG) can be understood through first and second order transitions For a spherical nucleus of radius r, the volume term is proportional to r³, while the surface term is proportional to r² The critical size (r_c) is defined as the point where these two terms balance, allowing for energy reduction when r exceeds r_c This phenomenon enables the first phase to coexist with the second phase in a metastable state, even above the critical temperature for the transition, exemplifying a first order phase transition Notable examples of this behavior include superheating and supercooling.
In a second-order phase transition, the entire system undergoes a simultaneous phase change, represented by the equation δG=V δg Despite the small difference δg between the properties of the two phases, the original phase cannot persist as a metastable state beyond the critical point, leading to its replacement by a new phase These phases exhibit distinct symmetries; for example, the paramagnetic state lacks a preferred direction, whereas the ferromagnetic state has a defined direction corresponding to the total magnetic moment At the critical point, a sudden change or discontinuity in properties occurs, marking the transition between these two phases.
Phase transitions can be classified based on the order of the derivative of free energy that becomes discontinuous at the transition temperature, as proposed by Ehrenfest In a first-order phase transition, it is the first derivative that exhibits this discontinuity, with a typical example being the transition between different states of matter.
Phase transitions can be categorized into first and second order transitions A first order phase transition, exemplified by water boiling into gas, is characterized by a discontinuity in density In contrast, second order phase transitions maintain continuity in properties like density and magnetic moment, but exhibit discontinuities in their derivatives, such as compressibility and magnetic susceptibility This book will focus primarily on second order phase transitions, which are associated with many intriguing and unusual properties.
Appearance of a Second Order Phase Transition
Before delving into a detailed mathematical analysis, it is important to qualitatively explore an example of a second-order phase transition This discussion will focus on the mean field theory of the paramagnetic-ferromagnetic phase transition in magnetic materials, a concept originally introduced by Pierre Weiss.
In 1907, a significant discovery was made regarding the behavior of magnetic materials, which revealed that as the temperature drops below a critical threshold known as T c, the magnetic moments within these materials suddenly align This phenomenon indicates that the materials are composed of particles, each possessing a magnetic dipole moment.
The maximum magnetic moment of a system with N particles is M₀ = Nμ, achievable when all particle moments are perfectly aligned This alignment occurs at absolute zero temperature (T = 0 K), where thermal energy does not disrupt the orientation of the moments In the presence of a weak magnetic field H, the energy associated with a dipole moment μ is defined by specific equations.
For simplicity, we focus on two orientations of dipoles: parallel (up) and anti-parallel (down) to the magnetic field, denoted as N+ and N−, respectively While more complex analyses can accommodate arbitrary orientations of dipoles, we derive the total magnetic moment of the system in the direction of the field as M = (N+ − N−)à The energy of the system can also be expressed in relation to these orientations.
E = −M H The main assumption of Weiss was that there is some internal magnetic field acting on each of the dipoles, and that this www.pdfgrip.com
The internal magnetic field experienced by a particle is proportional to the ratio of its magnetic moment to that of a reference particle (M/M0) This assumption is valid when considering that the field arises from the magnetic moments of neighboring particles However, it is important to note that this is an approximation, as each particle may not experience an identical magnetic environment.
In the absence of an external field, the effective field acting on each dipole is expressed as \( H_m = \frac{C_M}{M_0} \) According to Boltzmann's law, the number of particles \( N_{\pm} \) with magnetic moments oriented upwards and downwards at temperature \( T \) is proportional to \( \exp(\mp \frac{H_m}{kT}) \), where \( k \) represents the Boltzmann constant This relationship highlights the dependence of particle orientation on the effective magnetic field and temperature.
The critical temperature \( T_c \) is defined as \( T_c = Cà/k \) As illustrated in Fig 1.3, when \( T_c / T < 1 \), the only solution to the equation is the trivial one, \( M = 0 \), indicating that no spontaneous magnetic moment exists when \( T > T_c \) Conversely, if \( T < T_c \), the equation presents two solutions, with the solution where \( M > 0 \) being stable This implies that the system exhibits a spontaneous magnetic moment, characterizing it as ferromagnetic The transition from paramagnetism to ferromagnetism occurs at the critical temperature \( T_c \).
M 0 Fig 1.3 Graphical solution of Eq (1.2) for the magnetization M in the Weiss mean field model.
The Weiss equation, represented as T c = Cà/k, was formulated before the discovery of quantum mechanics, electronic spin, and exchange effects At that time, Weiss could only estimate the strength of the internal field based on known dipole-dipole interactions, resulting in an estimated T c of approximately 1 K Weiss recognized the significance of this estimate specifically in relation to iron.
At approximately 1000 K, T c was noted, leading the author to assert in his paper that while his theory contradicts experimental findings, future research must address this significant three-order magnitude discrepancy.
Despite the existing discrepancies, his paper was successfully accepted for publication, and it is now recognized that his concepts regarding the paramagnetic-ferromagnetic phase transition are fundamentally accurate.
Correlations
In a second-order phase transition, the second derivative of free energy diverges as the transition is approached, exemplified by the magnetic susceptibility, which approaches infinity as temperature T approaches the critical temperature T_c This phenomenon is described by the fluctuation-dissipation theorem, highlighting the relationship between fluctuations and response functions in the system.
Magnetic susceptibility is directly related to the integral of the average product of magnetic moments at two points separated by a distance, highlighting the correlation between these magnetic moments.
The magnetic moment at any given site generally aligns the spin of adjacent sites in the same direction to minimize energy However, this alignment is countered by the effects of entropy, resulting in a finite correlation length (ξ) that persists away from the critical point.
Here, the correlation length ξ has the following physical meaning.
The correlation length quantifies the alignment of spins in a specified direction, indicating how closely other spins align with it In a disordered state, spins are primarily influenced by adjacent random spins, resulting in a small correlation length As a phase transition approaches, the system prepares for a fully ordered state, leading to an increase in the correlation length Near the critical point, the correlation function does not diminish exponentially with distance but instead decays at most as an inverse power of the distance.
Near the critical point, the absence of a small energy parameter and the divergence of correlation length complicate the analysis, as all characteristic lengths become equally significant Unlike the motion of water in the ocean, where different phenomena correspond to various length scales—from Angstroms and microns for molecular interactions to meters for tides and kilometers for ocean currents—critical points do not allow for such a separation, making the study of these systems particularly challenging.
This chapter revisits the inquiry of how long-range correlations can arise from short-range interactions Mathematically, it explores the transfer of mutual influence among distant atoms through exponentially decaying correlations Stanley provides a qualitative answer, indicating that while correlations between two far-apart particles decay exponentially, the number of pathways facilitating these correlations increases exponentially as well This interplay between the decay of correlations and the growth of pathways highlights the complex nature of atomic interactions over long distances.
In phase transitions, long-range power law correlations emerge at the critical point, where negative factors compensate each other However, in one-dimensional systems, the exponential growth of possible paths is reduced to unity, resulting in a negative exponent that prevents ordering and eliminates phase transitions at non-zero temperatures.
Curiously enough, in the Red Army of the former Soviet Union the order given by a officer standing in front of the line of soldiers was
In a peculiar directive, soldiers are instructed to align themselves in a straight line by focusing on the chest of the fourth man This unusual decision stems from the determination that the correlation length is set to four, prompting each soldier to adjust their position according to the alignment of their fourth neighbor in the formation.
Conclusion
Phase transitions are universal phenomena observed in various systems under diverse conditions They are classified into first-order and second-order transitions based on anomalies in the derivatives of free energies at the transition point The fundamental concept of phase transitions was established a century ago through mean field theory.
Three key challenges complicate the theoretical understanding of phase transitions: the non-analytic nature of thermodynamic potentials, the lack of small parameters, and the equal significance of all length scales involved.
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We now consider a microscopic approach to phase transitions, in con- trast to the phenomenological approach used in the previous chapter.
This method, inspired by Gibbs, begins by examining the interactions among particles The initial task involves calculating the mechanical energy of the system, denoted as E_n, for each state n of all particles, a challenge that is typically complex for systems comprising 10^23 particles.
In the framework of classical and quantum mechanics we must then calculate the partition function
(2.1) respectively whereH is the system Hamiltonian, and the Helmholtz free energyF of the system,
In large systems, the summation over particles can be substituted with an integration over phase space, leading to the result that for non-interacting particles, the integral of the coordinates of all N particles equals V^N Consequently, the Helmholtz free energy is expressed as F = -N kT ln(CV), where C remains constant regardless of volume V Utilizing this relationship, we can derive further insights as indicated by Eq (1.1).
The equation of state for an ideal gas simplifies the complex behavior of a system by reducing it to just two degrees of freedom, contrasting sharply with the approximately 10^23 degrees of freedom in the original system This transition allows for a straightforward thermodynamic description, applicable primarily to systems in equilibrium However, the practical application of this method is complicated, as it requires calculating the mechanical energy of numerous many-particle states and summing or integrating over all possible configurations While exact calculations are feasible for only a few simple systems, studying these cases provides valuable insights into the underlying principles of thermodynamics.
Before Onsager's groundbreaking research, it was unclear if statistical mechanics could effectively explain phase transitions, particularly how temperature-related expressions could result in significant singularities at specific temperatures The key insight is that these singularities manifest only in infinite systems—those that reach the thermodynamic limit and possess an infinite number of configurations It is this infinite sum of terms that contributes to the emergence of singularities in the system.
The Ising model is one of the simplest systems used in statistical mechanics, and it will be explored in detail throughout this book This model is founded on three key assumptions that are essential for its analysis.
(1) The objects (which we call particles) are located on the sites of a crystal lattice.
(2) Each particle i can be in one of two possible states, which we call the particle’s spinS i , and we choose S i =±1.
(3) The energy of the system is given by
The Ising Model describes interactions among particles in a lattice, where J is a constant representing the interaction strength In a linear lattice, each particle interacts with its two nearest neighbors, while in a square lattice, it interacts with four nearest neighbors, and in a simple cubic lattice, with six nearest neighbors.
The Ising model, despite its simplicity, is widely applicable in scenarios where entities can exist in one of two states, such as atom occupancy, particle presence, or even binary voting in elections Its foundational assumptions can be expanded in the Potts model, which accommodates multiple states and considers interactions beyond just nearest neighbors, complicating the analysis However, for phase transition problems, the lattice approximation may be less significant, as longer distances, approximately equal to the correlation length, play a crucial role in these transitions.
The personal story of Ising is also of interest [6] In 1924–1926,
Ising was a doctoral student of the famous German physicist Wilhelm
Lenz proposed the model that Ising examined in his thesis, where he correctly demonstrated that one-dimensional systems do not undergo a phase transition However, his assertion regarding two-dimensional systems was later proven incorrect by Onsager in 1944 After completing his thesis, Ising became a high-school teacher and fled to Luxembourg during the Nazi rise to power In 1948, he relocated to the USA, where he taught at a small university and published only two scientific papers throughout his career Notably, upon his arrival in America, he encountered significant interest in the Ising model, which continues to be a crucial subject of research for understanding phase transitions in simple systems.
The one-dimensional Ising model serves as a foundational framework for calculating the partition function in statistical mechanics In this model, we consider a system comprising N particles, which allows us to express the partition function in a specific mathematical form.
For a system with N+ 1 particles, the sum contains one additional term, so that
Since S N S N +1 =±1, it follows that Z N +1 = 2 cosh(J/kT)Z N , and so (sinceZ 1 = 2), by induction,
Hence ifN 1 the free energy is
Since this is a monotonic function of T, with no singularity except atT = 0, the 1D Ising system cannot exhibit a phase transition at any finite temperature.
A notable category of systems features weak interactions between particles that have an infinite range, described by the equation ϕ(r) = −lim γ→ 0 γexp(−γr) This contrasts sharply with the Ising model, which is characterized by strong, short-range interactions.
This model exhibits a phase transition in one dimension, which will be explored in Chapter 7 Notably, a phase transition can occur without interactions among all spins in a system; even a limited number of random long-range interactions can be sufficient.
The Ising Model 17, along with short-range interactions, can effectively create a phase transition, similar to the dynamics observed in the small world model discussed in Chapter 8.
In 1944, Onsager provided the exact solution for the 2D Ising model in a complex and lengthy paper Despite simplifications, a recent proof still required seven pages in the concise classic text by Landau and Lifshitz This article aims to present qualitative arguments to analyze the phase transitions within the 2D Ising model.
Since we want to analyze the role of the energyE and the entropy
Sin order-disorder phase transitions, it is convenient to consider the
The Ising Model 13
Conclusion
Microscopical analysis of phase transitions is primarily applicable to simple models like the Ising model Findings indicate that while one-dimensional systems do not exhibit phase transitions at non-zero temperatures, two-dimensional systems do The onset of a phase transition can be qualitatively understood as an order-disorder transition, driven by the competition between the energy's ordering tendency and the entropy's disordering tendency Under certain simplified assumptions, these insights can yield quantitatively accurate results.
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In Chapter 1, we introduced the mean field approximation as the most straightforward method for facilitating interactions among particles This approach assumes that each particle's environment reflects the average state of the entire system, and this average state is determined in a self-consistent manner.
To grasp Weiss' mean field approximation discussed in Chapter 1, it's beneficial to analyze its application to the Ising model In this model, the energy of a system state, characterized by a specific arrangement of N spins (S1, S2, , SN), is determined on a lattice where each spin Sj can take values of ±1.
In Weiss’ mean field approximation for our system,
S i H m (3.2) whereH m is the effective magnetic field A comparison of Eqs (3.1) and (3.2) shows thatH m =JS j , so that
Mean Field Theory 25
Landau Mean Field Theory
Landau's thermodynamic approach to mean field theory introduces an order parameter, η, to differentiate between system phases, resulting in the Gibbs free energy expressed as G = G(P, T, η) In this framework, η = 0 indicates a disordered state at high temperatures (T > T_c), while η > 0 corresponds to an ordered state at low temperatures (T < T_c) The order parameter is dependent on pressure and temperature, represented as η = η(P, T) Landau's first assumption posits that the free energy has a minimum at ∂G/∂η = 0 and ∂²G/∂η² > 0 His second assumption states that at the phase transition, the order parameter equals zero (η = 0).
Therefore, close to the phase transition point he assumed thatGcan be expanded as a power series in η,
Mean Field Theory indicates that all coefficients are functions of pressure (P) and temperature (T) The condition ∂G/∂η = 0 can only be met universally if α equals zero In magnetic systems where the magnetic moment vector serves as the order parameter, symmetry considerations reveal that a scalar cannot be formed from η³, leading to β being zero Different systems exhibit various forms of order parameters; for example, in liquid-gas phase transitions, the order parameter is the density difference, a scalar quantity In contrast, superfluidity and superconductivity involve a wave function ψ with two components: its real and imaginary parts, or amplitude and phase Additionally, for the anisotropic Heisenberg ferromagnet, the order parameter η will also possess three independent components corresponding to M.
In this study, we focus on systems characterized by a single-component η, where the free energy near the phase transition point can be expressed in a specific mathematical form.
For the sake of simplicity, we consider systems at a constant pressure equal to the critical one,P =P c , and so omitP from the arguments.
Then for arbitrary η the requirement that ∂G/∂η = 0 means that
2Aη+ 4Bη 3 = 0, so thatη= 0 forT > T c , and there is an additional solution η 2 = −A/(2B) for T < T c From the second requirement
∂ 2 G/∂η 2 > 0 it follows that for T > T c , A > 0 while A < 0 for
T < T c It follows thatA(T c ) = 0, and the simplest form for A that satisfies these conditions is A = a(T −T c ), with a > 0 One then
finds that the order parameter η ∼
In this context, the term in η4 is essential because the term in η2 disappears at the critical point However, the temperature dependence of its coefficient B can be disregarded, allowing us to express B as B(Tc, Pc).
We now consider the implications of Eq (3.8) for the free energy.
From the formula for the entropy
The specific heat C=T(∂S/∂T), and so
In Eqs (3.9) and (3.10),S 0 (T) andC 0 (T) arise from the regular part
G 0 (P, T) of the free energy The resulting jump in the specific heat leads to aλshape of the curve ofC(T), and so this transition is also known as aλ-transition.
Finally, let us consider a magnetic system, for which we identify η 2 with the square of the magnetizationM 2 , and examine the effect of an external magnetic fieldH, so that
Since we require that ∂G/∂M = 0, while at T = T c we found that
A= 0, it follows that at the critical temperatureM 3 =H/(4B), i.e.,
The magnetic susceptibility is χ = ∂M/∂H = 1/(2A + 12BM 2 ), so that on substituting the above values for M 2 we find that the magnetic susceptibilityχ is given by
It follows that the magnetic susceptibility becomes infinite atT =T c
In all the above consideration we assumed that in Eq (3.5) the coefficientβ(P, T) vanishes because of symmetry requirements If this
Mean Field Theory 29 is not the case, all the above calculations are correct provided that in additionβ(P c , T c ) = 0 at the critical point, as well asA(P c , T c ) = 0.
However, these two conditions mean that the critical point is an isolated point in the T–P plane — a case which is of no special interest.
The results derived from Eqs (3.7)–(3.13) regarding the temperature dependence of thermodynamic parameters appear to neglect spatial dimensions, which is an oversight that will be addressed later This discrepancy is not unexpected, considering the implications of the mean.
field approach is based, in particular, on the assumption of analyt- icity, which is at least questionable for a function which may have singularities.
First Order Phase Transitions in Landau Theory
While our discussion has focused on continuous (second order) phase transitions, the Landau mean field theory can also effectively describe first order phase transitions, characterized by a jump in the order parameter This theory provides two distinct approaches for analyzing these transitions.
1 By adding a cubic term to the Landau expansion (3.8), one obtains
The assumption thatGas function ofηhas a minimum,∂G/∂η = 0, leads to η 1 = 0, η 2 =−3C
When the system approaches the temperature T 0 from above, the order parameter jumps discontinuously fromη 1 to η 2 , which means that a first order phase transition occurs there.
2 A first order phase transition can be obtained for negative values of the coefficientBin the Landau expansion (3.8) Then, for a minimum www.pdfgrip.com of G(M) to exist, one has to consider the sixth order term in the expansion ofG(M),
G=G 0 +a(T−T c )M 2 − |B|M 4 +DM 6 (3.17) The conditions for a minimum of Eq (3.17) lead to
By using a procedure identical to that used for the analysis of
Eq (3.16), we conclude that a first order phase transition occurs in this case as well.
Landau Theory Supplemented with Fluctuations
In this chapter, we highlighted that the discrepancies between mean field theory and exact results arise from fluctuations To address this, we extend the Landau theory by incorporating these fluctuations The initial step involves adopting a quasi-hydrodynamical approach, transitioning from a discrete lattice to a continuum Consequently, the free energy, G, and the order parameter, M, in ferromagnetism, are expressed as functions of continuous coordinates, G = G(r) and M = M(r).
(3.8) has to be replaced by a local one forG(r), while the global free energy and magnetic moment are obtained by integration over the wholed-dimensional system.
In non-homogeneous systems, the free energy expansions include both thermodynamic variables and their derivatives When considering small inhomogeneities, only the first gradient term (∇M)² is retained, while higher derivatives are disregarded Additionally, the length scale can be adjusted to set the constant to unity, leading to the simplest expression for the free energy density G(r).
According to Boltzmann’s formula, the probability for a fluctuation
P(M) is proportional to exp[−(G−G 0 )/kT], and so
In this analysis, we exclude the term BM 4, as it was introduced solely to prevent the vanishing of G−G 0 at the critical point (A = 0) Instead, we focus on the additional term (∇M) 2, which effectively prevents this issue Consequently, we adopt a Gaussian approximation that incorporates only quadratic terms in M and ∇M.
In order to calculate the above integral, it is convenient to use the
The radial distribution functiong(r) of M(r) is determined by the coefficientsM K [1], g(r) M(R)M (R+r)d R
According to the Gaussian approximation (3.20), the probability of a fluctuation described by a given set{M K }of Fourier coefficients is
, (3.23) where M K 2 = kT /[2(A+K 2 )] On substituting this expression in
Eq (3.22), we find that g(r) = kTexp(i K ã r)
It follows, e.g from integration in the complex plane and the residue theorem, that in three dimensions g(r)∼ exp(−r√
A) r (3.25) www.pdfgrip.com so that the correlation length ξ= 1
At the critical pointA= 0, and so atT =T c the correlation function becomes g(r)∼ 1 r (3.27)
Critical Indices
Understanding phase transitions requires analyzing the behavior of system properties near the critical point This behavior is often described by a critical index, denoted as x, which characterizes the function's singularity at the critical point, expressed as a ∼ |T − T c |^x When x equals zero, the critical index is determined by the formula x = ln(a)/ln|T − T c | In cases where x is not zero, two scenarios arise: one where a becomes a constant at the critical point, leading to a jump singularity, and another where a demonstrates a logarithmic singularity, expressed as a ∼ ln|T − T c | Critical indices also apply to other system properties near the critical point, with a set of these indices and their values for the mean field model and additional models presented in Table 4.1 of the following chapter.
Ginzburg Criterion
The accuracy of mean field theory is influenced by the fluctuations in the order parameter, with the expectation that the approximation holds true when these fluctuations are minimal Ginzburg suggested that mean field theory is applicable when these fluctuations are significantly smaller than the thermodynamic values By utilizing the relevant equation and the relationship M² ∼ A, one can derive results after integrating over angles.
Mean Field Theory 33 that this criterion is fulfilled if
If the dependence of the integral on its upper limit can be neglected, it follows from the substitution y = K√
In the context of critical phenomena, the mean field approximation is applicable when the spatial dimension \(d\) exceeds 4, establishing the upper critical dimension at four For \(d = 4\), the theory indicates that only logarithmic corrections to the mean field results occur, highlighting the nuanced behavior of systems near critical points.
Despite their simplicity, the estimates offer valuable insights By considering the coefficients in inequality (3.28) and substituting values for various phase transition types, we can qualitatively assess the area near the critical point where mean field theory holds true Notably, for the gas-liquid phase transition, the validity of mean field theory becomes evident.
Field theory is applicable within ten percent of the critical temperature, while mean field theory accurately describes superconducting transitions throughout the experimentally accessible region near the critical point This accuracy is linked to the significant correlation length observed in low-temperature superconductors.
Wilson’s -Expansion
In our analysis, we have focused on the local value η(r) at lattice points separated by a distance a, treating them as a continuum and often utilizing integrals instead of sums This coarse-graining method is effective in hydrodynamics, where the only significant length scale is a, provided that r > a However, as Wilson noted, this approach encounters issues in phase transition theory, particularly near the critical point where the correlation length ξ introduces an additional length scale Consequently, the coarse-graining assumption of r > a becomes inadequate unless it is also true that r > ξ, necessitating a new approach to handle the critical region where a < r < ξ, which is crucial as ξ approaches infinity near the critical point.
To address the issue, Wilson proposed that, unlike in hydrodynamics, the coefficients A and B in Eq (3.19) are influenced by the size of the coarse-graining region, denoted as L To determine these dependencies, we examine the change in free energy when transitioning from a region of size L (where a < L < ξ) to a region of size L + ∆L This comparison of free energies between two regions forms the foundation of the renormalization group approach, which will be explored in detail in Chapter 5.
Let us consider a value of ∆L so small that it introduces just one additional mode into a system If we label the modes by their
Fourier componentsK, this means [1] thatV 4πK d−1 ∆K/(2π) 3 = 1, or sinceK ∼1/L that, for V = 1, ∆L∼L d+1 Let us now write the free energyG L+∆L (P, T) in the Landau form (3.19),
G L+∆L (M) = dτ[G 0 +A L+∆L M 2 +B L+∆L M 4 +(∇M) 2 ] (3.29) and compare it with that obtained fromG L (P, T) supplemented by an additional fluctuating modemM 1 with a scaling factor m:
The contribution of the additional mode mM 1 will be taken into account not by the “hydrodynamic” average but by the correct
“Boltzmann” average Thus, one finds that exp
We focus exclusively on the Gaussian distribution, simplifying the exponent on the right side of Eq (3.31) to include only quadratic terms in m This leads us to the requirement for the fluctuating mode.
M 1 (r)dr= 0, M 1 2 (r)dr= 1 (3.32) For small ∆L one can write
A L+∆L =A L +dA dL∆L, B L+∆L =B L +dB dL∆L (3.33) The substitution of Eqs (3.32) and (3.33) into Eq (3.31) leads to the equation dA L dL ∆LM 2 +dB L dL ∆LM 4
On expanding the logarithm in (3.34) in a power series and equating the coefficients ofM 2 and M 4 , we find that dA dL∆L= 3BL 2 −3ABL 4 , dB dL∆L=−9B 2 L 4
Since ∆L∼L d+1 , this gives in terms of the parameter≡4−d,
Equations (3.36) must be valid fora < L < ξ, and so we substitute in themL=ξ On using Eq (3.26) forξ and A=a(T −T c ), one finds from Eq (3.36) that M ∼ [(T −T c )/T c ] β and ξ ∼ (|T −T c |/T c ) −ν where β = 1
The equation presented provides the critical indices' values, as defined previously and detailed in Table 4.1 These critical indices are expressed as power series in relation to a small parameter, leading to the method being referred to as Wilson’s ε-expansion.
Not by chance is Wilson’s article [13] called “Critical phenomena in
3.99 dimensions” If one is brave enough to set the “small” parameter www.pdfgrip.com equal to unity, = 1, then the one finds that for three-dimensional systems the critical indices areβ = 0.3 andν = 0.6.
The above approximate calculation, which gives the -corrections to the mean field critical indices, is a good introduction to the renor- malization group approach, where this transformation leading from
Conclusion
The phenomenological mean field description of phase transitions utilizes an order parameter η and expands the free energy G near the critical point By ensuring that G(η) reaches a minimum, we can derive critical indices that characterize the behavior of thermodynamic parameters close to the critical point A key aspect of mean field theory, as demonstrated by the comparison between Weiss’ mean free theory and the Ising model, is its neglect of fluctuations The Ginzburg criterion supports this approximation for spatial dimensions greater than four, known as the upper critical dimension To incorporate fluctuations into mean field theory, we add the simplest gradient terms to the free energy, allowing for a more comprehensive analysis of phase transitions.
find the temperature dependence of the correlation length and the form of the correlation function at the critical point.
An additional correction to mean field theory arises from the growing correlation length as one nears the critical point This adjustment is addressed by incorporating spatial dependence into the coefficients of the expansion G=G(η).
This leads to the dependence of the critical indices on the parameter
= 4−d, which defines the distance from the upper critical dimen- siond= 4 This -expansion is the simplest form of a renormalization group procedure.
Scaling is a widely recognized approach to understanding how a system reacts to disturbances In the context of physical systems, it is primarily examined through dimensional analysis and the development of dimensionless parameters This article will first explore these concepts through a few straightforward examples.
In analyzing the free fall of a mass m under gravitational force mg without air resistance, we expect the velocity v after falling from a height h to be influenced by the parameters m, h, and g, expressed as v = f(m, h, g) Assuming a power-law relationship, we represent velocity as v = am^α h^β g^γ, where a is a dimensionless constant To ensure dimensional consistency, we note that the dimensions must match on both sides of the equation, leading to the relationships [v] = LT^-1, [h] = L, [m] = M, and [mg] = MLT^-2.
LT −1 = M α L β (LT −2 ) γ (4.1) which has the unique solutionα= 0,β =γ = 1/2, so thatv=a√ gh.
We have established a functional relationship between velocity (v), gravitational acceleration (g), and height (h) without solving the problem directly Our findings indicate that if the height from which an object falls is increased by a factor of four, its resulting velocity will double.
Another example, which is very relevant for engineering applica- tions, is the forceF acting on a sphere of radius R moving through a fluid of viscosityη with velocityv We assume thatF =aR α v β η γ ,
To equate the dimensions in the equation, we analyze the units where force [F] is expressed as MLT−2, resistance [R] as L, velocity [v] as LT−1, and viscosity [η] as ML−1T−1 This leads to the conclusion that the exponents α, β, and γ are all equal to 1, resulting in the simplified equation F = aRvη.
Stokes' law allows for the prediction of air resistance faced by vehicles like cars and airplanes by using model systems with smaller objects or lower velocities However, it is important to consider that these vehicles are not spherical In this context, a key aspect is determining which of the four parameters—radius, velocity, mass, and medium viscosity—is irrelevant, as mechanical quantities are defined by three independent dimensions: mass (M), length (L), and time (T).
Scaling plays a crucial role in effectively displaying theoretical predictions alongside experimental data across varying parameter values on a unified curve This process is grounded in the concept of a homogeneous function f(x, y) of order p, which is characterized by the condition that f(λx, λy) = λ^p f(x, y).
Then, on choosingλ= 1/x, one finds thatf(1, y/x) = (1/x) p f(x, y), so that f(x, y) =x p f
Equation (4.3) indicates that a homogeneous function of two variables can be transformed into a function of a single variable By appropriately selecting variables such as f(x, y)/x^p and y/x, all data can be consolidated into a single curve.
One can also consider [14] a generalized homogeneous function f(λ a x, λ b y) =λ p f(x, y) (4.4) which for a=breduces to a standard homogeneous function (4.2).
Relations Between Thermodynamic Critical Indices
Critical indices, which were introduced in previous chapters, define the behavior of the thermodynamic parameters near the critical point A list of such indices, with their definitions, is given in
The definitions in Table 4.1, originally for a magnetic system, can be adapted to other systems, such as the liquid-gas system, where the magnetic field (H) is substituted with pressure (P) and the magnetic moment (M) with specific volume Thermodynamic parameters are interdependent, governed by relations that establish connections between various critical indices Two examples of these relationships will be examined.
The well-known formula [1] linking specific heats in magnetic sys- tems, c H and c M (or c p and c v for non-magnetic systems) has the following form: c H −c M = T(∂M/∂T) 2 H
Table 4.1 Critical indices for the specific heat (α), order parameter (β), susceptibility (γ), magnetic field (δ), correlation length (ν) and correlation func- tion (η) The indices α, γ, ν and α , γ , ν refer to T > T c and T < T c , respectively.
The results for the mean field and 2D Ising models are exact, while those for the
Exponent Definition Condition Mean field 2D Ising 3D Ising α C ∼ T−T T c c −α(−α
In order for a system to be stable, c M must be positive, and so it follows from Eq (4.5) that c H > T(∂M/∂T) 2 H
(∂M/∂H) T (4.6) or, on substituting the critical indices from Table 4.1, τ −α > τ 2(β−1)+γ (4.7) where τ = |T −T c |/T c Close to the critical point τ 1, and so
Eq (4.7) leads [15] to the following relation between the critical indices α+β+ 2γ ≥2 (4.8)
Another inequality can be obtained as follows [16] ForT 1 < T c ,
(4.9) However, since the stability condition requires that c M ∼ −
∂T must be a decreasing function of T, and so
M must be an increasing function of T It follows then from Eq (4.9) that
For T < T c , a magnetic moment M(T) appears spontaneously, so that G(T, M(T)) and S(T, M(T)) are fully determined by the temperatureT, and so Eq (4.10) can be rewritten in the form
On adding Eqs (4.12) and (4.11), we obtain
The inequalities (4.8) and (4.14) follow from thermodynamics, and so are rigorous In order to derive additional relations between critical indices one have to use some approximations.
Scaling Relations
This chapter explores various approximate methods for deriving scaling relations, focusing on Kadanoff's innovative approach of introducing a new "block" lattice connected to the original "site" lattice We apply this method within the context of thermodynamics, setting the stage for its application to Hamiltonians, which will be further examined in the subsequent chapter on renormalization group theory.
The Kadanoff construction shown in Fig 4.1 (“scaling hypothesis”) consists of the following three stages.
(1) Divide the original Ising site lattice with lattice constant of unit length into blocks of size L less than the correlation length ξ, i.e., 1< L < ξ.
In this approach, we simplify the model by replacing the L spins within each block with a single effective spin, denoted as i = ±1 The value of this spin is determined by the majority sign of the spins in the block Given that the size L of the block is smaller than the correlation length ξ, it is anticipated that the majority of spins in each block will align in the same direction, making this majority rule a valid and effective approximation.
(3) Return to the original site lattice by dividing all lengths byL. www.pdfgrip.com
Fig 4.1 The Kadanoff construction for a square lattice, with blocks of nine sites.
This procedure aims to reduce the system's degrees of freedom through coarse graining, averaging over smaller degrees of freedom The implications of this approach are significant, as comparing the site lattice and block lattice problems—even without knowing their exact solutions—can yield valuable insights.
The thermodynamic properties of a system are influenced solely by temperature and the external magnetic field, which dictate its free energy In this context, we examine a site lattice subjected to a dimensionless magnetic field \( h \) and in thermal contact with a bath at temperature \( T \), represented dimensionlessly as \( \tau = |T - T_c| / T_c \) Corresponding parameters for a block lattice are denoted as \( h \) and \( \tau \) It is evident that if the site lattice has no external field (\( h = 0 \)), the block lattice will also have no field (\( h = 0 \)), and similarly for \( \tau \) Considering this relationship, and recognizing that the size of the block lattice is characterized by \( L \), Kadanoff proposed the scaling relations \( \tau = L^x \tau \) and \( h = L^y h \).
The free energy per site for the site lattice is denoted as f(τ, h), while f(τ, h) represents the free energy per block of the block lattice It can be concluded that the free energy of a block is the aggregate of the free energy of the individual sites that comprise it, applicable in all dimensions.
The functional equation can be solved by substitution, yielding the solution f(τ, h) = τ d/x ψ(τ /h x/y ), where ψ represents an arbitrary function This characteristic allows us to establish the relationship between critical indices related to various properties of the system, even in the absence of knowledge about the function ψ.
M ∼[∂f /∂h] h=0 ∼τ d/x [τ /h 1+x/y ]ψ (z), where z = τ /h x/y The only way that this can be independent of h as h → 0 is that for z −→ ∞, ψ (z) ∼ z −1−y/x , in which case
M ∼ τ (d−y)/x Hence, the critical index β defined in Table 4.1, is given by β = d−y x (4.17)
It then follows that the magnetic susceptibility as h → 0, i.e., asz→ ∞, χ∼∂M/∂h∼τ d/x [τ /h 1+x/y ] 2 ψ (z), and thusψ (z)∼z −2−2y/x , so that χ∼τ d/x+2−2−2y/x , and hence γ = 2y−d x (4.18)
The specific heat ash →0, i.e., as z → ∞,C ∼∂ 2 f /∂τ 2 ∼τ d/x−2 , and so α= 2−d x (4.19)
At the critical point,τ = 0, the temperature-independent magnetic moment M ∼ τ d/x [τ /h 1+x/y ]ψ (z) requires ψ (z) ∼ z −(1+d/x) as z→0 A new critical index δ is defined by M ∼ h 1/δ at τ = 0.
It then follows that δ = y d−y (4.20) www.pdfgrip.com
To derive critical indices, we analyze the correlation function g(r, τ) within site and block lattices The initial step involves establishing the relationship between the average spins S of the site lattice and à of the block lattice This is achieved by comparing the average energy per block of L^d spins subjected to an external magnetic field h in the site lattice with the energy per block in the corresponding block lattice, where the external magnetic field is represented as h = L^y h.
It follows thatS ∼L y−d à , or for the correlation functionsg s and g à in the site and block lattices, respectively, g s (r, τ) =L 2(y−d) g à r
The solution to the functional equation (4.22) can be expressed as g_s(r, τ) = r²(y−d) Ψ(rτ^(1/x)), where Ψ is an arbitrary function At the critical point, τ = 0, Ψ(z) approaches a constant, leading to g_s(r, τ) ∼ r²(y−d) This relationship aligns with the definitions of critical indices outlined in Table 1.
The characteristic length enters Eq (4.23) in the formrτ 1/x , i.e., for the correlation lengthξ one obtainsξ ∼τ −1/x , or
The equations (4.17)–(4.20) and (4.24)–(4.25) establish six relationships among critical indices By eliminating the unknown parameters x and y, four key relationships emerge: α + β + 2γ = 2, βδ = β + γ, γ = ν(2 − η), and α = 2 − dν.
The analysis indicates that for temperatures above and below the critical temperature (T > Tc and T < Tc), the critical indices can be defined and calculations can be performed, demonstrating that the limiting behavior of thermodynamic functions is symmetric around Tc, specifically, α = α, ν = ν, and γ = γ.
Equations (4.26) and (4.27) show that, within the framework of the scaling hypothesis, there are seven relations between the nine ther- modynamic critical indices, i.e., only two indices are independent.
These two remaining indices will be found in the next chapter by the use of the renormalization group theory.
Since the scaling hypothesis is based on some postulates, it is worthwhile to check it using the values of critical indices listed in
Table 4.1 One can immediately see that the critical relations (4.26) are satisfied for the exact Onsager solution of the two-dimensional
The Ising model plays a crucial role in understanding phase transitions, particularly in the context of mean field theory Accurate experimental measurements of critical indices are essential for validating theoretical predictions and enhancing our comprehension of critical phenomena.
Dynamic Scaling
In addition to the scaling parameters that characterize a system's static properties near the critical point, its dynamic properties also display singularities linked to dynamic critical indices These dynamic properties illustrate how a system approaches its equilibrium state and cannot be analyzed solely through the mean field expression for the static order parameter Instead, it is essential to consider a system that incorporates fluctuations in the order parameter, utilizing the free energy framework established in Chapter 3 of Landau's theory, which is enhanced by these fluctuations.
In this section, we will focus on the (∇η)² term while disregarding the η⁴ term, as the former is adequate for establishing the dependence of free energy on η at T = T_c, particularly when A = 0 In the context of a static equilibrium system, where η remains constant over time, certain conditions must be met.
∂G/∂η = 0 and ∂ 2 G/∂η 2 > 0, and so we postulate that when the system is not far from equilibrium, dη dt =−ΓδG δη =−Γ
The Landau–Khalatnikov equation, represented as ∂(∇η), is justified by the observation that both sides equal zero at equilibrium, suggesting a proportional relationship in states near equilibrium By selecting a time scale where Γ = 1/2, the equation simplifies to dη/dt = ∇²η - Aη.
As in the previous chapter, we use the Fourier transform η K of η, so that η K (t) =η K (0) exp[−t(A+K 2 )] =η K (0) exp{−t[a(T c −T) +K 2 ]}
(4.31) where we have used the mean field value of A as derived in the
Landau theory If one denotes by τ K the relaxation time for mode
The relationship η K (t) = η K (0) exp[−t/τ K ] indicates that as K approaches 0, the relaxation time τ 0 becomes infinite as the temperature T nears the critical temperature T c For other values of K, τ K behaves as 1/K² as T approaches T c This results in a significantly slow approach to equilibrium near the critical temperature, a phenomenon referred to as critical slowing down Therefore, it is crucial to ensure that measurements of the thermodynamic properties in this temperature range are conducted on systems that are in equilibrium.
The results derived from the mean field approximation can be extended to a more general case by utilizing the equation τ K = f(K, τ) = τ z Ψ(Kτ − ν) Here, τ represents the reduced temperature, while z denotes the dynamic scaling exponent This formulation allows τ K to be expressed as a power proportionality, maintaining coherence with the foundational principles outlined in the previous equations.
Scaling 47 of τ, Ψ(0) must be a constant It then follows that for K = 0, τ 0 ∼ τ z For K= 0, the argument of Ψ tends to infinity as τ →0, i.e., as T →T c , and so τ K can only remain finite if Ψ(y) ∼ y z/υ , in which case τ K ∼ K z/ν In the mean field case, we found that τ 0 ∼1/τ as τ →0 and that τ K ∼1/K 2 , so that z=−1 andz/ν =−2, which leads to the resultν= 1/2, in agreement with what we found earlier Therefore, in addition to the thermodynamic critical indices, a new critical index z appears in the description of dynamic phenomena.
Conclusion
The scaling hypothesis simplifies the calculation of critical indices that characterize thermodynamic functions near critical points Near the critical point, the correlation length ξ becomes significantly larger than the inter-particle distance a, leading to the formation of "block" lattices of size L, where the condition a < L < ξ holds It is reasonable to assume that within each block, spins predominantly align in one direction, mirroring the behavior of the original site lattice Consequently, critical phenomena across these block lattices will exhibit similarity, provided that external parameters like temperature and magnetic field are adjusted using a power-law relationship with respect to L While the specific indices that govern these parameter changes remain unidentified, all other critical indices can be derived from these two fundamental indices.
The scaling relations derived from this hypothesis align with the findings from the exact solution of the two-dimensional Ising model and the mean field theory To effectively analyze dynamic phenomena near critical points, it is essential to understand an additional dynamic critical index, which indicates a deceleration in the approach to equilibrium.
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The Renormalization Group 49
Fixed Points of a Map
The renormalization group approach allows for the transformation of parameters K of the Hamiltonian H from a site lattice to those of a block lattice through the rule K = f(K) This process can be iteratively applied, treating the smaller block lattice as a site lattice for increasingly larger blocks, facilitating a systematic analysis of the system's behavior across different scales.
K =f(K ) = f[f(K)], and so on indefinitely for larger and larger
In the study of dynamical systems, a map defines a specific rule connecting two successive elements in a sequence, represented as x_{n+1} = f(x_n) As n approaches infinity, this process may reach a point where it stabilizes, resulting in a fixed point x* where x* = f(x*) A well-known example of a non-linear map is the logistic map, given by the equation x_{n+1} = f(x_n) = a x_n (1 - x_n) for 0 < x < 1 and a > 1 The fixed points of this map are determined by solving the equation x* = a x* (1 - x*).
The quadratic map features two fixed points, x* = 0 and x* = 1 - 1/a, as illustrated in Fig 5.1, where these points are found at the intersection of the parabola f(x_n) and the diagonal of the unit box Starting from an initial point x_0, a vertical shift to x_1 = f(x_0) occurs on the parabola, followed by a horizontal shift to the diagonal, establishing x_1 as the new starting point for subsequent approximations This process ultimately converges to the stable fixed point x* = 1 - 1/a To analyze the stability of these fixed points, we introduce a perturbation ξ_n around x*, represented as x_n = x* + ξ_n.
Fig 5.1 Approach to the fixed point for the logistic map.
The Renormalization Group method allows for the substitution of variables in Eq (5.1) By expanding f(x_n) in a Taylor series around x*, and retaining only the linear terms in the small parameters ξ_n, we derive the relationship ξ_{n+1} = (df/dx)_{x=x*} ξ_n.
In dynamical systems, the stability of a fixed point is determined by the derivative |df/dx| at that point If |df/dx| at the fixed point x* is less than 1, the fixed point is stable and will attract nearby points, causing the distance from the fixed point to decrease over time Conversely, if |df/dx| is greater than 1, the fixed point is unstable For the logistic map, represented by the equation (df/dx) = a(1 - 2x), the stability varies depending on the value of a Specifically, when a > 1, the fixed point at x* = 0 is unstable, while the fixed point at x* = 1 - 1/a is stable, as it results in |df/dx| being less than 1.
Basic Idea of the Renormalization Group
The above procedure is the basic idea of the renormalization group.
Assuming that a fixed point is reached at some step of the iteration procedure for
K =f(K ) =f[f(K)] =ã ã ã , namely K ∗ = f(K ∗ ), where K = J/(kT), one finds the value of
The critical temperature is defined by the equation K* = J/(kT_c), indicating that the accuracy of this estimate improves as the size of the block lattice approaches the correlation length, which is achieved through increased iteration steps.
The RG procedure is crucial as it allows for the determination of both the critical temperature and critical indices By applying the RG relation K = f(K) in proximity to a fixed point, one can derive significant equations that enhance our understanding of critical phenomena.
Instead of attempting to solve the problem for a single lattice, which is often unfeasible, we analyze the solutions for lattices of varying sizes The free energy per particle, denoted as g(K), remains consistent across different lattice sizes, specifically for a block of L^d sites in d dimensions, where g(K₁) = g(K₂) or Z(K₁, N) = Z(K₂, N/L^d) Similar properties among these lattices can only be expected if the block size is smaller than the system's correlation length At a fixed point, however, this similarity holds for blocks of any size, allowing us to associate the fixed point with the critical point of a phase transition, where the correlation length becomes infinite The presence of such a fixed point indicates that a phase transition occurs, and if a value K is near the fixed point K*, we can reformulate the equation for T = J/kK accordingly.
On comparing Eq (5.5) with the main scaling relation τ = L x τ, whereτ =|T −T c |/T c ,one finds that x= ln df dK
If there is more than one external parameter, sayK andH, then the functionf will depend on more than one parameter, f =f(K, H), and so we will have two RG equations
K =f 1 (K, H), H =f 2 (K, H) (5.7) and the fixed pointsare defined by the solutions of equations
On repeating all the procedure leading to (5.5), one finds that
Finally, on diagonalizing the 2×2 matrix and writing the diagonal elements in the formL x andL y , one finds the critical indicesxandy.
It should be noted that the procedure described here is the sim- plest real space form of the renormalization techniques In many
The Renormalization Group 53 applications, the renormalization group technique is used in momen- tum space rather than in real space.
RG: 1D Ising Model
We begin our analysis of the RG method by exploring the one-dimensional Ising model, recognizing that it lacks a phase transition.
For the original site lattice, the partition function (2.1) for the one- dimensional Ising model (2.3), withK =J/(kT),has the form
The partition function Z(K, N) is expressed as exp[K(S1S2 + S2S3 + + SN-1SN)], where the summation includes all neighboring pairs among N spins In a block lattice configuration, we simplify the model by omitting spins at even-numbered sites and combining two spins into one, resulting in N/2 effective spins We denote these spins as S2j-1 and focus on calculating the partition function Z(K, N/2), which sums exclusively over the odd spins.
The initial terms in the exponents of Eq (5.10), after summing over \( S_2 = \pm 1 \), can be expressed as \( \exp[K(S_1 + S_3)] + \exp[-K(S_1 + S_3)] = F \exp[K(S_1 S_3)] \) This approach can be similarly applied to each pair of subsequent terms, effectively eliminating all spins with even indices from the summation Consequently, this allows for a comparison of the partition functions for both site and block lattices.
2ln[cosh(2K)], F = 2 cosh(2K) (5.14) www.pdfgrip.com
Let us now compare the free energy per site, f 1 (K), for the site lattice and that,f 2 (K ), for the block lattice Since f 1 (K) = (1/N) ln[Z(K, N)] and f 2 (K ) = (2/N) ln[Z(K , N/2)], it follows from Eq (5.12)–(5.14) that f(K ) = 2f(K)−ln
By continuously doubling the size of the blocks, the interactions between distant spins can be minimized, leading to a very small value of interactions Consequently, when transitioning from small to large interactions, one can begin with a minimal interaction value, such as K = 0.01.
The relationship Z(K) is approximately equal to 2 raised to the power of N, while the function f is approximately the natural logarithm of 2 By applying these equations, it is possible to determine the renormalized interaction and the corresponding free energy Notably, research indicates that achieving the exact result from K = 0.01 requires eight distinct steps.
The one-dimensional Ising model lacks a phase transition at finite temperatures, serving primarily as a straightforward example to illustrate the functionality of the renormalization group (RG) method In contrast, we now focus on the two-dimensional Ising model, which presents a different scenario.
Chapter 2 that a phase transition does occur.
RG: 2D Ising Model for the Square Lattice (1)
In the 2D Ising model on a square lattice, the partition function accounts for interactions between a spin and its four nearest neighbors, complicating the analysis To simplify this, we apply a method similar to that used for the 1D Ising lattice by creating a block lattice that excludes the nearest neighbors of half the sites, as illustrated in Fig 5.2, and then summing the interactions.
The Ising model on a square lattice with alternate sites removed presents a complex interaction scenario The spin S₀ at site (0,0) interacts with spins S₁, S₂, S₃, and S₄ at neighboring sites (0,1), (1,0), (0,-1), and (-1,0), respectively This interaction leads to 16 possible configurations for S₁, S₂, S₃, and S₄, resulting in four distinct equations, in contrast to the two equations observed in a one-dimensional lattice To accommodate these interactions in the partition function of a block lattice, we introduce three different interaction terms, represented as exp[K(S₁ + S₂ + S₃ + S₄)] + exp[-K(S₁ + S₂ + S₃ + S₄)].
The choice of using K 1/2 instead of K 1 in the equation is due to the recurrence of spin products like S 1 S 2 in other terms of the block lattice's partition function To find the values of the four unknowns—f, K 1, K 2, and K 3—we equate both sides of the equation for four distinct combinations of (S 1, S 2, S 3, S 4) and then take the logarithm of each side Specifically, for the combination (S 1, S 2, S 3, S 4) = (1, 1, 1, 1), we derive from Eq (5.16) that ln[2 cosh(4K)] equals lnF plus 2K 1, 2K 2, and K 3.
Similarly, for (S 1 , S 2 , S 3 , S 4 ) = (1,1,1,−1) one obtains ln[2 cosh(2K)] = lnF −K 3 (5.18) while (S 1 , S 2 , S 3 , S 4 ) = (1,1,−1,−1) leads to ln 2 = lnF−2K 2 +K 3 (5.19) and (S 1 , S 2 , S 3 , S 4 ) = (1,−1,1,−1) to ln 2 = lnF−2K 1 + 2K 2 +K 3 (5.20)
This set of four linear equations for the four unknowns has the unique solution
Unlike the one-dimensional Ising model, the two-dimensional Ising model exhibits more complex interactions within a block lattice, making it unsuitable for precise renormalization group (RG) analysis.
To advance in our analysis, we begin with a basic approximation by neglecting the terms K2 and K3, which leads us back to the one-dimensional equations that do not exhibit phase transitions For a meaningful physical interpretation, we consider the four-spin interaction to be sufficiently weak to disregard The interactions represented by K1 and K2 pertain to nearest and next-nearest neighbor interactions within the block lattice, both of which promote parallel spin alignment Thus, we can express the combined effect as K1 + K2 = K.
Finally, as discussed above, a phase transition occurs when K is a fixed point of the transformation, i.e., K =K =K c For the 2D
Ising model, this leads toK c = 0.50698, while Onsager’s exact result was K c = 0.44069 [4], so that the error is about 13% This is not
The Renormalization Group 57 too bad for such an approximate theory, and suggests that its basic physical ideas are correct.
RG: 2D Ising Model for the Square Lattice (2)
In this article, we explore an alternative configuration of the block lattice by grouping five adjacent spins into a block, as illustrated in Fig 5.3 The overall spin of the block, denoted as \( a \), is established using the majority rule, defined mathematically as \( a = \text{sgn} \).
As can be seen, the block lattice is also a square lattice with a dis- tance between blocks of L = √
5 For à a = 1, there are sixteen possible values of S a , which can be grouped into the six different configurations shown in Fig 5.4, having characteristic energies 0, ±2K,and ±4K For à a =−1 one obtains the same energies as for à a = 1 with signs reversed.
In order to calculate the partition functionZ à for the Hamiltonian
H à of the block lattice it is convenient to writeH à =H 0 +V, where
H 0 contains the interactions between the spins within a block andV a
Fig 5.3 Ising model on a square lattice with five sites in a block. www.pdfgrip.com
Fig 5.4 The different spin configurations for the five sites in a block on the square lattice, where black and white circles represent spins of opposite signs.
The number of states in each configuration and their energy are shown. those between spins on adjacent blocks, so that
Z à exp[−βH à ] exp(−βH o ) exp(−βV) exp(−βH o ) exp(−βH o ) exp(−βV) exp(−βH o )
(5.24) where the first factor in the final expression in Eq (5.24) is related to the independent blocks, and therefore reduces to a multiple of the partition functionZ 0 (K) of a single block exp(−βH o ) = [Z 0 (K)] N (5.25)
The sixteen configurations that contribute to Z 0 (K), which are shown in Fig 5.4 forà a = 1, lead to
The second factor in Eq (5.24) is the weighted average of exp(−βV) with weight function exp(−βH o ), which will be denoted by exp(−βV).
The formula (5.24) is still exact However, we only evaluate exp(−βV) approximately, by the so-called cumulant expansion In
The Renormalization Group 59 this one writes e W
(5.27) For|x|