Abstract We study the phase diagram of the (1 + 1)-dimensional Gross-Neveu model with both $$ {g}_x^2{\left(\overline{\psi}\psi \right)}^2 $$ g x 2 ψ ¯ ψ 2 and $$ {g}_y^2{\left(\overline{\psi}i{\gamma}_5\psi \right)}^2 $$ g y 2 ψ ¯ i γ 5 ψ 2 interaction terms on a spatial lattice. The continuous chiral symmetry, which is present in the continuum model when $$ {g}_x^2={g}_y^2 $$ g x 2 = g y 2 , has a mixed ’t Hooft anomaly with the charge conservation symmetry, which guarantees the existence of a massless mode. However, the same ’t Hooft anomaly also implies that the continuous chiral symmetry is broken explicitly in our lattice model. Nevertheless, from numerical matrix product state simulations we find that for certain parameters of the lattice model, the continuous chiral symmetry reemerges in the infrared fixed point theory, even at strong coupling. We argue that, in order to understand this phenomenon, it is crucial to go beyond mean-field theory (or, equivalently, beyond the leading order term in a 1/N expansion). Interestingly, on the lattice, the chiral Gross-Neveu model appears at a Landau-forbidden second order phase transition separating two distinct and unrelated symmetry-breaking orders. We point out the crucial role of two different ’t Hooft anomalies or Lieb-Schultz-Mattis obstructions for this Landau-forbidden phase transition to occur.
Abstract We study the phase diagram of the (1 + 1)-dimensional Gross-Neveu model with both gx2(ψψ)2 g x 2 ψ ¯ ψ 2 and gy2(ψiγ5ψ)2 g y 2 ψ ¯ i γ 5 ψ 2 interaction terms on a spatial lattice. The continuous chiral symmetry, which is present in the continuum model when gx2=gy2 g x 2 = g y 2 , has a mixed ’t Hooft anomaly with the charge conservation symmetry, which guarantees the existence of a massless mode. However, the same ’t Hooft anomaly also implies that the continuous chiral symmetry is broken explicitly in our lattice model. Nevertheless, from numerical matrix product state simulations we find that for certain parameters of the lattice model, the continuous chiral symmetry reemerges in the infrared fixed point theory, even at strong coupling. We argue that, in order to understand this phenomenon, it is crucial to go beyond mean-field theory (or, equivalently, beyond the leading order term in a 1/N expansion). Interestingly, on the lattice, the chiral Gross-Neveu model appears at a Landau-forbidden second order phase transition separating two distinct and unrelated symmetry-breaking orders. We point out the crucial role of two different ’t Hooft anomalies or Lieb-Schultz-Mattis obstructions for this Landau-forbidden phase transition to occur.
The canonical Gross-Neveu model for N two-component Dirac fermions in 2+1 dimensions suffers a continuous phase transition at a critical interaction gc1∼1/N at large N , at which its continuous symmetry SO(2N) is preserved and a discrete (Ising) symmetry becomes spontaneously broken. A recent mean-field calculation, however, points to an additional transition at a different critical gc2∼−Ngc1 , at which SO(2N)→SO(N)×SO(N) . To study the latter phase transition we rewrite the Gross-Neveu interaction g(ψ¯ψ)2 in terms of three different quartic terms for the single ( L=1 ) 4N -component real (Majorana) fermion, and then extend the theory to L>1 . This allows us to track the evolution of the fixed points of the renormalization group transformation starting from L≫1 , where one can discern three distinct critical points which correspond to continuous phase transitions into (1) SO(2N) -singlet mass-order-parameter, (2) SO(2N) -symmetric-tensor mass-order-parameters, and (3) SO(2N) -adjoint nematic-order-parameters, down to L=1 value that is relevant to the standard Gross-Neveu model. Below the critical value of Lc(N)≈0.35N for N≫1 only the Gross-Neveu critical point (1) still implies a diverging susceptibility for its corresponding ( SO(2N) -singlet) order parameter, whereas the two new critical points that existed at large L ultimately become equivalent to the Gaussian fixed point at L=1 . We interpret this metamorphosis of the SO(2N) -symmetric-tensor fixed point from critical to spurious as an indication that the transition at gc2 in the original Gross-Neveu model is turned first-order by fluctuations. Published by the American Physical Society 2024
It was recently established that the paradigmatic Gross-Neveu model with N copies of two-dimensional Dirac fermions features an SO ( 2 N ) symmetry if certain interactions are suppressed. This becomes evident when the theory is rewritten in terms of 2 N copies of two-dimensional Majorana fermions. Mean-field theory for the SO ( 2 N ) model predicts, besides the chiral Ising transition at g c 1 , a second critical point g c 2 where SO ( 2 N ) is broken down to SO ( N ) × SO ( N ) . A subsequent Wilsonian renormalization group analysis directly in d = 3 supports its existence in a generalized theory, where N f copies of the 4 N -component Majorana fermions are introduced. This allows us to track the evolution of a (i) quantum anomalous Hall Gross-Neveu-Ising, (ii) symmetric-tensor, and (iii) adjoint-nematic fixed point separately. However, it turns out that (ii) and (iii) lose their criticality when approaching N f = 1 , suggesting that the transition is first order. In this work, we approach the problem from the lower-critical dimension of two. We construct a Fierz-complete renormalizable Lagrangian, compute the leading order β functions, fermion anomalous dimension, as well as the order parameter anomalous dimensions, and resolve the three universality classes corresponding to (i)–(iii). Before becoming equal to the Gaussian fixed point at N f = 1 , (ii) remains critical for all values of N f > N f , c ST ( N ) ≈ 0.56 + 1.48 N + O ( ε ) , which compares well with the estimate of previous studies. We further find that (iii) becomes equal to (i) when approaching N f = 1 . An instability is, however, only present in the susceptibility corresponding to the Gross-Neveu-Ising order parameter.
We study the critical properties of the QED3-Gross-Neveu model with 2N flavors of two-component Dirac fermions coupled to a massless scalar field and a U(1) gauge field. For N=1, this theory has recently been suggested to be dual to the SU(2) noncompact CP1 model that describes the deconfined phase transition between the N\'eel antiferromagnet and the valence bond solid on the square lattice. For N=2, the theory has been proposed as an effective description of a deconfined critical point between chiral and Dirac spin liquid phases, and may potentially be realizable in spin-1/2 systems on the kagome lattice. We demonstrate the existence of a stable quantum critical point in the QED3-Gross-Neveu model for all values of N. This quantum critical point is shown to escape the notorious fixed-point annihilation mechanism that renders plain QED3 (without scalar-field coupling) unstable at low values of N. The theory exhibits an upper critical space-time dimension of four, enabling us to access the critical behavior in a controlled expansion in the small parameter $\ensuremath{\epsilon}=4\ensuremath{-}D$. We compute the scalar-field anomalous dimension ${\ensuremath{\eta}}_{\ensuremath{\phi}}$, the correlation-length exponent $\ensuremath{\nu}$, as well as the scaling dimension of the flavor-symmetry-breaking bilinear $\overline{\ensuremath{\psi}}{\ensuremath{\sigma}}^{z}\ensuremath{\psi}$ at the critical point, and compare our leading-order estimates with predictions of the conjectured duality.
Abstract We construct a Hamiltonian lattice regularisation of the N-flavour Gross-Neveu model that manifestly respects the full O(2N) symmetry, preventing the appearance of any unwanted marginal perturbations to the quantum field theory. In the context of this lattice model, the dynamical mass generation is intimately related to the Coleman-Mermin-Wagner and Lieb-Schultz-Mattis theorems. In particular, the model can be interpreted as lying at the first order phase transition line between a trivial and symmetry-protected topological (SPT) phase, which explains the degeneracy of the elementary kink excitations. We show that our Hamiltonian model can be solved analytically in the large N limit, producing the correct expression for the mass gap. Furthermore, we perform extensive numerical matrix product state simulations for N = 2, thereby recovering the emergent Lorentz symmetry and the proper non-perturbative mass gap scaling in the continuum limit. Finally, our simulations also reveal how the continuum limit manifests itself in the entanglement spectrum. As expected from conformal field theory we find two conformal towers, one tower spanned by the linear representations of O(4), corresponding to the trivial phase, and the other by the projective (i.e. spinor) representations, corresponding to the SPT phase.
The fluctuations of massless Dirac fermion can not only turn a first-order bosonic phase transition (in the Landau sense) to a quantum critical point, but also work reversely to enhance the first-order transition itself, depending on the implementation of finite-size effects in the coupling corrections. Here, we report a case study of the latter by employing quantum Monte Carlo simulation upon a lattice model in which the bosonic part featuring the Landau-Devonshire first-order phase transition and Yukawa coupled to the Dirac fermions. We find that the parameter range for the first-order phase transition becomes larger as the Yukawa coupling increases, and the microscopic mechanism of this phenomena is revealed, at a quantitative level, as the interplay between the critical fluctuations and the finite-size effects. Moreover, the scaling behavior at the separation point between the first-order and the continuous phase transitions is found to belong to the chiral tricritical Gross-Neveu universality. Our results demonstrate that the interplay of massless Dirac fermions, critical fluctuations, and the finite-size effects could trigger a plethora of interesting phenomena, and therefore great care is called for when making generalizations.
This thesis explores the interaction of symmetries and quantum anomalies on the low-energy dynamics and phase transitions of strongly correlated quantum matter. Beyond the traditional Landau-Ginzburg-Wilson paradigm, we investigate how the embedding of microscopic lattice symmetries into emergent continuum symmetries governs the stability of quantum critical points in (2+1) dimensions. Through the lens of 't Hooft anomalies and anomaly matching, we establish a theoretical framework for understanding the fate of various exotic quantum phase transitions. A central application of this framework is understanding the stability of emergent conformal field theories (CFTs) that describe gapless quantum critical points. The exact nature of these phase transitions is chiefly governed by how the microscopic lattice symmetries impose stringent conditions on the relevant operators allowed in the low-energy effective theory. When physical symmetries suppress relevant perturbations, an interacting conformal fixed point, such as (2+1)-dimensional quantum electrodynamics (QED₃), can stably emerge. Chapters 2 and 3 explore the consequences of this symmetry embedding through the phenomenon of "unnecessary'' quantum critical points, which uniquely describe continuous transitions occurring entirely within a single phase of matter. We demonstrate that the Dirac spin liquid (DSL) in lattice magnets can surprisingly describe a stable continuous phase transition within a single, conventional, symmetry breaking phase. In chapter 4, we apply these tools to a class of quantum phase transitions between featureless bosonic atomic insulators. Despite lacking topological order or protected edge modes, these insulators represent distinct phases because they can be "obstructed,'' meaning their Wannier centers are not pinned to the physical atomic sites. Surprisingly, we find that the transition between these topologically trivial states can host a conformally invariant state described by QED₃, suggesting that even transitions between trivial phases can harbor rich, fractionalized physics. Lastly, chapter 5 analyzes transitions from a composite Fermi liquid (CFL) to a chiral superconductor, employing more traditional field-theoretic techniques and making concrete connections with recent experiments in moire materials.