Session Y41: Strongly Correlated Systems - Theory

Nonequilibrium electron spectroscopy of Luttinger liquids

We theoretically study a Luttinger liquid (LL) driven out of equilibrium by injection of high-energy electrons. The electrons enter the LL locally, far away from any contacts, and at a fixed energy. Their spectral properties are detected at another spatial point some distance away by evaluating the average tunneling current from the LL into a resonant level with tunable energy. For energies slightly below the injection energy, the dependence of the detected current on the difference between injection and detection energies is described by a power law whose exponent depends continuously on the Luttinger parameter. In contrast, for tunneling into the chiral LL edge of a fractional quantum Hall state from the Laughlin sequence, we find that the detected current grows linearly with the energy difference, independent of the LL parameter determined by the inverse filling fraction. We develop a diagrammatic approach for the standard (non-chiral) LL which provides an intuitive physical picture for how the electrons can relax inside the wire.   

Quantum Monte Carlo calculations on the 1-dimensional electron gas

I will present quantum Monte Carlo (QMC) calculations on the 1-dimensional electron gas. In particular, we have computed the ground state energy, momentum density, pair-correlation function and static structure factor of electrons in 1d for different electron-electron interactions. By fitting to the resulting correlation functions, we find the Luttinger model parameters for the system and compare with experiments on a number of quasi-1d systems.   

Nonlinear Behaviour in Long Range Integrable Models with Spin

We study nonlinear aspects of long range integrable models with spin by going beyond the Luttinger Liquid theory. We present here [1], the fully nonlinear dynamics of spin and charge in spin-Calogero model (sCM), an integrable 1D model of quantum spin-1/2 particles interacting through inverse square interaction and exchange. Hydrodynamic equations of motion are written for this model in the regime where gradient corrections to the exact theory may be neglected. In this approximation, variables separate in terms of dressed Fermi momenta of the model. Hydrodynamic equations reduce to a set of decoupled Riemann-Hopf equations for the dressed Fermi momenta. We study the dynamics of some non-equilibrium spin-charge configurations for times smaller than the time-scale of gradient catastrophe. We then show [2] how this field theory allows to calculate correlation functions that cannot be considered with conventional bosonization. We also highlight the connections between sCM, Haldane-Shastry model and $\lambda=2$ spin-less Calogero model. [1] M. Kulkarni, F. Franchini, A. G. Abanov, Phys. Rev. B 80, 165105 (2009) [2] F. Franchini, M. Kulkarni, Nucl. Phys. B, 825, 320 (2010)   

Extremely Correlated Quantum Liquids

The Extremely Correlated Quantum Liquid (ECQL) is obtained in the $U\rightarrow \infty$ limit of strongly correlated systems- so that double occupancy is prohibited[1]. The $t$-$J$ model, a standard example of this class is studied by the new formulation[1]. We present the exact Schwinger Dyson equation for its Greens function, in terms of singlet and triplet vertex functions. The vertices are expanded further in a set of non linear equations, and the existence of a low hole density expansion is demonstrated. Its Fermi surface volume differs from the Luttinger Ward volume {\em at all densities}. The excitations are fractionally charged particles, where the fraction is determined by the density. The fractionalization arises from a modified Pauli principle for filling of the Lower Hubbard band. Numerical results for the magnetic susceptibility and the spectral functions arising from systematic expansions are presented elsewhere in this conference [2-3]. [1] {\em Extremely Correlated Quantum Liquids}, B. S. Shastry Preprint (2009). [2] {\em Magnetic response of the Extremely Correlated Electron Liquid}, A. Garg, D. Hansen and B. S. Shastry (APS March meeting 2010). [3] {\em Spectral functions of the Extremely Correlated Quantum Liquid}, D. Hansen, A. Garg, and B. S. Shastry (APS March meeting 2010).   

Spectral Functions of the Extremely Correlated Quantum Liquid

The recently developed theory of an Extremely Correlated Quantum Liquid (ECQL)[1], provides an exact Schwinger Dyson equation of the $t$-$J$ model, in terms of triplet and singlet vertices. The lowest order scheme for the vertices leads to a self energy $\Sigma \sim GGG$, with coefficients such that it survives the limit of $n \to 1$. We present the resulting spectral functions for this case. We also consider a higher order approximation (akin to the RPA) with a bubble sum, this approximation is argued to be dominant near half filling. The resulting non Fermi liquid like spectral functions are displayed. Optical and Raman spectra are calculated within each scheme. A general presentation of the ECQL as well as related numerical studies of magnetic response within the $t$-$J$ model are presented elsewhere at this conference [1,2]. [1] Extremely Correlated Quantum Liquids, B. Sriram Shastry Preprint (2009).   

Magnetic response of the Extremely correlated electron liquid

We study the magnetic susceptibility and Nuclear magnetic resonance (NMR) relaxation rates for the $t$-$J$ model within the recently proposed theory of Extremely Correlated Quantum Liquid (ECQL) [1]. The basic theory of ECQL and its spectral functions are presented elsewhere [1-2]. This theory leads to a sequence of systematic conserving schemes that satisfy the constraint of rotation invariance and gauge invariance via the Nozieres relations and Ward identities. As a first example, we present the results of a non linear RPA type scheme valid near half filling. A characteristic feature of this scheme is that the bandwidth gets renormalized by a factor that is a function of static spin and charge correlations. The resulting dynamical magnetic susceptibilities and NMR relaxation rates are presented and compared with experiments on High Tc systems. [1] Extremely Correlated Quantum Liquids, B. Sriram Shastry Preprint (2009) and (APS March Meeting 2010). [2] Spectral functions of the Extremely correlated electron liquid, Daniel Hansen, Arti Garg, and B. Sriram Shastry (APS March meeting 2010).   

Universal and non-universal renormalizations in the Fermi-liquid theory

We discuss the interplay between the Landau Fermi liquid theory and a direct perturbative approach to a Fermi liquid. In a Fermi liquid theory for Galilean-invariant systems, mass renormalization $m^*/m$ comes exclusively from fermions at the Fermi surface. We show that in the perturbation theory the same $m^*/m$ comes partly from fermions at the Fermi surface and partly from fermions far away from the Fermi surface. We show that there exists a particular relation between the self-energy contributions from high- and low-energy fermions which allows one to reconcile diagrammatics with a Fermi-liquid theory. We argue that extra care has to be exercised in the renormalization group approach to a Fermi liquid in order not to miss high-energy contributions to $m^*/m$. We present the results for $m^*/m$ and the quasiparticle residue $Z$ for a 2D Fermi gas with short-range interaction and discuss the extension of an $SU(2)$ theory to $SU(N)$.   

Spin conservation and Fermi liquid near a ferromagnetic quantum critical point

We propose a new low-energy theory for itinerant fermions near a ferromagnetic quantum critical point. We show that the full low-energy model includes, in addition to conventional interaction via spin fluctuations, another type of interaction, whose presence is crucial for the theory to satisfy SU(2) spin conservation. We demonstrate the consistency between a loop-wise expansion and a Fermi liquid description for the full model. We further show that, prior to the ferromagnetic instability, the system develops a Pomeranchuk-type instability into a state with zero magnetization but with p-wave deformations of the Fermi surfaces of spin-up and -down electrons (a spin nematic).   

Quantum Criticality of $3+1$-dimensional massless Dirac fermions

We have studied the quantum critical behavior of $3+1$- dimensional massless Dirac fermions in the presence of long- range Coulomb interaction and random chemical potential using renormalization group analysis. We have addressed the stability of noninteracting fixed point and scaling properties of various physical quantities. The possibility of observing critical scaling phenomena in $\textrm{Bi}_{0.96}\textrm{Sb}_{0.04}$, $\textrm{Pb}_{1-x}\textrm{Sn}_{x}\textrm{Te}$, and $\textrm{Hg}_ {1-x}\textrm{Cd}_{x}\textrm{Te}$ is discussed.   

The effects of interactions on the topological classification of free fermion systems

We describe in detail a counterexample to the topological classification of free fermion systems. We deal with a one dimensional chain of Majorana fermions with an unusual T symmetry. The topological invariant for the free fermion classification is an integer, but with the introduction of interactions it becomes well defined only modulo 8. We illustrate this in the microscopic model of the Majorana chain by constructing an explicit path between two distinct free phases whose topological invariants are equal modulo 8, along which the system remains gapped. The path goes through a strongly interacting region. We also find the field theory interpretation of this phenomenon. There is a second order phase transition between the two phases in the free theory which can be avoided by going through the strongly interacting region. We show that this transition is in the 2D Ising universality class, where a first order phase transition line, terminating at a second order transition, can be avoided by going through the analogue of a high temperature paramagnetic phase. In fact, we construct the full phase diagram of the system as a function of the thermal operator (i.e. the mass term that tunes between the two phases in the free theory) and two quartic operators, obtaining a first order Peierls transition region, a second order transition region, and a region with no transition.   

Non-local Representation of n-point Fermi Functions

We construct a useful non-local representation of n-point Fermi functions in terms of correlation functions between local currents and densities in one and higher dimensions for a finite density one component Fermi system. We also define rigorously a non-local operator corresponding to particle-hole creation and show how this may be used to compute the momentum distribution of a Fermi liquid in three spatial dimensions leading to Galitskii's well known early result on the quasi-particle residue. The same method may be used to study the momentum distribution of a Luttinger liquid in one and two spatial dimensions, the latter involving long range interactions. The exponents are obtained in both these cases using both the non-local n-point function technique as well as the non-local particle-hole creation operator technique. Other aspects such as anomalous scaling of the polarization function as well as the behavior of the dynamical density of states are addressed in both one and two dimensional Luttinger liquids. A qualitatively favorable comparison is made between these results and those of Bares and Wen and Bartosch and Kopietz.   

Phase Diagram of the Bosonic 2D Coulomb System

The system of particles in two-dimensions which interact with the long range Coulomb potential is an important model for condensed matter. This system potentially supports a rich phase diagram including hexatic and mesoscopic phases (stripes, bubbles, etc). In this talk we will extend our recent work on the distinguishable particles to a system with Bosons. We explore how the superfluid phase intersects the hexatic and Wigner crystal region and how the introduction of statistics affects conclusions about the presence of mesoscopic phases.   

Measurement of dynamic critical exponents in strongly correlated systems

We propose a method of obtaining dynamic critical exponents by measuring the gap between the ground-state energy and the first-excited state energy. The finite-size scaling of the gap produces the dynamic critical exponent directly. For one-dimensional hardcore boson Hubbard model at half-filling, we applied our method to obtain $z=0.975 \pm 0.002$. For one-dimensional spinless fermion model, we obtained $z=0.95 \pm 0.02$. The ground-state energy is obtained by modified Lanczos method and the excited spectrum is obtained by Grosso's method. We expect our method can be very useful in obtaining dynamic critical exponent without using two-parameter finite-size scaling of order parameters such as superfluid density. Application to other models is also presented.   

Nonanalytic behavior of the spin susceptibility in interacting two dimensional electron gases in the presence of spin-orbit interaction

The issue of nonanalytic corrections due to electron-electron interactions to the electron spin susceptibility is investigated in the presence of spin-orbit interaction. We consider a two-dimensional interacting electron gas at finite temperature for two orientations of an external magnetic field: perpendicular and parallel to the plane of the gas (notably, for the perpendicular magnetic field we neglect orbital effects). At second order in the electron-electron interaction we predict strong anisotropy in the spin susceptibility: in the limit of strong spin orbit-coupling, $|\alpha|\gg T$, we show that the leading term in $\alpha$ is nonanalytic and scales linearly with $|\alpha|$ with a prefactor depending on the direction of the magnetic field; furthermore, the correction to this behavior is linear in $T$ for the in-plane magnetic field but cubic in temperature, i.e., it scales as $T^{3}/\alpha^{2}$, for the out-the-plane magnetic field; in the opposite limit of a small spin-orbit coupling, $|\alpha|\ll T$, the leading linear-in-T term does not depend on the direction of the field, however, it receives a~correction proportional to $\alpha^{2}/T$ with a prefactor depending again on the field orientation. An extension to higher order in the Coulomb interaction shows that the $|\alpha|$-nonanaliticity is~renormalized by terms logarithmic in temperature and for sufficiently low temperatures the $|\alpha|/\ln^2T$ behavior is found.   

Quantum frustration and dissipation in the XY-anisotropic spin-boson model

Using a generalization of Wilson's numerical renormalization-group method, we study quantum frustration and dissipation in the XY-anisotropic spin-boson model in which a spin-1/2 magnetic impurity can tunnel between its two spin-z eigenstates and is also coupled via its x and y spin components to two bosonic baths. For the Ohmic case of baths described by a spectral exponent $s = 1$, we confirm the result, found previously [1] by mapping onto a pure-fermionic problem, that even very strong bath coupling is insufficient to localize the impurity. By contrast, in sub-Ohmic cases $0 < s < 1$, we find that the system exhibits a continuous quantum phase transition between a delocalized phase in which the impurity dynamics are dominated by the tunneling and a localized phase in which the dynamics are controlled by the coupling to the dissipative baths. Critical properties in the vicinity of the quantum phase transition will be presented. \\[4pt] [1] A. H. Castro-Neto et al., Phys. Rev. Lett. 91, 096401 (2003).