Statistical Mechanics of Phase Transitions Book

1	Introduction	1
1.1	Phase transitions in other systems	4
1.1.1	A ferrimagnet: cerium antimonide	4
1.1.2	Surfactants in solution	7
1.2	A microscopic model	8
1.2.1	A renormalization group	13
2	Statistical mechanics and thermodynamics	15
2.1	Statistical mechanics	15
2.2	Thermodynamics	16
2.3	Convexity properties of the free energy	19
2.4	Correlation functions	20
2.5	First-order and continuous phase transitions	21
2.6	Critical point exponents	25
2.6.1	Universality	27
2.6.2	Exponent inequalities	29
2.7	Problems	30
3	Models	33
3.1	The spin-1/2 Ising model	35
3.1.1	Order-disorder transitions in binary alloys	36
3.1.2	Lattice gas models	39
3.2	The spin-I Ising model	41
3.3	The q-state Potts model	41
3.4	X-Y and Heisenberg models	43
3.5	Universality revisited	45
3.6	Discussion	47
3.7	Problems	48
4	Mean-field theories	50
4.1	Mean-field theory for the Ising model	50
4.1.1	Mean-field critical exponents	53
4.2	Landau theory	54
4.2.1	Mean-field critical exponents revisited	56
4.3	The correlation function	57
4.4	Classical mean-field theories	59
4.4.1	Van der Waals theory of fluids	60
4.4.2	Weiss theory of magnetism	61
4.5	The validity of mean-field theory	61
4.6	Problems	63
5	The transfer matrix	67
5.1	Setting up the transfer matrix	67
5.2	The free energy	69
5.3	The correlation function	70
5.4	Results for the Ising model	72
5.4.1	The free energy	73
5.4.2	The magnetization	73
5.4.3	The correlation function	73
5.4.4	The correlation length	74
5.5	Problems	74
6	Series expansions	79
6.1	High temperature series expansions	80
6.2	Low temperature series expansions	85
6.3	The one-dimensional Ising model	86
6.4	Analysis of series expansions	88
6.5	Problems	92
7	Monte Carlo simulations	95
7.1	Importance sampling	95
7.2	Practical details	97
7.3	Considerations in the data analysis	100
7.3.1	Influence of the starting configuration	100
7.3.2	Statistical errors	101
7.3.3	Finite-size corrections	102
7.4	Examples	102
7.4.1	The three-dimensional Ising model	102
7.4.2	More complicated systems	103
7.5	Problem	104
8	The renormalization group	105
8.1	Definition of a renormalization group transformation	106
8.2	Flows in parameter space	108
8.3	Universality	112
8.3.1	Crossover	112
8.4	An example	113
8.5	Scaling and critical exponents	115
8.6	Scaled variables	118
8.7	Conformal invariance	120
8.8	Problems	121
9	Implementations of the renormalization group	124
9.1	The one-dimensional Ising model	124
9.1.1	Derivation of the recursion equations	125
9.1.2	Fixed points	127
9.1.3	Fixed points and scaling	129
9.1.4	The free energy	130
9.2	Higher dimensions	132
9.3	The q-state Potts model	136
9.4	The Monte Carlo renormalization group	139
9.5	The [epsilon]-expansion	140
9.6	Problems	141
	Further reading	145
	Index	147