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{{Short description|Statistical model in quantum mechanics of magnetic materials
}}
{{Use American English|date=January 2019}}

The '''quantum Heisenberg model''', developed by [[Werner Heisenberg]], is a [[statistical mechanics|statistical mechanical]] [[mathematical model|model]] used in the study of [[critical point (thermodynamics)|critical point]]s and [[phase transition]]s of magnetic systems, in which the [[Spin (physics)|spin]]s of the magnetic systems are treated [[quantum mechanics|quantum mechanically]]. It is related to the prototypical [[Ising model]], where at each site of a lattice, a spin <math>\sigma_i \in \{ \pm 1\}</math> represents a microscopic magnetic dipole to which the magnetic moment is either up or down. Except the coupling between magnetic dipole moments, there is also a multipolar version of Heisenberg model called the [[multipolar exchange interaction]].

==Overview==
For quantum mechanical reasons (see [[exchange interaction]] or {{slink|Magnetism|Quantum-mechanical origin of magnetism}}), the dominant coupling between two dipoles may cause nearest-neighbors to have lowest energy when they are ''aligned''.  Under this assumption (so that magnetic interactions only occur between adjacent dipoles) and on a 1-dimensional periodic lattice, the [[Hamiltonian (quantum mechanics)|Hamiltonian]] can be written in the form

:<math>\hat H = -J \sum_{j =1}^{N} \sigma_j \sigma_{j+1} - h \sum_{j =1}^{N} \sigma_j </math>,

where <math>J</math> is the [[coupling constant]] and dipoles are represented by classical vectors (or "spins") σ<sub>j</sub>, subject to the periodic boundary condition <math>\sigma_{N+1} = \sigma_1 </math>.    
The Heisenberg model is a more realistic model in that it treats the spins quantum-mechanically, by replacing the spin by a [[quantum operator]] acting upon the [[tensor product]] <math>(\mathbb{C}^2)^{\otimes N}</math>, of dimension <math>2^N</math>. To define it, recall the [[Pauli matrices|Pauli spin-1/2 matrices]]

:<math>
\sigma^x = 
\begin{pmatrix}
0&1\\
1&0
\end{pmatrix}
</math>,

:<math>
\sigma^y = 
\begin{pmatrix}
0&-i\\
i&0
\end{pmatrix}
</math>,

:<math>
\sigma^z = 
\begin{pmatrix}
1&0\\
0&-1
\end{pmatrix}
</math>,

and for <math>1\le j\le N</math> and <math>a\in \{x,y,z\}</math> denote <math>\sigma_j^a = I^{\otimes j-1}\otimes \sigma^a \otimes I^{\otimes N-j}</math>, where <math>I</math> is the <math>2\times 2</math> identity matrix.
Given a choice of real-valued coupling constants <math>J_x, J_y,</math> and <math>J_z</math>, the Hamiltonian is given by

:<math>\hat H = -\frac{1}{2} \sum_{j=1}^{N} (J_x \sigma_j^x \sigma_{j+1}^x + J_y \sigma_j^y \sigma_{j+1}^y + J_z \sigma_j^z \sigma_{j+1}^z + h\sigma_j^{z}) </math>

where the <math>h</math> on the right-hand side indicates the external [[magnetic field]], with periodic [[boundary conditions]]. The objective is to determine the [[spectrum (functional analysis)|spectrum]] of the Hamiltonian, from which the [[partition function (statistical mechanics)|partition function]] can be calculated and the [[thermodynamics]] of the system can be studied.

It is common to name the model depending on the values of <math>J_x</math>, <math>J_y</math> and <math>J_z</math>: if <math>J_x \neq J_y \neq J_z</math>, the model is called the Heisenberg XYZ model; in the case of <math>J = J_x = J_y \neq J_z = \Delta</math>, it is the Heisenberg XXZ model; if <math>J_x = J_y = J_z = J</math>, it is the Heisenberg XXX model. The spin 1/2 Heisenberg model in one dimension may be solved exactly using the [[Bethe ansatz]].<ref name=bonechi1992>{{cite journal |last1=Bonechi |first1=F |last2=Celeghini |first2=E |last3=Giachetti |first3=R |last4=Sorace |first4=E |last5=Tarlini |first5=M |title=Heisenberg XXZ model and quantum Galilei group |journal=Journal of Physics A: Mathematical and General |date=7 August 1992 |volume=25 |issue=15 |pages=L939–L943 |doi=10.1088/0305-4470/25/15/007 |arxiv=hep-th/9204054 |bibcode=1992JPhA...25L.939B |s2cid=119046025 }}</ref> In the algebraic formulation, these are related to particular [[quantum affine algebras]] and [[Elliptic Quantum Group|elliptic quantum groups]] in the XXZ and XYZ cases respectively.<ref>{{cite arXiv|eprint=hep-th/9605187v1|first=L. D.|last=Faddeev|title=How Algebraic Bethe Ansatz works for integrable model|date=26 May 1996}}</ref> Other approaches do so without Bethe ansatz.<ref>{{cite journal |last1=Rojas |first1=Onofre |last2=Souza |first2=S.M. de |last3=Corrêa Silva |first3=E.V. |last4=Thomaz |first4=M.T. |title=Thermodynamics of the limiting cases of the XXZ model without Bethe ansatz |journal=Brazilian Journal of Physics |date=December 2001 |volume=31 |issue=4 |pages=577–582 |doi=10.1590/s0103-97332001000400008 |bibcode=2001BrJPh..31..577R |doi-access=free }}</ref>

=== XXX model ===
The physics of the Heisenberg XXX model strongly depends on the sign of the coupling constant
<math>J</math> and the dimension of the space. For positive <math>J</math> the ground state is always [[ferromagnetism|ferromagnetic]]. At negative <math>J</math> the ground state is [[antiferromagnetism|antiferromagnetic]] in two and three dimensions.<ref>{{cite web|url=http://www.math.ucdavis.edu/~bxn/qs.html|title=The Heisenberg Model - a Bibliography|author1=Tom Kennedy|author2=Bruno Nachtergaele|access-date=6 Jun 2019}}</ref> In one dimension the nature of correlations in the antiferromagnetic Heisenberg model depends on the spin of the magnetic dipoles. If the spin is integer then only [[short-range order]] is present. A system of half-integer spins exhibits [[quasi-long range order]].

A simplified version of Heisenberg model is the one-dimensional Ising model, where the transverse magnetic field is in the ''x''-direction, and the interaction is only in the ''z''-direction:

:<math>\hat H = -J \sum_{j =1}^{N} \sigma_j^z \sigma_{j+1}^z - gJ \sum_{j =1}^{N} \sigma_j^x </math>.

At small ''g'' and large ''g'', the ground state degeneracy is different, which implies that there must be a [[quantum phase transition]] in between. It can be solved exactly for the critical point using the duality analysis.<ref>{{cite book |doi=10.1007/978-1-4020-3463-3_13 |chapter=Duality in low dimensional quantum field theories |title=Strong interactions in low dimensions |series=Physics and Chemistry of Materials with Low-Dimens |year=2004 |last1=Fisher |first1=Matthew P. A. |volume=25 |pages=419–438 |isbn=978-1-4020-1798-8 }}</ref> The duality transition of the Pauli matrices is <math display="inline">\sigma_i^z = \prod_{j \leq i}S^x_j</math> and <math>\sigma_i^x = S^z_i S^z_{i+1}</math>, where <math>S^x</math> and <math>S^z</math> are also Pauli matrices which obey the Pauli matrix algebra.
Under periodic boundary conditions, the transformed Hamiltonian can be shown is of a very similar form:

:<math>\hat H = -gJ \sum_{j =1}^{N} S_j^z S_{j+1}^z - J \sum_{j =1}^{N} S_j^x </math>

but for the <math>g</math> attached to the spin interaction term. Assuming that there's only one critical point, we can conclude that the phase transition happens at <math>g=1</math>.

==Solution by Bethe ansatz==
{{Main article|Bethe ansatz}}

===XXX<sub>1/2</sub> model===
Following the approach of {{harvs|txt|authorlink = Ludwig Faddeev|first=Ludwig |last=Faddeev|year=1996}}, the spectrum of the Hamiltonian for the XXX model
<math display = block>H = \frac{1}{4}\sum_{\alpha, n}(\sigma^\alpha_{n}\sigma^\alpha_{n+1} - 1)</math>
can be determined by the Bethe ansatz. In this context, for an appropriately defined family of operators <math>B(\lambda)</math> dependent on a spectral parameter <math>\lambda \in \mathbb{C}</math> acting on the total Hilbert space <math>\mathcal{H} = \bigotimes_{n=1}^N h_n</math> with each <math>h_n \cong \mathbb{C}^2</math>, a '''Bethe vector''' is a vector of the form
<math display = block>\Phi(\lambda_1, \cdots, \lambda_m) = B(\lambda_1)\cdots B(\lambda_m)v_0</math>
where <math>v_0 = \bigotimes_{n=1}^N |\uparrow\,\rangle</math>.
If the <math>\lambda_k</math> satisfy the '''Bethe equation'''
<math display = block>\left(\frac{\lambda_k + i/2}{\lambda_k - i/2}\right)^N = \prod_{j \neq k}\frac{\lambda_k - \lambda_j + i}{\lambda_k - \lambda_j - i},</math>
then the Bethe vector is an eigenvector of <math>H</math> with eigenvalue <math>-\sum_k \frac{1}{2}\frac{1}{\lambda_k^2 + 1/4}</math>.

The family <math>B(\lambda)</math> as well as three other families come from a [[Transfer-matrix method (statistical mechanics)|transfer matrix]] <math>T(\lambda)</math> (in turn defined using a [[Lax matrix]]), which acts on <math>\mathcal{H}</math> along with an auxiliary space <math>h_a \cong \mathbb{C}^2</math>, and can be written as a <math>2\times 2</math> block matrix with entries in <math>\mathrm{End}(\mathcal{H})</math>,
<math display = block>T(\lambda) = \begin{pmatrix}A(\lambda) & B(\lambda) \\ C(\lambda) & D(\lambda)\end{pmatrix},</math>
which satisfies fundamental [[commutation relation]]s (FCRs) similar in form to the [[Yang–Baxter equation]] used to derive the Bethe equations. The FCRs also show there is a large commuting subalgebra given by the [[generating function]] <math>F(\lambda) = \mathrm{tr}_a(T(\lambda)) = A(\lambda) + D(\lambda)</math>, as <math>[F(\lambda), F(\mu)] = 0</math>, so when <math>F(\lambda)</math> is written as a [[polynomial]] in <math>\lambda</math>, the coefficients all commute, spanning a commutative subalgebra which <math>H</math> is an element of. The Bethe vectors are in fact simultaneous eigenvectors for the whole subalgebra.

===XXX<sub>s</sub> model===
For higher spins, say spin <math>s</math>, replace <math>\sigma^\alpha</math> with <math>S^\alpha</math> coming from the [[Lie algebra representation]] of the [[Lie algebra]] <math>\mathfrak{sl}(2, \mathbb{C})</math>, of dimension <math>2s + 1</math>. The XXX<sub>s</sub> Hamiltonian
<math display = block>H = \sum_{\alpha, n}(S^\alpha_{n}S^\alpha_{n+1} - (S^\alpha_{n}S^\alpha_{n+1})^2)</math>
is solvable by Bethe ansatz with Bethe equations
<math display = block>\left(\frac{\lambda_k + is}{\lambda_k - is}\right)^N = \prod_{j \neq k}\frac{\lambda_k - \lambda_j + i}{\lambda_k - \lambda_j - i}.</math>

===XXZ<sub>s</sub> model===
For spin <math>s</math> and a parameter <math>\gamma</math> for the deformation from the XXX model, the BAE (Bethe ansatz equation) is
<math display = block>\left(\frac{\sinh(\lambda_k + is\gamma)}{\sinh(\lambda_k - is\gamma)}\right)^N = \prod_{j \neq k}\frac{\sinh(\lambda_k - \lambda_j + i\gamma)}{\sinh(\lambda_k - \lambda_j - i\gamma)}.</math>
Notably, for <math>s = \frac{1}{2}</math> these are precisely the BAEs for the [[six-vertex model]], after identifying <math>\gamma = 2\eta</math>, where <math>\eta</math> is the '''anisotropy''' parameter of the six-vertex model.<ref name="lieb">{{cite journal 
|last1=Lieb |first1=Elliott H. 
|title=Exact Solution of the Problem of the Entropy of Two-Dimensional Ice 
|journal=Physical Review Letters |date=24 April 1967 
|volume=18 |issue=17 |pages=692–694 
|doi=10.1103/PhysRevLett.18.692|bibcode=1967PhRvL..18..692L 
}}</ref><ref name="DDT">{{cite journal 
|last1=Dorey |first1=Patrick 
|last2=Dunning |first2=Clare 
|last3=Tateo |first3=Roberto 
|title=The ODE/IM correspondence 
|journal=Journal of Physics A: Mathematical and Theoretical |date=10 August 2007 
|volume=40 |issue=32 |pages=R205–R283 
|doi=10.1088/1751-8113/40/32/R01 
|s2cid=14281617 
|url=https://iopscience.iop.org/article/10.1088/1751-8113/40/32/R01 |issn=1751-8113}}</ref> This was originally thought to be coincidental until Baxter showed the XXZ Hamiltonian was contained in the algebra generated by the transfer matrix <math>T(\nu)</math>,<ref name="baxterXXZ">{{cite journal 
|last1=Baxter |first1=Rodney J 
|title=One-dimensional anisotropic Heisenberg chain 
|journal=Annals of Physics |date=1 April 1972 
|volume=70 |issue=2 |pages=323–337 
|doi=10.1016/0003-4916(72)90270-9 
|bibcode=1972AnPhy..70..323B 
|url=https://dx.doi.org/10.1016/0003-4916%2872%2990270-9 
|language=en |issn=0003-4916}}</ref> given exactly by
<math display = block>H_{XXZ_{1/2}} = -i \sin 2\eta \frac{d}{d\nu}\log T(\nu)\Big|_{\nu = -i\eta} - \frac{1}{2}\cos 2\eta 1^{\otimes N}.</math>

==Applications==
* Another important object is [[Entropy of entanglement|entanglement entropy]]. One way to describe it is to subdivide the unique ground state into a block (several sequential spins) and the environment (the rest of the ground state). The entropy of the block can be considered as entanglement entropy. At zero temperature in the critical region (thermodynamic limit) it scales logarithmically with the size of the block. As the temperature increases the logarithmic dependence changes into a linear function.<ref>{{cite journal |last1=Korepin |first1=V. E. |title=Universality of Entropy Scaling in One Dimensional Gapless Models |journal=Physical Review Letters |date=5 March 2004 |volume=92 |issue=9 |pages=096402 |doi=10.1103/PhysRevLett.92.096402 |pmid=15089496 |arxiv=cond-mat/0311056 |bibcode=2004PhRvL..92i6402K |s2cid=20620724 }}</ref> For large temperatures linear dependence follows from the [[second law of thermodynamics]].
* The Heisenberg model provides an important and tractable theoretical example for applying [[DMRG of the Heisenberg model|density matrix renormalisation]].
* The [[Ice-type model|six-vertex model]] can be solved using the algebraic Bethe ansatz for the Heisenberg spin chain {{Harvard citations |last1=Baxter |year=1982}}.
* The half-filled [[Hubbard model]] in the limit of strong repulsive interactions can be mapped onto a Heisenberg model with <math>J<0</math> representing the strength of the [[superexchange]] interaction.
* Limits of the model as the lattice spacing is sent to zero (and various limits are taken for variables appearing in the theory) describes integrable field theories, both non-relativistic such as the [[nonlinear Schrödinger equation]], and relativistic, such as the <math>S^2</math> [[sigma model]], the <math>S^3</math> sigma model (which is also a [[principal chiral model]]) and the [[sine-Gordon model]].
* Calculating certain [[correlation functions]] in the planar or large <math>N</math> limit of [[N = 4 supersymmetric Yang–Mills theory]]<ref>{{cite journal |last1=Beisert |first1=Niklas |author-link=Niklas Beisert |title=The dilatation operator of N=4 super Yang–Mills theory and integrability |journal=Physics Reports |date=1 December 2004 |volume=405 |issue=1 |pages=1–202 |doi=10.1016/j.physrep.2004.09.007|arxiv=hep-th/0407277 |bibcode=2004PhR...405....1B |s2cid=118949332 }}</ref>

== Extended symmetry ==
The integrability is underpinned by the existence of large symmetry algebras for the different models. For the XXX case this is the [[Yangian]] <math>Y(\mathfrak{sl}_2)</math>, while in the XXZ case this is the [[quantum group]] <math>\hat{ \mathfrak{sl}_q(2)}</math>, the [[q-deformation]] of the [[affine Lie algebra]] of <math>\hat{\mathfrak{sl}_2}</math>, as explained in the notes by {{harvs|txt|last=Faddeev|year=1996}}.

These appear through the transfer matrix, and the condition that the Bethe vectors are generated from a state <math>\Omega</math> satisfying <math>C(\lambda) \cdot \Omega = 0</math> corresponds to the solutions being part of a [[highest-weight representation]] of the extended symmetry algebras.

==See also==
*[[Classical Heisenberg model]]
*[[DMRG of the Heisenberg model]]
*[[Quantum rotor model]]
*[[t-J model]]
*[[J1 J2 model]]
*[[Majumdar–Ghosh model]]
*[[AKLT model]]
*[[Multipolar exchange interaction]]

== References ==
* R.J. Baxter, ''Exactly solved models in statistical mechanics'', London, Academic Press, 1982
*{{cite journal |last1=Heisenberg |first1=W. |title=Zur Theorie des Ferromagnetismus |trans-title=On the theory of ferromagnetism |language=German |journal=Zeitschrift für Physik |date=1 September 1928 |volume=49 |issue=9 |pages=619–636 |doi=10.1007/BF01328601 |bibcode=1928ZPhy...49..619H |s2cid=122524239 }}
*{{cite journal |last1=Bethe |first1=H. |title=Zur Theorie der Metalle |trans-title=On the theory of metals |language=German |journal=Zeitschrift für Physik |date=1 March 1931 |volume=71 |issue=3 |pages=205–226 |doi=10.1007/BF01341708 |bibcode=1931ZPhy...71..205B |s2cid=124225487 }}

==Notes==
{{reflist}}

{{Statistical mechanics topics}}
{{Integrable systems}}

[[Category:Spin models]]
[[Category:Quantum magnetism]]
[[Category:Quantum lattice models]]
[[Category:Magnetic ordering]]
[[Category:Werner Heisenberg]]