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I am stuck on the interpretation/derivation of the 2-spin terms of the quantum Heisenberg model Hamiltonian.

In this model, our electrons, with spin up or down, are confined to sites on a lattice. The coupling constants $J$ describe the coupling strength between spin components in $x,y,z$ directions.

$$ \sum _ { \langle i , j \rangle } J _ { i j } \vec { S } _ { i } \vec { S } _ { j } = \sum _ { \langle i , j \rangle } J _ { i j } \left[ S _ { i } ^ { x } S _ { j } ^ { x } + S _ { i } ^ { y } S _ { j } ^ { y } + S _ { i } ^ { z } S _ { j } ^ { z } \right] $$

In the computational basis ${\uparrow, \downarrow}$, which we choose to be the eigenstates of the Pauli-z matrix, we can rewrite this, using the raising/lowering operators $S^+|\downarrow>=|\uparrow$>and $S^-|\uparrow>=|\downarrow>$ as

$$ H= \sum _ { \langle i , j \rangle } J _ { i j } \left[ \frac { 1 } { 2 } \left( S _ { i } ^ { + } S _ { j } ^ { - } + S _ { i } ^ { - } S _ { j } ^ { + } \right) + S _ { i } ^ { z } S _ { j } ^ { z } \right] $$

Now, the next step I don't see immediately. How do I get to this representation - and what is the intuitive interpretation of the matrix representation? We see coupling between aligned spins?

The matrix representation for one 2-spin term of the Hamiltonian is

$$ \left( \begin{array} { c c c c } { J _ { i j } / 4 } & { 0 } & { 0 } & { 0 } \\ { 0 } & { - J _ { i j } / 4 } & { J _ { i j } / 2 } & { 0 } \\ { 0 } & { J _ { i j } / 2 } & { - J _ { i j } / 4 } & { 0 } \\ { 0 } & { 0 } & { 0 } & { J _ { i j } / 4 } \end{array} \right) $$

in the basis $\{ | \uparrow \uparrow \rangle , | \uparrow \downarrow \rangle , | \downarrow \uparrow \rangle , | \downarrow \downarrow \rangle \}$

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Matrix representation of the operator $H$ is constructed from its matrix elements: $$ H_{\alpha \beta} = \langle\alpha|H|\beta\rangle. $$ In this case, for example, we have $$ H_{23} = \langle\uparrow\downarrow|H|\downarrow\uparrow\rangle. $$ From algebraic properties of spin $1/2$ operators and definition of two-spin state we obtain: $$ S^z_iS^z_j|\downarrow\uparrow\rangle = -\frac14 |\downarrow\uparrow\rangle, \qquad S^+_iS^-_j|\downarrow\uparrow\rangle = |\uparrow\downarrow\rangle, \qquad S^-_iS^+_j|\downarrow\uparrow\rangle = 0\quad \longrightarrow $$ $$ H|\downarrow\uparrow\rangle = -\frac{J_{ij}}4 |\downarrow\uparrow\rangle + \frac{J_{ij}}2 |\uparrow\downarrow\rangle $$ It is easy to see now, that $$ H_{23} = \langle\uparrow\downarrow|H|\downarrow\uparrow\rangle = \frac{J_{ij}}2 $$ And further: $$ H_{21} = \langle\uparrow\downarrow|H|\uparrow\uparrow\rangle = 0,\quad H_{24} = \langle\uparrow\downarrow|H|\downarrow\downarrow\rangle = 0,\quad H_{22} = \langle\uparrow\downarrow|H|\uparrow\downarrow\rangle = -\frac{J_{ij}}4 $$ Calculation of other matrix elements is similar.

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