The critical behavior of the $(n+1)$-states Potts model in $d$-dimensions is studied with functional renormalization group techniques. We devise a general method to derive $β$-functions for continuos values of $d$ and $n$ and we write the flow equation for the effective potential (LPA) when instead $n$ is fixed. We calculate several critical exponents, which are found to be in good agreement with Monte Carlo simulations and $ε$-expansion results available in the literature. In particular, we focus on Percolation $(n\to0)$ and Spanning Forest $(n\to-1)$ which are the only non-trivial universality classes in $d=4,5$ and where our methods converge faster.
We study the $q$-state Potts models on a cubic lattice in the thermodynamic limit using tensor renormalization group transformations with the triad approximation. By computing the thermodynamic potentials, we locate the first-order phase transitions for $10 < q \le 20$ which has not been explored using any method. We also examine the efficiency of the triad approximation method in obtaining the fixed-point tensor and comment on how this can be improved.
We derive an exact renormalization group recursion relation for the Loschmidt amplitude of the quantum $Q$-state clock model and the quantum $Q$-state Potts model in one dimension. The renormalization group flow is discussed in detail. The fixed-points of the renormalization group flow are found to be complex in general. These fixed-points control the dynamical phases of the two models, giving rise to non-analyticities in its Loschmidt rate function, for both the pure and the disordered system. For the quench protocols studied, dynamical quantum phase transitions are found to occur in the cloc…
We obtain the exact renormalization map and plots of Lee-Yang and Fisher zeros distributions for Potts models on a number of hierarchical lattices: the diamond hierarchical lattice, a lattice we call spider web, the Sierpinski gasket and cylinders. Such models are only examples among the ones we can study in the general framework of hierarchical lattices, developed in a previous paper.
We develop a dynamic field-theoretic renormalization-group (RG) theory for the cooling first-order phase transitions in the Potts model. It is suggested that the well-known imaginary fixed points of the $q$-state Potts model for $q>10/3$ in the RG theory are the origin of the dynamic scaling found recently, apart from the logarithmic corrections. This indicates that the real and imaginary fixed points of the Potts model are both physical and control the scalings of the continuous and discontinuous phase transitions, respectively, of the model. Our one-loop results for the scaling exponents are…
We extend the model of a 2$d$ solid to include a line of defects. Neighboring atoms on the defect line are connected by ?springs? of different strength and different cohesive energy with respect to the rest of the system. Using the Migdal-Kadanoff renormalization group we show that the elastic energy is an irrelevant field at the bulk critical point. For zero elastic energy this model reduces to the Potts model. By using Monte Carlo simulations of the 3- and 4-state Potts model on a square lattice with a line of defects, we confirm the renormalization-group prediction that for a defect interac…
We prove that the generator of the renormalization group of Potts models on hierarchical lattices can be represented by a rational map acting on a finite-dimensional product of complex projective spaces. In this framework we can also consider models with an applied external magnetic field and multiple-spin interactions. We use recent results regarding iteration of rational maps in several complex variables to show that, for some class of hierarchical lattices, Lee-Yang and Fisher zeros belong to the unstable set of the renormalization map.
The q-state ferromagnetic Potts model under a non-zero magnetic field coupled with the 0^th Potts state was investigated by an exact real-space renormalization group approach. The model was defined on a family of diamond hierarchical lattices of several fractal dimensions d_F. On these lattices, the renormalization group transformations became exact for such a model when a correlation coupling that singles out the 0^th Potts state was included in the Hamiltonian. The rich criticality presented by the model with q=3 and d_F=2 was fully analyzed. Apart from the Potts criticality for the zero fie…
This is the third in a series of papers in which we set up and discuss the functional relations for the ``split rapidity line'' correlation function in the N - state chiral Potts model. The order parameters of the model can be obtained from this function. Here we consider the case N = 3 and write the equations explicitly in terms of the hyperelliptic functions parametrization. We also present four-term low-temperature series expansions, which we hope will cast light on the analyticity properties needed to solve the relations. The problem remains unsolved, but we hope that this will prove to be…
We use scale invariant scattering theory to exactly determine the renormalization group fixed points of a $q$-state Potts model coupled to an $r$-state Potts model in two dimensions. For integer values of $q$ and $r$ the fixed point equations are very constraining and show in particular that scale invariance in coupled Potts ferromagnets is limited to the Ashkin-Teller case ($q=r=2$). Since our results extend to continuous values of the number of states, we can access the limit $r\to 1$ corresponding to correlated percolation, and show that the critical properties of Potts spin clusters cannot…
Spinodal and Equilibrium Global Phase Diagram of the d=3 Merged Potts-Cubic-Clock Model: First-Order Equilibrium and Second-Order Spinodal Boundaries with Hidden Topologies from Renormalization-Group Theory
Based on a Renormalization-Group picture of $Z_n$ symmetric models in three dimensions, we derive a scaling law for the $Z_n$ order parameter in the ordered phase. An existing Monte Carlo calculation on the three-state antiferromagnetic Potts model, which has the effective $Z_6$ symmetry, is shown to be consistent with the proposed scaling law. It strongly supports the Renormalization-Group picture that there is a single massive ordered phase, although an apparently rotationally symmetric region in the intermediate temperature was observed numerically.
We use scale invariant scattering theory to exactly determine the lines of renormalization group fixed points invariant under the permutational symmetry $S_q$ in two dimensions, and show how one of these scattering solutions describes the ferromagnetic and square lattice antiferromagnetic critical lines of the $q$-state Potts model. Other solutions we determine should correspond to new critical lines. In particular, we obtain that a $S_q$-invariant fixed point can be found up to the maximal value $q=(7+\sqrt{17})/2$. This is larger than the usually assumed maximal value 4 and leaves room for a…
Renormalization group and Coulomb gas mappings are used to derive theoretical predictions for the corrections to the exactly known asymptotic fractal masses of the hull, external perimeter, singly connected bonds and total mass of the Fortuin-Kasteleyn clusters for two-dimensional $q$-state Potts models at criticality. For q=4 these include exact logarithmic (as well as loglog) corrections.
Critical behaviour of a system, subjected to strongly anisotropic turbulent mixing, is studied by means of the field theoretic renormalization group. Specifically, relaxational stochastic dynamics of a non-conserved multicomponent order parameter of the Ashkin-Teller-Potts model, coupled to a random velocity field with prescribed statistics, is considered. The velocity is taken Gaussian, white in time, with correlation function of the form $\propto δ(t-t') /|{\bf k}_{\bot}|^{d-1+ξ}$, where ${\bf k}_{\bot}$ is the component of the wave vector, perpendicular to the distinguished direction ("dire…
All local bond-state densities are calculated for q-state Potts and clock models in three spatial dimensions, d=3. The calculations are done by an exact renormalization group on a hierarchical lattice, including the density recursion relations, and simultaneously are the Migdal-Kadanoff approximation for the cubic lattice. Reentrant behavior is found in the interface densities under symmetry breaking, in the sense that upon lowering temperature the value of the density first increases, then decreases to its zero value at zero temperature. For this behavior, a physical mechanism is proposed. A …
By computing the low-lying energy excitation spectra with the density matrix renormalization group algorithm we show that boundaries polarized in the direction of the transverse field lead to scale-invariant conformal towers of states at the critical point of the quantum four-state Potts model - a special symmetric case of the Ashkin-Teller model. Furthermore, by direct comparison of the excitation spectra we phenomenologically establish the duality between the transverse-polarized and three-state-mixed boundary conditions at the four-state Potts critical point. Finally, for completeness, we v…
We study the critical behavior of the random q-state Potts quantum chain by density matrix renormalization techniques. Critical exponents are calculated by scaling analysis of finite lattice data of short chains ($L \leq 16$) averaging over all possible realizations of disorder configurations chosen according to a binary distribution. Our numerical results show that the critical properties of the model are independent of q in agreement with a renormalization group analysis of Senthil and Majumdar (Phys. Rev. Lett.{\bf 76}, 3001 (1996)). We show how an accurate analysis of moments of the distri…
We present an extension of the density-functional theory (DFT) formalism for lattice gases to systems with internal degrees of freedom. In order to test approximations commonly used in DFT approaches, we investigate the statics and dynamics of occupation (density) profiles in the one-dimensional Potts model. In particular, by taking the exact functional for this model we can directly evaluate the quality of the local equilibrium approximation used in time-dependent density-functional theory (TDFT). Excellent agreement is found in comparison with Monte Carlo simulations. Finally, principal limi…
Tensor Renormalization Group Calculations of Partition-Function Ratios
Graduate School of Science and Technology, Keio University（庆应义塾大学理工学部）；Keio University Sustainable Quantum Artificial Intelligence Center (KSQAIC), Keio University（庆应义塾大学可持续量子人工智能中心）；Institute for Solid State Physics, The University of Tokyo（东京大学固体物理研究所）；Trans-scale Quantum Science Institute, The University of Tokyo（东京大学跨尺度量子科学研究所）
The phase transitions of random-field q-state Potts models in d=3 dimensions are studied by renormalization-group theory by exact solution of a hierarchical lattice and, equivalently, approximate Migdal-Kadanoff solutions of a cubic lattice. The recursion, under rescaling, of coupled random-field and random-bond (induced under rescaling by random fields) coupled probability distributions is followed to obtain phase diagrams. Unlike the Ising model (q=2), several types of random fields can be defined for q >= 3 Potts models, including random-axis favored, random-axis disfavored, random-axis ran…
A calculation method for higher-order moments of physical quantities, including magnetization and energy, based on the higher-order tensor renormalization group is proposed. The physical observables are represented by impurity tensors. A systematic summation scheme provides coarse-grained tensors including multiple impurities. Our method is compared with the Monte Carlo method on the two-dimensional Potts model. While the nature of the transition of the $q$-state Potts model has been known for a long time owing to the analytical arguments, a clear numerical confirmation has been difficult due …
We present a solution of the non-linear renormalization group equations leading to the dominant and subdominant singular behaviours of physical quantities (free energy density, correlation length, internal energy, specific heat, magnetization, susceptibility and magnetocaloric coefficient) at the critical temperature in a non- vanishing magnetic field. The solutions i) lead to exact cancellation of logarithmic corrections in universal amplitude ratios and ii) prove recently proposed relations among logarithmic exponents.
Three-dimensional (3D) $q$-state Potts models ($q$=3, 4, and 5) are studied by the tensor product variational approach (TPVA), which is a recently developed variational method for 3D classical lattice models. The variational state is given by a two-dimensional (2D) product of local factors, and is improved by way of self-consistent calculations assisted by the corner transfer matrix renormalization group (CTMRG). It should be noted that no a priori condition is imposed for the local factor. Transition temperatures and latent heats are calculated from the observations of thermodynamic functions…
Using a weak-disorder scheme and real-space renormalization-group techniques, we obtain analytical results for the critical behavior of various q-state Potts models with correlated disordered exchange interactions along d1 of d spatial dimensions on hierarchical (Migdal-Kadanoff) lattices. Our results indicate qualitative differences between the cases d-d1=1 (for which we find nonphysical random fixed points, suggesting the existence of nonperturbative fixed distributions) and d-d1>1 (for which we do find acceptable perturbartive random fixed points), in agreement with previous numerical calcu…
We investigate the two-dimensional $q=3$ and 4 Potts models with a variable interaction range by means of Monte Carlo simulations. We locate the phase transitions for several interaction ranges as expressed by the number $z$ of equivalent neighbors. For not too large $z$, the transitions fit well in the universality classes of the short-range Potts models. However, at longer ranges the transitions become discontinuous. For $q=3$ we locate a tricritical point separating the continuous and discontinuous transitions near $z=80$, and a critical fixed point between $z=8$ and 12. For $q=4$ the trans…
In the ferromagnetic phase of the q-state Potts model, switching on an external magnetic field induces confinement of the domain wall excitations. For the Ising model (q = 2) the spectrum consists of kink-antikink states which are the analogues of mesonic states in QCD, while for q = 3, depending on the sign of the field, the spectrum may also contain three-kink bound states which are the analogues of the baryons. In recent years the resulting "hadron" spectrum was described using several different approaches, such as quantum mechanics in the confining linear potential, WKB methods and also th…
We discuss recently generated high-temperature series expansions for the free energy and the susceptibility of random-bond q-state Potts models on hypercubic lattices. Using the star-graph expansion technique quenched disorder averages can be calculated exactly for arbitrary uncorrelated coupling distributions while keeping the disorder strength p as well as the dimension d as symbolic parameters. We present analyses of the new series for the susceptibility of the Ising (q=2) and 4-state Potts model in three dimensions up to order 19 and 18, respectively, and compare our findings with results …
The spin-glass $q$-state Potts model on $d$-dimensional diamond hierarchical lattices is investigated by an exact real space renormalization group scheme. Above a critical dimension $d_l(q)$ for $q > 2$, the coupling constants probability distribution flows to a low-temperature \emph{strange attractor} or to the high-temperature paramagnetic fixed point, according to the temperature is below or above the critical temperature $T_c(q,d)$. The strange attractor was investigated considering four initial different distributions for $q = 3$ and $d = 5$ presenting strong robustness in shape and tempe…
Quantum critical points in quasiperiodic magnets can realize new universality classes, with critical properties distinct from those of clean or disordered systems. Here, we study quantum phase transitions separating ferromagnetic and paramagnetic phases in the quasiperiodic $q$-state Potts model in $2+1d$. Using a controlled real-space renormalization group approach, we find that the critical behavior is largely independent of $q$, and is controlled by an infinite-quasiperiodicity fixed point. The correlation length exponent is found to be $ν=1$, saturating a modified version of the Harris-Luc…
One-dimensional edges of classical systems in two dimension sometimes show surprisingly rich phase transitions and critical phenomena, particularly when the bulk is at criticality. As such a model, we study the surface critical behavior of the 3-state dilute Potts model whose bulk is tuned at the tricritical point. To investigate it more precisely than in the previous works [Y. Deng and H. W. J. Blöte, Phys. Rev. E 70, 035107(R) (2004); Phys. Rev. E 71, 026109 (2005)], we analyze it from the viewpoint of the boundary conformal field theory (BCFT). The complete classification of the conformal b…
We study the critical behavior of the random q-state Potts model in the large-q limit on the diamond hierarchical lattice with an effective dimensionality $d_{\rm eff} > 2$. By varying the temperature and the strength of the frustration the system has a phase transition line between the paramagnetic and the ferromagnetic phases which is controlled by four different fixed points. According to our renormalization group study the phase-boundary in the vicinity of the multicritical point is self-similar, it is well represent ed by a logarithmic spiral. We expect infinite number of reentrances in t…
The two-dimensional ferromagnetic anisotropic Ashkin-Teller model is investigated through a real-space renormalization-group approach. The critical frontier, separating five distinct phases, recover all the known exacts results for the square lattice. The correlation length $(ν_T)$ and crossover $(ϕ)$ critical exponents are also calculated. With the only exception of the four-state Potts critical point, the entire phase diagram belongs to the Ising universality class.
Critical behaviour of a nearly critical system, subjected to vivid turbulent mixing, is studied by means of the field theoretic renormalization group. Namely, relaxational stochastic dynamics of a non-conserved order parameter of the Ashkin-Teller-Potts model, coupled to a random velocity field with prescribed statistics, is considered. The mixing is modelled by Kraichnan's rapid-change ensemble: time-decorrelated Gaussian velocity field with the power-like spectrum $\propto k^{-d-ξ}$. It is shown that, depending on the symmetry group of the underlying Potts model, the degree of compressibilit…
The $N$-state chiral Potts model in lattice statistical mechanics can be obtained as a ``descendant'' of the six-vertex model, via an intermediate ``$Q$'' or ``$τ_2 (t_q)$'' model. Here we generalize this to obtain a column-inhomogeneous $τ_2 (t_q)$ model, and derive the functional relations satisfied by its row-to-row transfer matrix. We do {\em not} need the usual chiral Potts relations between the $N$th powers of the rapidity parameters $a_p, b_p, c_p, d_p$ of each column. This enables us to readily consider the case of fixed-spin boundary conditions on the left and right-most columns. We t…
We develop a novel real-space renormalization group (RG) scheme which accurately estimates correlation length exponent $ν$ near criticality of higher-dimensional quantum Ising and Potts models in a transverse field. Our method is remarkably simple (often analytical), grouping only a few spins into a block spin so that renormalized Hamiltonian has a closed form. A previous difficulty of spatial anisotropy and unwanted terms is avoided by incorporating rotational invariance and internal $\mathbb{Z}_q$ symmetries of the Hamiltonian. By applying this scheme to the (2+1)-dim Ising model on a triang…
It is known that a trained Restricted Boltzmann Machine (RBM) on the binary Monte Carlo Ising spin configurations, generates a series of iterative reconstructed spin configurations which spontaneously flow and stabilize to the critical point of physical system. Here we construct a variety of Neural Network (NN) flows using the RBM and (variational) autoencoders, to study the q-state Potts and clock models on the square lattice for q = 2, 3, 4. The NN are trained on Monte Carlo spin configurations at various temperatures. We find that the trained NN flow does develop a stable point that coincid…
We consider the critical behavior of the random q-state Potts model in the large-q limit with different types of disorder leading to either the nonfrustrated random ferromagnet regime or the frustrated spin glass regime. The model is studied on the diamond hierarchical lattice for which the Migdal-Kadanoff real-space renormalization is exact. It is shown to have a ferromagnetic and a paramagnetic phase and the phase transition is controlled by four different fixed points. The state of the system is characterized by the distribution of the interface free energy P(I) which is shown to satisfy di…
The flow equations of the Functional Renormalization Group are applied to the O(N)-symmetric scalar theory, for N=1 and N=4, in four Euclidean dimensions, d=4, to determine the effective potential and the renormalization function of the field in the broken phase. In our numerical analysis, the infrared limit, corresponding to the vanishing of the running momentum scale in the equations, is approached to obtain the physical values of the parameters by extrapolation. In the N=4 theory a non-perturbatively large value of the physical renormalization of the longitudinal component of the field is o…
A Microcanonical Finite Site Ansatz in terms of quantities measurable in a Finite Lattice allows to extend phenomenological renormalization (the so called quotients method) to the microcanonical ensemble. The Ansatz is tested numerically in two models where the canonical specific-heat diverges at criticality, thus implying Fisher-renormalization of the critical exponents: the 3D ferromagnetic Ising model and the 2D four-states Potts model (where large logarithmic corrections are known to occur in the canonical ensemble). A recently proposed microcanonical cluster method allows to simulate syst…
Known results on two-dimensional quantum electrodynamics (QED_2) have been used to study the dependence of functional renormalization group equations on renormalization schemes and approximations applied for its bosonized version. It is demonstrated that the singularity of flow equations can be avoided in the optimized and power-law schemes for the bosonized model and the drawback of renormalization on bosonization is shown: it is indicated that renormalization of QED_2 possibly requires interaction terms corresponding to higher frequency modes of its bosonized version.
The Lifshitz critical behavior for a single component field theory is studied for the specific isotropic case in the framework of the Functional Renormalization Group. Lifshitz fixed point solutions of the flow equation, derived by using a Proper Time regulator, are searched at lowest and higher order in the derivative expansion. Solutions are found when the number of spatial dimensions d is contained within the interval 5.5 < d < 8.
Expansion for the free energy functionals of the Potts spin glass models with 3, 4 and 5 states up to the fourth order in $δq_{αβ}$ around the replica symmetric solution (RS) is investigated using a special quadrupole-like representation. The temperature dependence of the 1RSB order parameters is obtained in the vicinity of the point $T=T_c$ where the RS solution becomes unstable. The crossover from continuous to jumpwise behavior with increasing of number of states is derived analytically. The comparison is made of the free energy expansion for the Potts spin glass with that for other models.
At zero temperature, the 3-state antiferromagnetic Potts model on a square lattice maps exactly onto a point of the 6-vertex model whose long-distance behavior is equivalent to that of a free scalar boson. We point out that at nonzero temperature there are two distinct types of excitation: vortices, which are relevant with renormalization-group eigenvalue 1/2; and non-vortex unsatisfied bonds, which are strictly marginal and serve only to renormalize the stiffness coefficient of the underlying free boson. Together these excitations lead to an unusual form for the corrections to scaling: for ex…
A phenomenological approach to the ferromagnetic two dimensional Potts model on square lattice is proposed. Our goal is to present a simple functional form that obeys the known properties possessed by the free energy of the q-state Potts model. The duality symmetry of the 2D Potts model together with the known results on its critical exponent α allow to fix consistently the details of the proposed expression for the free energy. The agreement of the analytic ansatz with numerical data in the q=3 case is very good at high and low temperatures as well as at the critical point. It is shown that t…
Monte Carlo (MC) simulations and series expansions (SE) data for the energy, specific heat, magnetization, and susceptibility of the three-state and four-state Potts model and Baxter-Wu model on the square lattice are analyzed in the vicinity of the critical point in order to estimate universal combinations of critical amplitudes. We also form effective ratios of the observables close to the critical point and analyze how they approach the universal critical-amplitude ratios. In particular, using the duality relation, we show analytically that for the Potts model with a number of states $q\le …
Monte Carlo (MC) simulations and series expansion (SE) data for the energy, specific heat, magnetization and susceptibility of the ferromagnetic 4-state Potts model on the square lattice are analyzed in a vicinity of the critical point in order to estimate universal combinations of critical amplitudes. The quality of the fits is improved using predictions of the renormalization group (RG) approach and of conformal invariance, and restricting the data within an appropriate temperature window. The RG predictions on the cancelation of the logarithmic corrections in the universal amplitude ratios …
We numerically obtain the conformal spectrum of several classical spin models on a two-dimensional lattice with open boundaries, for every boundary fixed point obtained by the Cardy's derivation [J. L. Cardy, Nucl. Phys. B 324, 581 (1989)]. In order to extract accurate conformal data, we implement the tensor network renormalization algorithm [G. Evenbly and G. Vidal, Phys. Rev. Lett. 115, 180405 (2015)] extended so as to be applicable to a square lattice with open boundaries. We successfully compute the boundary conformal spectrum consistent with the underlying boundary conformal field theorie…
The chiral spin-glass Potts system with q=3 states is studied in d=2 and 3 spatial dimensions by renormalization-group theory and the global phase diagrams are calculated in temperature, chirality concentration p, and chirality-breaking concentration c, with determination of phase chaos and phase-boundary chaos. In d=3, the system has ferromagnetic, left-chiral, right-chiral, chiral spin-glass, and disordered phases. The phase boundaries to the ferromagnetic, left- and right-chiral phases show, differently, an unusual, fibrous patchwork (microreentrances) of all four (ferromagnetic, left-chira…
