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arXiv:2302.13243 · Physical review. B./Physical review. B

Quantum z=2 Lifshitz criticality in one-dimensional interacting fermion systems

Ke Wang

Published Aug 16, 2023• 2 citations
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Abstract

We consider Lifshitz criticality (LC) with the dynamical critical exponent $z=2$ in one-dimensional interacting fermion systems with a filled Dirac sea. We report that interactions have crucial effects on Lifshitz criticality. Single-particle excitations are destabilized by an interaction and decay into the particle-hole continuum, which is reflected in the logarithmic divergence in the imaginary part of one-loop self-energy. We show that the system is sensitive to the sign of the interaction. The random-phase approximation shows that the collective particle-hole excitations emerge only when the interaction is repulsive. The dispersion of collective modes is gapless and linear. If the interaction is attractive, the one-loop renormalization group (RG) shows that there may exist a stable RG fixed point described by two coupling constants. We also show that the on-site interaction (without any other perturbations at the UV scale) would always turn on the relevant velocity perturbation to the quadratic Lagrangian in the RG flow, driving the system flow to the conformal-invariant criticality. In the numerical simulations of the lattice model at half filling, we find that, for either on-site positive or negative interactions, the dynamical critical exponent becomes $z=1$ in the infrared limit and the entanglement entropy is a logarithmic function of the system size $L$. The work paves the way to study one-dimensional interacting LCs.

DOI
10.1103/physrevb.108.l081112

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We consider Lifshitz criticality (LC) with the dynamical critical exponent z=2z=2z=2 in one-dimensional interacting fermion systems with a filled Dirac sea. We report that interactions have crucial effects on Lifshitz criticality. Single-particle excitations are destabilized by an interaction and decay into the particle-hole continuum, which is reflected in the logarithmic divergence in the imaginary part of one-loop self-energy. We show that the system is sensitive to the sign of the interaction. The random-phase approximation shows that the collective particle-hole excitations emerge only when the interaction is repulsive. The dispersion of collective modes is gapless and linear. If the interaction is attractive, the one-loop renormalization group (RG) shows that there may exist a stable RG fixed point described by two coupling constants. We also show that the on-site interaction (without any other perturbations at the UV scale) would always turn on the relevant velocity perturbation to the quadratic Lagrangian in the RG flow, driving the system flow to the conformal-invariant criticality. In the numerical simulations of the lattice model at half filling, we find that, for either on-site positive or negative interactions, the dynamical critical exponent becomes z=1z=1z=1 in the infrared limit and the entanglement entropy is a logarithmic function of the system size LLL. The work paves the way to study one-dimensional interacting LCs.
Universal finite-size amplitude and anomalous entangment entropy of z=2z=2z=2 quantum Lifshitz criticalities in topological chains
We consider Lifshitz criticalities with dynamical exponent z=2 z = 2 that emerge in a class of topological chains. There, such a criticality plays a fundamental role in describing transitions between symmetry-enriched conformal field theories (CFTs). We report that, at such critical points in one spatial dimension, the finite-size correction to the energy scales with system size, L L , as \sim L^{-2} ∼ L − 2 , with universal and anomalously large coefficient. The behavior originates from the specific dispersion around the Fermi surface, \epsilon \propto \pm k^2 ϵ ∝ ± k 2 . We also show that the entanglement entropy exhibits at the criticality a non-logarithmic dependence on l/L l / L , where l l is the length of the sub-system. In the limit of l\ll L l ≪ L , the maximally-entangled ground state has the entropy, S(l/L)=S_0+(l/L)\log(l/L) S ( l / L ) = S 0 + ( l / L ) log ( l / L ) . Here S_0 S 0 is some non-universal entropy originating from short-range correlations. We show that the novel entanglement originates from the long-range correlation mediated by a zero mode in the low energy sector. The work paves the way to study finite-size effects and entanglement entropy around Lifshitz criticalities and offers an insight into transitions between symmetry-enriched criticalities.
This Thesis explores the formation of novel phases in Dirac systems that are driven by strong many-body interactions and collective quantum fluctuations. The central focus is the half-filled honeycomb lattice, which is the prototypical Dirac semimetal with low-energy excitations described by Dirac fermions. Of particular interest are exotic phases of matter that are not favoured at mean-field but are stabilised by fluctuations. Self-consistent path integral techniques are used to demonstrate that extended repulsive interactions favour charge order that breaks translational and other lattice symmetries. This charge order outcompetes interaction induced topological insulating phases, which are dominant at the mean-field level. The interacting quantum critical point was found to be near the critical point of a topological Lifshitz transition. Interacting Dirac fermions exhibit perhaps the simplest example of fermionic quantum criticality. Gross-Neveu-Yukawa field theories describe the universality class of semimetal-insulator transitions, which are driven by on-site and nearest neighbour repulsive interactions on the honeycomb lattice. The Dirac fermions couple to the dynamical order parameter fields and play a crucial role in determining the universal behaviour. Naturally, this variety of criticality is outside of the Ginzburg-Landau-Wilson paradigm. This idea is extended to include the translational symmetry breaking charge order. Here it is found that the breaking of crystal symmetries corresponds to Lorentz violating effective field theories. The criticality is analysed with the renormalisation group. The low energy excitations at the topological transition are gapless semi-Dirac fermionic quasiparticles that disperse linearly in one direction and quadratically in the other. The instabilities induced by generic short-range repulsions are investigated. The criticality of this anisotropic system is analysed with the renormalisation group in two spatial dimensions, and it is found to exhibit anisotropic order parameter correlations.
Critical points of classical and quantum lattice models are often described by scale-invariant Lifshitz theories which are anisotropic in the continuum limit, as characterized by a dynamical critical exponent z≠1z\neq1z=1. This type of critical behavior can in principle be studied by deforming ordinary z=1z=1z=1 conformal field theories (CFTs) by relevant vector operators breaking the rotational/Lorentz symmetry. In this short note, we consider a two-dimensional system of coupled minimal model CFTs Mm,m+1\mathcal{M}_{m,m+1}Mm,m+1​ which realizes this perspective in a controlled fashion via Zamolodchikov's large mmm expansion. The model turns out to exhibit interesting properties, including a manifold of interacting Lifshitz fixed points and emergent rotational symmetry in the infrared.
The roles of magnetic and electric perturbations in the quantum Lifshitz model in 2+12+12+1 dimensions are examined in this paper. The quantum Lifshitz model is an effective field theory for quantum multicritical systems, which include generalized two-dimensional (2D) quantum dimer models in bipartite lattices and their generalizations. It describes a class of quantum phase transitions between ordered and topological phases in 2+12+12+1 dimensions. Magnetic perturbations break the dimer conservation law. Electric excitations, the condensation of which leads to ordered phases, have been studied extensively both in the classical three-dimensional model and in the quantum 2D model. The role of magnetic vortex excitations, the condensation of which drives these systems into a Z2{\mathbb{Z}}_{2}Z2​ topological phase, has been largely ignored. Recent numerical studies claim that the quantum theory has a peculiar feature: the dynamical exponent zzz flows continuously and the quantum theory is hence unstable to magnetic vortices. To study the interplay of both excitations, we perform a perturbative renormalization group study to one-loop order and study the stability of the theory away from quantum multicriticality. This is done by generalizing the operator-product expansion to anisotropic models. It is found that the dynamical exponent does not appear to flow, in contrast to the classical Monte Carlo study. Possible reasons for this difference are discussed at length.
In Part I, the relationship between the topology of the Fermi surface and the entanglement entropy S is examined. Spinless fermionic systems on one and two dimensional lattices at fixed chemical potential are considered. The lattice is partitioned into sub-system of length L and environment, and the entanglement of the subsystem with the environment is calculated via the correlation matrix. S is plotted as a function of the next-nearest or next-next nearest neighbor hopping parameter, t. In 1 dimension, the entanglement entropy jumps at lifshitz transitions where the number of Fermi points changes. In 2 dimensions, a neck-collapsing transition is accompanied by a cusp in S, while the formation of electron or hole-like pockets coincides with a kink in the S as a function of the hopping parameter. The entanglement entropy as a function of subsystem length L is also examined. The leading order coefficient of the LlnL term in 2 dimensions was seen to agree well with the Widom conjecture. Of interest is the difference this coefficient and the coefficient of the term linear in L near the neck-collapsing point. The leading order term changes like |t-tc|1/2 whereas the first sub-leading term varies like |t-tc|1/3, where tc is the critical value of the hopping parameter at the transition. In Part II, we study the statistics of fractionalized excitations in a bosonic model which describes strongly interacting excitons in a N-band insulator. The elementary excitations of this system are strings, in a large N limit. A string is made of a series of bosons whose flavors are correlated such that the end points of a string carries a fractionalized flavor quantum number. When the tension of a string vanishes, the end points are deconfined. We determine the statistics of the fractionalized particles described by the end points of strings. We show that either bosons or Fermions can arise depending on the microscopic coupling constants. In the presence of the cubic interaction in the Hamiltonian as the only higher order interaction term, it was shown that bosons are emergent. In the presence of the quartic interaction with a positive coupling constant, it was revealed that the elementary excitations of the system possess Fermion statistics.
A formalism based on the fermionic functional-renormalization-group approach to interacting electron models defined on a lattice is presented. One-loop flow equations for the coupling constants and susceptibilities in the particle-particle and particle-hole channels are derived in weak-coupling conditions. It is shown that lattice effects manifest themselves through the curvature of the spectrum and the dependence of the coupling constants on momenta. This method is then applied to the one-dimensional extended Hubbard model; we thoroughly discuss the evolution of the phase diagram, and in particular the fate of the bond-centered charge-density-wave phase, as the system is doped away from half-filling. Our findings are compared to the predictions of the field-theory continuum limit and available numerical results.