Jordan–Wigner transformation ( $S_{n}^{+} = \prod_{m < n} (1 - 2 c_{m}^{†} c_{m}) c_{n}$ )

The Jordan–Wigner transformation represents a one-dimensional spin one-half chain by fermions with a parity string preceding each site. With

P_{n} = \prod_{m < n} (1 - 2 n_{m})

, one has

S_{n}^{+} = P_{n} c_{n}

and

S_{n}^{z} = 1/2 - n_{n}

. The string converts the off-site canonical anticommutation relations into commuting spin operators. Adjacent exchange terms become quadratic fermion expressions, while a periodic end bond depends on global fermion parity.

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Bondwise cancellation of a Jordan-Wigner string ( $4 S_{n}^{x} S_{n + 1}^{x} = (c_{n}^{†} - c_{n}) (c_{n + 1}^{†} + c_{n + 1})$ )

 0  0

For

S_{n}^{+} = P_{n} c_{n}

with

P_{n} = \prod_{m < n} (1 - 2 c_{m}^{†} c_{m})

, neighbouring Jordan–Wigner transformation strings differ by only the local parity factor. The canonical anticommutation relations give

(c_{n} + c_{n}^{†}) (1 - 2 c_{n}^{†} c_{n}) = c_{n}^{†} - c_{n}

, proving the displayed local spin identity. The closing bond of a periodic chain instead retains total fermion parity; cancellation on bulk bonds does not justify discarding that global sector.

Parity-dependent boundary condition of the Jordan-Wigner chain ( $c_{N + 1} = - (- 1)^{N_{f}} c_{1}$ )

 0  0

For

S_{n}^{+} = \prod_{m < n} (1 - 2 n_{m}) c_{n}

and periodic spins, the end exchange term carries the global fermion parity

P = (- 1)^{N_{f}}

. Replacing the end bond by a uniform hopping expression requires

c_{N + 1} = - P c_{1}

: even parity gives antiperiodic fermions, odd parity periodic fermions. Each resulting quadratic spectrum must be restricted to states of the parity used in its boundary condition.

Two-site Jordan-Wigner normalization and signs ( $s_{j}^{\pm} = (σ_{j}^{x} \pm i σ_{j}^{y}) /2, σ_{j}^{z} = 2 f_{j}^{†} f_{j} - 1$ )

 0  0

With occupied fermion sites representing up-spins, use

s_{1}^{-} = f_{1}

s_{2}^{-} = P_{1} f_{2}

and

P_{1} = (- 1)^{n_{1}} = 1 - 2 n_{1}

. The canonical anticommutation relations imply

P_{1}^{2} = I

P_{1} f_{1} = - f_{1} P_{1}

and

[P_{1}, f_{2}] = 0

. Consequently the two sites commute as spin operators, while

[s_{j}^{-}, s_{j}^{+}] = 1 - 2 n_{j} = - σ_{j}^{z}

. This normalization is essential: the unscaled combinations

σ_{j}^{x} \pm i σ_{j}^{y}

are twice

s_{j}^{\pm}

The two-spin XY Hamiltonian becomes

H = 2 h - 2 h (n_{1} + n_{2}) - 2 J (f_{1}^{†} f_{2} + f_{2}^{†} f_{1}) .

(1)

In particular,

f_{1}^{†} P_{1} = f_{1}^{†}

and

f_{1} P_{1} = - f_{1}

fix the hopping signs. The modes

b = (f_{1} + f_{2}) / 2

and

c = (f_{1} - f_{2}) / 2

preserve the canonical anticommutation relations and give

H = 2 h + (- 2 h - 2 J) b^{†} b + (- 2 h + 2 J) c^{†} c

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The Jordan-Wigner transformation is a mathematical technique used in quantum mechanics and condensed matter physics to map spin systems to fermionic systems. It provides a way to express operators of spin-1/2 systems (like those found in quantum spin chains) in terms of fermionic creation and annihilation operators.

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Jordan–Wigner transformation (Sn+​=∏m<n​(1−2cm†​cm​)cn​)

Bondwise cancellation of a Jordan-Wigner string (4Snx​Sn+1x​=(cn†​−cn​)(cn+1†​+cn+1​))

Parity-dependent boundary condition of the Jordan-Wigner chain (cN+1​=−(−1)Nf​c1​)

Two-site Jordan-Wigner normalization and signs (sj±​=(σjx​±iσjy​)/2,σjz​=2fj†​fj​−1)

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 Articles by others on the same topic (1)

Jordan–Wigner transformation ( $S_{n}^{+} = \prod_{m < n} (1 - 2 c_{m}^{†} c_{m}) c_{n}$ )

Bondwise cancellation of a Jordan-Wigner string ( $4 S_{n}^{x} S_{n + 1}^{x} = (c_{n}^{†} - c_{n}) (c_{n + 1}^{†} + c_{n + 1})$ )

Parity-dependent boundary condition of the Jordan-Wigner chain ( $c_{N + 1} = - (- 1)^{N_{f}} c_{1}$ )

Two-site Jordan-Wigner normalization and signs ( $s_{j}^{\pm} = (σ_{j}^{x} \pm i σ_{j}^{y}) /2, σ_{j}^{z} = 2 f_{j}^{†} f_{j} - 1$ )

 Discussion (0)