0.2 0.6 1.0 1.4 1.8  0  0.01  0.02  0.03  0.04  0.05  0.06  0.07  0.08  0.09 0.00 0.05 0.10 0.15 0.20 0.25 0.30 T/ broken Z$_{2}$ a 2  0  0.05  0.1  0.15  0.2  0.25  0.3  0.015 0.02 0.025 0.03 0.035 0.04 0.045 0.05 0.055 0.06 T/ a 2 broken Z$_{2}$ unbroken Z$_{2}$ FIG. 6. Finite-temperature phase diagram of the N = 1 Wess-Zumino model. Left panel: GLYPH<CI> 2 phase boundary in the space spanned by temperature T / Λ and the value of the couplings ( λ Λ , a 2 $_{Λ}$) at vanishing temperature. Right panel: Slice of the GLYPH<CI> 2 phase-boundary for fixed λ Λ = 0 . 8. is in accordance with our expectations from a scalar O (1) ≃ GLYPH<CI> 2 model since ( λ$_{∗}$a 2 k → $_{0}$)$_{T}$$_{=0}$ sets the scale at T = 0 and therefore plays a similar role as a finite expec- tation value of the fields in O ( N ) models, see e. g. [34]. the value of the couplings ( λ$_{Λ}$, a 2 $_{Λ}$) at the UV cutoff scale at vanishing temperature $^{10}$. The lower end of the phase boundary on the T = 0 plane in Fig. 6 corresponds to the phase-transition line which separates the phase with broken supersymmetry from the one associated with a su- persymmetric ground state, see also Fig. 3 (right panel). Choosing couplings ( λ$_{Λ}$, a 2 $_{Λ}$) at T = 0 associated with a supersymmetric ground state (and broken GLYPH<CI> 2 symmetry) we find always a second-order phase-transition tempera- ture at which the system enters the phase with restored GLYPH<CI> 2 symmetry. As discussed in Sect. IV A the thermo- dynamic properties of this GLYPH<CI> $_{2}$-symmetric phase are sim- ilar to the ones of a gas of massless bosons. We ex- pect that this finite-temperature phase transition falls into the Ising universality class with critical exponents determined by the underlying 2 d GLYPH<CI> 2 theory, namely by the exponents of Onsager’s solution of the corresponding lattice spin model. However, a quantitative analysis of the critical behavior would require the inclusion of the anomalous dimension in our studies. This is deferred to future work. In the right panel of Fig. 6, we show a slice of the GLYPH<CI> 2 phase boundary for fixed λ$_{Λ}$ = 0 . 8. We find that the phase-transition temperature increases with increas- ing ( a 2 $_{Λ}$)$_{T}$$_{=0}$ . Since ( a 2 k → $_{0}$)$_{T}$$_{=0}$ increases with increas- ing ( a 2 $_{Λ}$)$_{T}$$_{=0}$ , the phase-transition temperature increases with increasing ( a 2 k → $_{0}$)$_{T}$$_{=0}$ , i. e. in terms of the renor- malized quantity $^{11}$. Our observation of an increasing phase-transition temperature with increasing ( a 2 k → $_{0}$)$_{T}$$_{=0}$ At zero temperature we have found results quite sim- ilar to our findings for the two-dimensional model [13]. An investigation of the fixed point structure yields the su- persymmetric analogue of the Wilson-Fisher fixed-point for bosonic theories. It is maximally IR stable with one relevant direction only. As in the two-dimensional model the relevant direction is given by a $^{2}$. Again we find a scaling relation between the critical exponent 1 /ν$_{W}$ of the instable direction and the anomalous dimension η . The critical exponent ν$_{W}$ governs the freeze-out of the minimum of the superpotential W . We also find that the IR limit of the three-dimensional Wess-Zumino model is given by a conformally invariant theory. The critical ex- ponent ν$_{W}$ governs the vanishing of the mass with the RG scale. This is different from three-dimensional scalar O ( N ) theories since for the supersymmetric models the relevant coupling does not feed back in the flow equa- tions of higher n -point functions. As in two dimensions In this work we have employed the functional RG for a study of the three dimensional N = 1 Wess-Zumino model at zero and at finite temperature. Since the model exists only in Minkowski space we have worked with a for- mulation of the Wetterich equation in Minkowski-space and have Wick-rotated the momentum integrals. V. CONCLUSIONS 10 10 As initial conditions for the finite-temperature RG flows we use the zero-temperature values of the couplings ( λ$_{Λ}$, a 2 $_{Λ}$) at the cut- off scale. Therefore we have to restrict our study to temperatures significantly smaller than the UV cutoff in order to ensure that it is justified to use these zero-temperature values of the couplings at the UV starting point of the flow. 11 Recall that λ flows into its fixed-point value λ$_{∗}$ for T = 0 in- dependently of the initial condition λ$_{Λ}$ . Nevertheless λ$_{Λ}$ can be used to classify different theories at finite temperature.