Heisenberg model

The Heisenberg model (after Werner Heisenberg ) in the quantum mechanical formulation is a mathematical model that is widely used in theoretical physics to describe ferromagnetism (as well as antiferromagnetism and ferrimagnetism ) in solids . The aim of the observation is to model experimentally observed effects such as spontaneous magnetization and the critical exponents at the phase transitions .

The model is suitable for the qualitative description of ferromagnetism in insulators , but fails with most metals (the Hubbard model is more suitable here).

formulation

In 1928 Werner Heisenberg and Paul Dirac recognized that ferromagnetism in a solid can be described by an effective Hamilton operator that does not contain the quantum mechanical spatial functions, since it is only made up of interacting localized electron spins on the crystal lattice . The interaction is (initially) reduced to neighboring spins (closest-neighbor interaction). In contrast to the classical Heisenberg model, the spins are expressed by vector operators and obey the rules of quantum mechanics: ${\displaystyle H_{\text{Heis}}}$

${\displaystyle H_{\text{Heis}}=-J\sum _{\langle i,j\rangle }{\vec {S_{i}}}\cdot {\vec {S_{j}}}\qquad {\text{mit }}i,j\,{\text{n}}{\ddot {\rm {a}}}{\text{chste Nachbarn}}}$

There

• are and the quantum mechanical spin operators at a given spin quantum number ( ) ${\displaystyle {\vec {S}}_{i}}$ ${\displaystyle {\vec {S}}_{j}}$ ${\displaystyle s}$ ${\displaystyle s\in \{1/2,\,1,\,3/2,\,2,\,\ldots \}}$
• The indices and refer to the grid positions, where the grid can be a chain (one-dimensional Heisenberg model), a two-dimensional grid (e.g. a hexagonal grid) or a three-dimensional arrangement (e.g. a cubic grid ). The spin, however, is always three-dimensional in the Heisenberg model, making it as a special case of the n-vector model with is called. ${\displaystyle i}$ ${\displaystyle j}$ ${\displaystyle n=3}$
• the exchange interaction between the localized spins is caused by Coulomb repulsion and the Pauli principle and, when restricted to closest-neighbor interaction and isotropy (see below), is expressed with a single coupling constant , the exchange energy . ${\displaystyle J}$

The model can be justified by a generalization of the Heitler-London approximation for the formation of diatomic molecules (see the relevant subsection in magnetism ). For one-dimensional systems it can be solved exactly (see below); in two and three dimensions, however, there are only approximate solutions, e.g. B. with quantum Monte Carlo methods .

Explanations

The ferromagnetism of insulators is caused by localized magnetic moments that can be ascribed to an incompletely filled electron shell (3d, 4d, 4f or 5f). An angular momentum is assigned to these localized magnetic moments , which can be expressed with the respective spin : ${\displaystyle {\vec {m}}_{i}}$ ${\displaystyle {\vec {J}}_{i}}$ ${\displaystyle {\vec {S}}_{i}}$

${\displaystyle {\begin{aligned}{\vec {m_{i}}}&=\mu _{\mathrm {B} }\,g_{J}\,{\vec {J_{i}}}\\&=\mu _{\mathrm {B} }\,g_{J}\,{\frac {\vec {S_{i}}}{g_{J}-1}}\end{aligned}}}$

With

• the Bohr magneton ${\displaystyle \mu _{\mathrm {B} }}$
• the Landé factor ${\displaystyle g_{J}}$
• the spin vector given by the spin 1/2 operators . ${\displaystyle {\vec {S_{i}}}}$

The exchange interaction between the magnetic moments can thus be expressed in terms of the associated spins. The exchange interaction thus simulates the Coulomb repulsion and the Pauli principle. The coupling constants between the localized spins are therefore also called exchange integrals. It is assumed that the exchange integrals are noticeably different from zero only for neighboring spins. All in all, one obtains an effective Hamiltonian that is designed to only explain ferromagnetism in isolators: ${\displaystyle J}$

${\displaystyle {\begin{aligned}H_{\text{Heis}}&=-J\sum _{\langle i,j\rangle }{\vec {S_{i}}}\cdot {\vec {S_{j}}}\qquad {\text{mit }}i,j{\text{ nächste Nachbarn}}\\&=-J\sum _{\langle i,j\rangle }\left(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}+S_{i}^{z}S_{j}^{z}\right)\\&=-J\sum _{\langle i,j\rangle }\left[{\frac {1}{2}}\left(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+}\right)+S_{i}^{z}S_{j}^{z}\right]\end{aligned}}}$

Generalizations

The Heisenberg model can be generalized by making the coupling constant direction-dependent (i.e. by going from isotropic to anisotropic systems).

${\displaystyle {\begin{aligned}H_{\text{verallg. Heis}}&=-\sum _{\langle i,j\rangle }\left(J^{x}S_{i}^{x}S_{j}^{x}+J^{y}S_{i}^{y}S_{j}^{y}+J^{z}S_{i}^{z}S_{j}^{z}\right)\qquad {\text{mit }}i,j\;\mathrm {n{\ddot {a}}chste} {\text{ Nachbarn}}\end{aligned}}}$

A special case of the generalized Heisenberg model is the XXZ model, which gets its name from the fact that the coupling constant is the same in two directions (i.e. ) and deviates from it in the z direction ( ): ${\displaystyle J_{x}=J_{y}=J}$ ${\displaystyle J_{z}=\Delta }$

${\displaystyle {\begin{aligned}H_{\text{XXZ}}&=-\sum _{\langle i,j\rangle }\left[J\left(S_{i}^{x}S_{j}^{x}+S_{i}^{y}S_{j}^{y}\right)+\Delta S_{i}^{z}S_{j}^{z}\right]\qquad {\text{mit }}i,j\;\mathrm {n{\ddot {a}}chste} {\text{ Nachbarn}}\\&=-\sum _{\langle i,j\rangle }\left[{\frac {J}{2}}\left(S_{i}^{+}S_{j}^{-}+S_{i}^{-}S_{j}^{+}\right)+\Delta S_{i}^{z}S_{j}^{z}\right]\end{aligned}}}$

The Heisenberg model and its special cases are often considered in connection with an applied magnetic field in the z-direction. The Hamiltonian then reads: ${\displaystyle h=g_{J}\mu _{\mathrm {B} }B_{0}}$

${\displaystyle {\begin{aligned}H_{\text{verallg. Heis,h}}&=-\sum _{\langle i,j\rangle }\left(J^{x}S_{i}^{x}S_{j}^{x}+J^{y}S_{i}^{y}S_{j}^{y}+J^{z}S_{i}^{z}S_{j}^{z}\right)-h\sum _{i}S_{i}^{z}\end{aligned}}}$

A further generalization includes the inclusion of couplings not only between nearest neighbors as well as inhomogeneities : ${\displaystyle J\rightarrow J_{ij}}$

${\displaystyle {\begin{aligned}H_{\text{verallg. Heis, inhom.}}&=-\sum _{\langle i,j\rangle }\left(J_{ij}^{x}S_{i}^{x}S_{j}^{x}+J_{ij}^{y}S_{i}^{y}S_{j}^{y}+J_{ij}^{z}S_{i}^{z}S_{j}^{z}\right)\quad {\text{mit }}i,j\;\mathrm {Gitterpl{\ddot {a}}tze} \end{aligned}}}$

The transitions to the XY model and the Ising model can best be represented in the n-vector model .

Model in k-space

To analyze the model and to consider the suggestions, it makes sense to consider the model in k-space . The transform ( discrete Fourier transform ) for the spin operators is: ${\displaystyle a\in \{x,y,z,+,-\}}$

${\displaystyle S^{a}({\vec {k}})=\sum _{i}e^{i{\vec {k}}\cdot {\vec {R}}_{i}}S_{i}^{a}}$

The generalized Heisenberg model in the magnetic field with no directional dependence and can then be written as ${\displaystyle (J_{ij}^{x}=J_{ij}^{y}=J_{ij}^{z})}$ ${\displaystyle J_{ij}=J_{ji}}$ ${\displaystyle J_{ii}=0}$

${\displaystyle {\begin{aligned}H_{\text{heis,k}}&=-{\frac {1}{N}}\sum _{\vec {k}}J({\vec {k}})\left(S^{+}({\vec {k}})S^{-}(-{\vec {k}})+S^{z}({\vec {k}})S^{z}(-{\vec {k}})\right)-hS^{z}(0)\end{aligned}},}$

where the exchange integrals also depend on the circular wave number : ${\displaystyle {\vec {k}}}$

${\displaystyle J({\vec {k}})={\frac {1}{N}}\sum _{ij}J_{ij}e^{i{\vec {k}}\cdot ({\vec {R}}_{i}-{\vec {R}}_{j})}}$

Basic state

In this section, the ground state of the generalized Heisenberg model in a magnetic field without directional dependence is considered. The basic state is the eigenstate of the system with the lowest energy. This is strongly dependent on the sign of the coupling constants:

${\displaystyle {\begin{aligned}{\text{alle}}\quad J_{ij}>0:&\qquad {\text{Ferromagnet}}\\{\text{alle}}\quad J_{ij}<0:&\qquad {\text{Anti-Ferromagnet/Ferrimagnet}}\end{aligned}}}$

Ferromagnetic ground state

It is energetically more favorable for the spins to align themselves in the same direction, and one speaks of a ferromagnetic ground state . The Heisenberg model does not change when all spin vectors are rotated, so it is invariant when rotated . Due to the rotational invariance, no direction is marked, so the orientation in the z-direction is assumed. The direction in the solid is determined by anisotropies or by a weak magnetic field . You still specialize ${\displaystyle J>0}$ ${\displaystyle |F\rangle }$

${\displaystyle J_{0}=\sum _{i}J_{ij}=\sum _{j}J_{ij},}$

then the energy of the ground state can be given as:

${\displaystyle {\begin{aligned}H|F\rangle =&E_{0}|F\rangle \\{\text{mit}}\qquad &E_{0}=-NJ_{0}\hbar ^{2}S^{2}-NhS\end{aligned}}}$

The eigenvalue of the operator was used as. For the spin 1/2 Heisenberg model is . ${\displaystyle S_{i}^{z}}$ ${\displaystyle S_{i}^{z}|F\rangle =\hbar S|F\rangle }$ ${\displaystyle S=1/2}$

Ferric or antiferromagnetic ground state

For it is energetically more favorable when neighboring spins pointing in different directions. The ground state is therefore heavily dependent on the underlying crystal lattice. a. be antiferromagnetic or ferrimagnetic . Magnetic frustration can occur for special crystal lattices, see Geometric Frustration and Spin Glass . ${\displaystyle J<0}$

Magnons and spin waves

In this section, the excitations from the ferromagnetic ground state of the generalized Heisenberg model in the magnetic field without directional dependence are considered. The excitation states are assigned to the quasiparticle magnon . These are collective excitations of the entire crystal lattice, which are therefore also referred to as spin waves .

The one-time application of the operator to the ferromagnetic ground state gives an excited eigenstate of the Heisenberg model and is called the (normalized) one-magnon state : ${\displaystyle S^{-}({\vec {k}})}$

${\displaystyle |{\vec {k}}\rangle ={\frac {1}{\hbar {\sqrt {2SN}}}}S^{-}({\vec {k}})|F\rangle }$

The associated energy of the state is given as:

${\displaystyle E({\vec {k}})=E_{0}+\hbar \omega ({\vec {k}})\qquad {\text{mit}}\qquad \hbar \omega ({\vec {k}})=\hbar h+2S\hbar ^{2}(J_{0}-J({\vec {k}}))}$

The excitation energy is ascribed to the magnon quasiparticle. If one considers the expectation value of the operator on this state, one obtains: ${\displaystyle \hbar \omega ({\vec {k}})}$ ${\displaystyle S_{i}^{z}}$

${\displaystyle \langle {\vec {k}}|S_{i}^{z}|{\vec {k}}\rangle =\hbar \left(S-{\frac {1}{N}}\right)}$

The left side of the equation is no longer dependent on space  i . This clearly means that the excitation from the ground state (one-magnon state) is not generated by simply flipping a spin on a grid location, but that the one-magnon state is evenly distributed over the grid. Hence, the state is considered to be collective excitation and is called a spin wave . ${\displaystyle |{\vec {k}}\rangle }$

1D Heisenberg model

In the one-dimensional Heisenberg model, the spins are lined up on a chain. In the case of periodic boundary conditions , the chain is closed to form a ring. The eigenstates and eigenenergies for the one-dimensional Heisenberg model were determined exactly by Hans Bethe in 1931 using the Bethe approach .

Eigenvectors and eigenstates

Since the -operator commutes with the Hamilton operator, the whole Hilbert space is divided into different subspaces, which can be diagonalized individually. ${\displaystyle S_{z}^{\text{tot}}}$

${\displaystyle [S_{z}^{\text{tot}},H]=\sum _{i=1}^{N}[S_{i}^{z},H]=0}$

The different subspaces can be described by their quantum numbers. This means that the eigenvectors are superpositions from base states with the same quantum number. In the Bethe approach, these states are classified from the ferromagnetic ground state by means of the flipped states. For example, the state with two flipped spins (i.e. ) at the lattice sites and is given as: ${\displaystyle S_{z}^{\text{tot}}=-N\dots N}$ ${\displaystyle S_{z}^{\text{tot}}}$ ${\displaystyle S_{z}^{\text{tot}}=N-2}$ ${\displaystyle n_{1}}$ ${\displaystyle n_{2}}$

${\displaystyle |n_{1}n_{2}\rangle =|\uparrow \uparrow \underbrace {\downarrow } _{n_{1}}\uparrow \dots \uparrow \underbrace {\downarrow } _{n_{2}}\uparrow \dots \uparrow \rangle }$

The eigenvectors in a subspace with a quantum number are superpositions from all possible states ${\displaystyle S_{z}}$ ${\displaystyle S_{z}=N-r}$ ${\displaystyle |n_{1},n_{2},\dots ,n_{N-r}\rangle :}$

${\displaystyle |\Psi \rangle =\sum _{n1<n2<\dots <n_{r}}^{N}a(n_{1},n_{2},\dots ,n_{r})|n_{1},n_{2},\dots ,n_{r}\rangle }$

The coefficients are plane waves and are given by the Bethe approach:

${\displaystyle a(n_{1},\dots ,n_{r})=\sum _{P\in S_{r}}\exp \left(i\sum _{j=1}^{r}k_{P_{j}}n_{j}+i\sum _{i<j}\theta _{P_{i}P_{j}}\right)}$

The parameters can be determined using the equations of the Bethe approach :

${\displaystyle {\begin{alignedat}{2}2\cot {\frac {\theta _{ij}}{2}}&=\cot {\frac {k_{i}}{2}}-\cot {\frac {k_{j}}{2}}&\qquad {\text{mit}}\quad &i,j=1,\dots ,r\\Nk_{i}&=2\pi \lambda _{i}+\sum _{j\neq i}\theta _{ij}&&\lambda _{i}={1,\dots ,N-1}\end{alignedat}}}$

The eigenvectors are given by all combinations of the Bethe quantum numbers that satisfy the equations of the Bethe approach . A classification of the eigenvectors is therefore possible via the Bethe quantum numbers. The determination of all eigenvectors is however not trivial. The associated energy of the state is given as: ${\displaystyle (\lambda _{1},\dots ,\lambda _{r})}$

${\displaystyle (E-E_{0})=J\sum _{j=1}^{r}(1-\cos k_{j})}$

Jordan-Wigner Transformation

With periodic boundary conditions, the 1D Heisenberg model can be mapped to spinless fermions on a chain with only the closest neighbor interaction using a Jordan-Wigner transformation . The Hamiltonian of the 1D Heisenberg model can therefore be written as: ${\displaystyle H_{\text{Heis}}}$

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

They are creation and annihilation operators for spinless fermions. ${\displaystyle c_{i},c_{i}^{\dagger }}$

literature

• Wolfgang Nolting: Basic course in theoretical physics. Volume 7 - many-body theory. Springer publishing house.

Web links

• B. Nachtergaele: Bibliography on the Heisenberg model.

swell

1. ↑ W. Heisenberg: On the theory of ferromagnetism . In: Journal of Physics . tape 49 , no. 9 , 1928, pp. 619-636 , doi : 10.1007 / BF01328601 .
2. ^ Paul Dirac: On the Theory of Quantum Mechanics . In: Proc. Roy. Soc. London A . tape 112 , 1926, pp. 661-677 .
3. ↑ H. Bethe: On the theory of metals. I. Eigenvalues ​​and eigenfunctions of the linear atomic chain. (On the theory of metals. I. Eigenvalues ​​and eigenfunctions of the linear atom chain), Zeitschrift für Physik A, Vol. 71, pp. 205-226 (1931). doi: 10.1007 / BF01341708 .