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Does renormalization group flow have to decrease temperature when $T<T_C$, with $T_C$ being critical point temperature? I think not, but my professor suggests something like that. Maybe I misunderstood him. I am asking whether temperature decrease has to happen along renormalization group flow in all reasonable circumstances, not just for specific circumstances.

What would happen for $T>T_C$? Would there be monotonic temperature increase or decrease in reasonable circumstances?

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In the case of a single critical fixed point, I think that the statement is at least asymptotically correct.

On the one hand, renormalization flow cannot cross the critical manyfold. Therefore, if the flow starts from a point above the critical temperature of a system, it should remain above the critical temperature of all the systems described by couplings corresponding to each point along a flow line, and similarly for points starting below the critical temperature. On the other hand, soon or later, since the system does not start from the critical manyfold, it will move towards a trivial fixed point, high temperature or $T=0$, depending on where the flow started.

In the presence of more than a critical fixed point, I do not know if it is possible justify in general such a behavior.

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