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arXiv:2407.20704 · Physical review. D/Physical review. D.

Thermal and quantum phase transitions of the ϕ 4 model

István Gábor Márián, Andrea Trombettoni, I. Nándori

Published Sep 26, 2025• 1 citation
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Abstract

In this paper we discuss and revisit the finite temperature extension of the renormalization group (RG) treatment of T = 0 field theories, focusing as a case study on the ϕ 4 model. We first discuss the extension of RG equations of the very same model from T = 0 to finite T in the usual way by resorting to sums on the Matsubara frequencies and fixing the physical temperature parameter T . 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, T ≡ k T = τ k , where k T is the running cutoff for thermal and k is for the quantum fluctuations. Once this dimensionless temperature τ is introduced, we investigate the consequences on the thermal RG approach for the ϕ 4 model and construct its phase diagram. Finally, we formulate requirements for the phase diagram of the ϕ 4 theory based on known properties of the quantum and classical phase diagrams of the Ising model.

DOI
10.1103/5rfz-5lfy

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In this paper we discuss and revisit the finite temperature extension of the renormalization group (RG) treatment of T = 0 field theories, focusing as a case study on the ϕ 4 model. We first discuss the extension of RG equations of the very same model from T = 0 to finite T in the usual way by resorting to sums on the Matsubara frequencies and fixing the physical temperature parameter T . 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, T ≡ k T = τ k , where k T is the running cutoff for thermal and k is for the quantum fluctuations. Once this dimensionless temperature τ is introduced, we investigate the consequences on the thermal RG approach for the ϕ 4 model and construct its phase diagram. Finally, we formulate requirements for the phase diagram of the ϕ 4 theory based on known properties of the quantum and classical phase diagrams of the Ising model.
We revisit the two-dimensional quantum Ising model by computing renormalization group flows close to its quantum critical point. The low but finite temperature regime in the vicinity of the quantum critical point is squashed between two distinct non-Gaussian fixed points: the classical fixed point dominated by thermal fluctuations and the quantum critical fixed point dominated by zero-point quantum fluctuations. Truncating an exact flow equation for the effective action, we derive a set of renormalization group equations and analyze how the interplay of quantum and thermal fluctuations, both non-Gaussian in nature, influences the shape of the phase boundary and the region in the phase diagram where critical fluctuations occur. The solution of the flow equations makes this interplay transparent: we detect finite temperature crossovers by computing critical exponents and we confirm that the power law describing the finite temperature phase boundary as a function of control parameter is given by the correlation length exponent at zero temperature as predicted in an ϵϵϵ-expansion with ϵ=1ϵ=1ϵ=1 by Sachdev [Phys. Rev. B 55, 142 (1997)].
We study the critical dynamics of the quantum random-field Ising model using the nonperturbative functional renormalization group (NP-FRG). The static critical behavior is found to be controlled by the zero-temperature fixed point of the classical random-field Ising model, where both thermal and quantum fluctuations are dangerously irrelevant. Considering a family of quantum dynamical universality classes defined by a bare dynamical kernel FΛ(ω)∼∣ω∣σF_Λ(ω)\sim |ω|^σFΛ​(ω)∼∣ω∣σ, we show how this fluctuationless fixed point nevertheless controls the quantum dynamics by computing the full Matsubara-frequency dependence of the running dynamical kernel Fk(ω)F_k(ω)Fk​(ω). This is essential at zero temperature: a naive treatment of the dynamical kernel flow leads to a divergence at a finite length scale, resulting in apparent localization. In contrast, keeping the full frequency dependence of the dynamical kernel and choosing a regulator adapted to its running scale yields a controlled flow. The resulting dynamics is of activated form, with a relaxation time given by ln⁡τ∼ξΨ\ln τ\sim ξ^Ψlnτ∼ξΨ. The exponent ΨΨΨ is determined by the static RFIM fixed-point exponents and by σσσ. At finite temperature, the flow crosses over to the classical thermally activated scaling of the random-field Ising model. These results provide a quantitative field-theoretic realization of the heuristic activation scenario proposed earlier for the quantum random-field model and establish a framework for analyzing the dynamics of other disordered quantum systems that may exhibit similar tentative localization-like singularities.
We implement the temperature flow scheme first proposed by Honerkamp and Salmhofer in Phys.~Rev.~B 64, 184516 (2001) into the pseudo-Majorana functional renormalization group method for quantum spin systems. Since the renormalization group parameter in this approach is a physical quantity -- the temperature TTT -- the numerical efficiency increases significantly compared to more conventional renormalization group parameters, especially when computing finite temperature phase diagrams. We first apply this method to determine the finite temperature phase diagram of the J1J_1J1​-J2J_2J2​ Heisenberg model on the simple cubic lattice where our findings support claims of a vanishingly small nonmagnetic phase around the high frustration point J2=0.25J1J_2=0.25J_1J2​=0.25J1​. Perhaps most importantly, we find the temperature flow scheme to be advantageous in detecting finite temperature phase transitions as, by construction, a phase transition is never encountered at an artificial, unphysical cutoff parameter. Finally, we apply the temperature flow scheme to the dipolar XXZ model on the square lattice where we find a rich phase diagram with a large non-magnetic regime down to the lowest accessible temperatures. Wherever a comparison with error-controlled (quantum) Monte Carlo methods is applicable, we find excellent quantitative agreement with less than 5%5\%5% deviation from the numerically exact results.
A nonconventional renormalization-group (RG) treatment close to and below four dimensions is used to explore, in a unified and systematic way, the low-temperature properties of a wide class of systems in the influence domain of their quantum critical point. The approach consists in a preliminary averaging over quantum degrees of freedom and a successive employment of the Wilsonian RG transformation to treat the resulting effective classical Ginzburg-Landau free energy functional. This allows us to perform a detailed study of criticality of the quantum systems under study. The emergent physics agrees, in many aspects, with the known quantum critical scenario. However, a richer structure of the phase diagram appears with additional crossovers which are not captured by the traditional RG studies. In addition, in spite of the intrinsically static nature of our theory, predictions about the dynamical critical exponent, which parametrizes the link between statics and dynamics close to a continuous phase transition, are consistently derived from our static results.
The low-temperature phase diagram of one-dimensional weakly disordered quantum systems such as charge or spin density waves and Luttinger liquids is studied by a full finite temperature renormalization-group (RG) calculation. For vanishing quantum fluctuations this approach is amended by an exact solution in the case of strong disorder and by a mapping onto the Burgers equation with noise in the case of weak disorder, respectively. At zero temperature we reproduce the quantum phase transition between a pinned (localized) and an unpinned (delocalized) phase for weak and strong quantum fluctuations, respectively, as found previously by Fukuyama or Giamarchi and Schulz. At finite temperatures the localization transition is suppressed: the random potential is wiped out by thermal fluctuations on length scales larger than the thermal de Broglie wavelength of the phason excitations. The existence of a zero-temperature transition is reflected in a rich crossover phase diagram of the correlation functions. In particular we find four different scaling regions: a classical disordered, a quantum disordered, a quantum critical, and a thermal region. The results can be transferred directly to the discussion of the influence of disorder in superfluids. Finally we extend the RG calculation to the treatment of a commensurate lattice potential. Applications to related systems are discussed as well.
We implement the temperature flow scheme first proposed by Honerkamp and Salmhofer [Phys. Rev. B 64, 184516 (2001)] into the pseudo-Majorana functional renormalization group method for quantum spin systems. Since the renormalization group parameter in this approach is a physical quantity, the temperature TTT, the numerical efficiency increases significantly compared to more conventional renormalization group parameters, especially when computing finite-temperature phase diagrams. We first apply this method to determine the finite-temperature phase diagram of the ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the simple cubic lattice, where our findings support claims of a vanishingly small nonmagnetic phase around the high frustration point J2=0.25J1{J}_{2}=0.25{J}_{1}J2​=0.25J1​. Perhaps most importantly, we find the temperature flow scheme to be advantageous in detecting finite-temperature phase transitions as, by construction, a phase transition is never encountered at an artificial, unphysical cutoff parameter. Finally, we apply the temperature flow scheme to the dipolar XXZ model on the square lattice, where we find a rich phase diagram with a large nonmagnetic regime down to the lowest accessible temperatures. Wherever a comparison with error-controlled (quantum) Monte Carlo methods is applicable, we find excellent quantitative agreement with less than 55%5 deviation from the numerically exact results.