Renormalization Analysis Research Articles

Dominant superconducting fluctuations in the one-dimensional extended Holstein–extended Hubbard model

The search for realistic one-dimensional (1D) models that exhibit dominant superconducting (SC) fluctuations effects has a long history. In these 1D systems, the effects of commensurate band fillings--strongest at half-filling--and electronic repulsions typically lead to a finite charge gap and the favoring of insulating density wave ordering over superconductivity. Accordingly, recent proposals suggesting a gapless metallic state in the Holstein-Hubbard (HH) model, possibly superconducting, have generated considerable interest and controversy, with the most recent work demonstrating that the putative dominant superconducting state likely does not exist. In this paper we study a model with non-local electron-phonon interactions, in addition to electron-electron interactions, this model unambiguously possesses dominant superconducting fluctuations at half filling in a large region of parameter space. Using both the numerical multi-scale functional renormalization group for the full model and an analytic conventional renormalization group for a bosonized version of the model, we demonstrate the existence of dominant superconducting (SC) fluctuations. These dominant SC fluctuations arise because the spin-charge coupling at high energy is weakened by the non-local electron-phonon interaction and the charge gap is destroyed by the resultant suppression of the Umklapp process. The existence of the dominant SC pairing instability in this half-filled 1D system suggests that non-local boson-mediated interactions may be important in the superconductivity observed in the organic superconductors.

On the finite amplitudes for open graphs in Abelian dynamical colored Boulatov–Ooguri models

In the work (Ben Geloun and Bonzom 2011 Int. J. Theor. Phys. 50 2819), it has been proved that the radiative corrections of the 2-point function in the SU(2) Boulatov tensor model generates a marginal (in the renormalization group sense) contribution of the form of a Laplacian. Such a term, which was missing in the initial Boulatov model action, should be added in that action before discussing the renormalization analysis of this model. In this work, by linearizing the group manifold, we prove that the amplitudes associated with Feynman graphs with external legs of the colored Boulatov model over U(1)3 endowed with a Laplacian dynamics are all convergent. We conjecture that the same feature happens for the corresponding Boulatov model over SU(2). Higher rank models are also discussed.

Chaotic correlations in barrier billiards with arbitrary barriers

We study autocorrelation functions in symmetric barrier billiards for golden mean trajectories with arbitrary barriers. Renormalization analysis reveals the presence of a chaotic invariant set and thus that, for a typical barrier, there are chaotic correlations. The chaotic renormalization set is the analogue of the so-called orchid that arises in a generalized Harper equation.

Phase separation in the Edwards model

The nature of charge transport within a correlated background medium can be described by spinless fermions coupled to bosons in the model introduced by Edwards. Combining numerical density matrix renormalization group and analytical projector-based renormalization methods we explore the ground-state phase diagram of the Edwards model in one dimension. Below a critical boson frequency any long-range order disappears and the system becomes metallic. If the charge carriers are coupled to slow quantum bosons the Tomonaga-Luttinger liquid is attractive and finally makes room for a phase separated state, just as in the t-J model. The phase boundary separating repulsive from the attractive Tomonaga-Luttinger liquid is determined from long-wavelength charge correlations, whereas fermion segregation is indicated by a vanishing inverse compressibility. On approaching phase separation the photoemission spectra develop strong anomalies.

Sublattice interference in the kagome Hubbard model

We study the electronic phases of the kagome Hubbard model (KHM) in the weak coupling limit around van Hove filling. Through an analytic renormalization group analysis, we find that there exists a sublattice interference mechanism where the kagome sublattice structure affects the character of the Fermi surface instabilities. It leads to major suppression of Tc for d+id superconductivity in the KHM and causes an anomalous increase of Tc upon addition of longer-range Hubbard interactions. We conjecture that the suppression of conventional Fermi liquid instabilities makes the KHM a prototype candidate for hosting exotic electronic states of matter at intermediate coupling.

A Vanishing Theorem for Operators in Fock Space

We consider the bosonic Fock space over the Hilbert space of transversal vector fields in three dimensions. This space carries a canonical representation of the group of rotations. For a certain class of operators in Fock space we show that rotation invariance implies the absence of terms which either create or annihilate only a single particle. We outline an application of this result in an operator theoretic renormalization analysis of Hamilton operators, which occur in non-relativistic qed.

Universal behavior in the scattering of heavy, weakly interacting dark matter on nuclear targets

Particles that are heavy compared to the electroweak scale (M≫mW), and that are charged under electroweak SU(2) gauge interactions display universal properties such as a characteristic fine structure in the mass spectrum induced by electroweak symmetry breaking, and an approximately universal cross section for scattering on nuclear targets. The heavy particle effective theory framework is developed to compute these properties. As illustration, the spin independent cross section for low-velocity scattering on a nucleon is evaluated in the limit M≫mW, including complete leading-order matching onto quark and gluon operators, renormalization analysis, and systematic treatment of perturbative and hadronic-input uncertainties.

Two-channel pseudogap Kondo and Anderson models: Quantum phase transitions and non-Fermi liquids

We discuss the two-channel Kondo problem with a pseudogap density of states, $\rho(\w)\propto|\w|^r$, of the bath fermions. Combining both analytical and numerical renormalization group techniques, we characterize the impurity phases and quantum phase transitions of the relevant Kondo and Anderson models. The line of stable points, corresponding to the overscreened non-Fermi liquid behavior of the metallic $r=0$ case, is replaced by a stable particle-hole symmetric intermediate-coupling fixed point for $0<r<\rmax\approx0.23$. For $r>\rmax$, this non-Fermi liquid phase disappears, and instead a critical fixed point with an emergent spin--channel symmetry appears, controlling the quantum phase transition between two phases with stable spin and channel moments, respectively. We propose low-energy field theories to describe the quantum phase transitions, all being formulated in fermionic variables. We employ epsilon expansion techniques to calculate critical properties near the critical dimensions $r=0$ and $r=1$, the latter being potentially relevant for two-channel Kondo impurities in neutral graphene. We find the analytical results to be in excellent agreement with those obtained from applying Wilson's numerical renormalization group technique.

Uniqueness of the Ground State in the Feshbach Renormalization Analysis

In the operator theoretic renormalization analysis introduced by Bach, Froehlich, and Sigal we prove uniqueness of the ground state.

Dynamical conductivity at the dirty superconductor-metal quantum phase transition

We study the transport properties of ultrathin disordered nanowires in the neighborhood of the superconductor-metal quantum phase transition. To this end we combine numerical calculations with analytical strong-disorder renormalization group results. The quantum critical conductivity at zero temperature diverges logarithmically as a function of frequency. In the metallic phase, it obeys activated scaling associated with an infinite-randomness quantum critical point. We extend the scaling theory to higher dimensions and discuss implications for experiments.

Haldane, Large-D, and Intermediate-DStates in anS= 2 Quantum Spin Chain with On-Site andXXZAnisotropies

Using mainly numerical methods, we investigate the ground-state phase diagram of the S=2 quantum spin chain described by $H = \sum_j (S_j^x S_{j+1}^x + S_j^y S_{j+1}^y + \Delta S_j^z S_{j+1}^z) + D \sum_j (S_j^z)^2$, where $\Delta$ denotes the $XXZ$ anisotropy parameter of the nearest-neighbor interactions and $D$ the on-site anisotropy parameter. We restrict ourselves to the case with $\Delta \ge 0$ and $D \ge 0$ for simplicity. Each of the phase boundary lines is determined by the level spectroscopy or the phenomenological renormalization analysis of numerical results of exact-diagonalization calculations. The resulting phase diagram on the $\Delta$-$D$ plane consists of four phases; the XY 1 phase, the Haldane/large-$D$ phase, the intermediate-$D$ phase and the N\'eel phase. The remarkable natures of the phase diagram are: (1) the Haldane state and the large-$D$ state belong to the same phase; (2) there exists the intermediate-$D$ phase which was predicted by Oshikawa in 1992; (3) the shape of the phase diagram on the $\Delta$-$D$ plane is different from that believed so far. We note that this is the first report of the observation of the intermediate-$D$ phase.

Ground States in the Spin Boson Model

We prove that the Hamiltonian of the model describing a spin which is linearly coupled to a field of relativistic and massless bosons, also known as the spin-boson model, admits a ground state for small values of the coupling constant λ. We show that the ground-state energy is an analytic function of λ and that the corresponding ground state can also be chosen to be an analytic function of λ. No infrared regularization is imposed. Our proof is based on a modified version of the BFS operator theoretic renormalization analysis. Moreover, using a positivity argument we prove that the ground state of the spin-boson model is unique. We show that the expansion coefficients of the ground state and the ground-state energy can be calculated using regular analytic perturbation theory.

Regularization, renormalization, and dimensional analysis: Dimensional regularization meets freshman E&amp;M

We illustrate the dimensional regularization technique using a simple example from electrostatics. This example illustrates the virtues of dimensional regularization without the complications of a full quantum field theory calculation. We contrast the dimensional regularization approach with the cutoff regularization approach, and demonstrate that dimensional regularization preserves translational symmetry. We then introduce minimal subtraction and modified minimal subtraction schemes to renormalize the result. Finally, we consider dimensional transmutation as encountered in the case of compact “extra dimensions.”

Renormalization group flow, competing phases, and the structure of superconducting gap in multiband models of iron-based superconductors

We perform an analytical renormalization group (RG) study to address the role of Coulomb repulsion, the competition between extended s-wave superconducting order (s+-) and the spin-density wave (SDW) order and the angular dependence of the superconducting gap in multi-pocket models of Iron based superconductors. Previous analytic RG studies considered a toy model of one hole and one electron pocket. We consider more realistic models of two electron pockets and either two or three hole pockets, and also incorporate the angular dependence of interaction. We neglect for simplicity one of the two hole pockets centered at k=0, which is less nested with electron pockets, i.e., consider 3-pocket and 4-pocket models. In a toy 2-pocket model, SDW order always wins over s+- order at perfect nesting, and s+- order only appears at a finite doping, and only if RG flow extends long enough to overcome intra-pocket Coulomb repulsion. For multi-pocket models, we find two new effects. First, there always exists an attractive component of the interaction in s+- channel, such that the system necessary becomes a superconductor once it overcomes the competition from the SDW state. Second, in 3-pocket case (but not in 4-pocket case), there are situations when s+- order wins over SDW order even for perfect nesting, suggesting that SDW order is not a necessary pre-condition for the s+- order. Our results are in good agreement with recent numerical functional RG studies by Thomale et al. [arXiv:1002.3599]

Dynamical Conductivity at the Dirty Superconductor-Metal Quantum Phase Transition

We study the transport properties of ultrathin disordered nanowires in the neighborhood of the superconductor-metal quantum phase transition. To this end we combine numerical calculations with analytical strong-disorder renormalization group results. The quantum critical conductivity at zero temperature diverges logarithmically as a function of frequency. In the metallic phase, it obeys activated scaling associated with an infinite-randomness quantum critical point. We extend the scaling theory to higher dimensions and discuss implications for experiments.

Charge-density-wave formation in a half-filled fermion-boson transport model: A projective renormalization approach

We study the metal-insulator transition in a very general two-channel transport model, where charge carriers are coupled to a correlated background medium. The fluctuations of the background were described as bosonic excitations, having the ability to relax. Employing an analytical projector-based renormalization technique, we calculate the ground-state and spectral properties of this fermion-boson model and corroborate recent numerical results, which indicate---in dependence on the `stiffness' of the background medium---a Luttinger-liquid to charge-density-wave transition for the one-dimensional half-filled band case. In particular, we determine the renormalized electron and boson dispersion relations and show that the quantum phase transition is not triggered by a softening of the boson modes. Thus the charge density wave is different in nature from an usual Peierls distorted state.

DOMAIN WALL RENORMALIZATION GROUP ANALYSIS OF TWO-DIMENSIONAL ISING MODEL

Using a recently proposed new renormalization group method (tensor renormalization group), we analyze the Ising model on the two-dimensional square lattice. For the lowest-order approximation with two-domain wall states, it realizes the idea of coarse graining of domain walls. We write down explicit analytic renormalization transformation and prove that the picture of the coarse graining of the physical domain walls does hold for all physical renormalization group flows. We solve it to get the fixed point structure and obtain the critical exponents and the critical temperature. These results are very near to the exact values. We also briefly report the improvement using four-domain wall states.

Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of α3/2, α being the fine structure constant. A suitably chosen ground state vector depends analytically on α3/2 and it is twice continuously differentiable with respect to the nuclear coordinates.

Density-matrix renormalization study of frustrated fermions on a triangular lattice

We show that the two-dimensional density-matrix renormalization analysis is useful to detect the symmetry breaking in the fermionic model on a triangular lattice. Under the cylindrical boundary conditions with chemical potentials on edge sites, we find that the open edges work as perturbation to select the strongest correlations {\it only in the presence of a long range order}. We also demonstrate that the ordinary size scaling analysis on the charge gap as well as that of the local charge density under this boundary condition could determine the metal-insulator phase boundary, which scales almost perfectly with the density of states and the exact solutions in the weak and strong coupling region, respectively.

Theory of Smeared Quantum Phase Transitions

We present an analytical strong-disorder renormalization group theory of the quantum phase transition in the dissipative random transverse-field Ising chain. For Ohmic dissipation, we solve the renormalization flow equations analytically, yielding asymptotically exact results for the low-temperature properties of the system. We find that the interplay between quantum fluctuations and Ohmic dissipation destroys the quantum critical point by smearing. We also determine the phase diagram and the behavior of observables in the vicinity of the smeared quantum phase transition.