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I have seen arguments for why there is no symmetry breaking in the 1D Ising model--for example, using the transfer matrix method to explicitly solve the model, and another of energy-entropy arguments showing that a uniformly spin up or down state is unstable to the creation of domain walls.

However, I was wondering if there are explicit arguments like this except using the continuum version of the Ising model, given by the action:

$$S = \int d^dx \; [c_1 (\partial\phi)^2 + c_2\phi^2 + c_4\phi^4]$$

From reading Altland and Simons, I believe the argument involves showing that this theory allows for instantons, and in $d=1$, one can make a similar energy-entropy argument. However I am not sure how to do this explicitly using the functional formalism. Does anyone have any source recommendations on how to go through this argument, or some pointers on how to perform the argument?

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  • $\begingroup$ SSB of continuous symmetry? or discrete? in $d=1$ cont. symmetries cannot break, there is an old, very general argument due to Coleman. $\endgroup$
    – AccidentalFourierTransform
    Commented Nov 10, 2021 at 0:55
  • $\begingroup$ You can have a look here arxiv.org/pdf/1507.02498.pdf $\endgroup$
    – Adam
    Commented Nov 16, 2021 at 8:04
  • $\begingroup$ One route would be to use a "quantum-classical mapping" showing that the expectation value of $\phi$ in equilibrium for your theory maps to the expectation value $<x>$ in the ground state of a quantum Hamiltonian of the form $H = p^2 + c x^2 + c' x^4$. One can prove this has a unique non-degenerate ground state, which quickly leads to $<x> = 0$. $\endgroup$
    – Seth Whitsitt
    Commented Nov 17, 2021 at 6:12

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