Thermal and quantum phase transitions of the <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline"> <mml:msup> <mml:mi>ϕ</mml:mi> <mml:mn>4</mml:mn> </mml:msup> </mml:math> model

In this paper we discuss and revisit the finite temperature extension of the renormalization group (RG) treatment of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"> <a:mi>T</a:mi> <a:mo>=</a:mo> <a:mn>0</a:mn> </a:math> field theories, focusing as a case study on the <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"> <c:msup> <c:mi>ϕ</c:mi> <c:mn>4</c:mn> </c:msup> </c:math> model. We first discuss the extension of RG equations of the very same model from <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"> <e:mi>T</e:mi> <e:mo>=</e:mo> <e:mn>0</e:mn> </e:math> to finite <g:math xmlns:g="http://www.w3.org/1998/Math/MathML" display="inline"> <g:mi>T</g:mi> </g:math> in the usual way by resorting to sums on the Matsubara frequencies and fixing the physical temperature parameter <i:math xmlns:i="http://www.w3.org/1998/Math/MathML" display="inline"> <i:mi>T</i:mi> </i:math> . We show that this approach, although useful for a variety of applications, may lead to the disappearance of the critical points as extracted from the RG flow. Since the identification of fixed points is key in the study of classical and quantum phase transitions, we propose a modification of the usual finite-temperature RG approach by relating the temperature parameter to the running RG scale, <k:math xmlns:k="http://www.w3.org/1998/Math/MathML" display="inline"> <k:mi>T</k:mi> <k:mo>≡</k:mo> <k:msub> <k:mi>k</k:mi> <k:mi>T</k:mi> </k:msub> <k:mo>=</k:mo> <k:mi>τ</k:mi> <k:mi>k</k:mi> </k:math> , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="inline"> <m:msub> <m:mi>k</m:mi> <m:mi>T</m:mi> </m:msub> </m:math> is the running cutoff for thermal and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" display="inline"> <o:mi>k</o:mi> </o:math> is for the quantum fluctuations. Once this dimensionless temperature <q:math xmlns:q="http://www.w3.org/1998/Math/MathML" display="inline"> <q:mi>τ</q:mi> </q:math> is introduced, we investigate the consequences on the thermal RG approach for the <s:math xmlns:s="http://www.w3.org/1998/Math/MathML" display="inline"> <s:msup> <s:mi>ϕ</s:mi> <s:mn>4</s:mn> </s:msup> </s:math> model and construct its phase diagram. Finally, we formulate requirements for the phase diagram of the <u:math xmlns:u="http://www.w3.org/1998/Math/MathML" display="inline"> <u:msup> <u:mi>ϕ</u:mi> <u:mn>4</u:mn> </u:msup> </u:math> theory based on known properties of the quantum and classical phase diagrams of the Ising model.