基研研究会「場の理論と超弦理論の最前線」京都、 21 July, 2010

物性物理と場の理論物性物理と場の理論

Hideo Aoki Dept Physics, Univ Tokyo 　

My talk today --- A (personal) overview on condensed-matter problems that can be related to field theories.

(a) Gauge-symm broken states exemplified by superconductivity,

(b) Massless Dirac particles in graphene,

(c) Nonequilibrium phenomena. 　

CMP -- many world

i.e., diverse effective theories on orders of mag smaller E scales

CMP -- many world

i.e., diverse effective theories on orders of mag smaller E scales

HEP -- quantum field theoretic world HEP -- quantum field theoretic world

destined to be related due to deg of freedom

* gauge - superconductivity / superfluidity (U(1)) - QHE (Chern-Simons)* chiral - some phases of graphene ～ pion condens.? * T - ferromagnets, T-broken SC, …* P - ferroelectrics, noncentrosymm SC, …

Broken symmetriesBroken symmetries

Exotic SC:

・ T-rev broken (spin-triplet p+ip, singlet s+id, …)

・ FFLO (nonzero total momentum)

・ noncentrosymmetric (spin singlet mixes with triplet) ・ multi-band (as in the iron-based)

・ ferromagnetic SC (two broken gauge symms coexist)

* Integer QHE --- a topological phase

* Fractional QHE Chern-Simons gauge field in (2+1)D anyons, nonabelions, etc

* More generally, topological phases / insulators - anomalous Hall effect (AHE) - spin Hall effect (SHE) - quantum spin Hall effect (QSHE)

QHE

Topological phases(as opposed to broken symmetries)

Topological phases(as opposed to broken symmetries)

* topological phases -- no broken symmetries, no classical order p's; (Berry's phase relevant) -- still, a gap opens -- edge states hallmark the phase (boundary states in FT)

--- realisable in CMP as

* 1D: quantum wires, nanotubes Tomonaga-Luttinger* 2D: 　　 electron gas at interfaces QHE 　　 layered crystal structures high-Tc cuprates atomically single layer graphene massless Dirac * 3D: nonabelions, massless Dirac, etc possible as well?

Spatial dimensionalitySpatial dimensionality

* Quantum critical points

* Holographic methods (AdS/CFT, AdS/QCD)

Hartnoll, Class Quant Grav (hep-th:0903.3246)

* non-equilibrium phases / phenomena e.g., dielectric breakdown of a Mott insulator Schwinger's QED vacuum decay + Landau-Zener

* Collective modes in non-equilibrium --- Tomonaga-Luttinger mode in non-equilibrium

* non-equilibrium in strong ac fields --- dynamically generated mass gap, etc

Non-equilibriumNon-equilibrium

My talk today --- A (personal) overview on condensed-matter problems that can be related to field theories.

(a) Gauge-symm broken states exemplified by superconductivity,

(b) Massless Dirac particles in graphene (chiral symm, Nielson-Ninomiya, etc),

(c) Nonequilibrium phenomena (dielectric breakdown of Mott insulator, dymanically generated mass in graphene,

etc). 　

attraction

phononphonon

el-el repulsion(spin /charge)

el-el repulsion(spin /charge)

Tc ～ 0.1ωD

Tc ～ 0.1ωD

Tc ～ 0.01tTc ～ 0.01t

anisotropic pairing

100K 10000K10K 100K

isotropic pairing

* phonon mechanism

* electron mechanism

1. Heavy fermion

2. Superfluid 3He 　　　　　　　　　　　

3. High-Tc cuprates

4. SC in the Coulomb gas

Non-phonon mechanism SC/SFNon-phonon mechanism SC/SF

Kohn & Luttinger 1965; Chubukov 1993: Repulsively interacting fermions Attractive pairing channel exists for T 0 (weak-coupling, dilute case)

Vsinglet

:

Vtriplet

:

ー ＋

＋＋

Spin- and charge-fluctuation mediated pairing Spin- and charge-fluctuation mediated pairing

Attraction isotropic SCAttraction isotropic SC

V(k,k ’)

SC from repulsion: nothing strangeSC from repulsion: nothing strange

Repulsion anisotropic SCRepulsion anisotropic SC

attraction if changes sign

+- + -

eg, d wave pairing in cuprateseg, d wave pairing in cuprates

node

node

--

spin-fluctuation mediated pairing interaction

Tc in SC from repulsionTc in SC from repulsion(if duality, self-dual pt?)

half-filling

pairin

g co

rrela

tion

attra

ctio

n

(QMC: Kuroki et al,1999)

Strong coupling

(Tc ～ t/pairing int)T C

The closer to SC, the worse negative-sign problem in QMC ! --- Murphy's law in many-body physics, not an accident but strong Q fluctuations inherent ?

Weak coupling

(Tc ～ e-t/pairing int)

Fe-compoundFe-compound

Kuroki et al, PRL 101, 087004 (2008)

Kamihara et al, JACS 130, 3296 (2008)

+-

dx2-y2dx2-y2

dxzdxz

dyzdyz

Phase-sensitive measurementFlux detection(Chen et al, nature physics 2010)

+-

Fourier-transform STM spectroscopy(Hanaguri et al, Science 2010)

Multiband systems various pairings when materials / p are tuned

Degeneracy point s+id ？ (S.C. Zhang's group, PRL 2009)

dd

nodal snodal s

s±

(Kuroki et al, PRB 2009)

Collective (phase) modes in SCCollective (phase) modes in SC

Phase modes (Bololiubov 1959, Anderson 1958) = massless Goldstone mode for neutral SC massive for real (charged) SC (Anderson-Higgs mechanism 1963) as observed in NbSe2

Phase modes (Bololiubov 1959, Anderson 1958) = massless Goldstone mode for neutral SC massive for real (charged) SC (Anderson-Higgs mechanism 1963) as observed in NbSe2

Collective modes in one-band SCCollective modes in one-band SC

Out-of-phase (countersuperflow) mode (massive, Leggett 1966) as observed in MgB2 (Blumberg et al 2007) Out-of-phase (countersuperflow) mode (massive, Leggett 1966) as observed in MgB2 (Blumberg et al 2007)

Collective modes in multi-band SCCollective modes in multi-band SC

g23

　

g12 　

g31

　

+

+ +[class "even"]internal Josephson c's add

+

- + [class "odd"]Josephson c's subtract

even / odd (g -1)

ij

● Ohta et al (in prep)

Question here: 3-band = 2-band in terms of the collective modes ?Question here: 3-band = 2-band in terms of the collective modes ?

(Ohta et al, in prep)

When the pairing interactions gij are variedWhen the pairing interactions gij are varied

L+

L-

/

qz g23

　

g12

　g31

　

g3

1

　

g12

　

g23=-0.07

　

mass difference

repulsive gij's always class odd large mass difference frustration when |g12| ～ |g23| ～ |g31|repulsive gij's always class odd large mass difference frustration when |g12| ～ |g23| ～ |g31|

(Ohta et al, in prep)multiple Leggett modes

Three-band SC can accommodate complex i.e., spontaneously broken T

Three-band SC can accommodate complex i.e., spontaneously broken T

(Stanev & Tesanovic, PRB 2010)

1

2

3

ReIm

hole concentration

SC

T

quark chemical pot

Quark-gluon plasma

~ 1 GeV

~ 170 MeV

Colour SCHadronic fluid

Vacuum

pseudogap

SC in solids vs SC in hadron physicsSC in solids vs SC in hadron physics

Differences in SC in solid-state vs hadron phys

Energy scale ～ 0.01 eV ～ 100 MeV

Length scale 1 ～ 103 nm 1 ～ 10 fm

Particles involved e's with anisotropic FS relativistic quarks with isotr FS

Interaction e-e, e-phonon gluon-mediated long-range

Tc Tc ～ 0.01, 0.1 F Tc ～ 0.1 F

Internal deg spin, orbit colour, flavour, spin

Order of phase tr weakly 1st fl of EM 1st thermal fl of gluons

Fractional quantum Hall effect Fractional quantum Hall effect

1/ν (∝ B )0 1 2 3 4

2DEG

Rxy

RxxB

B (T)

Rxx (kΩ

）

N

S

Composite fermion pictureComposite fermion picture

A very neat way of incorporating (short-range) part of interactionA very neat way of incorporating (short-range) part of interaction

Speciality about 2+1D ?

© H Aoki

braid group rather than permutation groupbraid group rather than permutation group

Effective mass of the composite fermion(Onoda et al, 2001)

Various phases in the quantum Hall system

0 0.1 0.2 0.3 0.4 0.5

N =0

N =2

N =1

Compressible liquid

Stripe

Laughlin stateWigner crystal

BubbleDMRG result: Shibata & Yoshioka 2003

1

Triplet p-ip (Pfaffian state) non-abelions Trial wf: Moore-Read, Greiter-Wen-Wilczek 1991

Numerical: Morf 1998, Rezayi-Haldane 2000; Onoda-Mizusaki-Aoki 2003

Experiment: Willett-West-Pfeiffer 1998, 2002

CF

CF

BCS paired state at = 5/2 BCS paired state at = 5/2

B

p

T

A1

Solid

Superfluid

B A

superfluid 3He

Sr

Ru

O

Sr2RuO4

～ ～p+ip

© Bergemann

px+y

+i =

px-y

(Arita et al, PRL 2004; Onari et al PRB 2004)

Sr2RuO4

34

When T-broken pairing can occur? When T-broken pairing can occur?

When the space group of the pair has two-dimensional rep: as in

● Tetragonal systems: p + ip in Sr2RuO4

● Hexagonal systems: d + id (6+)(Onari et al, PRB

2002)

　　 p + ip (5-)

(Uchoa & Castro Neto, PRL 2007)

d1 d2

+ i

My talk today --- A (personal) overview on condensed-matter problems that can be related to field theories.

(a) Gauge-symm broken states exemplified by superconductivity,

(b) Massless Dirac particles in graphene (chiral symm, Nielson-Ninomiya, etc),

(c) Nonequilibrium phenomena (dielectric breakdown of Mott insulator, dymanically generated mass in graphene,

etc). 　

　

Graphene － massless Dirac Graphene － massless Dirac

(Geim)

K K’

two massless Dirac pointstwo massless Dirac points

HK = vF(x px + y py)

= vF

HK’ = vF(- x px + y py)

= vF -+

Effective-mass formalismEffective-mass formalism

Graphene － Dirac eqn Graphene － Dirac eqn

　

K'

K

Chirality in graphene

H = ck+g(k)ck

eigenvalues: ±m|g(k)|, m: integer degeneracy at g(k)=0.

Honeycomb lattice

(3+1) (2+1)0 1

1 2

2

3

4

5 3

(3+1) (2+1)0 1

1 2

2

3

4

5 3g(k)

HH 1 0, H

Chiral symmetry

12 ii cc 1 sublattice Aiii cc 1 sublattice Bi

Unitary operator

Pairs of eigenstates

:E

chiral symmetry Dirac cone (E =0 states can be made eigenstates of N = 0 Landau level at E =0 in B

Chiral symmetry Chiral symmetry

: E

40

QHE for massless Dirac QHE for massless Dirac

xy /(

-e2/h

)

Each Landau level carries 1/2 topological (Chern) #Each Landau level carries 1/2 topological (Chern) #

E

k

E

QHE in honeycomb lattice QHE in honeycomb lattice

xx

xy

(Novoselov et al ; Zhang et al, Nature 2005)

xy /(e2/h) = 1xy /(e2/h) = 1

33

55

77

-3-3

-1-1

-5-5

-7-7

(Hatsugai et al, 2006)

7531

xybulk

xyedge in graphenexy

bulk xy

edge in graphene(Hatsugai, 1993)

　

=1/50,t = 0.12, t = 0.00063, 500000 sites

Random Dirac field Random Dirac field

Correlated/a=1.5)

Uncorrelated(/a=0)

n=-1

Correlatedrandom bonds Uncorrelated

↑ preserves chiral symmetry

(Kawarabayashi et al 2009)(Kawarabayashi et al 2009)

n=0

n=1

a fixed-pt behaviour

　

Dirac cones in other / higher-D systems ?

Dirac cones in other / higher-D systems ?

Dispersion of the quasi-particlein the Bogoliubov Hamiltonian edge states

anisotropic (eg, d-wave)superconductors (Ryu & Hatsugai, PRL 2002)

　

D-dimensional diamondD-dimensional diamond1D (Se, Te) 2D (graphite) 3D (diamond) 4D (Creutz 2008)

t’t

“Massless Dirac” sequence “Massless Dirac” sequence

* Dirac cones seem to always appear in pairs -- Nielsen-Ninomiya* Dirac cones seem to always appear in pairs -- Nielsen-Ninomiya

t’=-1: -flux lattice t’=0: honeycomb t’=+1: square

* Flux phase cf Kogut-Susskind* Flux phase cf Kogut-Susskind

　

Can we have a single Dirac cone ?Can we have a single Dirac cone ?

(Aoki et al, 1996)

Flat-band (dp)model

B

E

massless DiracLandau levels

　

Can we manipulate two Dirac cones ?Can we manipulate two Dirac cones ?

(Watanabe et al, in prep)

(Haldane, 1988)

QHE without Landau levels

B

Dirac cones on surfaces of 3D crystals Dirac cones on surfaces of 3D crystals

Bi2Te3 (Liang et al 2008; Chen et al 2009)

(Alpichshev et al 2009)

Bi1-x Sbx

(Hsieh et al, 2008)

　

braid group in 2D rather than permutation gr in 3Dbraid group in 2D rather than permutation gr in 3D

Surface of a topological insulator single Dirac cone evading Nielsen-Ninomiya (1/2 ordinary fermion ～ Majorana fermion) Surface of a top insulator / SC can have p+ip 3D generalisation Majorana f at hedgehog top defects acts as nonabelions

Surface of a topological insulator single Dirac cone evading Nielsen-Ninomiya (1/2 ordinary fermion ～ Majorana fermion) Surface of a top insulator / SC can have p+ip 3D generalisation Majorana f at hedgehog top defects acts as nonabelions

１． QHE

２． Spin Hall effect in topological insulators 　

３． Photovoltaic Hall effect

Spin-orbit too small in graphene; rather, HgTe systems

(Geim; Kim)

(Oka & Aoki, 2009)

graphene

Wider Hall effects in Dirac systems Wider Hall effects in Dirac systems

Circularly-polarised light breaks time-reversal

mass term spin-orbit

(Kane & Mele, 2005)

My talk today --- A (personal) overview on condensed-matter problems that can be related to field theories.

(a) Gauge-symm broken states exemplified by superconductivity,

(b) Massless Dirac particles in graphene (chiral symm, Nielson-Ninomiya, etc),

(c) Nonequilibrium phenomena (dielectric breakdown of Mott insulator, dymanically generated mass in graphene,

etc). 　

Optical

Out of equilibrium Out of equilibrium

Transport

Strongly correlated system

Production of carriers in crossing the phase boundaries: Schwinger mechanism for QED vacuum decay * Mott insulator - metal (Oka & Aoki, 2003; 2005) * Superfluid - Mott insulator (Sondhi et al, 2005)

　

Dielectric breakdown

Mott`s gap

Mott transition

Two non-perturbative effects

Non-adiabatic quantum tunnelling

(Oka et al, 2003; 2005)

[Oka, Aoki et al, PRL 91 (2003); PRL 94 (2005); PRL 95 (2005); PRB (2009)]

✔ Analogy with breakdown of 2D QED vacuum ✔ Quantum walk on the many-body energy levels (exactly solvable toy model)

✔ (1+1)d Hubbard model + finite E (exactly solvable with Bethe-ansatz nonhermitian generalisation + imaginary t )

Dielectric-breakdown phase diagramDielectric-breakdown phase diagram

See Nakamura, this conferenceSee Nakamura, this conference

Floquet spectrum for massive Thirring + ac field

Photo-induced metallic state in the 1D Mott insulator

(Oka & Aoki, PRB (2009)

　time evolution in a honeycomb lattice

K’K

Wave propagation in graphene Wave propagation in graphene

B = 0B = 0

　

honeycomb lattice + circularly polarised light (B = 0)

Dynamical mass opens

Wavefunctions evolve more slowly,

Wave propagation in a circularly polarised light Wave propagation in a circularly polarised light

B = 0B = 0

© Oka 2009

　

DC response in AC fields － Geometric phase

ac field k-point encircles the Dirac points

Aharonov-Anandan phase Non-adiabatic charge pumping Photovoltaic Hall effect (Oka & Aoki 09)Can Berry`s phase still be defined in non-equilibrium?

--- Yes, Aharonov-Anandan phase (1987)Can Berry`s phase still be defined in non-equilibrium? --- Yes, Aharonov-Anandan phase (1987)

K

K'

(Oka & Aoki, 2009)

Aharonov-Anandan curvatureAharonov-Anandan curvature

An overview on condensed-matter problems ⇔ field theories.

(a) Superconductivity,

(b) Massless Dirac particles in graphene,

(c) Nonequilibrium phenomena. 　

SummarySummary

Future problems

Further realisations / bilateral applications field theories in CMP

Kazuhiko Kuroki Univ Electro-Commun Ryotaro Arita Univ Tokyo Seiichiro Onari, Yukio Tanaka, Hiroshi Kontani Nagoya Univ Shiro Sakai Univ. Tokyo, now at Vienna Masaki Tezuka Univ. Tokyo, now at Kyoto Karsten Held MPI Stuttgart, now at ViennaYukihiro Ota, Masahiko Machida Japan Atomic Energy Agency Tomio Koyama Tohoku Univ

Kazuhiko Kuroki Univ Electro-Commun Ryotaro Arita Univ Tokyo Seiichiro Onari, Yukio Tanaka, Hiroshi Kontani Nagoya Univ Shiro Sakai Univ. Tokyo, now at Vienna Masaki Tezuka Univ. Tokyo, now at Kyoto Karsten Held MPI Stuttgart, now at ViennaYukihiro Ota, Masahiko Machida Japan Atomic Energy Agency Tomio Koyama Tohoku Univ

Yasuhiro Hatsugai, Mitsuhiro Arikawa Univ Tsukuba Takahiro Fukui Ibaraki Univ Takahiro Morimoto Univ TokyoToru Kawarabayashi Toho Univ

Takashi Oka Univ Tokyo Naoto Tsuji Univ Tokyo Philipp Werner ETH Zurich