Self-consistent dynamical approximations for strongly correlated fermion systems are particularly successful in capturing the dynamical competition of local correlations. In these, the effect of spatially extended degrees of freedom is usually only taken into account in a mean field fashion or as a secondary effect. As a result, critical exponents associated with phase transitions have a mean field character. Here we demonstrate that diagrammatic multiscale methods anchored around local approximations are indeed capable of capturing the non-mean-field nature of the critical point of the lattice model encoded in a nonvanishing anomalous dimension and of correctly describing the transition to mean-field-like behavior as the number of spatial dimensions increases.
Self-consistent dynamical approximations for strongly correlated fermion systems are particularly successful in capturing the dynamical competition of local correlations. In these, the effect of spatially extended degrees of freedom is usually only taken into account in a mean field fashion or as a secondary effect. As a result, critical exponents associated with phase transitions have a mean field character. Here we demonstrate that diagrammatic multiscale methods anchored around local approximations are indeed capable of capturing the non-mean-field nature of the critical point of the lattice model encoded in a nonvanishing anomalous dimension and of correctly describing the transition to mean-field-like behavior as the number of spatial dimensions increases.
Abstract Understanding phase transitions requires not only identifying order parameters but also characterizing how their correlations behave across scales. By quantifying how fluctuations at distinct spatial or temporal points are related, correlation functions reveal the structural organization of complex systems. In this work, we reexamine the theoretical foundations of these correlations in systems undergoing second-order phase transitions, with emphasis on the Ising model extended to non-integer spatial dimensions. We revisit the hypotesis that, at criticality, the equilibrium dynamics become effectively confined to the fractal edge of spin clusters and redo the analysis using fractional calculus. Within this framework, the fractal dimension that governs the correlations in that subspace is directly related to Fisher’s exponent η η , which quantifies the singular behavior of the correlation function near criticality. Importantly, this correlation fractal dimension is distinct from the fractal dimension associated with the order parameter. The fractional approach allows us to directly compute the correlation fractal dimension and to establish an explicit geometrical relation connecting the two fractal dimensions. Moreover, the formulation naturally extends to non-integer spatial dimensions, remaining valid below the upper critical dimension and yielding the correct value of Fisher’s exponent η η for a continuous spatial dimension d . Within this framework, we also provide empirical functions describing how the main critical exponents vary continuously as a function of the space dimension.
Almost 50 years after the invention of the Hubbard model [1–3] and despite modern petaflop supercomputers, a precise analysis of the criticality of this most basic model for electronic correlations has not been achieved so far, at least not in three dimensions. Dynamical mean field theory (DMFT) [4–6] was a big step forward to quantitatively calculate the three dimensional Hubbard model since the major contribution of electronic correlations, i.e., the local ones, are well captured within this theory. Local correlations give rise to quasiparticle renormalization, the Mott-Hubbard transition, magnetism, and even more subtle issues such as kinks in a purely electronic models [7]. However, non-local spatial correlations are also naturally generated by a purely local Hubbard interaction, and, as it is well known, they become of essential importance in the vicinity of secondorder phase transitions. As these correlations are neglected in DMFT, this scheme can provide only for a conventional mean-field (MF) description of the critical properties. To overcome this shortcoming cluster extensions to DMFT such as the dynamical cluster approximation (DCA) and cluster-DMFT have been proposed, see Ref. 8 for a review. In these approaches non-local correlations beyond DMFT are taken into account, however only within the range of the cluster size; and due to computational limitations the actual size of the cluster is severely restricted to about 3-5 sites for each direction. Hence, short-range correlations are included by these approaches, whereas long-range ones are not. Nonetheless, Kent et al. [9] were able to extrapolate the cluster size of so-called Betts clusters to infinity, albeit assuming from the very beginning the critical exponents to be those of the Heisenberg model. This way they extrapolated the critical temperature of the paramagnetic-toantiferromagnetic phase transition which was found to be in agreement with earlier quantum Monte Carlo (QMC) simulations of Staudt et al. [10] on finite lattices.
We apply the dynamical mean field theory (DMFT) approximation to the real, scalar ${\ensuremath{\varphi}}^{4}$ quantum field theory. By comparing to lattice Monte Carlo calculations, perturbation theory and standard mean field theory, we test the quality of the approximation in two, three, four and five dimensions. The quantities considered in these tests are the critical coupling for the transition to the ordered phase and the associated critical exponents $\ensuremath{\nu}$ and $\ensuremath{\beta}$. We also map out the phase diagram in the most relevant case of four dimensions. In two and three dimensions, DMFT incorrectly predicts a first-order phase transition for all bare quartic couplings, which is problematic, because the second-order nature of the phase transition of lattice ${\ensuremath{\varphi}}^{4}$ theory is crucial for taking the continuum limit. Nevertheless, by extrapolating the behavior away from the phase transition, one can obtain critical couplings and critical exponents. They differ from those of mean field theory and are much closer to the correct values. In four dimensions the transition is second order for small quartic couplings and turns weakly first order as the coupling increases beyond a tricritical value. In dimensions five and higher, DMFT gives qualitatively correct results, predicts reasonable values for the critical exponents and considerably more accurate critical couplings than standard mean field theory. The approximation works best for small values of the quartic coupling. We investigate the change from first- to second-order transition in the local limit of DMFT which is computationally much cheaper. We also discuss technical issues related to the convergence of the nonlinear self-consistency equation solver and the solution of the effective single-site model using Fourier-space Monte Carlo updates in the presence of a ${\ensuremath{\varphi}}^{4}$ interaction.
Statistical methods are essential for understanding thermodynamic systems with many degrees of freedom. For systems in equilibrium, a very useful method is that of correlation functions, which establish a correlation between a field phi(x), which depends on the spatial position x, and the same field evaluated at another position, phi(x0). Fisher [Journal of Mathematical Physics 5, 944322 (1964)] introduced the autocorrelation function for fluctuations of the order parameter, which has been an important mathematical tool for understanding second-order phase transitions in equilibrium. However, his analysis is restricted to a Euclidean space of dimension d, and an exponent eta is introduced to correct the spatial behavior of the correlation function at T = Tc. In a recent work, Lima et al. [Phys. Rev. E 110, L062107 (2024)] demonstrated that a modern fractional differential analysis is necessary for a complete description of the correlation function at Tc. In this study, we highlight the deep connection among scaling behavior, critical exponents, and fractal geometry. Our results provide a unified geometric interpretation of critical exponents and fractal dimensions, broadly applicable to thermodynamic phase transitions. However, the approach does not apply to topological phase transitions, which lack local order parameters and the associated scale-invariant fractal geometry. We verify its predictions for several cornerstone thermodynamic models: the Ising, Potts, XY, and Heisenberg systems.
We show that mean-field-type symmetry-breaking approximations for correlated fermions can be constructed by performing vertex renormalization (VR) in the linked-cluster expansion, just as for the case of lattice spin models. In the limit of infinite spatial dimensions we sum up exactly the renormalized diagrams for the grand-canonical potential of the Hubbard model and, using the diagrammatic technique for Hubbard operators, we construct explicitly an approximate VR. The procedure leads to a self-consistent symmetry breaking analogous to that of the mean-field theory for spin systems. We present solutions for the case of half-filled energy band. They are valid for arbitrary temperatures and become exact for both large-U limit and for U=0. The antiferromagnetic order parameter is found to have the conventional mean-field form and, in the regime of strong correlations, the obtained critical temperature coincides with the mean-field-theory result for the N\'eel temperature of the effective Heisenberg Hamiltonian.
Quantum phase transitions in strongly correlated electron materials are one of the most intriguing phenomena in condensed matter physics. In fact, the rich phase-diagrams of these systems usually allow for the presence of several quantum phase transitions. At the same time, a comprehensive theoretical description of the quantum critical properties of correlated electrons has been not fully developed yet. For one of the most fundamental model in solid state physics, the 3D Hubbard model, quantum critical behaviour appears to violate the conventional Hertz-Millis- Moriya theory. This unexpected finding has been ascribed to the presence of specific features on the Fermi surface (FS), such as Kohn points and/or lines. If this is the case, the correct description of 3D correlated metal should be already accessible by means of Dynamical Mean Field Theory (DMFT) calculations. DMFT, in fact, preserves the information about the FS geometry and correctly captures temporal fluctuations, crucial for the description of quantum criticality. The main aim of this work is to test the hypothesis that DMFT treatment is sufficient to describe the quantum critical behaviour of 3D correlated metals, without resorting to more advanced (and much heavier!) quantum many-body schemes. Our DMFT results for the hole-doped 3D-Hubbard model have allowed to determine the location of the quantum-critical point of the magnetic transition and to highlight promising trends for the associated quantum critical exponents.