geometry and topology in condensed matter physics · geometry and topology in condensed matter...

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Geometry and Topologyin Condensed Matter Physics

Raffaele Resta

Istituto Officina dei Materiali, CNR, Trieste, Italy

Società Italiana di Fisica, 1050 Congresso Nazionale

L’Aquila, September 25th, 2019

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Outline

1 Introduction: the polarization dilemma

2 What does it mean “geometrical”?

3 Bloch orbitals & geometry in k space

4 Two paradigmatic observables: P and M

5 Observables defined modulo 2π

6 Observables exempt from 2π ambiguity

7 Geometry in r space vs. k space

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Elementary textbook definition

P =dV

=1V

∫dr r ρ(micro)(r)

P appears as dominated by surface contributions

Phenomenologically P is a bulk property

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Silicon in a capacitor: induced charge density

..

..

..

Sili

con

(pse

udo)

char

gede

nsity

,unp

ertu

rbed

Silicon

(pseudo)charge

density,unperturbed

min

max

Thecreativeroleofcomputationstounderstand

thepolarizationofsolids–p.24/??

. . . . . .

How is polarization retrieved?DIELECTRIC INSIDE A CAPACITOR

silicon polarization density

planar average

min max

How a series of computationschanged our viewof the polarization of solids – p. 24/61

. . . . . .

How is polarization retrieved?DIELECTRIC INSIDE A CAPACITOR

silicon polarization density

planar average

min max

How a series of computationschanged our viewof the polarization of solids – p. 24/61planar average

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Dilemma’s solution

Bulk macroscopic polarization P has nothing to do withthe charge density in the bulk of the material(contrary to what many texbooks pretend!)

Instead, P is a geometric phase (Berry’s phase) of theground electronic wavefunction

Breakthrough due to King-Smith & Vanderbilt, 1993

An early account (in Italian):R. Resta,Che cos’è la polarizzazione dielettrica macroscopica?Il Nuovo Saggiatore 9 (5/6), 79 (1993).

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Outline








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The simplest geometrical property: Distance

Two state vectors |Ψ1⟩ and |Ψ2⟩ in the same Hilbert space

D212 = − log |⟨Ψ1|Ψ2⟩|2

D212 = 0 if the two quantum states coincide

apart for an irrelevant phase: gauge-invariant

D212 = ∞ if the two states are orthogonal

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A second geometrical property: Connection

D212 = − log |⟨Ψ1|Ψ2⟩|2 = − log⟨Ψ1|Ψ2⟩ − log⟨Ψ2|Ψ1⟩

The two terms are not gauge-invariantEach of the two terms is a complex numberWhat is the meaning of Im log ⟨Ψ1|Ψ2⟩ ?

⟨Ψ1|Ψ2⟩ = |⟨Ψ1|Ψ2⟩|eiφ12

−Im log ⟨Ψ1|Ψ2⟩ = φ12, φ21 = −φ12

The connection fixes the phase differenceThe connection is arbitraryGiven that it is arbitrary, why bother?

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Change of paradigm due to Sir Michael Berry

The Royal Society is collaborating with JSTOR to digitize, preserve, and extend access toProceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences.

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The connection by itself cannot have any physicalmeaning, but it can be used to build a gauge-invariantquantityWithin QM, any gauge-invariant quantity is in principlemeasurableToday, Berry’s geometric phase enters all modernquantum-mechanics texbooks

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Differential forms in quantum geometry

The state vector |Ψκ⟩ depends on the continuous parameter κ

Quantum metric gαβ:

d D2 = D2κ,κ+dκ = gαβdκαdκβ

Berry connection Aα:

d φ = Aαdκα

Berry curvature Ωαβ = ∂καAβ − ∂κβAα

d × dφ = Ωαβ dκαdκβ

The above list is nonexahustive:other geometrical quantities can be defined

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Quantum metric :

d D2 = D2κ,κ+dκ = gαβdκαdκβ 2-form

Berry connection :

d φ = Aαdκα 1-form

Berry curvature

d × dφ = Ωαβ dκαdκβ 2-form


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Quantum metric :

d D2 = D2κ,κ+dκ = gαβdκαdκβ

Berry connection :

d φ = Aαdκα

Berry curvature

d × dφ = Ωαβ dκαdκβ


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Distance and metric

The insulating vs. metallic character is a property of theelectronic ground state (W. Kohn, 1964)

The quantum metric sharply discriminates betweeninsulators and conductors (1999 onwards)

Comprehensive account:R. Resta,Theory of the insulating state,Riv. Nuovo Cimento 41, 463 (2018).

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Outline








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Bloch orbitals

Band theory:Noninteracting electrons in a mean-field periodic potential(Hartree-Fock of Kohn-Sham DFT)[

12m

p2 + V (r)]|ψjk⟩ = ϵjk|ψjk⟩, |ψjk⟩ = eik·r|ujk⟩

[1

2m(p + ℏk)2 + V (r)

]|ujk⟩ = ϵjk|ujk⟩

Hk = e−ik·rHeik·r =1

2m(p + ℏk)2 + V (r)

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Geometry of Bloch orbitals (single band)

D212 = − log |⟨ψk|ψk′⟩|2 = ∞, any k = k′

D212 = − log |⟨uk|uk′⟩|2 = finite

Quantum metric tensor (2-form, gauge invariant):

gαβ(k) = Re ⟨∂kαuk|∂kβuk⟩ − ⟨∂kαuk|uk⟩⟨uk|∂kβuk⟩

Berry connection (1-form, gauge dependent):

Aα(k) = i ⟨uk|∂kαuk⟩ real

Berry curvature (2-form, gauge-invariant):

Ωαβ(k) = ∂kαAβ(k)− ∂kβAα(k)= i( ⟨∂kαuk|∂kβuk⟩ − ⟨∂kβuk|∂kαuk⟩ )

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Outline








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Electrical polarization and orbital magnetization

Elementary textbook definitions:

P =dV

=1V


M =mV

=1

2cV

∫dr r × j(micro)(r)

Similar definitions and the same problem: runbounded, not a periodic operatorCannot be evaluated within periodic boundary conditionsBoth P and M look as dominated by surface contributions

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P and M as reciprocal-space integrals

Polarization (King-Smith & Vanderbilt formula, 1993):

Pα = −2e∫

BZ

dk(2π)d i

nb∑j=1

⟨ujk|∂kαujk⟩+ P(nuclei)α

Orbital magnetization (2005-6):

Mγ = − e2ℏc

εγαβ

∫BZ

dk(2π)d i

∑εjk<µ

⟨∂kαujk| (Hk+ϵjk−2µ) |∂kβujk⟩

customarily written per spin channel

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P and M as reciprocal-space integrals

1993: Polarization insulators only

Pα = −2ienb∑

j=1

∫BZ

dk(2π)d ⟨ujk|∂kαujk⟩+ P(nuclei)

α

2005-06: Orbital magnetization: including metals

Mγ = − ie2ℏc

εγαβ∑εjk<µ

∫BZ

dk(2π)d ⟨∂kαujk| (Hk+ϵjk−2µ) |∂kβujk⟩

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Similar problems vs. similar solutions?

P(el)α = −2ie

nb∑j=1

∫BZ

dk(2π)d ⟨ujk|∂kαujk⟩ = −2e

∫BZ

dk(2π)d Aα(k)

Mγ = − ie2ℏc

εγαβ∑εjk<µ

∫BZ

dk(2π)d ⟨∂kαujk| (Hk + ϵjk − 2µ) |∂kβujk⟩

Are they really similar?Polarization

Gauge-dependent integrandBulk P defined modulo 2π in dimensionless unitsTinkering with the boundaries can alter P

Orbital MagnetizationGauge-invariant integrandM has no “modulo” ambiguityTinkering with the boundaries cannot alter M

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Similar problems vs. similar solutions?

P(el)α = −2ie

nb∑j=1

∫BZ

dk(2π)d ⟨ujk|∂kαujk⟩ = −2e

∫BZ

dk(2π)d Aα(k)

Mγ = − ie2ℏc

εγαβ∑εjk<µ

∫BZ

dk(2π)d ⟨∂kαujk| (Hk + ϵjk − 2µ) |∂kβujk⟩

Why are they so different?Polarization

It is essentially a 1d phenomenonIts expression is the integral of a 1-form

Orbital MagnetizationIt is essentially a 2d phenomenonIts expression is the integral of a 2-form

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Message from modern differential geometry(& algebraic topology)

Results due to Pontryagin, Cartan, Weyl, Chern, Simons....(≃ first half of 20th century)

Features in odd vs. even dimension are quite different2n-forms and (2n−1)-forms behave in quite different waysChern forms, and the (topological) Chern number onlyexists in dimension 2nChern-Simons forms are instead (2n−1)-forms

The Berry connection entering the P formula is aChern-Simons 1-formDoes any Chern-Simons 3-form have physical meaning?

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Outline








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The polarization “quantum”D. Vanderbilt and R. D. King-Smith, Phys. Rev. B 48, 4442 (1993)

Bulk polarization P is a lattice, not a vector!

The value of P remains ambiguous until the sampletermination is specified

For a 1d system polarization is defined modulo e

P =e

2πγ, the Berry phase γ is defined modulo 2π

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From geometrical to topological

The polarization of a linear polymer is defined modulo e

P =e

2πγ

For a centrosymmetric polymer:

Either γ = 0 or γ = π (mod 2π).Either P = 0 mod e or P = e/2 mod e

One-to-one mapping to the Z2 groupP is a Z2 topological observable,“protected” by centrosymmetry

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Polyacetylene, different terminations

Quantization of the dipole moment and of the end chargesin push-pull polymers

Konstantin N. Kudina! and Roberto CarDepartment of Chemistry and Princeton Institute for Science, and Technology of Materials (PRISM),Princeton University, Princeton, New Jersey 08544, USA

Raffaele RestaCNR-INFM DEMOCRITOS National Simulation Center, Via Beirut 2, I-34014 Trieste, Italyand Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, I-34014 Trieste, Italy

!Received 18 June 2007; accepted 24 September 2007; published online 15 November 2007"

A theorem for end-charge quantization in quasi-one-dimensional stereoregular chains is formulatedand proved. It is a direct analog of the well-known theorem for surface charges in physics. Thetheorem states the following: !1" Regardless of the end groups, in stereoregular oligomers with acentrosymmetric bulk, the end charges can only be a multiple of 1 /2 and the longitudinal dipolemoment per monomer p can only be a multiple of 1 /2 times the unit length a in the limit of longchains. !2" In oligomers with a noncentrosymmetric bulk, the end charges can assume any value setby the nature of the bulk. Nonetheless, by modifying the end groups, one can only change the endcharge by an integer and the dipole moment p by an integer multiple of the unit length a. !3" Whenthe entire bulk part of the system is modified, the end charges may change in an arbitrary way;however, if upon such a modification the system remains centrosymmetric, the end charges can onlychange by multiples of 1 /2 as a direct consequence of !1". The above statements imply that—in allcases—the end charges are uniquely determined, modulo an integer, by a property of the bulk alone.The theorem’s origin is a robust topological phenomenon related to the Berry phase. The effects ofthe quantization are first demonstrated in toy LiF chains and then in a series of trans-polyacetyleneoligomers with neutral and charge-transfer end groups. © 2007 American Institute of Physics.#DOI: 10.1063/1.2799514$

I. INTRODUCTION

Push-pull polymers have received much attention due totheir highly nonlinear electronic and optical responses. Suchmolecules usually contain a chain of atoms forming a conju-gated !-electron system with electron donor and acceptorgroups at the opposite ends. Upon an electronic excitation acharge is transferred from the donor to the acceptor group,leading to remarkable nonlinear properties. What is surpris-ing, however, is that—as will be shown in the presentwork—nontrivial features already appear when addressingthe lowest-order response of such molecules to the staticelectric fields, i.e., their dipole moment. A model push-pullpolymer is shown in Fig. 1. Note that instead of addressingcomputationally challenging excited states, we would rathermuch prefer to focus on the ground state properties. There-fore, in the case of the push-pull system shown in Fig. 1, wesimulate the charge transfer not by moving an electron but bymoving a proton from the COOH to NH2 groups located atthe opposite ends.

The most general system addressed here is, therefore, along polymeric chain, which is translationally periodic !ste-reoregular, alias “crystalline”" along, say, the z direction,with period a. We are considering insulating chains only, i.e.,chains where the highest occupied molecular orbital–lowestunoccupied molecular orbital gap stays finite in the long-

chain limit. The chain is terminated in an arbitrary way, pos-sibly with some functional group attached, at each of the twoends. In the case of a push-pull polymer, such groups are adonor-acceptor pair. Therefore, the most general system iscomprised of Nc identical monomers !“crystal cells”" in thecentral !“bulk”" region, augmented by the left- and right-endgroups. If the total length is L, the bulk region has a length

a"Electronic mail: [email protected]

FIG. 1. !Color online" Two states of a prototypical push-pull system. Thelong insulating chain of alternant polyacetylene has a “donor” !NH2" and“acceptor” !COOH" groups attached at the opposite ends. The charge trans-fer occurring in such systems upon some physical or chemical process issimulated here by moving a proton from the COOH to NH2 groups: in !a"we show the “neutral” structure and in !b" the “charge-transfer” one. Thetwo structures share the same “bulk,” where the cell !or repeating monomer"is C2H2, and the figure is drawn for Nc=5.

THE JOURNAL OF CHEMICAL PHYSICS 127, 194902 !2007"

0021-9606/2007/127"19!/194902/9/$23.00 © 2007 American Institute of Physics127, 194902-1

Downloaded 16 Nov 2007 to 147.122.10.31. Redistribution subject to AIP license or copyright, see http://jcp.aip.org/jcp/copyright.jsp

Bulk is centrosymmetricBoth molecules are not centrosymmetricWhat about the dipole per unit length for L → ∞ ?

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Dipole per monomerK. N. Kudin, R. Car, and R. Resta, J. Chem. Phys. 127, 194902 (2007)

final statement is that the end charges Qend of the most gen-eral polymeric chain, whose bulk region is centrosymmetric,may only assume !in the large-Nc limit" values which areinteger multiples of 1 /2. We have previously anticipated thisstatement !Sec. II" and demonstrated it heuristically !Sec. III"using a simple binary chain as test case. Although we usedfor pedagogical purposes a strongly ionic system, the theo-rem is general and holds for systems of any ionicity. Further-more, in all cases, the actual value of Qend is determined,within the set of quantized values, by the chemical nature ofthe system.

E. The correlated case

Throughout this work, we have worked at the level ofsingle-particle approaches, such as HF or DFT. The specifictools used in our detailed proof !i.e., localized Boys’/Wannier orbitals" prevent us from directly extending thepresent proof to correlated wave function methods. Nonethe-less, the exact quantization of end charges !in the large-system limit" still holds, as a robust topological phenom-enon, even for correlated wavefunctions. In this respect, thephenomenon is similar to the fractional quantum Hall effect,where correlated wavefunctions are an essential ingredient.16

We have stated above that the bulk dipole per cell !or permonomer" p0 is defined in terms of Berry phases; more de-tails about this can be found in our previous paper,26 where aQC reformulation of the so-called “modern theory ofpolarization”7–10 is presented. The ultimate reason for theoccurrence of charge quantization is the modulo 2! arbitrari-ness of any phase, as, e.g., in Eq. !17". A correlated wavefunction version of the modern theory of polarization, alsobased on Berry phases, does exist.10,27,28 The quantizationfeatures, as discussed here for polymeric chains, remain un-changed. While not presenting a complete account here, weprovide below the expression for p0 in the correlated case.

Suppose we loop the polymer onto itself along the zcoordinate, with the loop of length L, where L equals a timesthe number of monomers. Let "!r1 ,r2 , . . . ,rN" be the many-body ground state wave function, where spin variables areomitted for the sake of simplicity. Since z is the coordinatealong the loop, " is periodic with period L with respect tothe zi coordinate of each electron. We define the !unitary andperiodic" many-body operator

U = ei!2!/L"#i=1N zi, !18"

nowadays called the “twist” operator,28 and the dimension-less quantity

# = Im ln$"%U%"& . !19"

This #, defined modulo 2!, is a Berry phase in disguise,which is customarily called a “single-point” Berry phase.27

In order to get p0 in the correlated case, it is enough toreplace the sum of single-band Berry phases occurring in Eq.!17" with the many-body Berry phase #, as defined in Eq.!19".

Notice that the large-L limit of Eq. !19" is quite non-trivial, since as L increases, U approaches the identity, butthe number of electrons N in the wave function " increases;

nonetheless, this limit is well-defined in insulators !and onlyin insulators".29,30 In the special case where " is a Slaterdeterminant !i.e., uncorrelated single-particle approaches",the large-L limit of # converges to the sum of the Berryphases of the occupied bands, each given by Eq. !13". Thisresult is proved in Refs. 10 and 27. Therefore, for a single-determinant ", the correlated p0 defined via # in Eq. !19"coincides !in the large-L limit" with p0 discussed throughoutthis paper.

V. CALCULATIONS FOR A CASE OF CHEMICALINTEREST

Our realistic example is a set of fully conjugated trans-polyacetylene oligomers with the C2H2 repeat unit !a=4.670 114 817 4 a.u.", such as shown in Fig. 1. For themonomer unit, the bond distances and angles are r!CvC"=1.363Å, r!C–C"=1.428Å, r!C–H"=1.09Å, $!CCC"=124.6°, and $!CvC–H"=117.0°. Note that due to alter-nating single-double carbon bond length, such a system isinsulating. The chain with the equal carbon bonds would beconducting and, therefore, the theorem would not be appli-cable. The calculations were carried out at the RHF/30-21Glevel of the theory with the GAUSSIAN 03 code,6 up to Nc=257 C2H2 units in the largest oligomer !Fig. 4". In order tosave computational time, all the monomers were taken to beidentical, i.e., each one with the same geometry. For thestructure with the noncharged groups 'Fig. 1!a"(, we computep!257"=8.0%10!7, i.e., both p, and Qend vanish, with a verysmall finite-size error. The charge-transfer structure 'Fig.1!b"( yields instead p!257"=4.669 728 2, which correspondsto Qend=1 to an accuracy of 8.0%10!5. Thus, by modifyingthe end groups, one can observe the quantization theorem ina conjugated system, and again, the quantization is extremelyaccurate. For comparison, we have also carried out full peri-odic calculations31 of the dipole moment via the Berry-phaseapproach,26,32 utilizing 1024 k points in the reciprocal space.Since these calculations were closed shell, the electronic di-pole was computed for only one spin and then doubled. If the

FIG. 4. Longitudinal dipole moment per monomer p!Nc" of the trans-polyacetylene oligomers, exemplified in Fig. 1, as a function of Nc: dia-monds for the neutral structure 'NN( 'Fig. 1!a"( and squares for the charge-tranfer structure '&¯'( 'Fig. 1!b"(. The double arrow indicates theirdifference, which is exactly equal to one quantum.

194902-7 Dipole moment quantization in polymers J. Chem. Phys. 127, 194902 !2007"


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End charges for a polyacetylene chain

Quantization of the dipole moment and of the end chargesin push-pull polymers

Konstantin N. Kudina! and Roberto CarDepartment of Chemistry and Princeton Institute for Science, and Technology of Materials (PRISM),Princeton University, Princeton, New Jersey 08544, USA

Raffaele RestaCNR-INFM DEMOCRITOS National Simulation Center, Via Beirut 2, I-34014 Trieste, Italyand Dipartimento di Fisica Teorica, Università di Trieste, Strada Costiera 11, I-34014 Trieste, Italy

!Received 18 June 2007; accepted 24 September 2007; published online 15 November 2007"

A theorem for end-charge quantization in quasi-one-dimensional stereoregular chains is formulatedand proved. It is a direct analog of the well-known theorem for surface charges in physics. Thetheorem states the following: !1" Regardless of the end groups, in stereoregular oligomers with acentrosymmetric bulk, the end charges can only be a multiple of 1 /2 and the longitudinal dipolemoment per monomer p can only be a multiple of 1 /2 times the unit length a in the limit of longchains. !2" In oligomers with a noncentrosymmetric bulk, the end charges can assume any value setby the nature of the bulk. Nonetheless, by modifying the end groups, one can only change the endcharge by an integer and the dipole moment p by an integer multiple of the unit length a. !3" Whenthe entire bulk part of the system is modified, the end charges may change in an arbitrary way;however, if upon such a modification the system remains centrosymmetric, the end charges can onlychange by multiples of 1 /2 as a direct consequence of !1". The above statements imply that—in allcases—the end charges are uniquely determined, modulo an integer, by a property of the bulk alone.The theorem’s origin is a robust topological phenomenon related to the Berry phase. The effects ofthe quantization are first demonstrated in toy LiF chains and then in a series of trans-polyacetyleneoligomers with neutral and charge-transfer end groups. © 2007 American Institute of Physics.#DOI: 10.1063/1.2799514$

I. INTRODUCTION

Push-pull polymers have received much attention due totheir highly nonlinear electronic and optical responses. Suchmolecules usually contain a chain of atoms forming a conju-gated !-electron system with electron donor and acceptorgroups at the opposite ends. Upon an electronic excitation acharge is transferred from the donor to the acceptor group,leading to remarkable nonlinear properties. What is surpris-ing, however, is that—as will be shown in the presentwork—nontrivial features already appear when addressingthe lowest-order response of such molecules to the staticelectric fields, i.e., their dipole moment. A model push-pullpolymer is shown in Fig. 1. Note that instead of addressingcomputationally challenging excited states, we would rathermuch prefer to focus on the ground state properties. There-fore, in the case of the push-pull system shown in Fig. 1, wesimulate the charge transfer not by moving an electron but bymoving a proton from the COOH to NH2 groups located atthe opposite ends.

The most general system addressed here is, therefore, along polymeric chain, which is translationally periodic !ste-reoregular, alias “crystalline”" along, say, the z direction,with period a. We are considering insulating chains only, i.e.,chains where the highest occupied molecular orbital–lowestunoccupied molecular orbital gap stays finite in the long-

chain limit. The chain is terminated in an arbitrary way, pos-sibly with some functional group attached, at each of the twoends. In the case of a push-pull polymer, such groups are adonor-acceptor pair. Therefore, the most general system iscomprised of Nc identical monomers !“crystal cells”" in thecentral !“bulk”" region, augmented by the left- and right-endgroups. If the total length is L, the bulk region has a length

a"Electronic mail: [email protected]

FIG. 1. !Color online" Two states of a prototypical push-pull system. Thelong insulating chain of alternant polyacetylene has a “donor” !NH2" and“acceptor” !COOH" groups attached at the opposite ends. The charge trans-fer occurring in such systems upon some physical or chemical process issimulated here by moving a proton from the COOH to NH2 groups: in !a"we show the “neutral” structure and in !b" the “charge-transfer” one. Thetwo structures share the same “bulk,” where the cell !or repeating monomer"is C2H2, and the figure is drawn for Nc=5.

THE JOURNAL OF CHEMICAL PHYSICS 127, 194902 !2007"

0021-9606/2007/127"19!/194902/9/$23.00 © 2007 American Institute of Physics127, 194902-1


The dipole of a chain of length L can be 0 or ±Le:Polarization is not a local observable!The end charge can be 0 or ±e: topologicalSame reason as for the topological soliton charge:Su, Schrieffer, & Heeger, Phys. Rev. Lett. 42, 1698 (1979).

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Chern-Simons 1-form & 3-form

1d BZ integral of the Chern-Simons 1-form:

γ =

∫BZ

dkx tr Ax(kx) (physical meaning: polarization)

3d BZ integral of the Chern-Simons 3-form:

θ = − 14πεαγβ

∫BZdk tr

Aα(k)∂kβAγ(k)−

2i3

Aα(k)Aβ(k)Aγ(k)

Formula taken from the mathematical literatureθ gauge-invariant modulo 2πDoes θ have any physical meaning?Yes: “Axion” term in magnetoelectric response(X.-L. Qi, T.L.Hughes, & S.-C. Zhang, PRB 2008)

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Analogies between γ (polarization) and θ (axion)

Chern-Simons 1-form Polarization(insulators only)

Chern-Simons 3-form Axion term in magnetoelectrics(insulators only)

⟨O⟩ =∫

BZ

dk(2π)d f(k)

Integrand f(k) is gauge-dependentBulk ⟨O⟩ defined modulo 2π (in dimensionless units)For a bounded sample ⟨O⟩ depends on terminationIn presence of some “protecting” symmetryboth γ and θ are Z2 topological invariants

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Outline








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Reciprocal space integrals of 2-forms

Time-reversal odd Time-reversal evenAnomalous Hall conductivity SWM sum rule

Magneto-optical sum rule ??Orbital magnetization Drude weight

⟨O⟩ =∫

BZ

dk(2π)d f(k)

Integrand f(k) gauge-independentNo modulo 2π ambiguityFor a bounded sample ⟨O⟩ independent of terminationTinkering with the boundaries cannot change ⟨O⟩

Tinkering with the boundaries cannot change ⟨O⟩Do they admit a local representation?

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Reciprocal space integrals of 2-forms

Time-reversal odd Time-reversal evenAnomalous Hall conductivity SWM sum rule

Magneto-optical sum rule ??Orbital magnetization Drude weight

Re σ(−)αβ (0) = −e2

ℏ∑εjk<µ

∫BZ

dk(2π)d i(⟨∂kαujk|∂kβujk⟩−⟨∂kβujk|∂kαujk⟩)

Mγ = − e2ℏc

εγαβ∑εjk<µ

∫BZ

dk(2π)d i⟨∂kαujk| (Hk + ϵjk − 2µ) |∂kβujk⟩

Tinkering with the boundaries cannot change ⟨O⟩Do they admit a local representation?

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Local vs. nonlocal observables

Most nongeometrical observables are local(e.g. spin magnetization)They are meaningful for inhomogeneous systems

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Local vs. nonlocal observables

⟨O⟩ =∫

BZ

dk(2π)d f(k)

Implicitly requires:Crystalline systemHomogenous noncrystalline system (supercell)

Useless for macroscopically inhomogeneous systems

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Physical property of a given region

Any k-space approach is useless!Can we address the geometrical observables in r-space?

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Electrical polarization and orbital magnetization, again

P =dV

=1V


M =mV

=1

2cV


These r-space formulas are definitely NOT local!The multivalued nature of P rules out any local descriptionInstead, M can be recast in a local form(similar in spirit to an integration by parts):

R. Bianco & R. Resta, PRL 2013 (insulators)A. Marrazzo & R. Resta, PRL 2016 (metals)

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Outline








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Bounded samples with square-integrable orbitals

One-body density matrix, a.k.a. ground-state projector:

P =∑ϵj<µ

|φj⟩⟨φj | (spinless)

P should allow to evaluate any ground-state observable(for independent electrons)What about the geometrical observables in terms of P?

No: observables expressed as trace of 1-forms or 3-forms(i.e. those defined modulo 2π)

Yes: observables expressed as trace of 2-forms(i.e. those exempt from modulo ambiguity)

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Anomalous Hall conductivity and orbital magnetization

One-body density matrix, a.k.a. ground-state projector:

P =∑ϵj<µ

|φj⟩⟨φj | (spinless)

Tensor fields in r-space:

Fαβ(r) = Im ⟨r| P [rα,P] [rβ,P] |r⟩Mαβ(r) = Im ⟨r| |H − µ| [rα,P] [rβ,P] |r⟩.

In the bulk of a crystallite the two tensor fieldsF(r) and M(r) are lattice-periodicalWell defined even for disordered and/or inhomogeneousbounded samples

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Geometrical observables as traces per unit volume

Anomalous Hall conductivity:

σ(−)αβ = −2e2

ℏIm TrV Fαβ (insulators and metals)

Orbital magnetization:

Mγ =e

2ℏcεγαβTrV Mαβ (insulators and metals)

M = − ie2ℏc

TrV |H − µ| [r,P]× [r,P]

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Why is our formula for M better than the trivial one?RAPID COMMUNICATIONS

ANTIMO MARRAZZO AND RAFFAELE RESTA PHYSICAL REVIEW B 95, 121114(R) (2017)

where the BZ integral is actually a Fermi-volume integralin the metallic case, owing to the θ function in Eq. (2).Equation (4) as it stands holds for both d = 2 and d = 3; wefurther notice that σαβ—when expressed in e2/h units (alsoknown as klitzing−1)—is dimensionless for d = 2, while it hasthe dimensions of an inverse length for d = 3.

The position operator r is notoriously ill defined withinperiodic boundary conditions [8]; nonetheless its off-diagonalelements over the |ψjk⟩ and |ujk⟩ are well defined. Exploitingsome results from linear-response theory [9], one may provethat

Qkr|ujk⟩ = iQk|∂kujk⟩ (5)

whenever j labels an occupied state at the given k. We maythus write the Berry curvature as a trace:

'αβ(k) = −2 Im Tr PkrαQkrβ. (6)

Using then the definitions of P and Pk (and theircomplementary), Eqs. (1) and (2), it is easy to prove the identity

1Vcell

!

celldr ⟨r|PrαQrβ |r⟩ =

!

BZ[dk] Tr PkrαQkrβ. (7)

This identity has been known since a few years ago [6,10–14]for the insulating case—and for the insulating case only. Westress that the alternate proof provided here applies to themetallic case as well. The left-hand of Eq. (7) has two out-standing virtues: (i) it is expressed directly in the Schrodingerrepresentation, making no reference to reciprocal space, and(ii) it can be adopted as such for supercells of arbitrarily largesize, thus extending the concept of geometrical AHC to dis-ordered systems, such as alloys, as well as “dirty” metals andinsulators. We thus recast Eqs. (4) and (7) in the compact form,

σαβ = 2e2

hIm TrV PrαQrβ

= −2e2

hIm TrV P [rα,P] [rβ,P], (8)

where “TrV ” means trace per unit volume/area. The twoexpressions in Eq. (8) are formally equivalent; the secondone, being a P-only formula, is more suited to numericalimplementations.

We pause at this point to make contact with Ref. [5],where a supercell approach to dirty metals was actuallyproposed: in retrospect, the approach of Ref. [5] is equivalent toevaluating Eq. (8) over the folded BZ of the superlattice. IndeedEq. (8), when applied to a dirty metal, combines the nominallyintrinsic contribution—as defined for the clean metal—to someextrinsic contributions of geometrical nature. Following thearguments of Ref. [5] we argue here that Eq. (8) may yieldthe sum of the intrinsic and side-jump contributions to theAHC, while instead it may not include the skew scattering [2].

Our major result so far, Eq. (8), applies to either insulatorsor metals, either crystalline or disordered, but it has only beenproved for an unbounded and macroscopically homogeneoussystem within periodic boundary conditions. The next issueis whether one may adopt Eq. (8) locally, in order toaddress inhomogeneous systems (e.g., heterojunctions) oreven bounded samples (e.g., crystallites).

FIG. 1. A typical “Haldanium” flake. We have considered flakeswith up to 10 506 sites, all with the same aspect ratio; the one shownhere has 1190 sites. In order to probe the AHC locality we evaluatethe trace per unit area either on the central cell (two sites) or on the“bulk” region (1/4 of the sites). The grey horizontal line (black dots)highlights the sites chosen for drawing Fig. 4.

The locality of the AHC was investigated in Ref. [6],where it was shown—for the insulating case only—that thetopological AHC can indeed be evaluated from Eq. (8) forbounded and/or macroscopically inhomogeneous systems.The concept of “topological marker” was proposed therein;in the following we are going to show that Eq. (8) yields ananalogous “geometrical marker”, effective in the metallic caseas well. The very important feature pointed out by Ref. [6] isthat—when a bounded sample is addressed—the trace per unitvolume has to be evaluated using only some inner region of thesample, and not the whole sample. If the bounded system is acrystallite, one evaluates, e.g., the left-hand side of Eq. (7) overits central cell; in the large-crystallite limit one recovers thebulk value of the AHC. In all the cases dealt with in Ref. [6] theconvergence with size proved to be very fast: this was attributedto the exponential decay of the one-body density matrix in insu-lators (nearsightedness [7]), as already said in the introduction.For the metallic case we are going to explore in the followingan uncharted territory by means of case-study simulations.

The paradigmatic model for investigating issues of thepresent kind is the one proposed by Haldane in 1988 [1]. It isa tight-binding 2d Hamiltonian on a honeycomb lattice withon-site energies ±(, first-neighbor hopping t1, and second-neighbor hopping t2 = |t2|eiφ , which provides time-reversalsymmetry breaking. The model is insulating at half fillingand metallic at any other filling. Our bounded samples arerectangular Haldanium flakes such as the one shown in Fig. 1;the corresponding simulations for lattice-periodical samples,with Bloch orbitals, are performed by means of the PythTBcode [15]. Oscillations as a function of the flake size occurin the metallic case; as customary, we adopt a regularizing“smearing” technique.

In Fig. 2 we plot—as a function of the Fermi level µ—thedimensionless quantity

−4π Im TrAP [rα,P] [rβ ,P] = h

e2σxy, (9)

where “TrA” means trace per unit area. The quantity in Eq. (9)equals minus the Chern number C1 in the quantized insulating

121114-2

m =12c

∫dr r × j(r) = − e

2c

∫dr ⟨r|P r × v |r⟩

m = − ie2ℏc

∫dr ⟨r| |H − µ| [r,P]× [r,P] |r⟩

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The “Haldanium” paradigm (F.D.M. Haldane, 1988)

2/5/12 1:06 PMUltrathin hexagonal boron nitride films

Page 1 of 2http://www.physik.unizh.ch/groups/grouposterwalder/kspace/BNhome/BNhome.htm

home

ultrathin hexagonal boron nitride (h-BN)films on metals

Willi Auwärter, Matthias Muntwiler, Martina Corso, Thomas Greberand Jürg Osterwalder

Physics Institute, University of Zurich, 12/12/03

Boron nitrides represent a class of materials with promising properties[1]. They are thermally stable, chemically inert and insulating. Pairs ofboron and nitrogen atoms are isoelectronic to pairs of carbon atoms.Therefore, boron nitrides show a similar structural variety as carbonsolids, including graphitic hexagonalboron nitride (h-BN) and diamond-likecubic boron nitride (c-BN) [2], onion-like fullerenes [3], and multi- andsingle-wall nanotubes [4,5].Differences arise due to thereluctance, in boron nitrides, to form B-B or N-N bonds which excludespentagon formation and thus the synthesis of simple fullerenesanalogous to e.g. C60. In our work we concentrate on hexagonal boronnitride, that is often called "white graphite" due to its color and the layerstructure similar to graphite. The combination of being an electricinsulator and a good thermal conductor that is stable up to hightemperatures and the easy machinability makes h-BN an interestingmaterial for many technical applications. The photograph below showstwo boron nitride blocks. Weaklyphysisorbed layers of h-BN on metalsurfaces have been studied for about adecade [6]. Well-ordered films can begrown by thermal decomposition ofborazine (HBNH)3 on transition metal

surfaces [7]. In most cases studied sofar the film growth was observed to be self-limiting at one monolayer;beyond that the sticking coefficient of the precursor molecule becomesexceedingly small. Most of the work has been concentrated on the

introduction

h-BN on Nickel

h-BN on Rhodium

+ “some magnetism”

Tight-binding parameters:1st-neighbor hopping t1staggered onsite ±∆

complex 2nd-neighbor t2eiϕ

f!!"" =1

1 + exp#!" ! !"/#$. !54"

In all subsequent calculations, we set #=0.05 a.u., whichprovides good convergence.

We compute the orbital magnetization as a function of thechemical potential ! with $ fixed at % /3. Using the sameprocedure as in the previous section, we compute the orbitalmagnetization by the means of the heuristic k-space formula!48" and we compare it to the extrapolated value from finitesamples, from L=8 !289 sites" to L=16 !1089 sites". Weverified that a k-point mesh of 100&100 gives well con-verged results for the bulk formula !48".

The orbital magnetization as a function of the chemicalpotential for $=% /3 is shown in Fig. 5. The resulting valuesagree to a good level, and provide solid numerical evidencein favor of Eq. !48", whose analytical proof is still lacking.The orbital magnetization initially increases as the filling ofthe lowest band increases, and rises to a maximum at a !value of about !4.1. Then, as the filling increases, the first!lowest" band crosses the second band and the orbital mag-netization decreases, meaning that the two bands carryopposite-circulating currents giving rise to opposite contribu-tions to the orbital magnetization. The orbital magnetizationremains constant when ! is scanned through the insulatinggap. Upon further increase of the chemical potential, the or-bital magnetization shows a symmetrical behavior as a func-tion of !, the two upper bands having equal but oppositedispersion with respect to the two lowest bands !see Fig. 3".

C. Chern insulating case

In order to check the validity of our heuristic Eq. !48" fora Chern insulator, we switch to the Haldane modelHamiltonian11 that we used in a previous paper7 to addressthe C=0 insulating case. In fact, depending on the parameterchoice, the Chern number C within the model can be eitherzero or nonzero !actually, ±1".

The Haldane model is comprised of a honeycomb latticewith two tight-binding sites per cell with site energies ±',real first-neighbor hoppings t1, and complex second-neighborhoppings t2e±i(, as shown in Fig. 6. The resulting Hamil-

tonian breaks TR symmetry and was proposed !for C= ±1"as a realization of the quantum Hall effect in the absence ofa macroscopic magnetic field. Within this two-band model,one deals with insulators by taking the lowest band as occu-pied.

In our previous paper7 we restricted ourselves to C=0 todemonstrate the validity of Eq. !48", which was also analyti-cally proved. In the present work we address the C!0 insu-lating case, where instead we have no proof of Eq. !48" yet.We are thus performing computer experiments in order toexplore uncharted territory.

Following the notation of Ref. 11, we choose the param-eters '=1, t1=1, and %t2%=1/3. As a function of the fluxparameter $, this system undergoes a transition from zeroChern number to %C%=1 when %sin $%)1/&3.

First we checked the validity of Eq. !48" in the Cherninsulating case by treating the lowest band as occupied. Wecomputed the orbital magnetization as a function of $ by Eq.!48" at a fixed ! value, and we compared it to the magneti-zation of finite samples cut from the bulk. For the periodicsystem, we fix ! in the middle of the gap; for consistency,the finite-size calculations are performed at the same !value, using the Fermi-Dirac distribution of Eq. !54". Thefinite systems have therefore fractional orbital occupancyand a noninteger number of electrons. The biggest samplesize was made up of 20&20 unit cells !800 sites". The com-parison between the finite-size extrapolations and the dis-cretized k-space formula is displayed in Fig. 7. This heuris-tically demonstrates the validity of our main results, Eqs.!46" and !48", in the Chern-insulating case.

Next, we checked the validity of Eq. !48" for the mostgeneral case, following the transition from the metallic phaseto the Chern insulating phase as a function of the chemicalpotential !. To this aim we keep the model Hamiltonianfixed, choosing $=0.7%; for ! in the gap this yields a Cherninsulator. The behavior of the magnetization while ! variesfrom the lowest-band region, to the gap region, and then tothe highest-band region is displayed in Fig. 8, as obtainedfrom both the finite-size extrapolations and the discretizedk-space formula. This shows once more the validity of ourheuristic formula. Also notice that in the gap region the mag-netization is perfectly linear in !, the slope being determinedby the lowest-band Chern number according to Eq. !49".

FIG. 5. Orbital magnetization of the square-lattice model as afunction of the chemical potential ! for $=% /3. The shaded areascorrespond to the two groups of bands. Open circles: extrapolationfrom finite-size samples. Solid line: discretized k-space formula!48".

FIG. 6. Four unit cells of the Haldane model. Filled !open"circles denote sites with E0=!' !+'". Solid lines connecting near-est neighbors indicate a real hopping amplitude t1; dashed arrowspointing to a second-neighbor site indicates a complex hopping am-plitude t2ei$. Arrows indicate sign of the phase $ for second-neighbor hopping.

CERESOLI et al. PHYSICAL REVIEW B 74, 024408 !2006"

024408-10

Zero flux per cell (no Landau levels!)Insulating (either trivial or topological) at half fillingMetallic at any other filling

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AHC in metals

Extrinsic mechanisms:Side jumpSkew scattering

Since the early 2000’sAn important contribution is intrinsicGeometrical property of the ground state(Fermi-volume integral of the Berry curvature)Nonquantized version of QAHE in insulators

We have proved that it is local in r-space

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Haldanium flake (OBCs)

RAPID COMMUNICATIONS

ANTIMO MARRAZZO AND RAFFAELE RESTA PHYSICAL REVIEW B 95, 121114(R) (2017)

where the BZ integral is actually a Fermi-volume integralin the metallic case, owing to the θ function in Eq. (2).Equation (4) as it stands holds for both d = 2 and d = 3; wefurther notice that σαβ—when expressed in e2/h units (alsoknown as klitzing−1)—is dimensionless for d = 2, while it hasthe dimensions of an inverse length for d = 3.

The position operator r is notoriously ill defined withinperiodic boundary conditions [8]; nonetheless its off-diagonalelements over the |ψjk⟩ and |ujk⟩ are well defined. Exploitingsome results from linear-response theory [9], one may provethat

Qkr|ujk⟩ = iQk|∂kujk⟩ (5)

whenever j labels an occupied state at the given k. We maythus write the Berry curvature as a trace:

'αβ(k) = −2 Im Tr PkrαQkrβ. (6)

Using then the definitions of P and Pk (and theircomplementary), Eqs. (1) and (2), it is easy to prove the identity

1Vcell

!

celldr ⟨r|PrαQrβ |r⟩ =

!

BZ[dk] Tr PkrαQkrβ. (7)

This identity has been known since a few years ago [6,10–14]for the insulating case—and for the insulating case only. Westress that the alternate proof provided here applies to themetallic case as well. The left-hand of Eq. (7) has two out-standing virtues: (i) it is expressed directly in the Schrodingerrepresentation, making no reference to reciprocal space, and(ii) it can be adopted as such for supercells of arbitrarily largesize, thus extending the concept of geometrical AHC to dis-ordered systems, such as alloys, as well as “dirty” metals andinsulators. We thus recast Eqs. (4) and (7) in the compact form,

σαβ = 2e2

hIm TrV PrαQrβ

= −2e2

hIm TrV P [rα,P] [rβ,P], (8)

where “TrV ” means trace per unit volume/area. The twoexpressions in Eq. (8) are formally equivalent; the secondone, being a P-only formula, is more suited to numericalimplementations.

We pause at this point to make contact with Ref. [5],where a supercell approach to dirty metals was actuallyproposed: in retrospect, the approach of Ref. [5] is equivalent toevaluating Eq. (8) over the folded BZ of the superlattice. IndeedEq. (8), when applied to a dirty metal, combines the nominallyintrinsic contribution—as defined for the clean metal—to someextrinsic contributions of geometrical nature. Following thearguments of Ref. [5] we argue here that Eq. (8) may yieldthe sum of the intrinsic and side-jump contributions to theAHC, while instead it may not include the skew scattering [2].

Our major result so far, Eq. (8), applies to either insulatorsor metals, either crystalline or disordered, but it has only beenproved for an unbounded and macroscopically homogeneoussystem within periodic boundary conditions. The next issueis whether one may adopt Eq. (8) locally, in order toaddress inhomogeneous systems (e.g., heterojunctions) oreven bounded samples (e.g., crystallites).

FIG. 1. A typical “Haldanium” flake. We have considered flakeswith up to 10 506 sites, all with the same aspect ratio; the one shownhere has 1190 sites. In order to probe the AHC locality we evaluatethe trace per unit area either on the central cell (two sites) or on the“bulk” region (1/4 of the sites). The grey horizontal line (black dots)highlights the sites chosen for drawing Fig. 4.

The locality of the AHC was investigated in Ref. [6],where it was shown—for the insulating case only—that thetopological AHC can indeed be evaluated from Eq. (8) forbounded and/or macroscopically inhomogeneous systems.The concept of “topological marker” was proposed therein;in the following we are going to show that Eq. (8) yields ananalogous “geometrical marker”, effective in the metallic caseas well. The very important feature pointed out by Ref. [6] isthat—when a bounded sample is addressed—the trace per unitvolume has to be evaluated using only some inner region of thesample, and not the whole sample. If the bounded system is acrystallite, one evaluates, e.g., the left-hand side of Eq. (7) overits central cell; in the large-crystallite limit one recovers thebulk value of the AHC. In all the cases dealt with in Ref. [6] theconvergence with size proved to be very fast: this was attributedto the exponential decay of the one-body density matrix in insu-lators (nearsightedness [7]), as already said in the introduction.For the metallic case we are going to explore in the followingan uncharted territory by means of case-study simulations.

The paradigmatic model for investigating issues of thepresent kind is the one proposed by Haldane in 1988 [1]. It isa tight-binding 2d Hamiltonian on a honeycomb lattice withon-site energies ±(, first-neighbor hopping t1, and second-neighbor hopping t2 = |t2|eiφ , which provides time-reversalsymmetry breaking. The model is insulating at half fillingand metallic at any other filling. Our bounded samples arerectangular Haldanium flakes such as the one shown in Fig. 1;the corresponding simulations for lattice-periodical samples,with Bloch orbitals, are performed by means of the PythTBcode [15]. Oscillations as a function of the flake size occurin the metallic case; as customary, we adopt a regularizing“smearing” technique.

In Fig. 2 we plot—as a function of the Fermi level µ—thedimensionless quantity

−4π Im TrAP [rα,P] [rβ ,P] = h

e2σxy, (9)

where “TrA” means trace per unit area. The quantity in Eq. (9)equals minus the Chern number C1 in the quantized insulating

121114-2

Sample of 1190 sites

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AHC as a function of the Fermi levelA. Marrazzo and R. Resta, Phys. Rev. B 95, 121114(R) (2017)

Solid line:Usual k-space expression (Fermi-volume integral)

Symbols: Our r-space “geometrical marker”RAPID COMMUNICATIONS

LOCALITY OF THE ANOMALOUS HALL CONDUCTIVITY PHYSICAL REVIEW B 95, 121114(R) (2017)

FIG. 2. AHC as a function of the Fermi level µ for a 3422-siteflake. Top: trivial insulator when µ is in the gap; bottom: topologicalinsulator (C1 = −1) when µ is in the gap. See text about labels: Cell,Bulk, and PBCs. All calculations adopt a “smearing” s = 0.05.

case [16]: nonzero C1 reveals the nontrivial (topological)nature of the insulating ground state. Each panel displaysthe trace per unit area, Eq. (9), evaluated in three differentways: over the central two sites (labeled “Cell”), evaluatedover 1/4 of the sites (labeled “Bulk”), and evaluated as theusual integral of the Berry curvature for an unbounded sample(labeled “PBCs”). The plots show that averaging over the bulkregion provides a better convergence. The two plots refer totwo different sets of parameters: in both cases we set t1 = 1and φ = 0.25, while " = 2 the for top plot and " = 1/3 forthe bottom plot. It is perspicuous from the figure that when µis in the gap region the former choice yields a trivial insulator,and the latter a topological one (C1 = −1).

Figure 2 proves our major claim: the geometri-cal/topological AHC, for both metals and insulators, is indeeda local property of the electronic ground state and can beevaluated for a bounded sample, where the orbitals are squareintegrable and the concept of reciprocal space does not makeany sense. What differentiates insulators from metals is onlythe kind of convergence with the system size: exponentialin the former case, power law in the latter. We show atypical convergence study in Fig. 3, where we have chosena metallic flake with µ = −2.5 and the Hamiltonian for whichthe corresponding insulator is trivial: top panel of Fig. 2. As forthe previous figure, averaging over the bulk region provides abetter convergence than taking the trace on the central two-site

FIG. 3. Convergence of AHC evaluated locally as a function ofthe flake size. Parameters as in the top panel of Fig. 2, and µ = −2.5.The quantity σxy(∞) is obtained via extrapolation in the large flakelimit. A smearing s = 0.05 is adopted.

cell. Interpolations in both panels clearly show that the AHCconvergence to the bulk value is of the order L−3, where L isthe linear size of the flake.

FIG. 4. Local AHC for an heterojunction, where the left andright halves of the flake are two different metals (see text). For thiscalculation the flake has 10 506 sites; our local function is shown on aline of 102 sites (grey area in Fig. 1). The two horizontal lines (labeled“PBCs”) show the corresponding Berry-curvature calculations.

121114-3









121114-3

Trivial at half filling Topological at half filling

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AHC in Haldanium metal/metal heterojunctionsA. Marrazzo and R. Resta, Phys. Rev. B 95, 121114(R) (2017)









121114-3

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Orbital magnetization a a function of the Fermi levelA. Marrazzo and R. Resta, Phys. Rev. Lett. 116, 137201 (2016)

In either insulating or metallic systems, the integratedvalues provided by Eqs. (1), (3), and (4) are identical,but the integrands therein are quite different. This is similarto what happens when integrating a function by parts; wealso stress that any reference to microscopic currents hasdisappeared in Eq. (5).Only the insulating case has been addressed so far, where

it has been proved [6–9] that Eq. (4) has the outstandingvirtue of providing a local expression for M ¼ m=V:instead of evaluating the trace over the whole system, asin Eq. (4), we may evaluate the trace per unit volume in thebulk region of the sample. Notably, this converges (in thelarge system limit) much faster than the textbook definitionbased on Eqs. (1) and (3), where the boundary contributionto the integral is extensive (see also Fig. 4 below).The metallic case has not been addressed yet; in this

work we investigate the behavior of MðrÞ, Eq. (4), inmetallic 2D samples by means of simulations based ontight-binding model Hamiltonians. Our samples are finiteflakes within OBCs, where the volume V is replaced byarea A. We remind the reader that if one instead adoptsperiodic boundary conditions, M has a known expressionas a reciprocal-space integral [4], which, however, onlyapplies to magnetization in either a vanishing or commen-surate macroscopic B field. In this Letter we present OBCtest-case simulations for both B ¼ 0 and B ≠ 0; the formercase adopts rectangular flakes like the one shown in Fig. 1,while the latter adopts square flakes. For reasons thor-oughly discussed below, the two cases present completelydifferent features.The paradigmatic model for breaking time-reversal

symmetry without a macroscopic B field is the HaldaneHamiltonian [11], adopted here as well as by severalauthors in the past. Our choice of parameters is first-and second-neighbor hopping t1 ¼ 1 and t2 ¼ eiϕ=3, withϕ ¼ 0.25π, and onsite energies $Δ, with Δ ¼ 1.5. Withrespect to the insulating case, the metallic one is

computationally more demanding: in fact, finite-size effectsinduce large oscillations (as a function of the flake size)when the Fermi level μ is not in an energy gap. As usual, wedeal with this problem by adopting the “smearing” tech-nique: what we present here is the result of a combinedlarge-size and small-smearing finite-size analysis. Here, weadopt Fermi-Dirac smearing, although we stress that we arenot addressing M at finite temperature [12,13]: the smear-ing is a mere computational tool.For orientation, we start showing in Fig. 2 the converged

magnetization M of our Haldanium flake as a function of μover the whole range: M depends on μ in the metallic rangeand stays constant, while μ sweeps the gap [14]. Next, in ourmetallic test case we set μ ¼ −1.7, rather far from the bandedges (see Fig. 2); we therefore have a sizable Fermi surface(a Fermi loop in 2D), which in turn guarantees a nonzeroDrudeweight. As recognized by Haldane himself, this modelsystem is a good paradigm for the anomalous Hall effect inmetals [15]. Our simulations also confirm that the OBC’slocalization tensor diverges with the flake size [16,17].

FIG. 1. A typical “Haldanium” flake. We have consideredflakes with up to 8190 sites, all with the same aspect ratio; the oneshown here has 1806 sites. In order to probe locality, the fieldMðrÞ, Eq. (5), is averaged either on the central cell (two sites) oron the “bulk” region (1=4 of the sites).

−−−−−

− − − −

FIG. 2. The magnetization of a large flake (6162 sites) as afunction of the Fermi level μ. The valence-conduction gap isbetween ε ¼ −0.4 and ε ¼ −1.0; our metallic simulations are atμ ¼ −1.7, shown as a vertical line.

FIG. 3. Convergence with flake size of the standard formula,Eqs. (1) and (3), in log-log scale; a typical metallic (μ ¼ −1.7 inthe valence band) and a typical insulating (μ ¼ −0.7 at midgap)case are shown. The interpolating straight lines clearly show the1=L convergence.

PRL 116, 137201 (2016) P HY S I CA L R EV I EW LE T T ER Sweek ending1 APRIL 2016

137201-2

At convergence all formulas coincide:Textbook formula: 1

2cV


Mγ = − ie2ℏc εγαβ

∫FV dk ⟨∂kαujk| (Hk + ϵjk − 2µ) |∂kβujk⟩

Our novel formula: e2ℏc εγαβTrV Mαβ

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Fast convergence in both insulator and metal

We show next the convergence of the textbook definitionin Fig. 3. We switch to an obvious vector notation and weevaluate

MðNÞ ¼ mA

¼ 1

A

Z

flakedrMðrÞ ð6Þ

for N-site flakes: this is clearly identical to Eqs. (1) and (3).The log-log plot shows that ½MðNÞ −M%=M is proportionalto 1=

ffiffiffiffiN

p, i.e., to the inverse linear dimension L−1 of the

flake. Notably, this occurs for both insulating and metallicflakes.Our main aim is to assess the locality ofM. We therefore

compare MðNÞ, Eq. (6), to our local expressions

Mcell ¼1

Acell

Z

celldrMðrÞ;

Mbulk ¼1

Abulk

Z

bulkdrMðrÞ; ð7Þ

whereMðrÞ is integrated either on a single cell in the centerof the flake or on an inner rectangular region of area 1=4 ofthe total (see Fig. 1). Within our tight-binding Hamiltonian,Eq. (7) amounts to averaging either over two sites or overN=4 sites. The results for a typical insulating and metalliccase are shown in Figs. 4 and 5: they show once more thatm=A, Eq. (6), converges to the bulkM value as L−1. Instead,computations of either Mbulk or Mcell by means of ourlocal formulas converge to the bulk value much faster.Remarkably, this happens in both the insulating and metalliccases. This provides evidence for our major claim, i.e., thateven in metals the macroscopic magnetization M can beexpressed in terms of the one-body density matrix in the bulkof the sample, disregarding what happens at its boundary.Nonetheless, we also expect the convergence to be

qualitatively different in the two cases: in order to magnifythis, we plot both (the insulator and the metal) on a log scalein Fig. 6. The plots show that Eq. (7) does indeed converge

exponentially to the bulk M value in the insulating case.In the metallic case, however, the convergence is definitelyslower than exponential. It is not easy to assess thekind of convergence in the metallic case. We may onlyclaim—based on several results, such as those shown inFigs. 5 and 6—that the convergence is of the order L−α,with α definitely larger than 1.Next, we switch to magnetization in a finite macroscopic

B field. Here, our main requirement—namely, that we aredealing with a 2D metal—is much more delicate. Even ifwe choose a system that is a very good metal at B ¼ 0, theubiquitous presence of Landau levels (LLs) opens gaps inthe density of states (DOS), and the metallic nature of ourmodel system must be carefully checked. We therefore relyon some previous results from the literature, where themetallic nature of the model Hamiltonian has been checkedby independent means. Following Ref. [18], we adopt asimple square lattice with a nearest-neighbor interaction,setting t ¼ 1 in the following: a B flux ϕ equal to ϕ0=8—where ϕ0 ¼ e=ðhcÞ is the flux quantum—is included viaPeierls substitution.

FIG. 6. Convergence of magnetization as a function of theflake size (the same Mbulk as in Figs. 4 and 5) in a log scale.The interpolating line shows an exponential convergence ofMbulk in the insulating case, while the convergence is slowerin the metallic case.

FIG. 5. Magnetization as a function of the flake size, at aconstant aspect ratio, in the metallic case: μ ¼ −1.7 in the valenceband.

FIG. 4. Magnetization as a function of the flake size, at aconstant aspect ratio, in the insulating case: μ ¼ −0.7 atmidgap.


137201-3

Insulator

We show next the convergence of the textbook definitionin Fig. 3. We switch to an obvious vector notation and weevaluate

MðNÞ ¼ mA

¼ 1

A

Z

flakedrMðrÞ ð6Þ

for N-site flakes: this is clearly identical to Eqs. (1) and (3).The log-log plot shows that ½MðNÞ −M%=M is proportionalto 1=

ffiffiffiffiN

p, i.e., to the inverse linear dimension L−1 of the

flake. Notably, this occurs for both insulating and metallicflakes.Our main aim is to assess the locality ofM. We therefore

compare MðNÞ, Eq. (6), to our local expressions

Mcell ¼1

Acell

Z

celldrMðrÞ;

Mbulk ¼1

Abulk

Z

bulkdrMðrÞ; ð7Þ

whereMðrÞ is integrated either on a single cell in the centerof the flake or on an inner rectangular region of area 1=4 ofthe total (see Fig. 1). Within our tight-binding Hamiltonian,Eq. (7) amounts to averaging either over two sites or overN=4 sites. The results for a typical insulating and metalliccase are shown in Figs. 4 and 5: they show once more thatm=A, Eq. (6), converges to the bulkM value as L−1. Instead,computations of either Mbulk or Mcell by means of ourlocal formulas converge to the bulk value much faster.Remarkably, this happens in both the insulating and metalliccases. This provides evidence for our major claim, i.e., thateven in metals the macroscopic magnetization M can beexpressed in terms of the one-body density matrix in the bulkof the sample, disregarding what happens at its boundary.Nonetheless, we also expect the convergence to be

qualitatively different in the two cases: in order to magnifythis, we plot both (the insulator and the metal) on a log scalein Fig. 6. The plots show that Eq. (7) does indeed converge

exponentially to the bulk M value in the insulating case.In the metallic case, however, the convergence is definitelyslower than exponential. It is not easy to assess thekind of convergence in the metallic case. We may onlyclaim—based on several results, such as those shown inFigs. 5 and 6—that the convergence is of the order L−α,with α definitely larger than 1.Next, we switch to magnetization in a finite macroscopic

B field. Here, our main requirement—namely, that we aredealing with a 2D metal—is much more delicate. Even ifwe choose a system that is a very good metal at B ¼ 0, theubiquitous presence of Landau levels (LLs) opens gaps inthe density of states (DOS), and the metallic nature of ourmodel system must be carefully checked. We therefore relyon some previous results from the literature, where themetallic nature of the model Hamiltonian has been checkedby independent means. Following Ref. [18], we adopt asimple square lattice with a nearest-neighbor interaction,setting t ¼ 1 in the following: a B flux ϕ equal to ϕ0=8—where ϕ0 ¼ e=ðhcÞ is the flux quantum—is included viaPeierls substitution.

FIG. 6. Convergence of magnetization as a function of theflake size (the same Mbulk as in Figs. 4 and 5) in a log scale.The interpolating line shows an exponential convergence ofMbulk in the insulating case, while the convergence is slowerin the metallic case.

FIG. 5. Magnetization as a function of the flake size, at aconstant aspect ratio, in the metallic case: μ ¼ −1.7 in the valenceband.

FIG. 4. Magnetization as a function of the flake size, at aconstant aspect ratio, in the insulating case: μ ¼ −0.7 atmidgap.


137201-3

Metal

1/L convergence with size: 12cV


Much better convergence: e2ℏc εγαβTrV Mαβ

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Conclusions

Meaning of “geometrical” in electronic structure theory

Specializing to band insulators and band metals:two quite different families of geometrical observables

Geometrical/topological features are (also) localproperties of the ground-state wave function

Simulations on crystalline Haldanium c⃝ flakes:

Locality of anomalous Hall conductivityLocality of orbital (spontaneous) magnetization

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Main references

David Vanderbilt’s recent textbook:Berry Phases in Electronic Structure TheoryCambridge University Press 2018

My Lecture Notes:Geometry and Topology in Electronic Structure Theoryhttp://www-dft.ts.infn.it/˜gtse/draft.pdf

R. Resta, Il Nuovo Saggiatore (in preparation)

geometry and topology in condensed matter physics · geometry and topology in condensed matter...

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