Coulomb interactions make electrodynamic tansport of Dirac semi – metals nonlocal Baruch Rosenstein Collaborators: Theory: Hsien-Chong Kao (Nat. Taiwan Normal University) Meir Lewkowicz (Ariel University, Israel) Experiment: Wen – Bin Jian, Jian-Jhong Lai (Nat. Chiao Tung University) Tours, June, 13, 2016

Outline 1. Free relativistic massless fermions in graphene: finite conductivity without either carriers or impurities. 2. Problems with calculation of the electron – electron interaction effects and how they were resolved. 3. Nonlocal electrodynamics in graphene. 4. Experiment. 5. 3D Weyl - semi metals.

Carbon atoms in graphene are arranged in honeycomb 2D crystal by the covalent bonds between nearest neighbours. E. Andrei et al, Nature Nano 3, 491(08 )

Suspended samples are clearly undoped and exhibit the minimal resistivity at zero temperature. Andrei et al, Nature Nano 3, 491(08 ) On optical frequencies conductivity is the same Nair et al, Science (08 ) 1. Finite conductivity and no screening in non- interacting graphene

Tight binding model Spectrum consists of two bands with Fermi surface pinched right between them: not an obvious band insulator or a metal.

Around the Hamiltonian is that of the left and right two component Weyl spinors: Fermi surface consists of two non - equivalent points of the Brillouin zone with sufficient little group to support a two dimensional representation

The Dirac point in band structure was pointed out very early by Wallace: electrons in graphene should behave like a 2D analogue of relativistic massless particles. The L,R Weyl spinors can be combined into two 4- Dirac spinor described by that possesses the chiral (sublattice) symmetry

For the spectrum DOS of relativistic massless fermions At Dirac point the charge of carriers density flips sign: no “free” carriers at the neutrality (Dirac) point. the number of electrons is Wang et al, APL 101, (12)

Explanation: Schwinger’s electron – hole pair creation by electric field The basic picture of the pseudo – diffusive resistivity in pure graphene is the creation of the electron – hole pairs by electric field. The pairs carry current that can be further increased by reorientation of moving particles Linear response for current

So that the current is proportional to electric field, number of available states The pair can be created with back to back momenta Heuristic interpretation of the conductivity and to the life time of the virtual pair Therefore

Dynamical approach to linear response demonstrating the absence of scale separation One indeed observes that the current stabilizes at finite value. In DC this requires the use of tight binding model rather than its Dirac continuum theory. Lewkowicz, B.R., PRL102, (09) Let’s try to understand qualitatively how massless fermions react on electric field by just switching a homogeneous electric field and observing the creation of electron - hole pairs and the induced charge motion.

In addition to the finite term converging to, there is also a linear acceleration term that vanishes due to cancellation of part near the Dirac points and far away. It can be presented as a full derivative Therefore there is no perfect scale separation when massless fermions are involved. This is also the source of the chiral anomaly and doubling. Kao, Lewkowicz, B.R., PRB81, (2010)

Stays there till a crossover time to a linearly increasing Schwinger’s regime B.R, Lewkowicz, PRL 77, (09) ; B.R., Kao, Lewkowicz, PRB 81, R (10) Accidental nature of the “pseudo-Ohmic” linear response

At each one solves, using the WKB approximation, a tunneling problem similar to that in the Landau – Zener transition. Schwinger, PR (1962 ) The interband transition probability at large times is Gavrilov, Gitman, PRD53, 7162 (1986) Schwinger, PR82, 664 (1951) Hence

The vacuum polarization Is too weak to exponentially screen the 3D Coulomb with 2D Fourier transform The relation between the polarization and conductivity obeys the usual charge conservation relation that is sometimes naively written as. Coulomb repulsion potential remains long range

2. Electron – electron interactions effect on conductivity and screening They break the (pseudo) relativistic invariance and are still 3D, namely less long range in 2D than that of the 2+1 dimensional massless QED.

Even if screened this would lead to a strongly coupled electronic system that for example might exhibit the chiral (exciton) condensate Coulomb interactions naively are nonperturbatively strong due to coupling constant of order 1 Just rescale the field in How strong are the interactions?

This would make quasiparticle massive and the phase insulating. Drut, Lande, PRB 79, (09) Ulyubishev et al, PRL 111, ). … Lattice simulation indeed demonstrated formation of the condensate above This is very similar to the scalar relativistic interaction that exhibits chiral phase transition of non – Wislon universality class

The first definitive measurement of the coupling (on the BN substrate) gave the value. The actual expansion parameter is, small enough to use perturbation theory. Elias et al, Science 337, 1196 (12) Siegel et al, PRL. 110, (13) However in the previous discussion, quite complete understanding of transport was achieved ignoring the interactions. The interaction effects have been observed, but are small.

First attempt in which sharp momentum cutoff was used gave a disappointingly large correction indicating that interactions are not weak Conflicting perturbation theory results in continuum limit (Dirac approximation) Herbut, Juricic, Vafek, PRL100, (08 ) The leading correction to conductivity would be of a value, and can be directly calculated via Kubo formula by calculating diagrams

However subsequent calculation of the polarization provided a different value. The charge conservation leads to a general relation between the charge and current transport functions: Mishchenko, PRL98, )07); EPL100, (08) For a regular (means local) isotropic time reversal invariant (no magnetic field) system, one can expand

This value was considered to be favorable phenomenologically. However in order to force the two quantities, the polarization and the “direct” conductivities to be the same, the interaction was modified with the long range part modified to depend on UV cutoff. It was claimed however that this way the Ward identities of the charge conservation were violated by the cutoff procedure Juricic,Vafek, Herbut, PRB82, (10) The polarization was recalculated with Ward identities obeyed by using a variant of dimensional regularization with yet different answer The new calculations indicated problems with the dimensional regularization and reaffirmed the polarization value MacDonald et al, PRB84, (11); Sodemann, Fogler, PRB86, (12)

Note that unlike in relativistic QFT, in graphene (or numerous other “Dirac” systems in condensed matter the natural regularization exists: a microscopic theory. To resolve the ambiguity we have performed the calculation within the tight binding model (lattice 3D with Coulomb interactions between orbitals). B.R., Maniv, Lewkowicz, PRL110, (13); B.R., H.C. Kao, Lewkowicz, PRB 90, (2014) Calculation on the lattice The result for the conductivity is, while the polarization plus conservation of charge gives How to reconcile this?

The way out is to reconsider the basic assumptions involved in the derivation of the customary relations Undoped graphene is not a “regular” system due to lack of separation of scales and an additional nonanalytic term appears

For a very clean metal the situation might in principle become complex since on the free electron level there is no obvious scale. The strongly screened interactions might be sometimes neglected (or “renormalized away” in the Fermi liquid theory), but the electric response generally becomes nonlocal. If a conducting system has a space scale like the mean free path in metals with significant disorder, one expects smooth limit leading for an isotropic time reversal invariant (no magnetic field) system to 3. Local vs. nonlocal electrodynamics When and why the Ohm’s low (locality) is obeyed

Knows it all… However it is possible to show that in any free lattice model conductivity is local since the difference can be again written B.R., H.C. Kao, in preparation (2016) as a full derivative and thus vanishes

Short range interactions metals or semi-metals do not change the situation, the screening length providing the necessary length scale. However in Weyl semi – metals (at neutrality point), despite the pseudo-Ohmic conductivity, Coulomb interaction remains long range. This creates a unique situation of the interaction correction creating the nonlocal electric response characterized in the rotation, time reversal invariant case by two scalar constants Long range interactions are required for a conductor to ne nonlocal

with an exact relation to polarization being For actual computation the trace of the conductivity tensor generally is simpler The naïve locality assumption is thus violated

From the perturbative values one obtains and,

Extention to other Dirac semi-metals and universality How this additional term affects physics? For a Weyl semi-metal the contributions to nonlocality due to interactions is sum of all the nodes: graphene – like, double – graphene like… The integrals converge well in continuum after the appropriate surface term is identified. and, No known “topological explanation for these numbers For the double layer graphene node one obtains:

Generally a vector field can be decomposed using two potentials One deduces the voltage between two points via sources Potentials for the nonlocal electrodynamics

Anomalous dependence on geometry of electrical resistance in a rectangular flake The correction to the electrical resistance becomes dependent on geometry of the source, drain and points between which the voltage is measured V 21 Drain Source r2r2 r1r1 Without nonlocal electrodynamics due to interactionsinteraction 4. Experimental observation

The correction to the electrical resistance becomes site dependent X =L x y y=W/2 y=- W/2 X =0 B D C A

An anomalous Hall effect X =L x y y=W/2 y=- W/2 X =0 B D C A

VgVg I DS VdVd graphene on SiO2 substrate Constant Current Source & Voltage Meter V TS V BS 300 K Mobility: 4000 ~10000 cm 2 /V-s The asymmetric flake set up

SD B T SD T B Current (  A) Volage (mV) Voltage (mV) Total (mV) (V DS =9.04) 10V TS =4.87V DT = V BS =4.53V DB =4.479

Nonlocality voltage near the Dirac Point T=300K T=80K

Temperature Dependence temperature destroys the nonlocality lowT

Disorder also destroys the nonlocality wide carrier range narrow carrier range weak nonlocality strong nonlocality

Anomalous reflection and transmission The additional term changes in a different way the two polarizations of the electromagnetic wave passing (bouncing off) a graphene layer The and the polarization components have the following reflection coefficients

Topological transition from TI to nontopological insulator was shown to result in 3D Dirac semi – metal: 5. 3D Dirac semi-metals

Na 3 Bi Liu ZK et al, Science (2014)

Dirac or Weyl points in 3D appeared occasionally in band structure calculations away from Fermi surface, but not always. Properties of a (poor) metal with strong spin – orbit, Bi, led to its (approximately) Dirac Hamiltonian with pseudospin replaced by spin with all three Pauli matrices now presentpoint theory

B.R., Lewkowicz, PRB 88, (13) For free quasi-particles in addition to the real part In many cases of interest the imaginary part dominates over small dissipation.

Electrodynamics of such a material becomes rather peculiar

More importantly the difference between the longitudinal and the transverse conductivities is. The correction as in graphene renormalizes coefficients of both the real and the imaginary parts B.R., Lewkowicz, PRB 88, (13) The interaction correction

Fluxon induces charges proportional to The charging phenomenon

LT Incident light splits into the transversal and the longitudinal ones, both very weakly attenuated. Moreover at certain angle and frequency the monochromatic wave appears that is both transverse and longitudinal.

For certain angle the reflection disappears: total absorption phenomenon

Summary 1. Dirac semi-metal like graphene at Dirac point is a neutral plasma that has finite conductivity, but does not makes the interactions between quasiparticles short range. 2. There is no complete scale separation of the energy scales also in the interacting case and some quantities like the conductivity tensor are nonanalytic. Low energy continuum Weyl model requires regularization that is sufficiently precise to avoid effects of chiral symmetry violation by the cutoff. 3. Transport and electromagnetic properties of WSM are unusual including interaction effects leading to nonlocal electrodynamics resulting in several optical and electric peculiarities.

A radial and a rotational current in a graphene layer with Corbino geometry exhibits different electrical resistance: B(t) S D