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We introduce an algebraic methodology for designing exactly-solvable Lie model Hamiltonians.
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It is important to note that all of the steps invoked in the high-temperature expansion discussed below rely only on the operator algebra and the dimension of its representation. First, note that exp [ β ( A s + B p ) ] = exp ( β A s ) exp ( β B p ) \exp[\beta(A_{s}+B_{p})]=\exp(\beta A_{s})\exp(\beta B_{p}) . Then, the partition function is 𝒵 = ( cosh β ) 2 N s Tr { σ } ∑ G ( 1 + ( tanh β ) | S | ∏ s ∈ G A s ) \displaystyle{\cal Z}=(\cosh\beta)^{2N_{s}}{\rm Tr}_{\{\sigma\}}\sum_{G}(1+(\tanh\beta)^{|S|}\prod_{s\in G}A_{s}) × ( 1 + ( tanh β ) | P | ∏ p ∈ G B p ) , \displaystyle\times(1+(\tanh\beta)^{|P|}\prod_{p\in G}B_{p}), (90)
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A. Kay and R. Colbeck, arXiv:0810.3557
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R. Alicki, M. Fannes, and M. Horodecki, arXiv:0810.4584
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As a curiosity, it is worth noting that our bond algebraic mapping gives rise to a crossover for finite size systems. As we showed by the application of bond algebras, Kitaev’s Toric code model is isomorphic to two decoupled Ising chains. The correlation length within an Ising chain of exchange constant J = 1 J=1 as in Eq.( 18
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2036
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