Fano resonance through Higgs
bound states in tunneling ofNambu-Goldstone modes
Shunji Tsuchiya (Tohoku Institute of Technology)
Collaborators: Takeru Nakayama (ISSP, Tokyo Univ.)
Tetsuro Nikuni (Tokyo Univ. of Science)
Ippei Danshita (Yukawa Institute, Kyoto Univ.)
arXiv: 1503.01516
基研研究会「熱場の量子論とその応用」@YITP, 京都
Sep. 1, 2015

Spontaneous symmetry breaking and collective modesF( )Y. Nambu

NGmodeJ. GoldstoneRe( )Higgs mode

Higgs
mode

P. Higgs

NG modeIm( )Higgs and NG modes are ubiquitous associated withspontaneous symmetry breaking.
Nambu-Goldstone mode - massless phase modepions, magnons, phonons, Bogoliubov mode in BECs …
Higgs mode - massive amplitude modein Standard Model, SCs, SFs, magnets, CDW materials …
- growing interest in Higgs modes in condensed matter physics

Nature 487, 454 (2012)

Cold bosons in an optical lattice - Higgs mode excitedby modulating lattice depth in time

Observation of the Higgs mode in the vicinity ofthe SF-MI phase transition point in 2d

Superfluid-Mott insulator transition
Bose-Hubbard model
H=LaserJU†bi bj +Ji,j† †bi bi bi bi

2iµi

U > 0 : on-site interaction

J : hoppingU†bi bi

J/U can be tuned by light intensity

superfluidSFMIJUMott insulator

87RbM.Greiner et al., Nature 415, 39 (2002)

Effective low-energy theory
3D

2.0

1.5

Deep in the SF regime J
Gross-Pitaevskii eq.

1.0i

0.5

0.0-0.5

U

00.05

0.10

0.15

0.20

0.25

t=22mbi =+ g| |2

amplitude and phase modes havethe same gapless dispersion= NG (Bogoliubov) mode

No Higgs mode
In the vicinity of the 2nd order SF-MI transition
Time-dependent Ginzburg-Landau eq.

C. Varma, JLTP (2002)
Sachdev, “Quantum
Phase Transitions”: SF orderparameter

Higgs and NG modes at the SF-MI transitions
TDGL eq.
3D

2.0

particle-hole symmetric

1.5

TDGL eq. is invariant under

1.0

0.5

•

0.0

-0.5

00.05

0.10

0.15

0.20

0.25

Higgs mode(x, t) =Emergent Lorentz invariancer0 /u0 + n(x, t)ei

linearize w.r.t

(x,t)

0 =

r0 /u0n(x, t) : amplitude fluctuations(x, t) : phase fluctuations

amplitude (Higgs) :phase (NG) :NG mode
we cosider 3D wherein long-lived Higgs exits

Effects of disorder• Disorder plays a crucial role in various condensed
matter systems - Anderson localization, Bose glass, ..
• Elementary excitations localized around disorder
potentials - Andreev bound states in SCs, edge
modes and Majorana bound states in topologically
non-trivial systems (TIs, TSCs), …
• We study transport properties of collective modes
throughapotential
barrierthesimplestdisorder.
NATURE PHYSICSARTICLES

DOI: 10.1038/NPHYS3227

abV(ψ )0.5 × 4e2/hψ0Energy

Higgs conductivity σ1H0.500ϕIm ψRe ψ24Energy6ω /EJSCρsωpairINS∆

0.50
800.250.75

0der pDisorFigure 1 | Broken U(1)-symmetry phase and quantum Monte Carlo calculation of the Higgs conductivity. a, When symmetry is broken, the potential
acquires a Mexican hat shape, with a circle of potential minima along the brim (black solid circle). Transverse modes of the order parameter = Aei' along
the brim (red line) are Nambu–Goldstone (phase) modes, and longitudinal modes (blue line) are Higgs (amplitude) modes associated with a finite energy.
In superconductivity, the potential corresponds to the free energy. b, The Higgs mode gives rise to low-frequency conductivity (in units of 4e2 /h) thatgrows as disorder p (fraction of disconnected superconducting islands) is increased and remains finite through the quantum phase transition (orange line).

M. P. A. Fisher, et al.,
PRB (1989)

D. Sherman, et al., Nat. Phys.
(2015)

J-D. Pillet, et al., Nat. Phys.
(2010)

Disorder in the BH model
Bose-Hubbard modelTwo kinds of disorder in the BH model
1) diagonal disorder: inhomogeneous on-site potential Viµi = µ0Vi

M. P. A. Fisher, PRB 40, 546 (1989)

2) off-diagonal disorder: inhomogeneous hopping amplitude Jij

Jij = J + Jij
N. Prokof’ev and B. Svistunov, PRL 92, 15703 (2004)
P. Sengupta and S. Haas, PRL 99, 050403 (2007)• We propose to introduce two kinds of external potential that tune
the two kinds of disorder independently in cold-atom experiments.

(a)Vext (x)Vext (x)Vext (x)2) inhomogeneous hopping amplitude
(a)
(b)
(a)

Homogeneous
latticeOpotential x+Potential barrier
O

Vext (x)Vext (x)x

Combined potential
O

Vext (x)xmodulation inlattice height+(b)OO

Vext (x)xOx

OOxxxOx

Ox

Vext (x)Vext (x)additional latticepotential
no change of on-sitewith Gaussian profile
potential -> fixed µ+

Jij = J + Jij(a)Vext (x)Vext (x)Vext (x)2) inhomogeneous hopping amplitude
(a)
(b)
(a)

Homogeneous
latticeOpotential x+Potential barrier
O

Vext (x)Vext (x)x

Combined potential
O

Vext (x)xmodulation inlattice height+(b)O

Vext (x)

xxOOO

xx

Vext (x)Vext (x)additional latticepotential
no change of on-sitewith Gaussian profile
potential -> fixed µ+

Jij = J + Jijx
OTDGL
equationincludingeffectsof thex
Ox
O potential•

We assume absence of the 1st order time-derivative term
far from the potential barrier: particle-hole symmetry

leading contribution to the linear term vr (x) =2W0t=222m2J (x)+ r0 + vr + u0 | |2vr (x) : a standard potential term that does not break p-h symmetry

1) inhomogeneous on-site potential(a)(b)

Homogeneous
lattice potential

Potential barrier

Vopt (x)Vbar (x)Combined potential

Vopt (x) + Vbar(x)

+(b)

O

Vopt (x)µi = µ0x

O+bar (x)opticalVdipole
potentialxOOxx

O

no change inlattice height-> fixed J

Vmodulation
opt (x) + Vbar(x) of on-site potentialµi = µ0VixxO
TDGL equation including effects of thepotential

leading contribution to the 1st order time-derivative term vK (x) =2ivKt

W0t=
222m+ r0 + u0 | |2vK (x) breaks p-h symmetry!

2W0 V (x)Effective 1D settingivK (x)

2t

W0t

2

=
2+ r0 + vr (x) + u0 | |22m: coherence lengthivK (x)2tt=
2221 + vr (x) + | |2

we assume delta-function potentials :vK = VK (x)vr = Vr (x)Potentials varying in the order of latticespacing d can be well approximated by
the delta-function potentials because ofd near the phase boundaryLinearized TDGL equation
2ivKtt=22

2

1 + vr + | |2

we assume fluctuations of the order parameter only in the x direction

Static eq. (same as the static GP eq.)no effect ofvK (x)NG:
Higgs:Higgs and NG modes are locally coupled via vK (x)Higgs bound states
Static solution for vr (x) = Vr (x)0 (x) = tanh(|x| + x0 )we set vK (x) = 0= Shroeinger eq.

decoupled Higgs and NG mode due to p-h symmetry

Bound-state solutions of amplitude fluctuation T(x) below
the bulk Higgs gap due to the deep condensate potential

Higgs bound states
1 d2
2 dx2

1 + 3| 0 |2 + Vr (x) T (x) =0 (x) = tanh(|x| + x0 )2T (x)

bound state energy
1.40

2

1.35

1

odd
0-1-2-10

1.30

Vr = 1.5even
1.25-505

10: bulk Higgs gap

024

6810

Scattering of collective modes
Scattering of NG modes with energy
below the bulk Higgs gap E < = 2incident to the potential barriers
vK = VK (x) vr = Vr (x)( (x) + iks )eiks x + rng ( (x)NG: S(x) =Incident
tng ( (x)(x < 0)Reflectediks )eiks xTransmitted

Higgs: T (x) =iks )e iks x(x > 0)(x) = tanh(|x| + x0 )ks =rh (3 (x)2 + 3 t (x) +2t1)e t x(x < 0)th (3 (x)2 + 3 t (x) +2t1)e(x > 0)

decay at infinity

Boundary condition at x=0txt =42E2E 2

Tunneling property of NG modes
Transmission probability:
Effective potential:
1.0(Vr , VK ) = (4, 4)

VK2 f (E))VrKovrizhin, Phys. Lett. A, 287, 392 (2001)
Kagan, et al., PRL 90, 130402 (2003)

Characteristic asymmetricpeak near the energy of
the Higgs bound state E+

0.40Veff (E) = (1E<

anomalous tunneling

0.602E 2
1 + (2E 2 +1)2 Veff (E)2

Perfect transmission of NG
mode in the low energy limit

0.8

0.2

T (E) =1

(Vr , VK ) = (4, 2)
0.2

0.4

0.6

0.8

1.0

Fano resonanceTunneling property of NG modes
Transmission probability:
Effective potential:

Veff (E) = (1T (E) =VK2 f (E))Vr

12E 2
1 + (2E 2 +1)2 Veff (E)2Vr

VK2Vr
E E+EE+direct scatteringresonant scattering involving excitationof the even Higgs bound statef (EE+ )1

VK2 f (E) = 0

destructive interference0Veff

TVeff = 0

T =1Interference between directly scattered waves within continuum
and resonantly scattered waves mediated by discrete states- Fano resonance1.0750.8

30.610-10.4-30.2-5-7

Veff (E) = (100.2f (E10.4T (E) =VK2 f (E))Vr
0.60.8E+ )VK2 f (E) = 00

1.000.2

1221 + (2E2E2 +1)2 Veff (E)
0.4

0.6

0.8

1.00Veff

T

Veff = 0

T =1

The asymmetric peak is manifestation of the Fano resonanceof the NG mode mediated by the even Higgs bound state.

Summary• TDGL eq. including effects of two kinds of potential
barriers - inhomogeneous on-site potential andhopping amplitude.
ivK (x)2t

W0t=
222m+ r0 + vr (x) + u0 | |22• Localized Higgs bound states below
the bulk Higgs gap• Fano resonance of NG mode
mediated by the Higgs bound states

10-1-2-10-5

1.0

0

50.8

Outlook Higgs bound states in othercondensed matter systems,
e.g. disordered SCs
Sherman et al., Nat. Phys. (2015)

0.6

0.4

0.2

00

0.2

0.4

0.6

0.8

1.0

10

Higgs modes in condensed matter physics
Raman spectroscopy in NbSe2- First observation of Higgs mode!
NbSe2: CDW transition at 40K and SC transition at 7.2 KHiggs
A new peak at the frequency twice of
the SC gap arises below SC Tc.

2K

9K

Littlewood and Varma developed amicroscopic theory by extending theBCS-Nambu theory. They found thepeak due to amplitude oscillations ofsuperconducting gap - Higgs mode.
P. Littlewood and C. Varma, PRL 47, 811 (1980)

Higgs gap (mass) = 2

in BCS (s-wave)

Sooryakumar and Klein, PRL 45, 660 (1980)

longitudinal mode (Higgs)
paramagnet

Neeltriplonspin wave (NG)triplon (broken valence bondexcitations)condensation
Higgs + NG mode