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Title
Abstract
Figures
Statistical Theory
Summary of Exact Calculations
Summary
Scaling Hypotheses
Scaling Hypothesis for Order Parameter
Scaling Hypothesis for Free Energy Density
Scaling Hypothesis for Correlation Function
Scaling Hypotheses 71
Summary of RG Ideas
Diagramatic Method
Migdal -Kadanoff Method
References

Renormalization Group Theory of Critical Phenomena

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Renormalization group theory provides a framework for understanding phase transitions in various physical systems. Introduced by Kenneth Wilson, it addresses the universality seen in distinct systems at critical points and has become a primary tool in many-body physics. This monograph details the fundamental concepts of renormalization group theory, offers applications to magnetic critical phenomena, and aims to serve as an introductory resource for students and researchers in the field.

Figures (48)
Figure 1.1: P - T Phase Diagram for Gas-Liquid Transition.
Figure 1.1: P - T Phase Diagram for Gas-Liquid Transition.
At 31°C, one observes the phenomenon of critical opalescence. When COz, which is transparent to visible light, is illuminated, an orange tinge is found if viewed from the forward direction and a bluish tinge is observed from a normal direction. If T is raised by a small amount (point c in the figure), opalescence disappears and the two phases, vapor and liquid, also disappear leaving behind a homogeneous fluid. One can go from point-1 to point-2, in the phase diagram (of Figure 1.1), along a path of continuous density change or along a path with a discontinuous density change as the vapor pressure curve is crossed. As one approaches the critical point along the vapor pressure curve, liquid density (g,) and vapor density (p,) come closer as shown in Figure 1.4. Again, the dependence of p; — p, on T, — T is found to be
At 31°C, one observes the phenomenon of critical opalescence. When COz, which is transparent to visible light, is illuminated, an orange tinge is found if viewed from the forward direction and a bluish tinge is observed from a normal direction. If T is raised by a small amount (point c in the figure), opalescence disappears and the two phases, vapor and liquid, also disappear leaving behind a homogeneous fluid. One can go from point-1 to point-2, in the phase diagram (of Figure 1.1), along a path of continuous density change or along a path with a discontinuous density change as the vapor pressure curve is crossed. As one approaches the critical point along the vapor pressure curve, liquid density (g,) and vapor density (p,) come closer as shown in Figure 1.4. Again, the dependence of p; — p, on T, — T is found to be
Figure 1.5: Specific Heat Vs Temperature.
Figure 1.5: Specific Heat Vs Temperature.
Figure 1.6: Magnetization Vs Applied Field. 14 Renormalization Group Theory
Figure 1.6: Magnetization Vs Applied Field. 14 Renormalization Group Theory
where > denotes summation over the nearest neighbours of 7 and Zn, is the number of nearest neighbours of any lattice point. Then H, can be written as The effective hamiltonian can be rewritten in a slightly different form. Not« that the partition function can be expressed as
where > denotes summation over the nearest neighbours of 7 and Zn, is the number of nearest neighbours of any lattice point. Then H, can be written as The effective hamiltonian can be rewritten in a slightly different form. Not« that the partition function can be expressed as
40 Renormalization Group Theory can now be readily obtained and the result is
40 Renormalization Group Theory can now be readily obtained and the result is
Figure 3.1: Scaled Magnetization Vs Scaled Field. Then, for T + T.,, Widom’s hypothesis shows that
Figure 3.1: Scaled Magnetization Vs Scaled Field. Then, for T + T.,, Widom’s hypothesis shows that
Figure 3.2: Ising Lattice and Cells. laws. Consider the Ising model and imagine that the lattice is divided into cells as shown in Figure 3.2. The side length of a cell is gay where ag is the lattice spacing. The parameter qg is chosen such that ag < gay < &, the correlation length. Thus there are a number of lattice points in each cell.
Figure 3.2: Ising Lattice and Cells. laws. Consider the Ising model and imagine that the lattice is divided into cells as shown in Figure 3.2. The side length of a cell is gay where ag is the lattice spacing. The parameter qg is chosen such that ag < gay < &, the correlation length. Thus there are a number of lattice points in each cell.
Figure 4.1: Steps involved in RG Transformation. -l1 RG for 1-D Ising Model The hamiltonian for the nearest neighbor 1-D Ising model is
Figure 4.1: Steps involved in RG Transformation. -l1 RG for 1-D Ising Model The hamiltonian for the nearest neighbor 1-D Ising model is
where the summation in the last term has been replaced by a volume integral. The last expression gives 2 Recursion Formulae
where the summation in the last term has been replaced by a volume integral. The last expression gives 2 Recursion Formulae
The factor a(q) = q@ and it is known that a and the exponent 77 are related as Thus the first step of RG transformation is completed. Now, the unit of length has to be changed by a factor q and the scale factor a(q) for changing the magnitude of s’(x) is to be introduced. These changes lead to the following replacements. the magnitude of s’(x) is to be introduced. ‘These changes lead to the following
The factor a(q) = q@ and it is known that a and the exponent 77 are related as Thus the first step of RG transformation is completed. Now, the unit of length has to be changed by a factor q and the scale factor a(q) for changing the magnitude of s’(x) is to be introduced. These changes lead to the following replacements. the magnitude of s’(x) is to be introduced. ‘These changes lead to the following
Figure 5.2: Fixed Point Function. 116 Renormalization Group Theory is determined by solving the non-linear integral equation
Figure 5.2: Fixed Point Function. 116 Renormalization Group Theory is determined by solving the non-linear integral equation
The linearised transformation equation then becomes
The linearised transformation equation then becomes
124 Renormalization Group Theory Therefore, the coefficients t; and tj can be expressed as
124 Renormalization Group Theory Therefore, the coefficients t; and tj can be expressed as
With these results, the renormalised hamiltonian in the first order approxi- mation can be calculated easily. 6.2 First Order Approximation to H's]
With these results, the renormalised hamiltonian in the first order approxi- mation can be calculated easily. 6.2 First Order Approximation to H's]
Since the variables in the summation are dummy variables, one finds that the first and last terms are the same while the remaining four are similar to each other. That means Since the variables in the summation are dummy variables, one finds that the
Since the variables in the summation are dummy variables, one finds that the first and last terms are the same while the remaining four are similar to each other. That means Since the variables in the summation are dummy variables, one finds that the
In this case, there are two s° in group-1 and group-2. A typical diagram for H? is shown in Figure 6.5. First of all, the s” terms in group-1 and group-2 can be connected among themselves. Various connected diagrams obtained in this way are shown in Figure 6.6. As seen from the calculation of < H; >, each diagram contributes a factor 144 Renormalization Group Theory
In this case, there are two s° in group-1 and group-2. A typical diagram for H? is shown in Figure 6.5. First of all, the s” terms in group-1 and group-2 can be connected among themselves. Various connected diagrams obtained in this way are shown in Figure 6.6. As seen from the calculation of < H; >, each diagram contributes a factor 144 Renormalization Group Theory
Now, consider the case when two s” in group-1 is at one end of the broker line and the two s” in group-2 are at different ends of the broken line. The first arrangement can be done in 2 ways while in the second there are 4 ways Thus there are 2 x 4 x 2 = 16 (the last 2 is for interchange of ends of the joining lines) ways. In a similar way, the s” in group-1 can be put at different ends of the broken line while the s” in group-2 are at the same end of the broken line. This gives another 16 ways of making connections. Thus a tota of 32 ways of connections are obtained. Eight of these cases are shown ir Figure 6.8. Another 8 diagrams arise by choosing for s” the solid lines at th left of the broken line of group-1. Then a similar 16 diagrams result with twe s” at the same end of the broken line of group-2. Nisur Ehe#beazn e!! af orn) and orn? ean he nivteatk differant anne of thr 146 Renormalization Group Theory
Now, consider the case when two s” in group-1 is at one end of the broker line and the two s” in group-2 are at different ends of the broken line. The first arrangement can be done in 2 ways while in the second there are 4 ways Thus there are 2 x 4 x 2 = 16 (the last 2 is for interchange of ends of the joining lines) ways. In a similar way, the s” in group-1 can be put at different ends of the broken line while the s” in group-2 are at the same end of the broken line. This gives another 16 ways of making connections. Thus a tota of 32 ways of connections are obtained. Eight of these cases are shown ir Figure 6.8. Another 8 diagrams arise by choosing for s” the solid lines at th left of the broken line of group-1. Then a similar 16 diagrams result with twe s” at the same end of the broken line of group-2. Nisur Ehe#beazn e!! af orn) and orn? ean he nivteatk differant anne of thr 146 Renormalization Group Theory
Figure 6.9: Diagrams in < H,[s’, s!”? >
Figure 6.9: Diagrams in < H,[s’, s!”? >
is 4x 4 = 16 times the factor obtained from a particular choice of s” in group-1 and group-2. The diagrams resulting from a specific choice of s” are shown in Figure 6.10. Here, there are two closed loops and four open loops thus giving a factor (2n + 4). Then the total factor is 16(2n + 4). The first diagram of Figure 6.10 can be evaluated as
is 4x 4 = 16 times the factor obtained from a particular choice of s” in group-1 and group-2. The diagrams resulting from a specific choice of s” are shown in Figure 6.10. Here, there are two closed loops and four open loops thus giving a factor (2n + 4). Then the total factor is 16(2n + 4). The first diagram of Figure 6.10 can be evaluated as
the diagrams of Figure 6.15 which again give the factor 2n + 4. One more possibility of similar connection yields another factor 2n + 4. Thus the weight obtained is 2(2n?+4n)+4(2n+4) = 4(n+2)?. However, an additional factor 2 is found by interchanging the ends of the joining lines in each of the diagrams. Hence the total weight is 8(n + 2)?. The first diagram from Figure 6.15 can be evaluated as 152 Renormalization Group Theory
the diagrams of Figure 6.15 which again give the factor 2n + 4. One more possibility of similar connection yields another factor 2n + 4. Thus the weight obtained is 2(2n?+4n)+4(2n+4) = 4(n+2)?. However, an additional factor 2 is found by interchanging the ends of the joining lines in each of the diagrams. Hence the total weight is 8(n + 2)?. The first diagram from Figure 6.15 can be evaluated as 152 Renormalization Group Theory
The expression for the exponent 7 obtained earlier is 7 = u* 2E. Substituting
The expression for the exponent 7 obtained earlier is 7 = u* 2E. Substituting
have been used. k3; (1 < i < 4) are the cartesian components of k3 and 6, is the angle between kg and the unit vector iy. Hence kz; is k3 cos(0,). The projection of k3 perpendicular to the i, axis is k3sin(@,) and it is a 3- dimensional vector which is resolved in the usual manner. In writing the integral J;, ko is taken along the i, axis and the four dimensional volume element has been used. On multiplying the r.h.s with JJ sin?(0,)d0, = 7/2, the angular part of the integral can be factored out. Then, with the usual notation, one
have been used. k3; (1 < i < 4) are the cartesian components of k3 and 6, is the angle between kg and the unit vector iy. Hence kz; is k3 cos(0,). The projection of k3 perpendicular to the i, axis is k3sin(@,) and it is a 3- dimensional vector which is resolved in the usual manner. In writing the integral J;, ko is taken along the i, axis and the four dimensional volume element has been used. On multiplying the r.h.s with JJ sin?(0,)d0, = 7/2, the angular part of the integral can be factored out. Then, with the usual notation, one
162 Renormalization Group Theory
162 Renormalization Group Theory
which cancels with the first term in the curly bracket in J;. Hence for small k,, one gets
which cancels with the first term in the curly bracket in J;. Hence for small k,, one gets
Figure 7.2: Partitioning of Triangular Lattice.
Figure 7.2: Partitioning of Triangular Lattice.
Figure 7.4: Cells for Second Order Calculation. 176 Renormalization Group Theory
Figure 7.4: Cells for Second Order Calculation. 176 Renormalization Group Theory
Thus it has been shown that n.n.n and n.n.n.n interactions are generated by the partial summation procedure. Therefore, to obtain the transformation laws, the initial site hamiltonian also should have such interactions. It is easy to incorporate these terms and find their contributions to first order accuracy. Referring to Case-1 of Figure 7.4, the contributions from n.n.n and n.n.n.n interactions to V;; can be written as
Thus it has been shown that n.n.n and n.n.n.n interactions are generated by the partial summation procedure. Therefore, to obtain the transformation laws, the initial site hamiltonian also should have such interactions. It is easy to incorporate these terms and find their contributions to first order accuracy. Referring to Case-1 of Figure 7.4, the contributions from n.n.n and n.n.n.n interactions to V;; can be written as
Figure 7.5: Division of Square Lattice in to Cells.
Figure 7.5: Division of Square Lattice in to Cells.
Figure 8.1: Cells in a Triangular Lattice.
Figure 8.1: Cells in a Triangular Lattice.
Feigenbaum also showed that the limiting function
Feigenbaum also showed that the limiting function

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