Jordan-Wigner Transformation

With the help of the Jordan-Wigner transformation , different one-dimensional quantum mechanical systems can be mapped onto one another. More precisely, it is possible to use the transformation to map one-dimensional spin -1 / 2 chains to fermions on a chain.

The Jordan-Wigner transformation maps the spin 1/2 operators and their algebra (algebra of the Pauli matrices ) to the creation and annihilation operators for fermions and their algebra. With the help of the transformation, the equivalence between the one-dimensional Heisenberg model and fermions on a one-dimensional grid with the closest neighbor interaction can be shown.

The transformation was published in 1928 by Pascual Jordan and Eugene Wigner in the Zeitschrift für Physik . In 1961 Elliott Lieb , T. Schultz, D. Mattis used the transformation when introducing their exactly solvable one-dimensional spin 1/2-xy model.

The Jordan-Wigner transformation has also been applied to two-dimensional spin systems and to three-dimensional systems. The application to two-dimensional systems was one of the first to be discussed by Eduardo Fradkin in 1989.

Elliott Lieb , T. Schultz, Daniel Mattis applied the transformation in 1964 to the transfer matrix in the two-dimensional Ising model and thus derived the exact solution previously found by Lars Onsager .

Basic idea

If one considers spin-1/2 operators in place , one finds that they obey the basic canonical (anti) commutator relations (anti-commutator relations) for fermions: ${\ displaystyle j}$

{\ displaystyle \ {S_ {j} ^ {+}, S_ {j} ^ {-} \} = 1, \ qquad \ {S_ {j} ^ {+}, S_ {j} ^ {+} \} = 0 = \ {S_ {j} ^ {-}, S_ {j} ^ {-} \},}

whereby . The idea is therefore to consider the spin 1/2 operators as fermionic operators. However, the spin 1/2 operators do not fulfill anti-commutator relations, but commutator relations on different grid positions and : ${\ displaystyle \ {A, B \} = AB + BA}$ ${\ displaystyle j}$ ${\ displaystyle k}$

{\ displaystyle [S_ {j} ^ {+}, S_ {k} ^ {-}] = 0 = [S_ {j} ^ {+}, S_ {k} ^ {+}] = [S_ {j} ^ {-}, S_ {k} ^ {-}],}

whereby . ${\ displaystyle [A, B] = AB-BA}$

Jordan and Wigner recognized that this can, however, be remedied by introducing a phase operator before the spin 1/2 operators. A path orientation is defined with a phase factor that is dependent on the number of up spins before the spin under consideration.

{\ displaystyle c_ {j} = e ^ {i \ phi _ {j}} S_ {j} ^ {-} \ qquad {\ text {with}} \ quad \ phi _ {j} = \ pi \ sum _ {k <j} S_ {k} ^ {+} S_ {k} ^ {-}}

If there is an up-spin at this point , a phase factor (−1) is “picked”, nothing happens with a down-spin (phase factor 1): ${\ displaystyle j}$

${\ displaystyle e ^ {i \ pi S_ {j} ^ {+} S_ {j} ^ {-}} = e ^ {i \ pi n_ {j}} = 1-2n_ {j} \ qquad {\ text {with}} \ quad n_ {j} = S_ {j} ^ {+} S_ {j} ^ {-}}$

The fermionic operators defined in this way fulfill the anti-commutator relations in different places and : ${\ displaystyle j}$ ${\ displaystyle k}$

{\ displaystyle \ {c_ {j}, c_ {k} ^ {\ dagger} \} = \ delta _ {jk}, \ qquad \ {c_ {j} ^ {\ dagger}, c_ {k} ^ {\ dagger} \} = 0 = \ {c_ {j}, c_ {k} \}}

The following relationships are particularly helpful for mapping between different models:

{\ displaystyle S_ {j} ^ {+} S_ {j + 1} ^ {-} = \ pm c_ {j} ^ {\ dagger} c_ {j + 1}}

{\ displaystyle S_ {z} = S_ {j} ^ {+} S_ {j} ^ {-} - {\ frac {1} {2}} = c_ {j} ^ {\ dagger} c_ {j} - {\ frac {1} {2}}}

Applications

1D Heisenberg model

To illustrate the Jordan-Wigner transformation, it is applied to the one-dimensional Heisenberg model . The products required by the various operators are already listed in the previous section. The Hamiltonian of the 1D Heisenberg model can therefore be written as: ${\ displaystyle H _ {\ text {Heis}}}$

{\ displaystyle {\ begin {aligned} H _ {\ text {Heis}} & = - J \ sum _ {n = 1} ^ {N} {\ vec {S}} _ {n} \ cdot {\ vec { S}} _ {n + 1} = - J \ sum _ {n = 1} ^ {N} \ left [{\ frac {1} {2}} (S_ {n} ^ {+} S_ {n + 1} ^ {-} + S_ {n} ^ {-} S_ {n + 1} ^ {+}) + S_ {n} ^ {z} S_ {n + 1} ^ {z} \ right] \\ & = - J \ sum _ {i = 1} ^ {N} \ left [{\ frac {1} {2}} \ left (c_ {i} ^ {\ dagger} c_ {i + 1} + {\ text {hc}} \ right) + \ left ((c_ {i} ^ {\ dagger} c_ {i} - {\ frac {1} {2}}) (c_ {i + 1} ^ {\ dagger} c_ {i + 1} - {\ frac {1} {2}}) \ right) \ right] \\ & = H_ {0} + H_ {J} \ end {aligned}}}

The transformation shows the equivalence of the 1D Heisenberg model with spinless fermions on the lattice with periodic boundary conditions and only the closest neighboring interaction. The first term describes interaction-free fermions and the second term is the interaction term with an interaction given by the coupling constant of the Heisenberg model. ${\ displaystyle H_ {0}}$ ${\ displaystyle H_ {J}}$ ${\ displaystyle U = -J}$

1D XY model

Another example is the one-dimensional XY model as a special case of the 1D Heisenberg model. The Hamiltonian of the XY model can be written as: ${\ displaystyle H _ {\ text {Heis}}}$

{\ displaystyle {\ begin {aligned} H _ {\ text {XY}} & = - J \ sum _ {n = 1} ^ {N} \ left [{\ frac {1} {2}} (S_ {n } ^ {+} S_ {n + 1} ^ {-} + S_ {n} ^ {-} S_ {n + 1} ^ {+}) \ right] \\ & = - J \ sum _ {i = 1} ^ {N} {\ frac {1} {2}} \ left (c_ {i} ^ {\ dagger} c_ {i + 1} + c_ {i + 1} ^ {\ dagger} c_ {i} \ right) = H_ {0} \ end {aligned}}}

The Jordan-Wigner transformation maps the spin system to non-interacting, spinless fermions. For this system, the sum of functions can be specified exactly.

Quantum information theory

The transformation was used in quantum information theory to map a system of interacting qubits to an equivalent system of interacting fermions and vice versa. In addition, Raymond Laflamme and colleagues were able to solve the problem of simulating fermionic quantum mechanical systems in quantum computers, a problem that was still open in the pioneering work of Richard Feynman from 1982.

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↑ P. Jordan and E. Wigner, On the Paulische Äquivalenzverbot , Zeitschrift für Physik 47, No. 9. (1928), pp. 631-651, doi : 10.1007 / BF01331938 .
^ Lieb, Schultz, Mattis, Annals of Physics, Volume 16, 1961, p. 407
↑ Oleg Derzho, Jordan-Wigner fermionization for spin-1/2 systems in two dimensions: A brief review, Journal of Physical Studies, Volume 5, 2001, pp. 49-64, Arxiv
^ Lieb, Schultz, Mattis, Review of Modern Physics, Volume 36, 1964, p. 856
↑ Michael Nielsen , The fermionic canonical commutation relations and the Jordan-Wigner transform, 2005 Online as Complete notes on fermions and the Jordan-Wigner transform.
^ R. Somma, G. Ortiz, JE Gubernatis, E. Knill, R. Laflamme, Simulating physical phenomena by quantum networks , Physical Review A, Volume 65, 2002, p. 042323, Arxiv
^ Richard Feynman, Simulating physics with computers, Int. J. Theor. Phys., Vol. 21, 1982, pp. 467-488

[1] P. Jordan and E. Wigner, On the Paulische Äquivalenzverbot , Zeitschrift für Physik 47, No. 9. (1928), pp. 631-651, doi : 10.1007 / BF01331938 .

[2] Lieb, Schultz, Mattis, Annals of Physics, Volume 16, 1961, p. 407

[3] Oleg Derzho, Jordan-Wigner fermionization for spin-1/2 systems in two dimensions: A brief review, Journal of Physical Studies, Volume 5, 2001, pp. 49-64, Arxiv

[4] Lieb, Schultz, Mattis, Review of Modern Physics, Volume 36, 1964, p. 856

[5] Michael Nielsen , The fermionic canonical commutation relations and the Jordan-Wigner transform, 2005 Online as Complete notes on fermions and the Jordan-Wigner transform.

[6] R. Somma, G. Ortiz, JE Gubernatis, E. Knill, R. Laflamme, Simulating physical phenomena by quantum networks , Physical Review A, Volume 65, 2002, p. 042323, Arxiv

[7] Richard Feynman, Simulating physics with computers, Int. J. Theor. Phys., Vol. 21, 1982, pp. 467-488