View source for Mermin-Wagner Theorem

The <strong>Mermin-Wagner theorem</strong> exemplifies the crucial influence of both the dynamical symmetry and the spatial dimensionality on thermal phase transitions in many-body systems. It says: ''At finite temperatures, the quantum spin-<math>S</math> Heisenberg model with isotropic and finite-range exchange interactions on one- or two-dimensional lattices can be neither ferro- nor antiferromagnetic.''
 
==Heisenberg Model, Symmetry==

(Units are chosen as <math>\hbar=1\ ,</math> <math>k_B=1\ ,</math> <math>g\mu_B=1\ .</math>)

This is a model for magnetic insulators and is specified by the Hamiltonian
:<math>\label{eq:hamiltonian}
\begin{array}{lcl}
\hat{H}_\Lambda(h) &=&  - \sum_{{\mathbf x},{\mathbf y} \in \Lambda}
J({\mathbf x}-{\mathbf y}) \vec{S}({\mathbf x}) \cdot \vec{S}({\mathbf y}) - 
\sum_{{\mathbf x} \in \Lambda} h({\mathbf x}) \hat{S}_3 ({\mathbf x}) \\
&=& \hat{H}^{(0)} + \hat{H}^{(h)} ,

\end{array}
</math>

where <math>\vec{S}({\mathbf x}) = (\hat{S}_1 ({\mathbf x}),\hat{S}_2 ({\mathbf x}),\hat{S}_3 ({\mathbf x}))</math> denotes 3-vectorial spin operators localized at the <math>N</math> lattice sites <math>{\mathbf x} = (x_1,x_2, \ldots, x_d)</math> of a finite <math>d</math>-dimensional cubic box <math>\Lambda \subset \mathbb{Z}^d</math> with volume <math>|\Lambda| = N</math> (i.e.&nbsp;unit lattice constant) and periodic boundary conditions. The spin components <math>\hat{S}_i({\mathbf x})</math> obey the usual commutation relations,
:<math>
[ \hat{S}_1  ({\mathbf x}), \hat{S}_2 ({\mathbf y})] = {\rm i} \delta_{{\mathbf x},{\mathbf y} }  S_3 ({\mathbf x}) , {\rm etc.}
</math>
and <math>\vec{S}^2 = S(S+1)I_{(2S+1)}\ ,</math> <math>I_{(2S+1)}</math> the identity in the spin space, <math>S=\frac{1}{2}, 1, \frac{3}{2}, \ldots\ .</math>

The exchange couplings <math>J({\mathbf x}-{\mathbf y})=J({\mathbf y}-{\mathbf x})\ ,</math> <math>J({\mathbf 0})=0</math> and <math>J({\mathbf x}-{\mathbf y})\geq 0 (\leq 0)</math> for (anti-)ferromagnets are assumed for the moment to have finite range. 

The operators <math>\hat{S}_i({\mathbf x})\ ,</math> <math>i=1,2,3\ ,</math> act in a <math>(2S+1)</math>-dimensional Hilbert space <math>{\mathcal H}({\mathbf x})\ ;</math> the total <math>(2S+1)^N</math>-dimensional Hilbert space is given by <math>{\mathcal H}_\Lambda = \otimes_{{\mathbf x} \in \Lambda} {\mathcal H}({\mathbf x})\ .</math> 

Let <math>{\mathcal A}_\Lambda^{Spin} = {\mathcal A}_\Lambda</math> denote the polynomial algebra of spin operators in <math>{\mathcal H}_\Lambda\ .</math> The <math>SU(2)</math>-symmetry of the Hamiltonian <math>\hat{H}^{(0)}</math> may be described by one-parameter groups of unitary automorphisms <math>\gamma_n(\phi) : {\mathcal A}_\Lambda \rightarrow {\mathcal A}_\Lambda\ ,</math> corresponding to simultaneous global rotations <math>0 \leq \phi \leq 2\pi</math> around fixed but arbitrary axes <math>\vec{n}\ ,</math> <math>\vec{n}^2 =1\ ,</math> of all <math>N</math> spins in the box <math>\Lambda\ .</math> In particular,

:<math>
\gamma_n(\phi) \hat{P} = \exp (-{\rm i} \phi \hat{G}_n) \hat{P}  \exp ({\rm i} \phi \hat{G}_n),
</math>
with generators <math>\hat{G}_n = \sum_{{\mathbf x} \in \Lambda} \vec{n}\cdot\vec{S}({\mathbf x})\ ,</math> and <math>\hat{P} \in {\mathcal A}_\Lambda\ .</math>

The continuous symmetry <math>\gamma_n(\phi) \hat{H}^{(0)} = \hat{H}^{(0)}</math> is ''explicitly broken'' by the external field <math>\vec{h}({\mathbf x})=(0,0,h({\mathbf x}))\ ,</math> <math>h({\mathbf x})= h \exp ({\mathrm i} {\mathbf q}\cdot {\mathbf x})\ .</math> In the case of a ferromagnet, we take <math>{\mathbf q} = 0\ .</math> To deal with conventional two-sublattice <math>(\Lambda_\pm)</math> Neel antiferromagnetism we choose '''q''' such that <math>e^{{\rm i} {\mathbf q} {\mathbf x} } = +1 (-1)</math> when <math>{\mathbf x} \in \Lambda_+ (\Lambda_-)\ .</math>

==Magnetisation==

The external field <math>h({\mathbf x})</math> induces a (sublattice) magnetisation per site as expressed by the thermal average (<math>\beta^{-1} := T</math>):
:<math>\label{eq:magnetisation}
\begin{array}{rcl}
m_{{\mathbf q} }(T,h,N) &=& \frac{1}{N Z} {\rm Tr} \left(e^{-\beta \hat{H}_\Lambda(h)} \sum_{{\mathbf x} \in \Lambda} \hat{S}_3 ({\mathbf x}) e^{{\rm i} {\mathbf q}\cdot {\mathbf x} } \right) \\
&=& \frac{1}{N} \left\langle \hat{M}_{{\mathbf q} } \right\rangle  =: \frac{1}{N} \omega( \hat{M}_{{\mathbf q} }),  \\
Z &=& Z (T,h,N) = {\rm Tr} \exp (-\beta \hat{H}_\Lambda (h)) .
\end{array}
</math>

''Definition:'' The Heisenberg model describes (anti-)ferromagnetism at <math>T>0</math> if it exhibits a finite ''spontaneous'' (sublattice) magnetisation in the ''thermodynamic'' limit <math>|\Lambda|=N \rightarrow \infty\ :</math>
:<math>
\begin{array}{rcl}
{\rm thermodynamic\ limit:} & & m_{{\mathbf q} } (T,h) := \lim_{N\rightarrow\infty} m_{{\mathbf q} } (T,h,N) \\
{\rm spontaneous\ magnetisation} & & \sigma_{{\mathbf q} } := \lim_{h\rightarrow 0} m_{{\mathbf q} } (T,h) 
\end{array}
</math>

==Spin Waves==

The absence of ferromagnetism in the 2d-Heisenberg model has been already anticipated by Felix Bloch (Bloch 1930), with physically plausible arguments, based on elementary ''spin wave theory''.
In the isotropic Heisenberg ferromagnet at temperatures <math>T\ll T_{{\textstyle Curie} }</math> the energetically low-lying thermal excitations may be described approximately as an ideal Bose gas of magnons (quantum spin waves) in the periodic box <math>\Lambda\subset \mathbb{Z}^d\ .</math> A magnon with wave vector <math>{\mathbf k}</math> has the energy <math>\epsilon_f ({\mathbf k}) \simeq \eta k^2\ ,</math> <math>\eta>0\ ,</math> for small values of <math>k\ .</math> These thermal excitations cause a <math>T</math>-dependent reduction <math>\Delta  \sigma_f(T) = \sigma_f (T=0) - \sigma_f (T)</math> of the average spontaneous magnetisation from its saturation value. In the spin wave approximation one finds 
:<math>
\Delta  \sigma_f(T) \propto \int_{k\geq 0} \frac{k^{d-1} {\rm d} k}{\exp(\beta \epsilon_f(k))-1} .
</math>
For small values of <math>k\ ,</math> the integrand goes as <math>k^{d-3}\ ;</math> therefore the integral diverges if <math>d\leq 2\ .</math> This result indicates a breakdown of the spin wave approximation and has been interpreted to mean that any spontaneous magnetisation is ultimately wiped out by the thermal excitation of too many magnons.

A similar argument has been put forward (Auerbach 1994) to exclude isotropic Heisenberg antiferromagnetism in <math>d\leq 2\ .</math> In this case, spin wave modes with small <math>k</math> have an energy <math>\epsilon_{af} = \gamma k\ ,</math> <math>\gamma>0\ .</math> Their thermal excitations are found in linear order of spin wave theory to reduce the sublattice magnetisation according to 
:<math>\label{eq:spinwavereductionaf}
\Delta \sigma_{af} (T) \propto \int_{k\geq 0} \frac{k^{d-1} a(k) {\rm d} k}{\exp(\beta \epsilon_{af}(k))-1}.

</math>

The factor <math>a(k) = const./\epsilon_{af}(k)</math> is due to quantum zero-point motions, which originate from the fact that the alternating magnetisation is not a conserved quantity, in contrast to the ferromagnetic case, where <math>\sum_{{\mathbf x} \in \Lambda} \hat{S}_i({\mathbf x})\ ,</math> <math>\alpha=1,2,3\ ,</math> commutes with the Hamiltonian <math>\hat{H}_\Lambda^{(0)}\ .</math> Since one finds <math>a(k) \propto k^{-1}</math> for small values of <math>k\ ,</math> the integral \eqref{eq:spinwavereductionaf} displays the same singularity as in the ferromagnetic case and leads to the same conclusion that 
<math> \Delta \sigma_{af} (T) =0</math> for <math>T>0</math> and <math>d\leq 2\ .</math>

The Mermin-Wagner theorem validates these intuitive spin wave arguments with rigorous bounds for the magnetisation.

==Theorem==

Let <math>m_{{\mathbf q} } (T,h)</math> be the magnetisation defined in \eqref{eq:magnetisation}, where <math>\hat{H}_\Lambda(h)\ ,</math> with <math>\sum_{{\mathbf R} } | J({\mathbf R})| {\mathbf R}^2 =: \overline{J} <\infty\ ,</math> is given by \eqref{eq:hamiltonian}.

Then we have 
:<math>\label{eq:theorem}
\sigma_{{\mathbf q} } (T) = \lim_{h\rightarrow 0} m_{{\mathbf q} } (T,h) = 0

</math>

if <math>d \leq 2</math> and <math>T>0</math> (Mermin and Wagner 1966).

==Proof==

In order to establish the proposition \eqref{eq:theorem}, we try to probe the stability of the spontaneous magnetisation against thermal fluctuations arising from energetically low-lying elementary excitations. In particular, our aim is to find an upper bound <math>\delta(T,h)\geq |m_{{\mathbf q} }(T,h))|</math> such that <math>\delta(T,h) \rightarrow 0</math> when <math>h \rightarrow 0\ ,</math> if <math>d \leq 2\ .</math>

To carry out this plan, we adopt an idea due to P. Hohenberg (1967) and utilize the Bogoliubov inequality,
:<math>\label{eq:bogoliubov}
\frac{\beta}{2} \omega (\hat{A}^\dagger \hat{A} + \hat{A} \hat{A}^\dagger) \geq \frac{|\omega([\hat{C},\hat{A}^\dagger])|^2}{\omega([\hat{C}[\hat{H}_\Lambda(h),\hat{C}^\dagger]])}

</math>

for thermal averages <math>\omega(\hat{Q})\ ,</math> <math>\hat{Q} \in {\mathcal A}_\Lambda\ ,</math> as introduced in \eqref{eq:magnetisation}. (For convenience, we include below an elementary, model-independent derivation of this inequality.)

The crucial task is to choose suitable operators <math>\hat{A}</math> and <math>\hat{C}\ .</math> For this purpose, we consider locally twisted spin configurations, generated by
:<math>
\hat{G}_n := \sum_{{\mathbf x} } f({\mathbf x}) \vec{n} \cdot \vec{S}({\mathbf x}),
</math>
:<math>\label{eq:gamma1}
\left. i \frac{d}{d\phi} \gamma_n(\phi, f) \hat{Q} \right|_{\phi=0} = [ \hat{G}_n(f), \hat{Q}] =: i \dot{\gamma}_n(f) \hat{Q}, \quad \hat{Q}\in {\mathcal A}_\Lambda ,

</math>

with smoothly varying mappings <math>f : \mathbb{Z}^d \rightarrow \mathbb{C}</math> (here and subsequently, sums over <math>{\mathbf x}</math> are understood to run over <math>\Lambda</math>).

In particular, 

:<math>\label{eq:gamma2}
\dot{\gamma}_n(f) ( (\dot{\gamma}_n(f) \hat{H}_\Lambda)^\dagger ) = [\hat{G}_n(f) , [\hat{H}_\Lambda,\hat{G}^\dagger_n(f)]] .

</math>


With the spin wave argument in mind, we expect that such thermally fluctuating twists may be sufficiently violent to prevent the appearance of a spontaneous magnetisation at low spatial dimensions. 

By comparing \eqref{eq:gamma1} and \eqref{eq:gamma2} with the commutators in \eqref{eq:bogoliubov}, it is suggestive to choose 
:<math>
\hat{C} = \sum_{{\mathbf x} } e^{- i {\mathbf k}\cdot {\mathbf x} } \hat{S}_1 ({\mathbf x}) =: \hat{S}_1 ({\mathbf k}) ,
</math>
with <math>{\mathbf k} \neq 0</math> an arbitrary vector from the first Brillouin zone <math>\Lambda^*\ .</math> By taking
:<math>
\hat{A} = \sum_{{\mathbf x} } e^{- i ({\mathbf k}+{\mathbf q})\cdot {\mathbf x} } \hat{S}_2  ({\mathbf x}) =: \hat{S}_2 ({\mathbf k}+{\mathbf q}),
</math>
we find immediately
:<math>
\begin{array}{rcl}
|\omega( [\hat{C},\hat{A}^\dagger]) | &=& N | m_{{\mathbf q} } (T,h,N)| , \\
\frac{1}{2} \omega (\hat{A}^\dagger \hat{A} + \hat{A} \hat{A}^\dagger) &=& \omega (\hat{S}_2^\dagger ({\mathbf k} + {\mathbf q}) \hat{S}_2 ({\mathbf k} + {\mathbf q})).
\end{array}
</math>
The evaluation of <math>\omega ( [ \hat{C}, [ \hat{H}_\Lambda, \hat{C}^\dagger ]])</math> is a bit more tedious but elementary. By using translational symmetry and \eqref{eq:magnetisation} we obtain

:<math>
D_{{\mathbf k} } = 2 \sum_{{\mathbf x}, {\mathbf y} } J ({\mathbf x} - {\mathbf y})
(1 - \cos ({\mathbf k} \cdot ({\mathbf x} - {\mathbf y}))) \omega
(\hat{S}_2 ({\mathbf x}) \hat{S}_2 ({\mathbf y}) +  \hat{S}_3 ({\mathbf x}) \hat{S}_3 ({\mathbf y})) + Nhm_{{\mathbf q} } .
</math> 
We note that <math>D_{{\mathbf k} } > 0</math> by construction.

The inequality \eqref{eq:bogoliubov} now takes the form 
:<math>\label{eq:bogoliubov2}
\frac{\beta}{N^2} \omega (| \hat{S}_2^\dagger ({\mathbf k}+{\mathbf q}) \hat{S}_2 ({\mathbf k}+{\mathbf q})  |) \geq \frac{|m_{{\mathbf q} }|^2}{D_{{\mathbf k} } } .

</math>

With <math>1 - \cos {\mathbf k}\cdot {\mathbf R} \leq \frac{1}{2} {\mathbf k}^2 {\mathbf R}^2\ ,</math> the Schwarz inequality <math>|\omega( \hat{S}_i ({\mathbf x})\hat{S}_i ({\mathbf y}) | 
\leq ( \omega( \hat{S}^2_i ({\mathbf x}))  \omega( \hat{S}^2_i ({\mathbf y})) )^{1/2}\ ,</math> 
and with the following upper bound for <math>D_{{\mathbf k} }\ ,</math>
:<math>
D_{{\mathbf k} } \leq N \sum_{{\mathbf x} } | J({\mathbf x})| {\mathbf x}^2 {\mathbf k}^2 
( \omega ( \hat{S}^2_2 ({\mathbf x})) + \omega ( \hat{S}^2_3 ({\mathbf x}))) + N |hm_{{\mathbf q} }|,
</math> 
the inequality \eqref{eq:bogoliubov2} turns into 
:<math>\label{eq:bogoliubov3}
\frac{\beta}{N^2} \omega  (\hat{S}_2^\dagger ({\mathbf k} + {\mathbf q}) \hat{S}_2 ({\mathbf k} + {\mathbf q})) \geq \frac{1}{N} \frac{|m_{{\mathbf q} }|^2}{\rho k^2 + | h m_{{\mathbf q} }|} ,

</math>

where <math>\rho := \overline{J} S(S+1) > 0\ .</math> Since \eqref{eq:bogoliubov3} holds for any <math>{\mathbf k} \in \Lambda^*\ ,</math> we may sum both sides over <math>{\mathbf k}\ ,</math> 
:<math>
\frac{\beta}{N^2} \sum_{{\mathbf k} \in \Lambda^*} \omega  (\hat{S}_2^\dagger ({\mathbf k} + {\mathbf q}) \hat{S}_2 ({\mathbf k} + {\mathbf q})) = \frac{\beta}{N} \sum_{{\mathbf x} } \omega (\hat{S}^2 _2 ({\mathbf x})) ,
</math>
:<math>
\frac{\beta}{N} \sum_{{\mathbf x} } \omega (\hat{S}^2 _2 ({\mathbf x})) \geq
\frac{|m_{{\mathbf q} }|^2}{N} \sum_{{\mathbf k} \in \Lambda^*} \frac{1}{\rho k^2 + | h m_{{\mathbf q} }|} .
</math>
After replacing the left-hand side by the upper bound <math>\beta S(S+1)</math> and going to the TD limit, we arrive at 
:<math>\label{eq:bogolfinal}
\frac{1}{T} S(S+1) \geq |m_{{\mathbf q} }(T,h)|^2 \frac{\Omega_d d}{(2\pi)^d} \int_0^{k_0} dk 
\frac{k^{d-1}}{\rho k^2 + | h m_{{\mathbf q} }|} .

</math>

We have strengthened the inequality by restricting the <math>k</math>-integration to run over the volume <math>\Omega_d k_0^{d}</math> of a ball with radius <math>0 < k_0 < \pi\ .</math> The integration in \eqref{eq:bogolfinal} is elementary and yields for <math>d=1\ :</math>
:<math>\label{eq:mw1}
S(S+1) \geq \frac{T|m_{\mathbf q} (T,h) |^2 }{\pi \sqrt{\rho  | h m_{{\mathbf q} }|} } \arctan \left( \sqrt{\frac{ | h m_{{\mathbf q} }|}{\rho k_0^2} } \right)

</math>

and for <math>d=2\ :</math>
:<math>\label{eq:mw2}
S(S+1) \geq \frac{T|m_{\mathbf q} (T,h) |^2 }{4\pi\rho} \ln \left( 1 + \frac{\rho k_0^2}{ | h m_{{\mathbf q} }| }\right) .

</math>

These results imply <math>|m_{\mathbf q} (T,h) | \rightarrow 0</math> when <math>h \rightarrow 0</math> for all finite values of <math>T, \overline{J}, S\ .</math>

Explicitly, \eqref{eq:mw1} and \eqref{eq:mw2} yield upper bounds on <math>|m_{\mathbf q} (T,h) |\ ,</math> which hold asymptotically when <math>h\rightarrow 0\ :</math>
:<math>
|m_{\mathbf q} (T,h) | \leq \left\{ 
\begin{array}{ll}
c_1 \overline{J}^{1/3} S(S+1) | h T^{-2} |^{1/3}, & d=1 \\
c_2 \overline{J}^{1/2} S(S+1) | T \ln |h| |^{-1/2}, & d=2
\end{array}
\right.
</math>
with numerical constants <math>c_{1,2}\ .</math>

Consequently, <math>|\sigma_{\mathbf q} (T) | =0</math> for <math>T>0</math> in <math>d\leq 2\ .</math>

==Complementary remarks==

  
* If the exchange coupling is anisotropic, then Bogoliubov's inequality yields inconclusive results. In the uniaxial case <math>J_1 = J_2 \neq J_3\ ,</math> <math>\vec{h}=(0,h,0)\ ,</math> however the proposition \eqref{eq:theorem} remains true.  
* The theorem does not rule out the existence of thermal equilibrium phases with a singular admittance (see e.g. Stanley and Kaplan 1966), caused by a slow algebraic decay of corresponding spatial correlations, Kosterlitz and Thouless 1973.  
* The conclusion \eqref{eq:theorem} has also been shown (Mermin 1967) to hold for the ''classical'' Heisenberg model defined by the lattice Hamiltonian  :<math>\label{eq:hamclass}
H_\Lambda (h^{cl}) = - \sum_{{\mathbf x}, {\mathbf y} } J^{cl} ({\mathbf x} - {\mathbf y}) \vec{n}({\mathbf x}) \cdot \vec{n} ({\mathbf y}) - \sum_{{\mathbf x} } h^{cl} ({\mathbf x}) n_3 ({\mathbf x}) ,</math>
with an SO(3) invariant coupling of classical unit vectors <math>\vec{n}\ .</math>     Formally, we obtain \eqref{eq:hamclass} via the following scaling procedure where   :<math>\label{eq:classscaling}
\lambda(S) := \sqrt{S(S+1)}, \quad \frac{1}{\lambda(S)} \vec{S} =: \vec{n}, \quad \lambda^2 (S) J =: J^{cl}, \quad \lambda(S) h =: h^{cl} .</math>
In the limit <math>S\rightarrow\infty</math> with <math>J^{cl}</math> and <math>h^{cl}</math> fixed, the 3-vector components <math>n_i\ ,</math> <math>i=1,2,3\ ,</math> turn into commuting degrees of freedom. Also the inequality \eqref{eq:bogolfinal} survives the limit with the replacements \eqref{eq:classscaling} together with <math>m_{{\mathbf q} } / \lambda(S) =: m^{cl}_{{\mathbf q} } (T, h^{cl})\ .</math>     A variety of extensions and further applications of the theorem \eqref{eq:theorem} are reviewed in Gelfert and Nolting (2001).    
* Consider a geometrically disordered classical or quantum system with a compact continuous   symmetry group <math>{\mathcal G}\ ,</math> where the internal degrees of freedom are localized at the vertices of a ''graph''. As an extension of previous work by D. Cassi (Cassi 1992) it can be shown (Merkl and Wagner 1994) that ''equilibrium states'' <math>\omega_\beta</math> ''remain invariant'' at any temperature <math>\beta^{-1}>0</math> under the group action <math>\gamma_G\ ,</math> <math>\omega_\beta \circ \gamma_G = \omega_\beta\ ,</math> if a ''random walk'' on the graph is ''recurrent''. The absence of long-range order and of an order parameter (such as a spontaneous magnetisation) follows as a special case. Thereby, the random walk device serves as a substitute for translational invariance.      The above result generalizes theorem \eqref{eq:theorem}, since random walks on <math>d</math>-dimensional ''regular'' lattices are known to be recurrent when <math>d \leq 2\ .</math> The result also illustrates the fact (Klein et al 1981), that thermal equilibrium states of <math>(d\leq 2)</math>-systems with short-ranges interaction exhibit all the symmetries of their Hamiltonians.

==Supplement: Bogoliubov Inequality==

Be <math>{\mathcal H}</math>  a Hilbert space with a (general) self-adjoint Hamiltonian <math>H</math> such that <math>\exp (-\beta H)</math> is in the trace class. At the same time, we consider <math>{\mathcal B}({\mathcal H}) \ni A, B, C, \ldots\ ,</math> the space of bounded linear operators on <math>{\mathcal H}\ .</math> 

We can then define thermal equilibrium states, which we denote by <math>\omega_\beta = \omega\ ,</math> as linear functionals on the (<math>C^*</math>-) algebra <math>{\mathcal A}</math> of the operators in <math>{\mathcal B}({\mathcal H})\ :</math>

:<math>\label{eq:thermalstates}
\omega : {\mathcal A} \rightarrow \mathbb{C}, \quad {\mathcal A} \mapsto \omega({\mathcal A}) .

</math>

In particular, <math>\omega_\beta(A) = {\rm Tr}(\rho_\beta A)\ ,</math> <math>\rho_\beta = Z^{-1} e^{-\beta H}\ ,</math> <math>{\rm Tr} \rho_\beta = 1\ .</math> We set <math>\beta H =: K</math> and define
:<math>
A_\tau := e^{\tau K} A e^{-\tau K}, \quad 0 \leq \tau \leq 1.
</math>
Note that
:<math>\label{eq:aadjoint}
(A^\dagger)_\tau = (A_{-\tau})^\dagger .

</math>

The thermal states \eqref{eq:thermalstates} fulfill the fundamental Kubo-Martin-Schwinger (KMS) relation (Bratteli and Robinson 1981)
:<math>\label{eq:kmsrelation}
\omega(A_\tau B) = \omega (A B_{-\tau}) ,

</math>

which follows from the cyclic invariance of the trace operation. 

At last, we introduce the sesquilinear Mori product that serves as an inner product on the operator algebra <math>A\ :</math>
:<math>\label{eq:moriproduct}
(A| B) := \int_0^1 {\rm d} \tau \, \omega ( (A^\dagger)_\tau B) = (B | A)^* = (B^\dagger | A^\dagger).
 
</math>

The following relations are elementary consequences of the Eqs.&nbsp;\eqref{eq:aadjoint}, \eqref{eq:kmsrelation} and \eqref{eq:moriproduct}:

First,
:<math>\label{eq:derivative}
\frac{{\rm d} }{{\rm d} \tau} \omega ( (A^\dagger)_\tau C) = \omega ((A^\dagger)_\tau [C, K]),

</math>

and 
:<math>
(A | [C, K]) = \int_0^1 {\rm d}\tau \frac{{\rm d} }{{\rm d} \tau} \omega ((A^\dagger)_\tau C) = \omega ([C, A^\dagger]).
</math>
Second, 
:<math>
\frac{{\rm d}^2}{{\rm d} \tau^2} \omega ( (A^\dagger)_\tau A) = \omega ((Q_{-\tau/2})^\dagger Q_{-\tau/2}) \geq 0,
</math>
where <math>Q=[A,K]\ .</math> Hence <math>\omega ((A^\dagger)_\tau A) =: \Omega_A (\tau)</math> is convex in <math>\tau\ .</math> 

Therefore 
:<math>\label{eq:semipositivity}
0 \leq (A | A) = \int_0^1 {\rm d}\tau\, \Omega_A(\tau) \leq \frac{1}{2} (\Omega_A(1) + \Omega_A(0)) =
\frac{1}{2} \omega (A^\dagger A + AA^\dagger) .

</math>

The Mori product satisfies all the properties of an inner product required to establish the Schwarz inequality:
:<math>\label{eq:schwarzinequality}
| ( A| B) |^2 \leq (A|A) (B| B) .

</math>

After setting <math>B = [C, K]</math> and using Eqs.&nbsp;\eqref{eq:derivative} and \eqref{eq:semipositivity} we finally arrive at the Bogoliubov inequality:
:<math>\label{eq:bogoliubovapp}
| \omega([C,A^\dagger])|^2 \leq \frac{\beta}{2} \omega(A^\dagger A + AA^\dagger) \omega([C, [H,C^\dagger]]) .

</math>

The above choice for <math>B</math> is made in view of later applications. 

The original derivation of the Bogoliubov inequality can be found in Bogoliubov 1971. The inequality was used for the first time by P. Hohenberg (Hohenberg 1967) to rule out the existence of long-range order in translationally invariant Bose and Fermi systems at <math>d\leq 2\ ,</math> <math>T>0\ .</math>

The mathematical status of Bogoliubov's inequality and its role as a tool in statistical physics is documented, for instance, in Bratteli and Robinson 1981 and Simon 1993.

==References==
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*F. Bloch, Zur Theorie des Ferromagnetismus, Z. Physik '''61''', 206 (1930).

*N. N. Bogoliubov, Lectures on quantum statistics, Vol. 2, London (1971).

*O. Bratteli, D. W. Robinson, Operator algebras and quantum statistical mechanics II, New York, Berlin (1981).

*D. Cassi, Phase transitions and random walks on graphs: A generalization of the Mermin-Wagner theorem to disordered lattices, fractals and other structures, Phys. Rev. Lett. '''68''', 3631 (1992)

*A. Gelfert, W. Nolting, The absence of finite-temperature phase transitions in low-dimensional many-body models: a survey and new results, J. Phys. Cond. Matt. 13, 505 (2001). 

*P. Hohenberg, Existence of long-range order in one and two dimensions, Phys. Rev. '''158''', 383 (1967).

*A. Klein, L. J. Landau, D. S. Shucker, On the absence of spontaneous breakdown of symmetry for equilibrium states in two dimensions, J. Stat. Phys. '''26''', 505 (1981).

*J. M. Kosterlitz, D. J. Thouless, Ordering, metastability, and phase transitions in two-dimensional systems, J. Phys. C: Solid State Phys. '''6''', 1181 (1973).

*N. D. Mermin, Absence of ordering in certain classical systems, J. Math. Phys. '''8''', 1061 (1967).

*N. D. Mermin, H. Wagner, Absence of ferromagnetism and antiferromagnetism in one- or two-dimensional Heisenberg models, Phys. Rev. Lett. '''17''', 1133 (1966); erratum: Phys. Rev. Lett. '''17''', 1307 (1966).

*F. Merkl, H. Wagner, Recurrent random walks and the absence of continuous symmetry breaking on graphs, J. Stat. Phys. '''75''', 153 (1994).

*H. E. Stanley, T. Kaplan, Possibility of a phase transition for the two-dimensional Heisenberg model, Phys. Rev. Lett. '''17''', 913 (1966).

*B. Simon, The statistical mechanics of lattice gases I, Princeton (1993).

<!-- Authors, please check this list and remove any references that are irrelevant. This list is generated automatically to reflect the links from your article to other accepted articles in Scholarpedia. -->
<b>Internal references</b>

* Eugene M. Izhikevich (2007) [[Equilibrium]]. Scholarpedia, 2(10):2014.

* Giovanni Gallavotti (2008) [[Fluctuations]]. Scholarpedia, 3(6):5893.

* Philip Holmes and Eric T. Shea-Brown (2006) [[Stability]]. Scholarpedia, 1(10):1838.  

== See also ==

''(pending)''

[[Category:Statistical Mechanics]]
[[Category:Multiple_Curators]]
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