models:toy_models:sine_gordon
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models:toy_models:sine_gordon [2018/03/17 15:24] jakobadmin [Layman] |
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| - | ====== Sine-Gordon Model: $ \quad \mathcal{L} = \frac{1}{2} \left( (\partial_\mu \phi)^2 \right) - (1-cos \phi) $ ====== | + | <WRAP lag>$ \mathcal{L} = \frac{1}{2} \left( (\partial_\mu \phi)^2 \right) - (1-cos \phi) $</WRAP> |
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| - | <tabbox Why is it interesting?> | + | |
| - | + | ||
| - | The Sine-Gordon model is a toy model that helps to understand fundamental notions like [[advanced_notions:quantum_field_theory:duality|duality]] in a simplified setup. | + | |
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| - | In addition, the equations of the theory permit topological non-trivial solutions called solitons. Many basic features of such solitons can be studied in the Sine-Gordon model in a simplified setup. | + | |
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| - | + | ||
| - | The name "Sine-Gordon model" is a wordplay, because the equation of motion of the model is similar to the famous [[equations:klein-gordon_equation|Klein-Gordon equation]] and contains a [[basic_tools:trigonometric_functions|sinus function]]. | + | |
| + | ====== Sine-Gordon Model ====== | ||
| - | <tabbox Layman> | + | <tabbox Intuitive> |
| The Sine-Gordon model describes a coupled set of pendulums that hang from a common support rod. | The Sine-Gordon model describes a coupled set of pendulums that hang from a common support rod. | ||
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| - | <tabbox Student> | + | <tabbox Concrete> |
| **Specification of the Model** | **Specification of the Model** | ||
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| So in the simplest case we have simply $\phi(\infty)= \phi(-\infty)=0$. In words, this means that there is no twist in the system. However, for all other possible values of the field at the boundary, we have one or more twists in the system. For example, in the example above we twisted the support rod at $x=- \infty$ once by $360^\circ = 2\pi$ and left it at $x= \infty$ as it was. This means the boundary conditions of this field configuration are | So in the simplest case we have simply $\phi(\infty)= \phi(-\infty)=0$. In words, this means that there is no twist in the system. However, for all other possible values of the field at the boundary, we have one or more twists in the system. For example, in the example above we twisted the support rod at $x=- \infty$ once by $360^\circ = 2\pi$ and left it at $x= \infty$ as it was. This means the boundary conditions of this field configuration are | ||
| - | $$ \phi(\infty)0 0, \quad \phi(-\infty)= 2\pi . $$ | + | $$ \phi(\infty) = 0, \quad \phi(-\infty)= 2\pi . $$ |
| This is what characterizes soliton. A soliton is a topological nontrivial field configuration and we can now understand why it is stable. To get rid of such a twist in the system we have to move all pendulums from $x=-\infty$ to $x = \infty$. There are infinitely many pendulums that need to be moved and hence the energy that would be required to get rid of such a twist is infinity. However, the energy stored in the twist is finite. This follows because the field configuration is in the ground state at infinity. The thing is that it is at different ground states at $x=-\infty$ to $x = \infty$. | This is what characterizes soliton. A soliton is a topological nontrivial field configuration and we can now understand why it is stable. To get rid of such a twist in the system we have to move all pendulums from $x=-\infty$ to $x = \infty$. There are infinitely many pendulums that need to be moved and hence the energy that would be required to get rid of such a twist is infinity. However, the energy stored in the twist is finite. This follows because the field configuration is in the ground state at infinity. The thing is that it is at different ground states at $x=-\infty$ to $x = \infty$. | ||
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| **Solitonic Solutions in Detail** | **Solitonic Solutions in Detail** | ||
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| Since solitons are stable and static, we now try to find such a solution of the field equations. Static means that a soliton is time-independent and therefore we can make the ansatz that our soliton solution is time independent. This yields the __static Sine-Gordon equation__ | Since solitons are stable and static, we now try to find such a solution of the field equations. Static means that a soliton is time-independent and therefore we can make the ansatz that our soliton solution is time independent. This yields the __static Sine-Gordon equation__ | ||
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| Since we know that at infinity we must have $\partial^x \phi \to 0$ and $\cos \phi \to 1$ in order to have a configuration with finite energy, we can deduce $A=1$. | Since we know that at infinity we must have $\partial^x \phi \to 0$ and $\cos \phi \to 1$ in order to have a configuration with finite energy, we can deduce $A=1$. | ||
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| + | {{ :models:kinksol2.png?nolink&400|}} | ||
| Now, using $\cos 2\phi = 1-\sin \phi$ and integrating yields | Now, using $\cos 2\phi = 1-\sin \phi$ and integrating yields | ||
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| $$ \phi(x)_k = 4\arctan \left( e^{\pm (x-x_)} \right) \tag{Eq. 1} $$ | $$ \phi(x)_k = 4\arctan \left( e^{\pm (x-x_)} \right) \tag{Eq. 1} $$ | ||
| - | {{ :models:kinksol.png?nolink&400|}} | + | |
| The $\pm$ in the exponent corresponds to solitons with winding number $n=\pm 1$. The solution with $n=1$ is known as the __kink solution__ and the solution with $n=-1$ as the __antikink solution__. | The $\pm$ in the exponent corresponds to solitons with winding number $n=\pm 1$. The solution with $n=1$ is known as the __kink solution__ and the solution with $n=-1$ as the __antikink solution__. | ||
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| + | {{ :models:kinkenergy.png?nolink&400|}} | ||
| When we plot the kink solution we can see that it is a wave-packet of finite energy that is centered around $x_0$. In other words, a kink is like a potential barrier in the system. | When we plot the kink solution we can see that it is a wave-packet of finite energy that is centered around $x_0$. In other words, a kink is like a potential barrier in the system. | ||
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| If we now want to calculate the mass of the kink, we have to integrate the Hamiltonian density since it simply represents the energy. We get the Hamiltonian density, as usual, from the Lagrangian density and the integrated result is | If we now want to calculate the mass of the kink, we have to integrate the Hamiltonian density since it simply represents the energy. We get the Hamiltonian density, as usual, from the Lagrangian density and the integrated result is | ||
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| - | <tabbox Researcher> | + | <tabbox Abstract> |
| + | The Sine-Gordon model is [[advanced_notions:duality|dual]] to the massive Thirring model. | ||
| - | <note tip> | + | This is surprising since the Sine-Gordon model is a purely scalar theory, while the massive Thirring model is a purely fermionic theory. However, the solitons in the Sine-Gordon model are fermionic, while the solitons in the Thirring model are scalars. |
| - | The motto in this section is: //the higher the level of abstraction, the better//. | + | |
| - | </note> | + | ^ ^ massive Thirring model ^ Sine-Gordon model | |
| + | | **fermions** | fundamental | solitons | | ||
| + | | **bosons** | solitons | fundamental | | ||
| | | ||
| + | Explicitly this means, that although the Lagrangian of the massive Thirring model and the Sine-Gordon model look completely different, they ultimately describe the same physics. The only thing that is different is the perspective. | ||
| + | The duality can be shown by verifying that the currents in both theories are equal. Expressed differently, if the greens functions are equal the theories are equal. | ||
| + | |||
| + | ---- | ||
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| + | The duality was shown by Coleman in | ||
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| + | * Coleman, S. (1975). [[https://journals.aps.org/prd/abstract/10.1103/PhysRevD.11.2088|"Quantum sine-Gordon equation as the massive Thirring model"]]. Physical Review D. 11 (8): 2088. | ||
| + | |||
| + | <tabbox Why is it interesting?> | ||
| + | |||
| + | The Sine-Gordon model is a toy model that helps to understand fundamental notions like [[advanced_notions:duality|duality]] in a simplified setup. | ||
| + | |||
| + | |||
| + | In addition, the equations of the theory permit topological non-trivial solutions called solitons. Many basic features of such solitons can be studied in the Sine-Gordon model in a simplified setup. | ||
| + | |||
| + | |||
| + | |||
| + | The name "Sine-Gordon model" is a wordplay, because the equation of motion of the model is similar to the famous [[equations:klein-gordon_equation|Klein-Gordon equation]] and contains a [[basic_tools:trigonometric_functions|sinus function]]. | ||
| - | <tabbox FAQ> | ||
| - | | ||
| <tabbox History> | <tabbox History> | ||
| The Sine-Gordon equation was originally | The Sine-Gordon equation was originally | ||
models/toy_models/sine_gordon.1521296659.txt.gz · Last modified: 2018/03/17 14:24 (external edit)
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