Jordan wigner transform and periodic boundary condition

In summary, the Jordan-Wigner transform can simplify a spin 1/2 system to a free fermion system when applied to an open boundary system. However, there is a difficulty when dealing with periodic boundary conditions, as evidenced by the presence of a phase term in the equation. Despite this, E.H. Lieb, T.D. Schultz, and D.C. Mattis were able to handle this problem in their research. Further insights on this topic can be found in Nielson's work.
  • #1
wdlang
307
0
i think jordan wigner transform, when applied to open boundary system, can simplify a spin 1/2 system to a free fermion system

but there is a difficulty in the case of periodic boundary condition

in this case, we have to deal with terms like

[tex] S_N^+S_1^-=(-)^{\sum_{k=1}^{N-1}n_k} a_N^\dagger a_1[/tex]

the phase term cannot drop out!
 
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  • #2
E.H. Lieb, T.D. Schultz, and D.C. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).
http://dx.doi.org/10.1016/0003-4916(61)90115-4
 
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  • #3
I thought Nielson's would help you!
http://www.qinfo.org/people/nielsen/blog/archive/notes/fermions_and_jordan_wigner.pdf
Yes, it is just an idea, you should calculate it by youself!
 
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  • #4
And I cannot understand the relationship between the periodicity(periodic boundary or antiperiodic boundary) and the parity!
 
  • #5
peteratcam said:
E.H. Lieb, T.D. Schultz, and D.C. Mattis, Ann. Phys. (N.Y.) 16, 407 (1961).
http://dx.doi.org/10.1016/0003-4916(61)90115-4

yes, they also have to deal with this problem

but to my surprise, they can handle it!
 
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1. What is the Jordan-Wigner transform?

The Jordan-Wigner transform is a mathematical transformation used in quantum mechanics to map fermionic operators to spin operators. It is often used in the study of quantum spin systems and can be used to solve certain physical systems that involve fermionic particles.

2. How does the Jordan-Wigner transform work?

The Jordan-Wigner transform works by representing fermionic operators in terms of spin operators. This is achieved by introducing a string of creation and annihilation operators, which are then transformed into spin operators using the Jordan-Wigner formula. This allows for the study of fermionic systems using techniques developed for spin systems.

3. What are periodic boundary conditions?

Periodic boundary conditions are a set of boundary conditions that are often used in the study of physical systems, including those that involve the Jordan-Wigner transform. They impose a periodicity on the system, such that any particle or field that crosses one boundary reappears on the opposite boundary. This allows for the study of systems that would otherwise be infinite in size.

4. How are periodic boundary conditions applied in the Jordan-Wigner transform?

In the Jordan-Wigner transform, periodic boundary conditions are applied by representing the fermionic operators in terms of spin operators on a periodic lattice. This allows for the study of systems with a finite number of particles, while still maintaining the periodicity of the system. The periodic boundary conditions are crucial for obtaining accurate results in the study of physical systems.

5. What are the applications of the Jordan-Wigner transform and periodic boundary conditions?

The Jordan-Wigner transform and periodic boundary conditions have a wide range of applications in the study of quantum spin systems, condensed matter physics, and quantum information theory. They are particularly useful for studying systems with strong correlation effects and have been used to solve various physical systems, including the Ising model, the Hubbard model, and the Kitaev model.

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