Jordan wigner transform and periodic boundary condition

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SUMMARY

The Jordan-Wigner transform simplifies a spin 1/2 system to a free fermion system in open boundary conditions. However, challenges arise under periodic boundary conditions due to the presence of phase terms, specifically represented by S_N^+S_1^-=(-)^{\sum_{k=1}^{N-1}n_k} a_N^\dagger a_1. This phase term cannot be disregarded, as highlighted in the work of E.H. Lieb, T.D. Schultz, and D.C. Mattis in their 1961 paper. The relationship between periodicity and parity remains a complex issue that requires further exploration.

PREREQUISITES
  • Understanding of the Jordan-Wigner transform
  • Familiarity with spin 1/2 systems
  • Knowledge of free fermion systems
  • Basic concepts of periodic boundary conditions in quantum mechanics
NEXT STEPS
  • Study the implications of the Jordan-Wigner transform in different boundary conditions
  • Examine the paper by E.H. Lieb, T.D. Schultz, and D.C. Mattis for deeper insights
  • Research the relationship between periodicity and parity in quantum systems
  • Explore advanced topics in fermionic systems and their applications in quantum mechanics
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Quantum physicists, researchers in condensed matter physics, and students studying quantum mechanics who are interested in the applications of the Jordan-Wigner transform and boundary conditions.

wdlang
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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

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

the phase term cannot drop out!
 
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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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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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And I cannot understand the relationship between the periodicity(periodic boundary or antiperiodic boundary) and the parity!
 
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!
 
Last edited by a moderator:

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