Introduction to Insulators

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Introduction to Insulators Johar M. Ashfaque There are two basic mechanisms for obtaining a metal-insulator phase transition. One is the mechanism described by Mott, leading to what is known as the Mott insulator which is an extension of the analysis of the fermionic Hubbard model. The other is the idea of Anderson localization due to the scattering from a random potential generated by impurities. Finally, we will look at topological insulators with properties dictated by topology. A.

Band Insulator

Perhaps the simplest example of an insulator is the band insulator. We can have partially filled conduction band allowing for metallic transport (electrons are free to move onto unoccupied levels and hop between atoms) or we can have fully filled conduction band, separated by a gap from the next band not allowing for electron transport thereby obtaining an insulator. In this case, however, the system is always an insulator and does not present a phase transition. B.

Electric Transport: The Drude Model

To characterize insulators, we must first understand electric transport. This is in general defined by a matrix relation between the current and the applied electric field valid at each momentum and frequency jα (ω, ~k) = σαβ (ω, ~k)Eβ (ω, ~k). For a metal, the usual transport theory (Drude model) gives σαβ (ω) =

σDC δαβ 1 − iωτ

where σDC =

ne2 τ . m∗

That then gives for the real part of the zero temperature, DC(ω → 0) conductivity Re[σαβ (T = 0, ω → 0] = (Dc )αβ

τ π(1 + ω 2 τ 2 )

where from the above Dc is given as a function of the effective mass m∗ of the electron by (Dc )αβ =

πne2 δαβ m∗

and as the relaxation time τ becomes large (for a highly conductive metal), we obtain the Drude peak at zero frequency Re[σαβ (T = 0, ω → 0, τ −1 → 0] = (Dc )αβ δ(ω). This Drude peak is a result of the translational invariance in the extreme IR (for ω = 0) leading to an infinite conductivity and is a characteristic of metals. In fact, we could define metals by its presence. We see that the key to the finiteness of σ is the momentum dissipation. On the contrary, an insulator would be an object with negligible zero temperature DC conductivity lim lim lim [σαβ (T, ω, ~q)] = 0.

T →0 ω→0 q ~→0

To obtain the insulator, we must break the translational invariance of the system even in the extreme IR that is to say obtain momentum dissipation at all scales.

2 I.

FERMIONIC HUBBARD MODEL AND METAL-INSULATOR TRANSITION

The fermionic Hubbard model describes the features of the transition from metal to insulator. In this case, the metal will be the weakly coupled state, which is disordered, whereas the insulator will be the strongly coupled state which will be ordered. One considers the tight-binding approximation, when the electrons belong to each atom (site on the lattice) but they hop between sites. The model has the Hamiltonian given by X X X †ni↑ ni↓ − Âľ ni ciĎƒ cjĎƒ + U H = −t i

hiji,Ďƒ

i

where niĎƒ = c†iĎƒ ciĎƒ and ni = ni↑ + ni↓ . Here hiji signifies that the sum is taken only over the nearest neighbours; the first term is then the hopping term for an electron to hop from site j to site i, U is an on-site interaction to have two electrons (of opposite spins, thus obeying the Pauli exclusion principle) at the same site and Âľ is the chemical potential. A.

Weak Coupling

Consider the U → 0 limit (U t). Go to the momentum basis (reciprocal space) 1 X i~k¡~l c~k,Ďƒ = √ e c~lĎƒ . N l Then the Hamiltonian becomes diagonal H≥

X

~k c~†c~kĎƒ kĎƒ

~ k

where the eigenenergies are ~kĎƒ = −2t(cos kx + cos ky ) − Âľ and we have added the chemical potential for completeness. Note that in one dimension we have 2Ď€n N and in general the momenta are discrete. So we obtain an electron energy band of width 4t and the electrons are approximately free. One can go further and analyze the Hamiltonian precisely to find that the true ground state has antiferromagnetic order. k=

B.

Strong Coupling

Now consider the limit U t. Since we are at small t/U , we can do perturbation theory in it. Consider an effective Heisenberg Hamiltonian for the model X ~i ¡ S ~j . Hef f = J S hiji

If we have two electrons in a singlet state (↑↓) and an unoccupied neighbour, virtual hopping (tunneling) followed by return to the original state will give an energy of −2t2 /U where the 2 comes from the two possible configurations, −t from the energy of the hopping and t/U from the probability. This energy is to be compared with JS1− S2+ 2 whuch fixes J=

4t2 . U

3 II.

DEFECTS AND ANDERSON LOCALIZATION

The insight of Anderson was to see that disorder, in the form of a random potential, induces an Anderson transition from delocalization to localization, in any spatial dimension higher than 2. In d = 1 and d = 2 spatial dimension, any amount of disorder localizes all the electronic states. Since the Anderson transition occurs in the strong-disorder limit, perturbation theory is of little use. For the random potential, the model Anderson took is a tight-binding model with electrons at site i in electronic orbital states with a probability for transition (hopping) given by V H=

X

n a†n an + V

n

X

a†n am .

hnmi

This Hamiltonian corresponds to conduction in an impurity band where the site energy n is attributed randomly. Randomness in the value of V does not change the problem qualitatively. One can extend the site labels j to site and orbital levels (jÂľ). The random site energies are distributed with some continuous probability which in the original Anderson model was taken to be a simple Heaviside function i.e. a uniform distribution of width W 1 W P ( j ) = θ − | j | . W 2 Note that if W = 0, all states have the same energy, and transport is ballistic in this regime. But when V = 0, the sites are not connected, so transport stops. Therefore the localization/delocalization transition should depend on the ratio W/V . In d > 2, as expected there is a critical ratio (W/V )c at which we have a change between localization and delocalization. But in d = 1 and d = 2, a single impurity implies localization. Note. There is no rigorous proof of these statements only various arguments and numerical simulations to back these statements.

A.

Transport

Transport is determined by the transition probability of an electron of energy E from position x to position x0 P (x, x0 ; E) = |cxx0 (E)|2 where cxx0 is the probability amplitude.

B.

Scaling Arguments

Consider the conductance (inverse resistance) G = 1/R, a function of the linear dimension L of the sample, and rescaled conductance g(L) =

2~ G(L). e2

In d spatial dimensions, Ohm’s law is written as G(L) = ĎƒLd−2 where Ďƒ is the conductivity, so that for d = 3, we have the well-known linear relation. Considering g(L) as a coupling constant, we can define a beta function for it β=

d ln g(L) . d ln L

4 It will have zeros g = gc , β(gc ) = 0, which will correspond to transition points and we can linearize it around them β(g) '

g − gc νgc

which stands as a definition of ν. This can be integrated to give L = L0

g − gc g0 − gc

ν

with g0 = g(L0 ). Defining the length scale Ξ = L0

gc g0 − gc

ν

≥ L0 | |−ν

which defines as well. We can write the scaling relation for the conductance near the transition point gc ν1 L g = gc 1 − . Ξ As can be seen from its definition, Ξ diverges as g0 → gc .

III.

MOTT INSULATORS

The Hubbard model predicts the existence of a type of insulator at strong coupling via an antiferromagnetic state, that is called the Mott insulator. The Mott insulator therefore has a strong electron coupling (the electrons are strongly correlated) and is very difficult to describe analytically. Under the name, Mott insulator, we can find objects with presence or absence of spontaneous symmetry breaking, gapped or gapless neutral particle low-energy excitations and the presence or absence of topological order or fractionally charged states. The Mott phase transition between the metal and the insulator is a quantum phase transition since it occurs at T = 0. It is generally first-order, though second-order examples do exist. In a (T, p) phase diagram, we have a phase transition line that goes all the way to T = 0, p = pc . Mott proposed a pseudo-order parameter , as the difference between the ionization energy I of an electron in the crystal and the electron affinity E of one atom in the crystal = I − E. If there are electron-electron interactions, we have by definition I > E → > 0 otherwise I = E, = 0. This will be a function of the lattice spacing d. As a function of 1/d, drops toward small values and can either • Have a finite value at the transition, then drop to zero for 1/d < 1/dc , giving a first-order phase transition or • Go to zero smoothly, obtaining a second-order phase transition.

IV.

TOPOLOGICAL INSULATORS

Topological insulators are solids for which the bulk behaves like an insulator having fully gapped states but the surface contains conducting states. In the bulk, we can have a band insulator i.e. of the usual (not Mott or Anderson) type, where strong interactions are not necessary to describe it. However, the surface states need to be protected by symmetries, especially time-reversal invariance. They are characterized by a topologically invariant index (Z2 index). Topological insulators were experimentally discovered and have potential application in quantum computation.

5

As a function of momentum, the valence and conduction bands of insulators are oppositely curved being closest at k = 0. The surface states form a double cone tangent to the bands, which therefore makes them touch each other at the tip of the cones allowing transport. The Fermi surface, which is situated between the valence and the conduction bands in the insulator, crosses the surface state bands. The 2-dimensional topological insulator also known as the quantum spin Hall insulator can be realized, though in the very weak sense, in graphene. It can also be realized in some quantum well systems like HgT e/CdT e which are made by a layer of material sandwiched between two slabs of a different material with a wider bandgap. The 3-dimensional topological insulators have surface states that are 2-dimensional massless Dirac fermions, and whose dispersion relation ω(k) forms a Dirac cone, on which a single massless Dirac fermion lives.

6 A.

2 + 1-Dimensional Topological Field Theory

For a band insulator with M occupied bands, we consider the Abelian part of the Berry connection

X ∂

Îą~k aj = −i Îą~k

j = x, y. ∂kj

Îąâˆˆoccupied

For the insulator, the longitudinal conductivity vanishes Ďƒxx = 0 but the off-diagonal Hall conductivity is given by the first Chern number of the Berry connection ĎƒH = Ďƒxy =

e2 C1 ~

where C1 is the first Chern number C1 =

1 2Ď€

Z

dkx dky fxy (~k) ∈ Z.

Therefore the Hall conductivity is quantized in units of e2 ~ i.e. we have the quantum Hall effect. The Hall response is written more completely as Ďƒ0 =

ji = ĎƒH ij Ej . Moreover, using the currrent conservation and the Maxwell equations Ď (B) = ĎƒH B + Ď 0 . Note that in 2-dimensions, we have that F0i = Ei and Fij = B ij . The equations can therefore be encoded into j Âľ as the Lorentz covariant equation e2 C1 ÂľÎ˝Ď âˆ‚ν AĎ . ~ 2Ď€ This is the response equation and considering that in general we can define the current as j Âľ = δS/δAmu , the response is encoded in the topological Chern-Simons field theory Z e2 C 1 Sef f = d2+1 x ÂľÎ˝Ď AÂľ ∂ν AĎ . ~ 4Ď€ jÂľ =

All topological responses of the quantum Hall state are encoded in this topological field theory which is the product of the first Chern number in two spatial directions of the Berry connection of the state with the Chern-Simons action for the electromagnetic field AÂľ . B.

4 + 1-Dimensional Topological Field Theory

One can do something similar and define a fictitious 4 + 1-dimensional topological insulator and then dimensionally reduce to 3 + 1-dimensions to find the properties of a real topological insulator. The Chern-Simons effective action in this case is Z C2 Sef f = d4+1 x ÂľÎ˝Ď ĎƒĎ„ AÂľ ∂ν AĎ âˆ‚Ďƒ AĎ„ . 24Ď€ 2 Here C2 is the second Chern number of the now non-Abelian Berry connection, being an integral over the four spatial dimensions Z 1 C2 = d4 x ijkl Tr[fij fkl ] 32Ď€ 2 ιβ where fij is the field strength of the non-Abelian Berry connection. The topological response of the system to the electromagnetic field is then

jÂľ =

C2 ÂľÎ˝Ď ĎƒĎ„ ∂ν AĎ âˆ‚Ďƒ AĎ„ . 8Ď€ 2

7 V.

SUMMARY

• Regular, perturbative insulators are band insulators, due to a fully occupied band. • A metal-insulator phase transition is described by the fermionic Hubbard model, with a hopping term t and an interaction U , as a function of U/t. It has antiferromagnetic states and transitions between a metallic state with a band structure at small coupling to a Mott insulator state at large coupling. • Another mechanism for phase transition between metal and insulator is the Anderson localization due to impurities. In d = 1 and d = 2 spatial dimensions, any impurity leads to Anderson localization and in d > 2 we have a phase transition, though there are no rigorous proofs of these statements. • The localization length, defined as the scale of the exponential decay of the transition amplitude, diverges at the phase transition point, approached from the insulator side. • When approaching from the metallic side of the Anderson transition, the conductivity vanishes smoothly. • A Mott insulator can be defined independently of the Hubbard model, as a zero temperature (quantum) phase transition between metal and insulator and is due to a strong interaction between electrons. It is generally first-order, although second-order examples also exist. • The Mott phase transition is difficult to obtain, though approximation methods like functional density theory can be used. • A topological insulator is an insulator in the bulk, with fully gapped states, but having surface conducting states protected by topology and symmetries, especially time-reversal invariance. They are also characterized by a Z2 topological invariant index. • The surface states are cones, extending out of the bulk bands, touching each other’s tips and thus allowing transport. • A 2 + 1-dimensional (time-reversal invariant) topological insulator has an electromagnetic ChernSimons topological field theory encoding its response. • A 3 + 1-dimensional (time-reversal invariant) topological insulator is obtained from dimensional reduction of a 4+1-dimensional insulator with a Chern-Simons effective action encoding its response. One obtains an axionic (theta term) topological field theory encoding its response.

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