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0ô©ˆ8Š0º___PPT10‹ ÀÀ?Ù Ú %ð†óŸ¨,Chap.3 Introduction to Critical Phenomena¡--(Ÿª óS%óT&Ÿª Ÿ¨ÏLet denote the density at the critical point, along the blue curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility óŸª Ÿ¨ÁFor ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: And susceptibility ,h is the external field¡$¨òóŸ 0ßpyó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó=Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-ó%ó&ó4ó5ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations ¡(]4¶GóP#Ÿ¨Dimensional Analysis ¡ Ÿ¨We getóR$Ÿ¨Dimensional Analysis ¡ Ÿª êîFï€ 0 ¶ð®pð¼ðFð( ð ð¼ð¹¢ ð¼ ƒ ð0€ô¿�ƒ¿Àÿðð � ðYŸ¨)Let start with the phase diagram of water¡**ðí¢ ð¼ £ ð<€…‡¿�ƒ¿Àÿð P5& ð�Ÿ¨QPhenomena observed near the critical point are referred to as Critical phenomena.¡RRðX² ð¼ ƒ ð0A9�ƒ¿Àÿ?ð`ànF ðH ð¼ ƒ ð0�ƒ“ŽŸ‹”Þ½h¿ÿ ?ð ÿÿÿ€€€»àã33™™™™Ìˆ8Š0º___PPT10‹ë.`sÊСj[rà®Söõ%™¼®Uù]Å1è1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó=Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-ó%ó&ó4ó5ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó=Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-ó%ó&ó4ó5ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-ó%ó&ó4ó5ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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Phenomena¡--(Ÿª óS%óT&Ÿª Ÿ¨ÊLet denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó=Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-ó%ó&ó4ó5ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó=Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-ó%ó&ó4ó5ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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0h–<ý4!d!d4„³x 0h–ˆ8Š0º___PPT10‹ ÀÀ?Ù Ú %ð›óŸ¨,Chap.3 Introduction to Critical Phenomena¡--(Ÿª óS%óT&Ÿª Ÿ¨ÊLet denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. 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2ð$SÁ\Ù0G9Qÿ½C�z"„ÿ’ó* 2ð$ (]®´Bô:åK[8ŽÖÿcÓz bð$Áü"­4÷sA)èêßâ²ÿb6} ð$ÿ2ð$b6~ìCê`Ã�âÁ@�ÿ„Oß ƒ ð0�ƒ†A¿ÀÅAÿ@ñ÷ðó€Ðwj[ Êš;&4r8Êš;úgþý4LdLd€‚«„É 02þÿÿ¦ÿÿÿpûppû@ ÿ <ý4dddd4„³x 0h–<ý4!d!d4„³x 0h–ˆ8Š0º___PPT10‹ ÀÀ?Ù Ú %ðóŸ¨,Chap.3 Introduction to Critical Phenomena¡--(Ÿª óS%óT&Ÿª Ÿ¨ÊLet denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations ¡(]4¶GóP#Ÿ¨Dimensional Analysis ¡ Ÿ¨We getóR$Ÿ¨Dimensional Analysis ¡ Ÿª óc*Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-êîèï€ 0 ÿð÷0ðÜð�ð( ð ðÜðX² ðÜ ƒ ð0A�ƒ¿Àÿ?ð ð¯ ð÷¢ ðÜ £ ð<€„…‡¿�ƒ¿ÀÿðG ò5 ð‹Ÿ¨ This table summarizes the experimental values of the critical exponents as well as the results from some theoretical models. 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y’´ÿ)àÔ2ð$dï’'ÖáRñî_6Ȧÿ. ×2ð$&îñWk§1?Á 2«!ãÿè7Ù2ð$¿¬«Do&q-¥~~Œt­ÿ£Ü2ð$ˆÌ_ã6˜n*ÊŒØÓRÑ<ÿ;ÂÞ2ð$–¶Ä¡!«è�jJPkFÿ¡ýá2ð$—cïKq4¿s›Ä¸ðX\^ÿþ@!çbð$µUðø�¢v™•c!"`Ä ÿq^ÿ 2ð$SÁ\Ù0G9Qÿ½C�z"„ÿ’ó* 2ð$ (]®´Bô:åK[8ŽÖÿcÓz bð$Áü"­4÷sA)èêßâ²ÿb6} ð$ÿ2ð$b6~ìCê`Ã�âÁ@�ÿ„Oß ƒ ð0�ƒ†A¿ÀÅAÿ@ñ÷ðó€Ðwj[ Êš;&4r8Êš;úgþý4LdLd€‚«„É 02þÿÿ¦ÿÿÿpûppû@ ÿ <ý4dddd4„³x 0h–<ý4!d!d4„³x 0h–ˆ8Š0º___PPT10‹ ÀÀ?Ù Ú %ðóŸ¨,Chap.3 Introduction to Critical Phenomena¡--(Ÿª óS%óT&Ÿª Ÿ¨ÊLet denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨Correlation function¡$Ÿª óŸ¨ Then we have¡  Ÿª óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations ¡(]4¶GóP#Ÿ¨Dimensional Analysis ¡ Ÿ¨We getóR$Ÿ¨Dimensional Analysis ¡ Ÿª óc*Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-êåxœíYËo›EŸÝÏÎÃ1Øy5i(åk$8T8 ¡„H‰ë8´1Qâ‚Ä%¸Ô£Ønâ„ ÔHH/)üHP.¹@%Šè©‡ö„ÔFNpªè!H@-OM–|•þS4 iüLƒ aH!NÁ\©)¬K{À]ØsÙÀ m!W4{ÈÙןŸ¸ýmæš ›š¯56õ L@¼"™Nò€$ÛØ^9cîÆÒÂõ0L Äpå ÔC õá§WËz+=©Ïr䓆N]B¹I8‹z8¯”3|5ý ùD F7–¿ãs\HG¹uìÆ‡|ÚÂ�Ú_•¶—¶Òÿžûwòñw¹ÿGwˆÿ›qõ°22ñÀ :Ü…ØU¿ÿ¿�9¿‘ÿ�ÿR> üÜú|Gváá=' ý÷ñ<B£²A€fÐ>Õ Ú°´cÙ‹åî¿ñ^X?¯TikIrüÁȇ t«OùÔŽñ×ü. ùÊC¡À]��;�s=›7ëî�/Ð-�ž£4w“Ǧ_ŽÎ��Û½]Ývxr&6�H§T$(pÂC*"f×àØpô0d�\œt ²æÝ#¨ñÁ¾‚¥ø�b‚„O¯Å�oÀ¿0ªº‚u·ÀPD³õb|y.‘ NL‹e/†Ò§ã#±—âhtGÉxÆŽÄgíÑt2–‚Fkl.y*=�¬Pzf*Ÿ"&4º†£vøµé©˜™É1IÍõË?`«;p(¼8x(8’õûžŽüvß~¬·ú‚� gïÏ.Çšß"X gŽÏ.ª–ÄsDZ<8>8ÎÒ_Ðç¨]!‘p…½ÄÂgFX�À†7C ] ýÐoQlô 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bð$Áü"­4÷sA)èêßâ²ÿb6} ð$ÿ2ð$b6~ìCê`Ã�âÁ@�ÿ„Oß ƒ ð0�ƒ†A¿ÀÅAÿ@ñ÷ðó€Ðwj[ Êš;&4r8Êš;úgþý4LdLd€‚«„É 02þÿÿ¦ÿÿÿpûppû@ ÿ <ý4dddd4„³x 0h–<ý4!d!d4„³x 0h–ˆ8Š0º___PPT10‹ ÀÀ?Ù Ú %ð‡óŸ¨,Chap.3 Introduction to Critical Phenomena¡--(Ÿª óS%óT&Ÿª Ÿ¨ÊLet denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. Near the critical point, the correlation length  is the only characteristic length of the system, in terms of which all other lengths must be measured. ó? Ÿ¨Dimensional Analysis ¡ Ÿ¨One way to implement the scaling hypothesis is to assume that a quantity of dimension is proportional to  near the critical point. First we note that is dimensionless. Hence is of dimension Note that G is the Gibbs Free Energy.óF!Ÿ¨Dimensional Analysis ¡ Ÿ¨}Normalizing the correlation function, we have By definition this has the same dimension as , hence And we know thatóM"Ÿ¨Dimensional Analysis ¡ Ÿ¨FThe dimension of the conjugate field can be obtained from the relation as ,thus Now replace the length L in the formulas above by . Using  , we obtain all the critical exponents. Comparing them with their definitions leads to the relations ¡(]4¶GóP#Ÿ¨Dimensional Analysis ¡ Ÿ¨We getóR$Ÿ¨Dimensional Analysis ¡ Ÿª óc*Ÿª Ÿ¨ËOnly two of the six critical exponents defined in the preceding discussion are independent, because of the following "scaling laws": Where in last relation, d is the dimensionality of space. ¡*�?Ÿ-êåxœíYËo›EŸÝÏÎÃ1Øy5i(åk$8T8 ¡„H‰ë8´1Qâ‚Ä%¸Ô£Ønâ„ ÔHH/)üHP.¹@%Šè©‡ö„ÔFNpªè!H@-OM–|•þS4 iüLƒ aH!NÁ\©)¬K{À]ØsÙÀ m!W4{ÈÙןŸ¸ýmæš ›š¯56õ L@¼"™Nò€$ÛØ^9cîÆÒÂõ0L Äpå ÔC õá§WËz+=©Ïr䓆N]B¹I8‹z8¯”3|5ý ùD F7–¿ãs\HG¹uìÆ‡|ÚÂ�Ú_•¶—¶Òÿžûwòñw¹ÿGwˆÿ›qõ°22ñÀ :Ü…ØU¿ÿ¿�9¿‘ÿ�ÿR> üÜú|Gváá=' ý÷ñ<B£²A€fÐ>Õ Ú°´cÙ‹åî¿ñ^X?¯TikIrüÁȇ t«OùÔŽñ×ü. ùÊC¡À]��;�s=›7ëî�/Ð-�ž£4w“Ǧ_ŽÎ��Û½]Ývxr&6�H§T$(pÂC*"f×àØpô0d�\œt ²æÝ#¨ñÁ¾‚¥ø�b‚„O¯Å�oÀ¿0ªº‚u·ÀPD³õb|y.‘ NL‹e/†Ò§ã#±—âhtGÉxÆŽÄgíÑt2–‚Fkl.y*=�¬Pzf*Ÿ"&4º†£vøµé©˜™É1IÍõË?`«;p(¼8x(8’õûžŽüvß~¬·ú‚� gïÏ.Çšß"X gŽÏ.ª–ÄsDZ<8>8ÎÒ_Ðç¨]!‘p…½ÄÂgFX�À†7C ] ýÐoQlô ;/Y¤�ê,á­Ôƒ{ùøg'µo c·`[0¿ÑŸÅ@þ2r.âa$·ÎIêù±â’J±ÝëS_LM–ÆfÆÆVÆ=ŒmŒíÖ�õSœ¿�­Ï«tÄÇ/•ãþ(\jh=p$ÙåŸÒú˜z™z,àZ;Çæ2Óñ$@wۙʌà5ÍùôOM<ÞRø‹�`­áw‘`ä„©?Ä�zf<Ìøc/㣌}ŒGc|œñ Æ~Æ£ŒO2XÅû›­Ìÿû¡zØ)´]ù?RÍÿ;BþÎÈÿ?_íC#:íÈÿ¯ïšü�ßM�N\Ô¾ÁêùÿÝ:�ÿ½n“ÿTþ_R\�ÿ·{�«¥êJò…Ç¥‚ŒÍ= ,×�ùb³N[ŸÿÉ„$—¤Z—ØËm JTóJÛu¥v5ÿÅh�õ%,t£ÿ¥åÔ¬‰QŒf›µh©5£go‘rÙEµf¬Åj©VO²¥æJñ¹[ï@'Ç·÷õ†Ið—úÿwþ:{ ‹mßž¥]1;Hç˜:Ȩ÷Àe¡ç¶­‹5›[òî®-ゲŒùœ‘!X†püKÍP‚0 aèWwƒ)ÌÊi¾#Õw…IÕJ; å®ßXàÊwDW±ô;~$»`­w¬Tž‚ ZZ(ú‚e¤,[çÝ“²ÖJ¢Q#ジé*•¡:¡ Î ð†×úëfS…;z)ZÔÖbå*ûŠâ*t•Œ €µÄ~1(Œ_œ<Ù'¶;Çý²—Ǽexœí[ˆ[EôÌL’î&ÑÜì¦/½-X¡Ðm©ôcÅt÷®õ#kØÝZ„BšnoÝHÝ$Ûm@° ì‡kAÄ?ÑOñG?Š~Ô/E¥Õ? 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. 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If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ól-Ÿ¨ Definition of critical exponent ¡!!$Ÿª óm.Ÿ¨ Then we have¡  Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ól-Ÿ¨ Definition of critical exponent ¡!!$Ÿª óm.Ÿ¨ Then we have¡  Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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%ðÖóŸ¨,Chap.3 Introduction to Critical Phenomena¡--(Ÿª óS%óT&Ÿª Ÿ¨ÊLet denote the density at the critical point, along the curve, observations show that for very small If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ól-Ÿ¨ Definition of critical exponent ¡!!$Ÿª óm.Ÿ¨ Then we have¡  Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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that for very small If is measured along a line through the critical point, one finds For compressibility ó^'óŸª Ÿ¨‡For ferromagnetism, When T is approaching the Curie temperature, the specific heat typically has a power law behavior: ¡$ˆnòóŸ 0ßpyó`(óa)ó Ÿ¨2Critical exponent (In the language of Ising model)¡33 ª&Ÿª ól-Ÿ¨ Definition of critical exponent ¡!!$Ÿª óm.Ÿ¨ Then we have¡  Ÿª ó%ó&ó6Ÿª Ÿ¨{The significance of the critical exponents lie in their universality. As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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As experiments have shown, widely different systems, with critical temperatures differing by orders of magnitudes, approximately share the same critical exponents. Their definitions have been dictated by experimental convenience. Some other linear combinations of them have more fundamental significance, ¡ ?=|óŸ¨(Chap.4 Mean-field theory of Ising model¡))(ªŸª ó Ÿª Ÿ¨_The main idea of mean field theory is to replace all interactions to any one body (here one spin) with an average or effective interaction. This reduces any multi-body problem into an effective one-body problem. The ease of solving mean field theory problems means that some insight into the behavior of the system can be obtained at a relatively low ¡``óŸ¨/Recall the Hamiltonian of the Ising model to be¡00ª Ÿª ó ó! Ÿ¨Graphical solution when K<1¡Ÿª ó" Ÿ¨Graphical solution when K>1¡Ÿª ó# Ÿ¨Graphical solution when K=Kc=1¡ªŸª ó(ó,ó/ó1ó2ó'Ÿ¨ Chap.5 Universality and Scaling¡!!(Ÿª óf+ó8Ÿ¨Universality class¡ Ÿ¨>As we can see, different systems at critical region, share the same critical exponents. However, not all the models have the same exponents, they depend on at least: 1>the lattice dimensionality 2>the spin dimensionality 3>the fallout exponent p of the long range interactions such as¡¦™?ó9Ÿ¨Explanation of universality ¡ Ÿ¨¼In the critical region, the correlation length is very large, approaching infinity. Only the large scale behavior is relevant. The details of the interaction at short distances are not. Such details will obviously affect the strength of interaction required to produce a phase transition, but the have no effect on the behavior in the critical region itself. This heuristic argument is made precise and quantitative by the renormalization group¡½Z½ó:ó<Ÿ¨Scaling hypothesis¡ Ÿ¨šThe scaling hypothesis has something to do with how various quantities change under a change of length scale. The value of a quantity with dimension must be expressed in terms of a standard unit of length, and it changes when that standard is changed. 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