In many text books on one dimensional quantum magnetic systems, it's said there is no orderd state for one dimensional magnetic systems. I understand that the one dimensional spin half Antiferromagnetic chain has no truly long range order since its correlation function decays exponentially at large distance. How about the ferromagnetic chain, especially when all the spins are in the same up state?
Ferromagnetic chain or Ising chain with transverse field has ordered states at zero tempreture. There is also other ordered states proved to be exist when T=0, such as the off-diagnol long range order(https://journals.aps.org/prb/cited-by/10.1103/PhysRevB.47.1113).
So I think the text books I mention only meant for the disorderd chains, such as antiferromagnetic chain.
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Yes and no. The quantum Ising model in 1+1 dimensions has a phase transition as you vary the coupling in the Hamiltonian. This model is directly related to the statistical mechanics of the classical Ising model in 2 spatial dimensions via the usual correspondence between quantum field theory and classical statistical mechanics.
But this is only a phase transition for zero temperature. For any finite temperature there is no phase transition. Going back to the statistical model in 2d, finite temperature corresponds to one of the two spatial dimensions being finite and periodic. At large scales that finite dimension looks thin, and the system behaves like the classical Ising model with 1 spatial dimension which has no phase transition.
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$\begingroup$ I understand the tranverse field Ising chain you mention here. But in Nagaosa's "Quantum Field Theory in Strongly Correlated Electronic Systems", Nagaosa wrote on the firs page, "Owing to strong quantum fluctuations, the system is a quantum liquid, and down to zero temperature". So I'm still confused whether there is true long range order at zero tempreture for 1D quantum chain. Would you like to give a comment on this? Thanks. $\endgroup$– mr.noCommented Nov 4, 2017 at 6:45
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$\begingroup$ Yes, there is definitely long range order in the quantum Ising model at zero temperature in one spatial dimension. By saying zero temperature I am talking about the ground state. For certain values of the parameters in the Hamiltonian there is only one ground state and 0 magnetization. For other parameters there are two ground states (spontaneous symmetry breaking) and nonzero magnetization. But as soon as you turn on temperature the Ising model can fluctuate between the two ordered states over large scales and in the thermodynamic limit there is no magnetization. $\endgroup$– octonionCommented Nov 4, 2017 at 6:50
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$\begingroup$ By the way Sachdev's book 'Quantum Phase Transitions' has a pretty good explanation of this if you didn't understand my brief explanation why in my answer and don't want to just take my word for it. $\endgroup$– octonionCommented Nov 4, 2017 at 6:58