Landau's phenomenological theory of Fermi liquids is a fundamental paradigm in many-body physics that has been remarkably successful in explaining the properties of a wide range of interacting fermion systems, such as liquid helium-3, nuclear matter, and electrons in metals. The d-dimensional boundaries of (d+1)-dimensional topological phases of matter such as quantum Hall systems and topological insulators provide new types of many-fermion systems that are topologically distinct from conventional d-dimensional many-fermion systems. We construct a phenomenological Landau theory for the two-dimensional helical Fermi liquid found on the surface of a three-dimensional time-reversal invariant topological insulator. In the presence of rotation symmetry, interactions between quasiparticles are described by ten independent Landau parameters per angular momentum channel, by contrast with the two (symmetric and antisymmetric) Landau parameters for a conventional spin-degenerate Fermi liquid. We then project quasiparticle states onto the Fermi surface and obtain an effectively spinless, projected Landau theory with a single projected Landau parameter per angular momentum channel that captures the spin-momentum locking or nontrivial Berry phase of the Fermi surface. As a result of this nontrivial Berry phase, projection to the Fermi surface can increase or lower the angular momentum of the quasiparticle interactions. We derive equilibrium properties, criteria for Fermi surface instabilities, and collective mode dispersions in terms of the projected Landau parameters. We briefly discuss experimental means of measuring projected Landau parameters.
The metallic surface states of a topological insulator support helical Dirac fermions protected by topology with their spin locked perpendicular to their momentum. They can acquire mass through magnetic doping or through hybridization of states on opposite faces of a thin sample. In this case there can be a component of electron spin oriented perpendicular to the surface plane. The electron-phonon interaction renormalizes the dynamics of the charge carriers through their spectral density. It also modifies the gap channel and a second spectral function enters which, not only determines the out …
We construct continuum models of 3D and 4D topological insulators by coupling spin-1/2 fermions to an SU(2) background gauge field, which is equivalent to a spatially dependent spin-orbit coupling. Higher dimensional generalizations of flat Landau levels are obtained in the Landau-like gauge. The 2D helical Dirac modes with opposite helicities and 3D Weyl modes with opposite chiralities are spatially separated along the third and fourth dimensions, respectively. Stable 2D helical Fermi surfaces and 3D chiral Fermi surfaces appear on open boundaries, respectively. The charge pumping in 4D Landa…
Majorana fermions were proposed to occur at edges and interfaces of gapped one-dimensional systems where phases with different topological character meet due to an interplay of spin-orbit coupling, proximity-induced superconductivity and external magnetic fields. Here we investigate the effect of strong particle interactions, and show that the helical liquid offers a mechanism that protects the very existence of Majorana edge states: whereas moderate interactions close the proximity gap which supports the edge states, in helical liquids the gap re-opens due to two-particle processes. However, …
Following our previous work, we study the quantum phase transitions which spontaneously develop ferromagnetic spin order in helical fermi liquids which breaks continuous spin-space rotation symmetry, with application to the edge states of 3d topological band insulators. With finite fermi surface, the critical point has both z = 3 over-damped and z = 2 propagating quantum critical modes, and the z = 3 mode will lead to non-fermi liquid behavior on the entire fermi surface. In the ordered phase, the Goldstone mode is over-damped unless it propagates along special directions, and quasiparticle is…
We formulate the lattice version of the three-dimensional SU(2) Landau level problem with time reversal invariance. By taking a Landau-type gauge, the system is reduced into the one-dimensional SU(2) Harper equation characterized by a periodic spin-dependent gauge potential. The surface spectra indicate the spatial separation of helical states with opposite eigenvalues of the lattice helicty operator. The band topology is investigated from both the analysis of the boundary helical Fermi surfaces and the calculation of the Z2-index based on the bulk wavefunctions. The transition between a 3D we…
We construct explicit lowest-Landau-level wave functions for the composite-fermion Fermi sea and its low energy excitations following a recently developed approach [Pu, Wu and Jain, Phys. Rev. B 96, 195302 (2018)] and demonstrate them to be very accurate representations of the Coulomb eigenstates. We further ask how the Berry phase associated with a closed loop around the Fermi circle, predicted to be $π$ in a Dirac composite fermion theory satisfying particle-hole symmetry [D. T. Son, Phys. Rev. X 5, 031027 (2015)], is affected by Landau level mixing. For this purpose, we consider a simple mo…
We investigate the quasiparticle excitations in the FFLO- type helical state of a superconductor with inversion-symmetry breaking and strong Rashba spin-orbit coupling. We restrict to a state with single finite momentum of Cooper pairs in the helical phase that is determined by minimization of the condensation energy. We derive the dependence of quasiparticle dispersions on the Rashba coupling strength and external field. It leads to a peculiar momentum-space segmentation of the corresponding Rashba Fermi surface sheets which has not yet been observed experimentally. We show that it may be dir…
We present a theory of spontaneous Fermi surface deformations for half-filled Landau levels (filling factors of the form $ν=2n+1/2$). We assume the half-filled level to be in a compressible, Fermi liquid state with a circular Fermi surface. The Landau level projection is incorporated via a modified effective electron-electron interaction and the resulting band structure is described within the Hartree-Fock approximation. We regulate the infrared divergences in the theory and probe the intrinsic tendency of the Fermi surface to deform through Pomeranchuk instabilities. We find that the correspo…
We discuss a platform for the synthetic realization of key physical properties of helical Tomonaga Luttinger liquids (HTLLs) with ultracold fermionic atoms in one-dimensional optical lattices. The HTLL is a strongly correlated metallic state where spin polarization and propagation direction of the itinerant particles are locked to each other. We propose an unconventional one-dimensional Fermi-Hubbard model which, at quarter filling, resembles the HTLL in the long wavelength limit, as we demonstrate with a combination of analytical (bosonization) and numerical (density matrix renormalization gr…
Topological fermions as excitations from multi-degenerate Fermi points have been attracting increasing interests in condensed matter physics. They are characterized by topological charges, and magnetic fields are usually applied in experiments for their detection. Here we present an index theorem that reveals the intrinsic connection between the topological charge of a Fermi point and the in-gap modes in the Landau band structure. The proof is based on mapping fermions under magnetic fields to a topological insulator whose topological number is exactly the topological charge of the Fermi point…
We demonstrate the possibility to induce and control peculiar spin textures in a helical Luttinger liquid, by means of a time-dependent magnetic scatterer. The presence of a perturbation that breaks the time-reversal symmetry opens a gap in the spectrum, inducing single-particle backscattering and a peculiar spin response. We show that in the weak backscattering regime asymmetric spin textures emerge at the left and right side of the scatterer, whose spatial oscillations are controlled by the ratio between the magnetization frequency and the Fermi energy and by the electron interaction. This p…
The physics of the state at even denominator fractional fillings of Landau levels depends on the Coulomb pseudopotentials, and produces, in different GaAs Landau levels, a composite fermion Fermi sea, a stripe phase, or, possibly, a paired composite fermion state. We consider here even denominator fractions in graphene, which has different pseudopotentials as well as a possible four fold degeneracy of each Landau level. We test various composite fermion Fermi sea wave functions (fully polarized, SU(2) singlet, SU(4) singlet) as well as the paired composite fermion states in the n=0 and $n=1$ L…
We consider the two lowest Landau levels at half filling. In the higher Landau level (nu =5/2), we find a first order phase transition separating a compressible striped phase from a paired quantum Hall state, which is identified as the Moore-Read state. The critical point is very near the Coulomb potential and the transition can be driven by increasing the width of the electron layer. We find a much weaker transition (either second order or a crossover) from pairing to the composite fermion Fermi liquid behavior. A very similar picture is obtained for the lowest Landau level but the transition…
Cold interacting fermions in two dimensions form exactly solvable Luttinger liquids, whose characteristic scaling exponents differ from those of conventional Fermi liquids. We use the AdS/CFT correspondence to discuss an equivalence between a class of helical, strongly coupled Luttinger liquids and fermions propagating in the background of a 3D black hole. The microscopic Lagrangian is explicitly known and the construction is fully embeddable in string theory. The retarded Green function at low temperature and energy arises from the geometry very near the black hole horizon. This structure is …
We study the 3D topological insulators in the continuum by coupling spin-1/2 fermions to the Aharonov-Casher SU(2) gauge field. They exhibit flat Landau levels in which orbital angular momentum and spin are coupled with a fixed helicity. The 3D lowest Landau level wavefunctions exhibit the quaternionic analyticity as a generalization of the complex analyticity of the 2D case. Each Landau level contributes one branch of gapless helical Dirac modes to the surface spectra, whose topological properties belong to the Z2-class. The flat Landau levels can be generalized to an arbitrary dimension. Int…
We re-examine the nature of the ground states of bilayer graphene at odd integer filling factors within a simplified model of nearly degenerate $n=0$ and $n=1$ Landau levels. Previous Hartree-Fock studies have found that ferroelectric states with orbital coherence can be stabilized by tuning the orbital splitting between these levels. These studies indicated that, in addition to a uniform ferroelectric state, a helical ferroelectric phase with spontaneously broken translational symmetry is possible. By performing exact diagonalization on the torus, we argue that the system does not have a unif…
The electrical resistivity of several relatively clean metallic ferromagnets, as well as the helimagnet MnSi, is commonly observed to exhibit non-Fermi-liquid behavior at low temperatures. This behavior, which is found in both ordered and disordered phases, and both near and away from the magnetic transition, remains a major unsolved problem. We derive and discuss three novel mechanisms underlying such behavior that are based on electron scattering mediated by the exchange of (1) ferromagnons or (2) skyrmionic fluctuations, both in conjunction with weak disorder, or (3) helimagnons in clean sy…
We consider a helical system of fermions with a generic spin (or pseudospin) orbit coupling. Using the equation of motion approach for the single-particle distribution functions, and a mean-field decoupling of the higher order distribution functions, we find a closed form for the charge and spin density fluctuations in terms of the charge and spin density linear response functions. Approximating the nonlocal exchange term with a Hubbard-like local-field factor, we obtain coupled spin and charge density response matrix beyond the random phase approximation, whose poles give the dispersion of fo…
The Ginzburg-Landau Hamiltonian for incommensurate frustrated classical spin systems is analyzed. The coupling to phonons through the Dzyaloshinskii-Moriya interaction and/or the four-spin exchange interaction of the Coulomb origin under the egde-sharing network of magnetic and ligand ions drive chiral spin piarings, introducing two successive second-order phase transitions upon cooling. First, a vector spin-chiral order appears with an either parity, leaving an O(2) chiral spin liquid. Then, the O(2) symmetry is broken by the spin ordering into a helical magnetic state. Possible candidate mat…
We generalize the Landau levels of two-dimensional Dirac fermions to three dimensions and above with the full rotational symmetry. Similarly to the two-dimensional case, there exists a branch of zero energy Landau levels of fractional fermion modes for the massless Dirac fermions. The spectra of other Landau levels distribute symmetrically with respect to the zero energy scaling with the square root of the Landau-level indices. This mechanism is a nonminimal coupling of Dirac fermions to the background fields. This high dimensional relativistic Landau-level problem is a square-root problem of …
We study the Ginsburg-Landau expansion for the non-Fermi model proposed by Anderson. We analyze the deviations of the main properties of a non-Fermi superconductor from the isotropic s-wave bidimensiona superconductor.
We show that a strongly correlated Fermi system with the fermion condensate, which exhibits strong deviations from Landau Fermi liquid behavior, is driven into the Landau Fermi liquid by applying a small magnetic field $B$ at temperature T=0. This field-induced Landau Fermi liquid behavior provides the constancy of the Kadowaki-Woods ratio. A reentrance into the strongly correlated regime is observed if the magnetic field $B$ decreases to zero, then the effective mass $M^*$ diverges as $M^*\propto 1/\sqrt{B}$. At finite temperatures, the strongly correlated regime is restored at some temperatu…
The properties of the strongly interacting edge states of two dimensional topological insulators in the presence of two particle backscattering are investigated. We find an anomalous behavior of the density-density correlation functions, which show oscillations that are neither of Friedel nor of Wigner type: they instead represent a Wigner crystal of fermions of fractional charge e/2, with e the electron charge. By studying the Fermi operator, we show that the state characterized by such fractional oscillations still bears the signatures of spin momentum locking. Finally, we compare the spin-s…
Motivated by recent experiments in the Kitaev honeycomb lattice, Kondo insulators, and the "Luttinger's theorem-violating" Fermi liquid phase of the underdoped cuprates, we extend the theoretical machinery of Landau-Fermi liquid theory to a system of itinerant, interacting Majorana-like particles. Building upon a previously introduced model of "nearly self-conjugate" fermionic polarons, a Landau-Majorana kinetic equation is introduced to describe the collective modes and Fermi surface instabilities in a fluid of particles whose fermionic degrees of freedom obey the Majorana reality condition. …
We calculate the Landau interaction function f(k,k') for the two-dimensional t-t' Hubbard model on the square lattice using second and higher order perturbation theory. Within the Landau-Fermi liquid framework we discuss the behavior of spin and charge susceptibilities as function of the onsite interaction and band filling. In particular we analyze the role of elastic umklapp processes as driving force for the anisotropic reduction of the compressibility on parts of the Fermi surface.
Beyond spin-charge separation: Helical modes and topological quantum phase transitions in one-dimensional Fermi gases with spin-orbit and Rabi couplings
One of the most successful theories of a non-Fermi liquid metallic state is the composite Fermi liquid (CFL) theory of the half-filled Landau level. In this paper, we study continuous quantum phase transitions out of the CFL state and into a Landau Fermi liquid, in the limit of no disorder and fixed particle number. This transition can be induced by tuning the bandwidth of the Landau level relative to the interaction energy, for instance through an externally applied periodic potential. We find a transition to the Landau Fermi liquid through a gapless Mott insulator with a Fermi surface of neu…
Landau-Fermi liquid theory is conventionally believed to hold whenever the interacting single-particle density of states develops a $δ$-like component at the Fermi surface, which is associated with quasiparticles. Here we show that a microscopic justification can be actually achieved under more general circumstances, even in case coherent quasiparticles are totally missing and the interacting single-particle density of states vanishes at the chemical potential as consequence of a pole singularity in the self-energy.
We study the intrinsic effects of dimensional reduction on the transport equation of a perfectly two-dimensional Landau-Fermi liquid. By employing the orthogonality condition on the 2D analog of the Fourier-Legendre expansion, we find that the equilibrium and non-equilibrium properties of the fermionic system differ from its three-dimensional counterpart, with the latter changing drastically. Specifically, the modified Landau-Silin kinetic equation is heavily dependent on the solution of a non-trivial contour integral specific to the 2D liquid. We find the solution to this integral and its gen…
We study Fermi liquids with a Fermi surface that lacks continuous rotational invariance, and in the presence of an arbitrary quartic interaction. We obtain the expressions of the generalized static susceptibilities that measure the linear response of a generic order parameter to a perturbation of the Hamiltonian. We apply our formulae to the spin and charge susceptibilities. Based on the resulting expressions, we make a proposal for the definition of the Landau parameters in non-isotropic Fermi liquid.
In this paper we study the Ginzburg-Landau (GL) equation for Fermi liquid superconductors with strong Landau interactions $F_0$ and $F_1$. We show that Landau interactions renormalize two parameters entering the GL equation leading to renormalization of superfluid density and separation of length and energy scales for amplitude and phase fluctuations. The separation of energy/length scales for amplitude and phase fluctuations in term leads to unconventional (2D) KT transition and vortex liquid phase. Application of the GL equation to describe underdoped high-$T_c$ cuprates is being discussed.
One challenge in contemporary condensed matter physics is to understand unconventional electronic physics beyond the paradigm of Landau Fermi-liquid theory. Here, we present a perspective that posits that most such examples of unconventional electronic physics stem from restrictions on the degrees of freedom of quantum electrons in Landau Fermi liquids. Since the degrees of freedom are deeply connected to the system's symmetries and topology, these restrictions can thus be realized by external constraints or by interaction-driven processes via the following mechanisms: (i) symmetry breaking, (…
A lattice model of spinless interacting electrons is used to formulate the Landau theory of the Fermi liquid to electron glass quantum phase transition. We demonstrate that the presence of additional random site energies does not affect the character of the transition, once the replica symmetry breaking is considered self-consistently at the mean-field level. Inside the glass phase, the low temperature conductivity assumes a non-Fermi liquid ~ T^{3/2} form, in agreement with recent experiments.
We study the Fermi liquid properties of the cold atomic dipolar Fermi gases with the explicit dipolar anisotropy using perturbative approaches. Due to the explicit dipolar anisotropy, Fermi surfaces exhibit distortions of the $d_{r^2-3z^2}$-type in three dimensions and of the $d_{x^2-y^2}$-type in two dimensions. The fermion self-energy, effective mass, and Fermi velocity develop the same anisotropy at the Hartree-Fock level proportional to the interaction strength. The Landau interaction parameters in the isotropic Fermi liquids become the tri-diagonal Landau interaction matrices in the dipol…
We study the field theory dual to a charged gravitational background in which the low temperature entropy scales linearly with the temperature. We exhibit the existence of a sound mode which is described by hydrodynamics, even at energies much larger than the temperature, and explain how this, and other properties of the field theory, are consistent with those of a (3+1)-dimensional Landau Fermi liquid, finely tuned to the Pomeranchuk critical point. We also discuss how one could engineer a higher-derivative gravitational Lagrangian which reproduces the correct low temperature behavior of shea…
We consider the behavior of strongly correlated electron liquid in high-temperature superconductors within the framework of the fermion condensation model. We show that at low temperatures the normal state recovered by the application of a magnetic field larger than the critical field can be viewed as the Landau Fermi liquid induced by the magnetic field. In this state, the Wiedemann-Franz law and the Korringa law are held and the elementary excitations are the Landau Fermi Liquid quasiparticles. Contrary to what might be expected from the Landau theory, the effective mass of quasiparticles de…
Majorana fermions were originally proposed as elementary particles acting as their own antiparticles. In recent years, it has become clear that Majorana fermions can instead be realized in condensed-matter systems as emergent quasiparticles, a situation often accompanied by topological order. Here we propose a physical system which realizes Landau levels - highly degenerate single-particle states usually resulting from an orbital magnetic field acting on charged particles - for Majorana fermions. This is achieved in a variant of a quantum spin system due to Kitaev which is distorted by triaxia…
