Luttinger Hamiltonian Research Articles

Fine structure of electron-hole complexes in trigonal quantum dots

A theory of the Zeeman effect and fine structure of the energy spectrum of electron-hole complexes in highly symmetric quantum dots grown along the [111] direction from materials with a zinc blende lattice has been presented. In the studied quantum dots with C3v point symmetry, the Zeeman effect for a heavy hole in a magnetic field B ‖ [111] has an unusual form: in addition to the diagonal component (gh1), the effective tensor of g-factors contains the off-diagonal element (gh2). For gh2 ≠ 0, there is a magnetically induced mixing of heavy-hole states, which explains two additional lines observed in experimental photoluminescence spectra of excitons and trions. A microscopic theory of the effective g-factors gh1 and gh2 within the Luttinger Hamiltonian in the spherical approximation has been discussed, as well as additional contributions to the off-diagonal element gh2 due to the warping of the spectrum of holes. The results of theoretical calculations of the g-factors gh1 and gh2 for quantum dots based on GaAs have been compared with the experimental data.

Particle partition entanglement of bosonic Luttinger liquids

We consider the R\'{e}nyi entanglement entropy of bosonic Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher order corrections involve a microscopic length scale, the leading order scaling depends only on this sole dimensionless parameter which characterizes the low energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Luttinger liquid theory for the Lieb-Liniger model of $\delta$-function interacting bosons in the one dimensional spatial continuum.

Phase diagram of the one-dimensional t-J model with long-range dipolar interactions

We study systematically the effect of long-range dipolar interactions on the ground-state phase diagram of the one-dimensional model by means of the density matrix renormalization group. While the basic phases described by the Luttinger parameter , namely, the repulsive Luttinger liquid (metallic phase, ), attractive Luttinger liquid (superconducting phase, ), and the phase separation , are similar to those of the conventional model, the presence of the long-range dipolar interactions leads to significant differences. i) At high-density regime, the phase boundaries of these three phases are even pushed to the large-J region; ii) at low-density regime, these phase boundaries shift toward the small-J region and, most importantly, iii) the spin-gap region spreads across the boundary of , suggesting an exotic metallic phase with spin-gap, which is absent in the conventional model. The result indicates that the long-range dipolar interactions have a significant influence on the ground-state properties of the one-dimensional model. Its implication on the pseudogap phenomenon of the hole-doped cuprates is briefly discussed.

Nontrivial contributions to the magnetoconductivity due to anomalous g-factor in the Luttinger Hamiltonian

The effective g-factor and the selection rules for the matrix elements of the velocity operator are investigated for the Luttinger Hamiltonian under a magnetic field. It is shown that the g-factor has a strong kz-dependence, where kz is the wave vector along the magnetic field. This anomalous kz-dependence is due to the interband effect of the magnetic field between the light and heavy hole bands. It is found that a nontrivial contribution to the longitudinal magnetoconductivity arises also due to the interband effects by analyzing the selection rules. This nontrivial interband contribution can be a source of the longitudinal spin current.

Predicting excitonic gaps of semiconducting single-walled carbon nanotubes from a field theoretic analysis

We demonstrate that a non-perturbative framework for the treatment of the excitations of single walled carbon nanotubes based upon a field theoretic reduction is able to accurately describe experiment observations of the absolute values of excitonic energies. This theoretical framework yields a simple scaling function from which the excitonic energies can be read off. This scaling function is primarily determined by a single parameter, the charge Luttinger parameter of the tube, which is in turn a function of the tube chirality, dielectric environment, and the tube's dimensions, thus expressing disparate influences on the excitonic energies in a unified fashion. We test this theory explicitly on the data reported in NanoLetters 5, 2314 (2005) and Phys. Rev. B. 82, 195424 (2010) and so demonstrate the method works over a wide range of reported excitonic spectra.

Scaling of temporal correlations in an attractive Tomonaga-Luttinger spin liquid

We report temperature-dependent neutron scattering measurements of the local dynamic structure factor in the quantum spin ladder (C$_{7}$H$_{10}$N)$_{2}$CuBr$_{4}$ in a magnetic field $H=9$~T, in its gapless quantum-critical phase. We show that the measured quantity has a scaling form consistent with expectations for a Tomonaga-Luttinger liquid with attraction. The measured Luttinger parameter $K\approx1.25$ and scaling function are in excellent agreement with density matrix renormalization group numerical calculations for the underlying spin Hamiltonian.

Quantum criticality in a Kondo quantum dot coupled to helical edge states of interacting 2D topological insulators

We investigate theoretically the quantum phase transition (QPT) between the one-channel Kondo (1CK) and two-channel Kondo (2CK) fixed points in a quantum dot coupled to helical edge states of interacting 2D topological insulators (2DTI) with Luttinger parameter . The model was studied by Law et al (2010 Phys. Rev. B 81 041305(R)), and was mapped onto an anisotropic two-channel Kondo model via bosonization. For , the strong coupling 2CK fixed point was argued to be stable for infinitesimally weak tunnelings between the dot and the 2DTI based on a simple scaling dimensional analysis (Law et al 2010 Phys. Rev. B 81 041305(R). We re-examine this model beyond the bare scaling dimension analysis via a one-loop renormalization group (RG) approach combined with bosonization and re-fermionization techniques near weak-coupling and strong-coupling (2CK) fixed points. We find for a fixed value of that the 2CK fixed point can be unstable towards the 1CK fixed point and the system is expected to undergo a quantum phase transition between 1CK and 2CK fixed points with changing Kondo couplings. Our RG approach is controlled near K = 1. In general, this QPT can also occur upon tuning the Luttinger parameter K to a critical value smaller than unity () for fixed Konodo couplings. The QPT in our model comes as a result of the combined Kondo and the helical Luttinger physics in 2DTI, and it serves as the first example of the 1CK-2CK QPT that is accessible by the controlled RG approach. We extract quantum critical and crossover behaviors from various thermodynamical quantities near the transition. Our results are robust against particle-hole asymmetry for .

Identifying a Bath-Induced Bose Liquid in Interacting Spin-Boson Models

We study the ground state phase diagram of a one-dimensional hard-core bosonic model with nearest-neighbor interactions (XXZ model) where every site is coupled Ohmically to an independent but identical reservoir, hereby generalizing spin-boson models to interacting spin-boson systems. We show that a bath-induced Bose liquid phase can occur in the ground state phase diagram away from half filling. This phase is compressible, gapless, and conducting but not superfluid. At half filling, only a Luttinger liquid and a charge density wave are found. The phase transition between them is of Kosterlitz-Thouless type where the Luttinger parameter takes a nonuniversal value. The applied quantum MonteCarlo method can be used for all open bosonic and unfrustrated spin systems, regardless of their dimension, filling factor, and spectrum of the dissipation as long as the quantum system couples to the bath via the density operators.

Luttinger-liquid behavior of one-dimensionalHe3

The ground-state properties of one-dimensional $^{3}\mathrm{He}$ are studied using quantum Monte Carlo methods. The equation of state is calculated in a wide range of physically relevant densities and is well interpolated by a power-series fit. The Luttinger liquid theory is found to describe the long-range properties of the correlation functions. The density dependence of the Luttinger parameter is explicitly found, and interestingly it shows a nonmonotonic behavior. Depending on the density, the static structure factor can be a smooth function of the momentum or might contain a peak of a finite or infinite height. Although no phase transitions are present in the system, we identify a number of physically different regimes, including an ideal Fermi gas, a ``Bose gas.'' a ``super-Tonks-Girardeau'' regime, and a ``quasicrystal.'' The obtained results are applicable to unpolarized, partially, or fully polarized $^{3}\mathrm{He}$.

Effect of geometry and composition on the intraband transitions of holes in quantum dots

The effect of shape and size anisotropy on unipolar intraband transitions of holes in quantum dots (QDs) is studied. The optical matrix elements are calculated for transitions of holes in valence band. To get the optical matrix elements, energy eigenvalues and eigenvectors are calculated using 4 × 4 Luttinger Hamiltonian in the effective mass approximation. The formulation is applied to InGaAs/GaAs QD with parabolic confinement potential in xy-plane. The optical matrix elements for intraband hole transitions are calculated for x and y polarised light. The transitions are considered from ground state to other excited states. The effect of In concentration on optical matrix elements is also investigated. It is important to note that the transitions of holes are governed by the character of initial and final states for different light polarisations that give specific transition selection rules. It is found that the polarisation is strongly dependent on the in-plane anisotropy of the QDs.

Generic Features of an Electron Injected into the Luttinger Liquid

An extra electron injected into the Luttinger liquid (LL) is not completely separated into spin and charge excitations, but it provides a peak of an electron-like excitation in addition to the two prominent spin- and charge-excitation peaks in the one-electron spectral function of an interacting one-dimensional (1D) system. By a scaling argument based on a renormalization-group scheme, we obtain universal features of this electron-like excitation for low energies and in the vicinity of the Fermi point, which are entirely characterized by the Luttinger parameter Kc and the renormalized Fermi velocity vF at the LL fixed point. The latter quantity vF is shown to be given by the slope of the lowest excitation spectrum near the Fermi point in the system with large but finite size. Some of exact results are presented for the 1D Hubbard model at n=0.59 filling.

Strain-induced non-linear optical properties of straddling-type indium gallium aluminum arsenic/indium phosphide nanoscale-heterostructures

Non-linear optical properties of indium gallium aluminum arsenic/indium phosphide (InGaAlAs/InP) material system based unstrained and strained lasing nano-heterostructures of straddled type (type-I) under TE (transverse electric) polarization mode have been studied. To produce the different states of strain at the heterointerfaces in the structure, the different layers of the active region of InGaAlAs (width ~6nm) with different lattice constants (due to different mole fractions) have been assumed to be deposited in between the barriers of In0.41Ga0.34Al0.25As followed by claddings of In0.52Al0.48As. The entire nano-structure is a type of step SCH (separate confinement heterostructure) and has been assumed to grow over InP substrate. To study the strain induced non-linear optical properties of the structure, single effective mass and Kohn–Luttinger Hamiltonian equations have been solved to obtain quantum states and envelope wave functions in the heterostructure. In addition, the optical gain, differential gain, and the anti-guiding factor in order to support the gain calculation have been computed and reported. On behalf of simulation and previously reported experimental results, it has been suggested that either unstrained or compressive strained InGaAlAs/InP nano-heterostructures are more suitable for novel applications in the emerging areas of nano-opto-electronics due to the maximum gain occurring at lasing wavelength of 1.55µm which is the wavelength of minimum optical attenuation.

Qualitative analysis of gain spectra of InGaAlAs/InP lasing nano-heterostructure

This paper deals with the studies of lasing characteristics along with the gain spectra of compressively strained and step SCH based In 0.71 Ga 0.21 Al 0.08 As/InP lasing nano-heterostructure within TE polarization mode, taking into account the variation in well width of the single quantum well of the nano-heterostructure. In addition, the compressive conduction and valence bands dispersion profiles for quantum well of the material composition In 0.71 Ga 0.21 Al 0.08 As at temperature 300 K and strain ~1.12% have been studied using 4 × 4 Luttinger Hamiltonian. For the proposed nano-heterostructure, the quantum well width dependence of differential gain, refractive index change and relaxation oscillation frequency with current density have been studied. Moreover, the G–J characteristics of the nano-heterostructure at different well widths have also been investigated, that provided significant information about threshold current density, threshold gain and transparency current density. The results obtained in the study of nano-heterostructure suggest that the gain and relaxation oscillation frequency both are decreased with increasing quantum well width but the required lasing wavelength is found to shift towards higher values. On behalf of qualitative analysis of the structure, the well width of 6 nm is found more suitable for lasing action at the wavelength of 1.55 μm due to minimum optical attenuation and minimum dispersion within the waveguide. The results achieved are, therefore, very important in the emerging area of nano-optoelectronics.

Magnetic field spectral crossings of Luttinger holes in quantum wells

We develop an analytic approach to two-dimensional (2D) holes in a magnetic field that allows us to gain insight into physics of measuring the parameters of holes, such as cyclotron resonance, Shubnikov-de Haas effect and spin resonance. We derive hole energies, cyclotron masses, and the $g$ factors in the semiclassical regime analytically, as well as analyze numerical results outside the semiclassical range of parameters, qualitatively explaining the experimentally observed magnetic field dependence of the cyclotron mass. In the semiclassical regime with large Landau level indices, and for size quantization energy much bigger than the cyclotron energy, the cyclotron mass coincides with the in-plane effective mass, calculated in the absence of a magnetic field. The hole $g$ factor in a magnetic field perpendicular to the 2D plane is defined not only by the constant of direct coupling of the angular momentum of the holes to the magnetic field, but also by the Luttinger constants defining the effective masses of holes. We find that the $g$ factor for quasi-2D holes with heavy mass in the [001] growth direction in GaAs quantum well is $g=4.05$ in the semiclasssical regime. Outside the semiclassical range of parameters, holes behave as a species completely different from electrons. Spectra for size- and magnetic-field-quantized holes are nonequidistant, not fanlike, and exhibit multiple crossings, including crossing in the ground level. We calculate the effect of Dresselhaus terms, which transform some of the crossings into anticrossings, and the effects of the anisotropy of the Luttinger Hamiltonian on the 2D hole spectra. Dresselhaus terms of different symmetries are taken into account, and a regularization procedure is developed for the ${k}_{z}^{3}$ Dresselhaus terms. Control of the nonequidistant levels and crossing structure by the magnetic field can be used to control the Landau level mixing in hole systems, and thereby control hole-hole interactions in the magnetic field.

Spectral-edge mode in interacting one-dimensional systems

A continuum of excitations in interacting one-dimensional systems is bounded from below by a spectral edge that marks the lowest possible excitation energy for a given momentum. We analyse short-range interactions between Fermi particles and between Bose particles (with and without spin) using Bethe-Ansatz techniques and find that the dispersions of the corresponding spectral edge modes are close to a parabola in all cases. Based on this emergent phenomenon we propose an empirical model of a free, non-relativistic particle with an effective mass identified at low energies as the bare electron mass renormalised by the dimensionless Luttinger parameter $K$ (or $K_\sigma$ for particles with spin). The relevance of the Luttinger parameters beyond the low energy limit provides a more robust method for extracting them experimentally using a much wide range of data from the bottom of the one-dimensional band to the Fermi energy. The empirical model of the spectral edge mode complements the mobile impurity model to give a description of the excitations in proximity of the edge at arbitrary momenta in terms of only the low energy parameters and the bare electron mass. Within such a framework, for example, exponents of the spectral function are expressed explicitly in terms of only a few Luttinger parameters.

Zeeman splitting of light hole in quantum wells: Comparison of theory and experiments

The theory for light-hole Zeeman splitting developed in [Physica E 44, 797 (2012)] is compared with experimental data found in literature for GaAs/AlGaAs, InGaAs/InP and CdTe/CdMgTe quantum wells. It is shown that the description of experiments is possible with account for excitonic effects and peculiarities of the hole energy spectrum in a quantum well including complex structure of the valence band and the interface mixing of light and heavy holes. It is demonstrated that the absolute values and the sign of the light-hole $g$-factor are extremely sensitive to the parametrization of the Luttinger Hamiltonian.

Hole spin helix: Anomalous spin diffusion in anisotropic strained hole quantum wells

We obtain the spin-orbit interaction and spin-charge coupled transport equations of a two-dimensional heavy hole gas under the influence of strain and anisotropy. We show that a simple two-band Hamiltonian can be used to describe the holes. In addition to the well-known cubic hole spin-orbit interaction, anisotropy causes a Dresselhaus-like term, and strain causes a Rashba term. We discover that strain can cause a shifting symmetry of the Fermi surfaces for spin up and down holes. We predict an enhanced spin lifetime associated with a spin helix standing wave similar to the Persistent Spin Helix which exists in the two-dimensional electron gas with equal Rashba and Dresselhaus spin-orbit interactions. These results may be useful both for spin-based experimental determination of the Luttinger parameters of the valence-band Hamiltonian and for creating long-lived spin excitations.

Edge physics of the quantum spin Hall insulator from a quantum dot excited by optical absorption.

The gapless edge modes of the quantum spin Hall insulator form a helical liquid in which the direction of motion along the edge is determined by the spin orientation of the electrons. In order to probe the Luttinger liquid physics of these edge states and their interaction with a magnetic (Kondo) impurity, we consider a setup where the helical liquid is tunnel coupled to a semiconductor quantum dot that is excited by optical absorption, thereby inducing an effective quantum quench of the tunneling. At low energy, the absorption spectrum is dominated by a power-law singularity. The corresponding exponent is directly related to the interaction strength (Luttinger parameter) and can be computed exactly using boundary conformal field theory thanks to the unique nature of the quantum spin Hall edge.

Asymptotically exact scenario of strong-disorder criticality in one-dimensional superfluids

We present a controlled rare-weak-link theory of the superfluid-to-Bose/Mott glass transition in one-dimensional disordered systems. The transition has Kosterlitz-Thouless critical properties but may occur at an arbitrary large value of the Luttinger parameter $K$. In contrast to the scenario by Altman {\it et al.} [Phys. Rev. B {\bf 81}, 174528 (2010)], the hydrodynamic description is valid under the correlation radius and defines criticality via the renormalization of microscopically weak links, along the lines of Kane and Fisher [Phys. Rev. Lett. {\bf 68}, 1220 (1992)]. The hallmark of the theory is the relation $K^{(c)}=1/\zeta$ between the critical value of the Luttinger parameter at macroscopic scales and the microscopic (irrenormalizable) exponent $\zeta$ describing the scaling $\propto 1/N^{1-\zeta}$ for the strength of the weakest link among the $N/L \gg 1$ disorder realizations in a system of fixed mesoscopic size $L$.

Low-energy properties of fractional helical Luttinger liquids

We investigate the low-energy properties of (quasi) helical and fractional helical Luttinger liquids. In particular, we calculate the Drude peak of the optical conductivity, the density of states, as well as charge transport properties of the interacting system with and without attached Fermi liquid leads at small and large (compared to the gap) frequencies. For fractional wires, we find that the low-energy tunneling density of states vanishes. The conductance of a fractional helical Luttinger liquid is noninteger. It is independent of the Luttinger parameters in the wire, despite the intricate mixing of charge and spin degrees of freedom, and only depends on the relative locking of charge and spin degrees of freedom.