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 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 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 dimensionality

* Quantum critical points * Holographic methods (AdS/CFT, AdS/QCD) Hartnoll, Class Quant Grav (hep-th: )

* 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-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 phonon el-el repulsion (spin /charge) Tc ～ 0.1ω D Tc ～ 0.01t anisotropic pairing 100K  10000K 10K  100K isotropic pairing * phonon mechanism * electron mechanism

1. Heavy fermion 2. Superfluid 3 He 3. High-Tc cuprates 4. SC in the Coulomb gas Non-phonon mechanism SC/SF Kohn & Luttinger 1965; Chubukov 1993: Repulsively interacting fermions  Attractive pairing channel exists for T  0 (weak-coupling, dilute case)

V singlet : V triplet : ー＋ ＋＋     Spin- and charge-fluctuation mediated pairing  

Attraction  isotropic SC V(k,k ’) SC from repulsion:  nothing strange Repulsion  anisotropic SC attraction if  changes sign eg, d wave pairing in cuprates node - spin-fluctuation mediated pairing interaction

Tc in SC from repulsion (if duality, self-dual pt?) half-filling pairing correlation attraction (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 ) -  0 +  BCS unitarity BEC 1/(ak F )

Fe-compound Kuroki et al, PRL 101, (2008) Kamihara et al, JACS 130, 3296 (2008) + -

dx2-y2 dxz dyz

Phase-sensitive measurement Flux 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) d d nodal s s± (Kuroki et al, PRB 2009)

Collective (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 NbSe 2 Collective modes in one-band SC Out-of-phase (countersuperflow) mode (massive, Leggett 1966) as observed in MgB 2 (Blumberg et al 2007) Collective modes in multi-band SC

g 23     g 12 g 31   [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 ? (Ohta et al, in prep)

When the pairing interactions g ij are varied  L+  L- // qzqz g 23     g 12 g 31   g 31 g 12 g 23 =-0.07 mass difference repulsive g ij 's  always class odd  large mass difference  frustration when |g 12 | ～ |g 23 | ～ |g 31 | repulsive g ij 's  always class odd  large mass difference  frustration when |g 12 | ～ |g 23 | ～ |g 31 | (Ohta et al, in prep) multiple Leggett modes

Three-band SC can accommodate complex  i.e., spontaneously broken T (Stanev & Tesanovic, PRB 2010) 11 22 33 Re Im

hole concentration SC T quark chemical pot  Quark-gluon plasma ~ 1 GeV ~ 170 MeV Colour SC Hadronic fluid Vacuum pseudogap SC 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 ～ 10 3 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 1/ν ( ∝ B ) DE G R xy R xx B B (T) R xx (kΩ ）

N S Composite fermion picture A very neat way of incorporating (short-range) part of interaction

Speciality about 2+1D ? © H Aoki braid group rather than permutation group

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

Various phases in the quantum Hall system  N =0 N =2 N =1 Compressible liquid Stripe Laughlin state Wigner crystal Bubble DMRG result: Shibata & Yoshioka

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 BCS paired state at = 5/2 B p T A1A1 Solid Superfluid B A superfluid 3 He Sr Ru O L: p+ip S: triplet Sr 2 RuO 4 ～ ～ p+ip

© Bergemann p x+y +i+i = p x-y (Arita et al, PRL 2004; Onari et al PRB 2004) Sr 2 RuO 4

34 When T-broken pairing can occur? When the space group of the pair has two-dimensional rep: as in ● Tetragonal systems: p + ip in Sr 2 RuO 4 ● Hexagonal systems: d + id (  6 + ) (Onari et al, PRB 2002) p + ip (  5 - ) (Uchoa & Castro Neto, PRL 2007) d1d1 d2d2 + 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 (Geim)

K K’K’ two massless Dirac points H K = v F (  x p x +  y p y ) = v F H K’ = v F (-  x p x +  y p y ) = v F - + Effective-mass formalism Graphene － Dirac eqn

K' K Chirality in graphene H =  c k + g  (k)   c k  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  3 g(k)g(k)

Chiral symmetry Unitary operator Pairs of eigenstates chiral symmetry  Dirac cone (E =0 states can be made eigenstates of   N = 0 Landau level at E =0 in B  Chiral symmetry

40 QHE for massless Dirac  xy /(-e 2 /h) Each Landau level carries 1/2 topological (Chern) # E k E

QHE in honeycomb lattice  xx  xy (Novoselov et al ; Zhang et al, Nature 2005)  xy /(e 2 /h) =

(Hatsugai et al, 2006)  xy bulk  xy edge in graphene (Hatsugai, 1993)

  =1/50,  t = 0.12,  t = , sites Random Dirac field Correlated  /a=1.5) Uncorrelated (  /a=0) n=-1 Correlated random bonds Uncorrelated ↑ preserves chiral symmetry (Kawarabayashi et al 2009) n=0 n=1 a fixed-pt behaviour

Dirac cones in other / higher-D systems ? Dispersion of the quasi-particle in the Bogoliubov Hamiltonian  edge states anisotropic (eg, d-wave) superconductors (Ryu & Hatsugai, PRL 2002)

D-dimensional diamond 1D (Se, Te) 2D (graphite) 3D (diamond) 4D (Creutz 2008)

t’t’ t “ Massless Dirac ” sequence * 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

Can we have a single Dirac cone ? (Aoki et al, 1996) Flat-band (dp)model B E massless Dirac Landau levels

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 Bi 2 Te 3 (Liang et al 2008; Chen et al 2009) (Alpichshev et al 2009) Bi 1-x Sb x (Hsieh et al, 2008)

braid 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

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 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 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 diagram See 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 B = 0

honeycomb lattice + circularly polarised light (B = 0) Dynamical mass opens Wavefunctions evolve more slowly, Wave propagation in a circularly polarised light B = 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) K K' (Oka & Aoki, 2009) Aharonov-Anandan curvature

 An overview on condensed-matter problems ⇔ field theories. (a) Superconductivity, (b) Massless Dirac particles in graphene, (c) Nonequilibrium phenomena. Summary 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 Vienna Yukihiro Ota, Masahiko Machida Japan Atomic Energy Agency Tomio Koyama Tohoku Univ Yasuhiro Hatsugai, Mitsuhiro Arikawa Univ Tsukuba Takahiro Fukui Ibaraki Univ Takahiro Morimoto Univ Tokyo Toru Kawarabayashi Toho Univ Takashi Oka Univ Tokyo Naoto Tsuji Univ Tokyo Philipp Werner ETH Zurich