Critical Number Of Flavors Research Articles

Bootstrapping conformal QED3 and deconfined quantum critical point

We bootstrap the deconfined quantum critical point (DQCP) and 3D Quantum Electrodynamics (QED3) coupled to Nf flavors of two-component Dirac fermions. We show the lattice and perturbative results on the SO(5) symmetric DQCP are excluded by the bootstrap bounds with an assumption that the lowest singlet scalar is irrelevant. Remarkably, we discover a new family of kinks in the 3D SO(N) vector bootstrap bounds with N ⩾ 6. We demonstrate coincidences between SU(Nf) adjoint and textrm{SO}left({N}_f^2-1right) vector bootstrap bounds due to a novel algebraic relation between the crossing equations. By introducing gap assumptions breaking the textrm{SO}left({N}_f^2-1right) symmetry, the SU(Nf) adjoint bootstrap bounds with large Nf converge to the 1/Nf perturbative results of QED3. Our results provide strong evidence that the SO(5) DQCP is not continuous and the critical flavor number of QED3 is slightly above 2: {N}_f^{ast}in left(2,4right) . Bootstrap results near {N}_f^{ast } are well consistent with the merger and annihilation mechanism for the loss of conformality in QED3.

Color-flavor dependence of the Nambu-Jona-Lasinio model and QCD phase diagram

We study the dynamical chiral symmetry breaking/restoration for various numbers of light quarks flavors and colors using the Nambu-Jona-Lasinio (NJL) model of quarks in the Schwinger-Dyson equation framework, dressed with a color-flavor dependence of effective coupling. For fixed and varying , we observe that the dynamical chiral symmetry is broken when exceeds its critical value . For a fixed and varying , we observe that the dynamical chiral symmetry is restored when reaches its critical value . Strong interplay is observed between and , i.e., larger values of tend to strengthen the dynamical generated quark mass and quark-antiquark condensate, while higher values of suppress both parameters. We further sketch the quantum chromodynamics (QCD) phase diagram at a finite temperature T and quark chemical potential μ for various and . At finite T and μ, we observe that the critical number of colors is enhanced, whereas the critical number of flavors is suppressed as T and μ increase. Consequently, the critical temperature , , and co-ordinates of the critical endpoint in the QCD phase diagram are enhanced as increases and suppressed when increases. Our findings agree with the lattice QCD and Schwinger-Dyson equations predictions.

Symmetries of Thirring Models on 3D Lattices

We review some recent developments about strongly interacting relativistic Fermi theories in three spacetime dimensions. These models realize the asymptotic safety scenario and are used to describe the low-energy properties of Dirac materials in condensed matter physics. We begin with a general discussion of the symmetries of multi-flavor Fermi systems in arbitrary dimensions. Then we review known results about the critical flavor number Ncrit of Thirring models in three dimensions. Only models with a flavor number below Ncrit show a phase transition from a symmetry-broken strong-coupling phase to a symmetric weak-coupling phase. Recent simulations with chiral fermions show that Ncrit is smaller than previously extracted with various non-perturbative methods. Our simulations with chiral SLAC fermions reveal that for four-component flavors Ncrit=0.80(4). This means that all reducible Thirring models with Nr=1,2,3,… show no phase transition with order parameter. Instead, we discover footprints of phase transitions without order parameter. These new transitions are probably smooth and could be used to relate the lattice Thirring models to Thirring models in the continuum. For a single irreducible flavor, we provide previously unpublished values for the critical couplings and critical exponents.

Two-loop mass anomalous dimension in reduced quantum electrodynamics and application to dynamical fermion mass generation

We consider reduced quantum electrodynamics ( {mathrm{RQED}}_{d_{gamma },{d}_e} ) a model describing fermions in a de-dimensional space-time and interacting via the exchange of massless bosons in dγ-dimensions (de ≤ dγ). We compute the two-loop mass anomalous dimension, γm, in general {mathrm{RQED}}_{4,{d}_e} with applications to RQED4,3 and QED4. We then proceed on studying dynamical (parity-even) fermion mass generation in {mathrm{RQED}}_{4,{d}_e} by constructing a fully gauge-invariant gap equation for {mathrm{RQED}}_{4,{d}_e} with γm as the only input. This equation allows for a straightforward analytic computation of the gauge-invariant critical coupling constant, αc, which is such that a dynamical mass is generated for αr> αc, where αr is the renormalized coupling constant, as well as the gauge-invariant critical number of fermion flavours, Nc, which is such that αc → ∞ and a dynamical mass is generated for N < Nc. For RQED4,3, our results are in perfect agreement with the more elaborate analysis based on the resolution of truncated Schwinger-Dyson equations at two-loop order. In the case of QED4, our analytical results (that use state of the art five-loop expression for γm) are in good quantitative agreement with those obtained from numerical approaches.

Critical behavior in the single flavor Thirring model in 2+1D

Results of a lattice field theory simulation of the single-flavor Thirring model in 2+1 spacetime dimensions are presented. The lattice model is formulated using domain wall fermions as a means to recover the correct U(2) symmetries of the continuum model in the limit where wall separation $L_s\to\infty$. Simulations on $12^3, 16^3\times L_s$, varying self-interaction strength $g^2$ and bare mass $m$ are performed with $L_s = 8, \ldots 48$, and the results for the bilinear condensate $\langle\bar\psi\psi\rangle$ fitted to a model equation of state assuming a U(2)$\to$U(1)$\otimes$U(1) symmetry-breaking phase transition at a critical $g_c^2$. First estimates for $g^{-2}a$ and critical exponents are presented, showing small but significant departures from mean-field values. The results confirm that a symmetry-breaking transition does exist and therefore the critical number of flavors for the Thirring model $N_c > 1$. Results for both condensate and associated susceptibility are also obtained in the broken phase on $16^3\times48$, suggesting that here the $L_s\to\infty$ extrapolation is not yet under control. We also present results obtained with the associated 2+1$d$ truncated overlap operator DOL demonstrating exponential localisation, a necessary condition for the recovery of U(2) global symmetry, but that recovery of the Ginsparg-Wilson condition as $L_s\to\infty$ is extremely slow in the broken phase.

Absence of chiral symmetry breaking in Thirring models in 1+2 dimensions

The Thirring model is an interacting fermion theory with current-current interaction. The model in $1+2$ dimensions has applications in condensed-matter physics to describe the electronic excitations of Dirac materials. Earlier investigations with Schwinger-Dyson equations, the functional renormalization group and lattice simulations with staggered fermions suggest that a critical number of (reducible) flavors $N^{\mathrm{c}}$ exists, below which chiral symmetry can be broken spontaneously. Values for $N^{\mathrm{c}}$ found in the literature vary between $2$ and $7$. Recent lattice studies with chirally invariant SLAC fermions have indicated that chiral symmetry is unbroken for all integer flavor numbers [Wellegehausen et al., 2017]. An independent simulation based on domain wall fermions seems to favor a critical flavor-number that satisfies $1<N^{\mathrm{c}}<2$ [Hands, 2018]. However, in the latter simulations difficulties in reaching the massless limit in the broken phase (at strong coupling and after the $L_s\to\infty$ limit has been taken) are encountered. To find an accurate value $N^{\mathrm{c}}$ we study the Thirring model (by using an analytic continuation of the parity even theory to arbitrary real $N$) for $N$ between $0.5$ and $1.1$. We investigate the chiral condensate, the spectral density of the Dirac operator, the spectrum of (would-be) Goldstone bosons and the variation of the filling-factor and conclude that the critical flavor number is $N^{\mathrm{c}}=0.80(4)$. Thus we see no chiral symmetry breaking in all Thirring models with $1$ or more flavors of ($4$-component) fermions. Besides the artifact transition to the unphysical lattice artifact phase we find strong evidence for a hitherto unknown phase transition that exists for $N>N^{\mathrm{c}}$ and should answer the question of where to construct a continuum limit.

Critical flavor number in the 2+1D Thirring model

The Thirring model in 2+1 spacetime dimensions, in which $N$ flavors of relativistic fermion interact via a contact interaction between conserved fermion currents, is studied using lattice field theory simulations employing domain wall fermions, which furnish the correct U(2N) global symmetry in the limit that the wall separation $L_s\to\infty$. Attention is focussed on the issue of spontaneous symmetry breakdown via a non-vanishing fermion bilinear condensate $\langle\bar\psi\psi\rangle\not=0$. Results from quenched simulations are presented demonstrating that a non-zero condensate does indeed form over a range of couplings, provided simulation results are first extrapolated to the $L_s\to\infty$ limit. Next, results from simulations with $N=1$ using an RHMC algorithm demonstrate that U(2) symmetry is unbroken at weak coupling but plausibly broken at strong coupling. Correlators of mesons with spin zero are consistent with the Goldstone spectrum expected from U(2)$\to$U(1)$\otimes$U(1). We infer the existence of a symmetry-breaking phase transition at some finite coupling, and combine this with previous simulation results to deduce that the critical number of flavors for the existence of a quantum critical point in the Thirring model satisfies $0<N_c<2$, with strong evidence that in fact $N_c>1$.

Scale-invariance and scale-breaking in parity-invariant three-dimensional QCD

We present a numerical study of three-dimensional two-color QCD with $N=0, 2, 4, 8$ and 12 flavors of massless two-component fermions using parity-preserving improved Wilson-Dirac fermions. A finite volume analysis provides strong evidence for the presence of $Sp(N)$ symmetry-breaking bilinear condensate when $N \le 2$ and its absence for $N \ge 8$. A weaker evidence for the bilinear condensate is shown for $N=4$. We estimate the critical number of flavors below which scale-invariance is broken by the bilinear condensate to be between $N=4$ and 6.

Critical flavor number of the Thirring model in three dimensions

The Thirring model is a four-fermion theory with a current-current interaction and $U(2N)$ chiral symmetry. It is closely related to three-dimensional QED and other models used to describe properties of graphene. In addition it serves as a toy model to study chiral symmetry breaking. In the limit of flavour number $N \to 1/2$ it is equivalent to the Gross-Neveu model, which shows a parity-breaking discrete phase transition. The model was already studied with different methods, including Dyson-Schwinger equations, functional renormalisation group methods and lattice simulations. Most studies agree that there is a phase transition from a symmetric phase to a spontaneously broken phase for a small number of fermion flavours, but no symmetry breaking for large $N$. But there is no consensus on the critical flavour number $N^\text{cr}$ above which there is no phase transition anymore and on further details of the critical behaviour. Values of $N$ found in the literature vary between $2$ and $7$. All earlier lattice studies were performed with staggered fermions. Thus it is questionable if in the continuum limit the lattice model recovers the internal symmetries of the continuum model. We present new results from lattice Monte Carlo simulations of the Thirring model with SLAC fermions which exactly implement all internal symmetries of the continuum model even at finite lattice spacing. If we reformulate the model in an irreducible representation of the Clifford algebra, we find, in contradiction to earlier results, that the behaviour for even and odd flavour numbers is very different: For even flavour numbers, chiral and parity symmetry are always unbroken. For odd flavour numbers parity symmetry is spontaneously broken below the critical flavour number $N_\text{ir}^\text{cr}=9$ while chiral symmetry is still unbroken.

Chiral and deconfinement phase transitions in QED3 with finite gauge boson mass

Based on the experimental observation that there is a coexisting region between the antiferromagnetic (AF) and d-wave superconducting (dSC) phases, the influences of gauge boson mass m a on chiral symmetry restoration and deconfinement phase transitions in QED3 are investigated simultaneously within a unified framework, i.e., Dyson–Schwinger equations. The results show that the chiral symmetry restoration phase transition in the presence of the gauge boson mass m a is a typical second-order phase transition; the chiral symmetry restoration and deconfinement phase transitions are coincident; the critical number of fermion flavors N c f decreases as the gauge boson mass m a increases, which implies that there exists a boundary that separates the N c f –m a plane into chiral symmetry breaking/confinement region for (N c f , m a ) below the boundary and chiral symmetry restoration/deconfinement region for (N c f , m a ) above it.

Excitonic gap generation in thin-film topological insulators

In this work, we analyze the excitonic gap generation in the strong-coupling regime of thin films of three-dimensional time-reversal-invariant topological insulators. We start by writing down the effective gauge theory in 2 + 1-dimensions from the projection of the 3 + 1-dimensional quantum electrodynamics. Within this method, we obtain a short-range interaction, which has the form of a Thirring-like term, and a long-range one. The interaction between the two surface states of the material induces an excitonic gap. By using the large-N approximation in the strong-coupling limit, we find that there is a dynamical mass generation for the excitonic states that preserves time-reversal symmetry and is related to the dynamical chiral-symmetry breaking of our model. This symmetry breaking occurs only for values of the fermion-flavor number smaller than . Our results show that the inclusion of full dynamical interaction strongly modifies the critical number of flavors for the occurrence of exciton condensation, and therefore cannot be neglected.

Erratum: Quantum superconducting criticality in graphene and topological insulators [Phys. Rev. B87, 041401(R) (2013)

We correct our previous conclusion regarding the fate of a charged quantum critical point across the superconducting transition for two dimensional massless Dirac fermion. Within the leading order $\epsilon$ expansion, we now find that the requisite number of four-component Dirac fermion flavors ($N_f$) for the continuous phase transition through a charged critical point is $N_f>18.2699$. For $N_f\geq1/2$, the critical number of bosonic flavors for this transition is significantly reduced as compared to the value determined in the absence of the Dirac fermions in the theory.

Influence of boson mass on chiral phase transition inQED3

Based on the truncated Dyson-Schwinger equations for the fermion propagator with $N$ fermion flavors at zero temperature, the chiral phase transition of quantum electrodynamics in $2+1$ dimensions (${\mathrm{QED}}_{3}$) with boson mass\char22{}which is obtained via the Anderson-Higgs mechanism\char22{}is investigated. In the chiral limit, we find that the critical behavior of ${\mathrm{QED}}_{3}$ with a massless boson is different from that with a massive boson: the chiral phase transition in the presence of a nonzero boson mass reveals the typical second-order phase transition, at either the critical boson mass or a critical number of fermion flavors, while for a vanishing boson mass it exhibits a higher than second-order phase transition at the critical number of fermion flavors. Furthermore, it is shown that the system undergoes a crossover behavior from a small number of fermion flavors or boson mass to its larger one beyond the chiral limit.

Fixed-point structure of low-dimensional relativistic fermion field theories: Universality classes and emergent symmetry

We investigate a class of relativistic fermion theories in 2<d<4 space-time dimensions with continuous chiral U(Nf)xU(Nf) symmetry. This includes a number of well-studied models, e.g., of Gross-Neveu and Thirring type, in a unified framework. Within the limit of pointlike interactions, the RG flow of couplings reveals a network of interacting fixed points, each of which defines a universality class. A subset of fixed points are "critical fixed points" with one RG relevant direction being candidates for critical points of second-order phase transitions. Identifying invariant hyperplanes of the RG flow and classifying their attractive/repulsive properties, we find evidence for emergent higher chiral symmetries as a function of Nf. For the case of the Thirring model, we discover a new critical flavor number that separates the RG stable large-Nf regime from an intermediate-Nf regime in which symmetry-breaking perturbations become RG relevant. This new critical flavor number has to be distinguished from the chiral-critical flavor number, below which the Thirring model is expected to allow spontaneous chiral symmetry breaking, and its existence offers a resolution to the discrepancy between previous results obtained in the continuum and the lattice Thirring models. Moreover, we find indications for a new feature of universality: details of the critical behavior can depend on additional "spectator symmetries" that remain intact across the phase transition. Implications for the physics of interacting fermions on the honeycomb lattice, for which our theory space provides a simple model, are given.

Running coupling in the conformal window of large-Nf QCD

Quantum Chromodynamics with a relatively large number of fundamentally charged quark flavours in the chiral limit is considered. A self-consistent solution of the quark, gluon and ghost propagator Dyson-Schwinger equations in Landau gauge exhibits a phase transition. Above the critical number of fermion flavours the non-perturbative running coupling develops a plateau over a wide momentum range, and the propagators follow a power law behaviour for these momenta. Hereby, the critical number of quark flavours depends crucially on the beyond-tree-level tensor structures of the quark-gluon vertex.

Nature of chiral phase transition inQED3at zero density

Based on the feature of chiral susceptibility and thermal susceptibility at finite temperature, the nature of chiral phase transition around the critical number of fermion flavors ($N_c$) and the critical temperature ($T_c$) at a fixed fermion flavors number in massless QED$_3$ are investigated. It is showed that, at finite temperature the system exhibits a second-order phase transition at $N_c$ or $T_c$ and each of the estimated critical exponents is less than 1, while it reveals a higher-order continuous phase transition around $N_c$ at zero temperature.

Phase structure of many-flavorQED3

We analyze the many-flavor phase diagram of quantum electrodynamics (QED) in $2+1$ (Euclidean) space-time dimensions. We compute the critical flavor number above which the theory is in the quasiconformal massless phase. For this, we study the renormalization group fixed-point structure in the space of gauge interactions and pointlike fermionic self-interactions, the latter of which are induced dynamically by fermion-photon interactions. We find that a reliable estimate of the critical flavor number crucially relies on a careful treatment of the Fierz ambiguity in the fermionic sector. Using a Fierz-complete basis, our results indicate that the phase transition towards a chirally broken phase occurring at small flavor numbers could be separated from the quasiconformal phase at larger flavor numbers, allowing for an intermediate phase which is dominated by fluctuations in a vector channel. If these interactions approach criticality, the intermediate phase could be characterized by a Lorentz-breaking vector condensate.

The chiral phase transition of QED3 around the critical number of fermion flavors

At zero temperature and density, the nature of the chiral phase transition in QED3 with Nf massless fermion flavors is investigated. To this end, in Landau gauge, we numerically solve the coupled Dyson–Schwinger equations for the fermion and boson propagator within the bare and simplified Ball–Chiu vertices separately. It is found that, in the bare vertex approximation, the system undergoes a high-order continuous phase transition from the Nambu–Goldstone phase into the Wigner phase when the number of fermion flavors Nf reaches the critical number Nf,c, while the system exhibits a typical characteristic of second-order phase transition for the simplified Ball–Chiu vertex.

Vacuum polarization and dynamical chiral symmetry breaking: Phase diagram of QED with four-fermion contact interaction

We study chiral symmetry breaking for fundamental charged fermions coupled electromagnetically to photons with the inclusion of a four-fermion contact self-interaction term, characterized by coupling strengths $\ensuremath{\alpha}$ and $\ensuremath{\lambda}$, respectively. We employ multiplicatively renormalizable models for the photon dressing function and the electron-photon vertex that minimally ensures mass anomalous dimension ${\ensuremath{\gamma}}_{m}=1$. Vacuum polarization screens the interaction strength. Consequently, the pattern of dynamical mass generation for fermions is characterized by a critical number of massless fermion flavors ${N}_{f}={N}_{f}^{c}$ above which chiral symmetry is restored. This effect is in diametrical opposition to the existence of criticality for the minimum interaction strengths, ${\ensuremath{\alpha}}_{c}$ and ${\ensuremath{\lambda}}_{c}$, necessary to break chiral symmetry dynamically. The presence of virtual fermions dictates the nature of phase transition. Miransky scaling laws for the electromagnetic interaction strength $\ensuremath{\alpha}$ and the four-fermion coupling $\ensuremath{\lambda}$, observed for quenched QED, are replaced by a mean field power law behavior corresponding to a second-order phase transition. These results are derived analytically by employing the bifurcation analysis and are later confirmed numerically by solving the original nonlinearized gap equation. A three-dimensional critical surface is drawn in the phase space of $(\ensuremath{\alpha},\ensuremath{\lambda},{N}_{f})$ to clearly depict the interplay of their relative strengths to separate the two phases. We also compute the $\ensuremath{\beta}$ functions (${\ensuremath{\beta}}_{\ensuremath{\alpha}}$ and ${\ensuremath{\beta}}_{\ensuremath{\lambda}}$) and observe that ${\ensuremath{\alpha}}_{c}$ and ${\ensuremath{\lambda}}_{c}$ are their respective ultraviolet fixed points. The power law part of the momentum dependence, describing the mass function, implies ${\ensuremath{\gamma}}_{m}=1+s$, which reproduces the quenched limit trivially. We also comment on the continuum limit and the triviality of QED.

Characteristic of chiral phase transition inQED3at zero density

Based on the truncated Dyson-Schwinger equation for the fermion propagator, the Cornwall-Jackiw-Tomboulis effective potential near the critical point is investigated. We show that, at zero temperature, the system undergoes a continuous phase transition into the chiral symmetric phase at a critical number of fermion flavors, while undergoing a second-order phase transition at high temperature.