Mark Kelbert, Yurii Suhov
This is the second in a series of papers on symmetry properties of bosonic quantum systems over 2D graphs, with continuous spins. Here the phase space of a single spin is $\cH_1={\rm L}_2(M)$ where $M$ a $d-$dimensional unit torus $M=\bbR^d/\bbZ^d$. The phase space of spins assigned to a vertex $i$ of the graph is the bosonic Fock space $\cH(i)\simeq \cH=\operatornamewithlimits{\oplus}\limits_{k=0,1,...} L_2^{\rm{sym}}(M^k)$. The kinetic energy part of the Hamiltonian includes (i) $-\Delta /2$, the minus a half of the Laplace operator on $M$, and (ii) an integral term describing possible `jumps'. The potential part is an operator of multiplication by a function which is a sum of (a) one-body potentials $U^{(1)}(x)$, $x\in M$, describing a field generated by a heavy atom, (b) two-body potentials $U^{(2)}(x,y)$, $x,y\in M$, showing the interaction between pairs of particles belonging to the same atom, and (c) two-body potentials $V(x,y)$, $x,y\in M$, scaled along the graph distance ${\ttd}(i,j)$ between vertices $i,j\in\Gam$. We assume that a compact Lie group ${\tt G}$ acts on $M$, represented by a torus of dimension $d'\leq d$), preserving the metric and the volume in $M$. Furthermore, we suppose that the potential $U^{(1)}$, $U^{(2)}$ and $V$ are ${\tt G}$-invariant. The result of the paper is that any (appropriately defined) Gibbs state generated by the above Hamiltonian is ${\tt G}$-invariant. This extends the Mermin-Wagner-type theorems established in Mermin and Wagner (1966) for a narrower class of quantum 2D models. As in Mermin and Wagner (1966), the definition of a Gibbs state (and its analysis) is based on the Feynman--Kac representation for the density matrices. (Following Mermin and Wagner (1966), we call such a Gibbs state an FK-DLR state, marking a connection with DLR-type equations.)
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http://arxiv.org/abs/1210.8344
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