Nonunitary Scaling Theory ofNon-Hermitian Localization
Kohei Kawabata (University of Tokyo)
ID rId2

August 27th, 2020
ID rId2

Collaborator: Shinsei Ryu (Chicago → Princeton)

arXiv: 2005.00604

Outline
1. Introduction and motivation
・Non-Hermitian localization

2. Scaling theory of non-Hermitian localization
・Nonunitary scattering matrices
・Scaling equations
・Two-parameter scaling

3. Symmetry and non-Hermitian localization
・No delocalization due to reciprocity (class AI†)・Bidirectional delocalization protected by symplectic
reciprocity (class AII†) xScaling theory of localization

1/20

Anderson localization: disorder-induced localization of coherent waves

Anderson, PR 109, 1492 (1958)

Unified understanding of localization: scaling theory
Abrahams et al., PRL 42, 673 (1979)

3D: localization transition
1D: absence of delocalization
Assumption of the scaling theory: one-parameter scaling
is given solely by: conductance: length scaleNon-Hermitian physics

2/20

Despite the enormous success, Anderson localization has been
mainly investigated in closed and conservative (Hermitian) systems.
Richer properties appear in non-Hermitian systems.
(non-Hermiticity can induce decoherence)☆ Non-Hermiticity arises from non-conservation of energy and/orparticles (e.g. non-equilibrium open systems).
・Photonic lattices with gain/loss
Unidirectional light transport.

Regensburger et al.,Nature 488, 167 (2012)

・Finite-lifetime quasiparticles
Bulk Fermi arc due to nonHermitian self-energy.

Kozii & Fu, arXiv:
1708.05841

Hatano-Nelson model

3/20

Delocalization is impossible in Hermitian systems in 1D.
Delocalization is possible in the presence of non-Hermiticity!
Hatano & Nelson, PRL 77, 570 (1996)

asymmetric hopping

Gong et al., PRX 8, 031079 (2018)

Strong non-Hermiticity leads to delocalization even in 1D.
Many subsequent works:
Efetov, PRL 79, 491 (1997); Feinberg et al., Nucl. Phys. B 504, 579 (1997); Brouwer et al., PRB 56, R4333(R) (1997);
Goldsheid et al., PRL 80, 2897 (1998); Mudry et al., PRL 80, 4257 (1998); Yurkevich et al., PRL 82, 5080 (1999)

Random laser

4/20

Random lasers can be described by non-Hermitian disordered systems:
complex potential(gain and/or loss)
No delocalization occurs in 1D and 2D,in contrast to the Hatano-Nelson model.
Delocalization occurs in 3D, and its
criticality seems to be the same as theHermitian Anderson model.

Wiersma, Nat. Phys. 4, 359 (2008)

e.g. Tzortzakakis et al., PRB 101, 014202 (2020); Huang et al., PRB 101, 014204 (2020)(subsequent talk)Motivation
・Non-Hermitian disordered systems can exhibit delocalizationeven in 1D (e.g. Hatano-Nelson model).
・For Hermitian systems, scaling theory forbids delocalizationin 1D.
Scaling theory of localization should be changed
in non-Hermitian systems.
・While the Hatano-Nelson model exhibits delocalization,
random lasers do not exhibit delocalization.
What determines the universality of
non-Hermitian localization?

Results
・We develop a scaling theory of localization in non-Hermitian systems.
・Non-Hermiticity introduces a new scale
and breaks down the one-parameter scaling.
Two-parameter scaling(origin of delocalization in 1D)(Hermitian)10

・Threefold universality based on reciprocity

localization(Hermitian)1

delocalization(non-Hermitian)Outline
2. Scaling theory of non-Hermitian localization
・Nonunitary scattering matrices
・Scaling equations
・Two-parameter scalingxNonunitary scattering matrices

7/20

The conductance is obtained by the scattering matrix (Landauer formula)(reflection from right to left)(transmission from right to left)(transmission from left to right)(reflection from left to right)

Hermiticity of the Hamiltonian
Unitarity of the scattering matrix☆ The conductance is not necessarily finite in non-Hermitian systems.
(direct consequence of non-conservation of particles)

Unitarity of(transmission and reflectionamplitudes)

Nonunitary regime

8/20: conductance: length scale

Scaling function

Abrahams et al., PRL 42, 673 (1979)

Localized regimelocalization
Diffusive regime
Ohm’s law

no delocalizationdelocalization (transition)
Non-unitary regime
non-conservationof particles(amplification)☆ Non-unitarity enables delocalization even in 1D.

Thouless criterion

9/20
Thouless, Phys. Rep. 13, 93 (1974)☆ Thouless time: time for a particle to reach one end from the other end
diffusive transport
To realize this diffusive transport, we need(energy-level spacing)

Diffusion is impossible in 1D:☆ Non-unitarity enables a different type of transport.
particle inflow from the environment > localization
ballistic transport
can be smaller even in 1D

R. Hamazaki (private communication)

Continuum model

10/20

Hatano-Nelson model

To understand a universal feature of non-Hermitian localization,
we consider a continuum model of the Hatano-Nelson model.
right-moving

left-moving

disorder (Gaussian)(“Fermi surface”)
Hatano & Nelson, PRL 77, 570 (1996)☆ Independent from specific detailsof the modelScaling equations

11/20

We obtain the scaling equations for the continuum model(non-unitary generalization of the DMPK equations)
Dorokhov, JETP Lett. 36, 318 (1982);
Mello et al., Ann. Phys. 181, 290 (1988)

non-Hermiticity,・For

mean-free pathis amplified, andis attenuated・transmission amplitude in Hermitian systems

Around,exhibits criticality.

.

Two-parameter scaling

12/20

Scaling equation
・Hermitian・Non-Hermitian: the scaling equation is described solely by(one-parameter scaling): two length scales(Hermitian)The unusual delocalization originates fromthe breakdown of the one-parameter scaling.

10localization(Hermitian)☆ Two-parameter scaling

1delocalization(non-Hermitian)Outline

3. Symmetry and non-Hermitian localization
・No delocalization due to reciprocity (class AI†)・Bidirectional delocalization protected by symplectic
reciprocity (class AII†) x

Time reversal and reciprocity

13/20☆ Symmetry can change universality of Anderson localization.

Threefold way by time-reversal symmetry (reciprocity)
Class A: no symmetry
Class AI:Class AII:・Localization is enhanced in class AI, while it is suppressed in class AII.
Hikami et al., PTP 63, 707 (1980)
e.g. Delocalization in 2D class AII
NO delocalization in 2D class A

Does symmetry change universality of localizationeven in non-Hermitian systems?

Reciprocity in non-Hermitian systems☆ Symmetry ramifies in non-Hermitian systems.

(Hermitian)(non-Hermitian)Kawabata et al.,
PRX 9, 041015 (2019)

time-reversal symmetry
reciprocity
Two symmetry is different in non-Hermitian systems,
while it is equivalent in Hermitian systems.
・Why is the latter symmetry reciprocity?
(reciprocal transport)☆ Reciprocity leads to threefold universality of localization.

No delocalization (class AI†)

15/20

・Continuum model in class AI†
class AI†
(Hermitian)reciprocity:

The universality is the same as
the Hermitian counterpart.

10☆ The absence of delocalization in
random lasers originates from reciprocity.

localization(Hermitian)

1delocalization(non-Hermitian)Hermitian Hamiltonian in class AI + complex onsite potential→ non-Hermitian Hamiltonian in class AI†

Bidirectional delocalization (class AII†)・Continuum model in class AII†
orbit

spin

reciprocity:
Kramers degeneracy(Hermitian)Kramers
partner

Kramerspair

10localization(Hermitian)

1a modedelocalization(non-Hermitian)☆ Symplectic reciprocity gives rise to a new type of delocalization.

Threefold universality

17/20☆ Threefold universality of non-Hermitian localization by reciprocity(Hatano-Nelson)(random laser)(symplectic Hatano-Nelson)☆ Localization is enhanced in class AI†, while it is suppressed in class AII†.

Similar to the Hermitian counterparts,
but the influence pf reciprocity is more dramatic☆ Time-reversal symmetry does not change universality.

Lattice models (class A)

18/20

Numerical calculations of localization lengths (transfer-matrix method).
e.g. Kramer et al., Int. J. Mod. Phys. B 24, 1841 (2010)☆ Two localization lengths in non-Hermitian systems(non-Hermitian)(Hermitian)・Hatano-Nelson model (class A)
localization length10

γ=0.4 (left)
γ=0.4 (right)

8642002468

disorder strength10Lattice models (class AII†)

19/20

・Symplectic Hatano-Nelson model (class AII†) cf. Okuma et al., PRL 124, 086801 (2020)

Reciprocity leads to☆ The delocalization is protected by reciprocity.
A small symmetry-breaking perturbation destroys the delocalization.
cf. Okuma & Sato, PRL 123, 097701 (2019)

Remarks

20/20

・Other symmetry classes
Dyson, PR 92, 1331 (1953)
In the presence of chiral (sublattice) symmetry,delocalization can appear even in Hermitian systems in 1D.

nonunitary delocalization(sublattice symmetry)
unitary delocalization(chiral symmetry)
・Random-matrix theory

Reciprocity leads to threefold universalityof non-Hermitian random matrices.
(Tomorrow’s talk by Hamazaki)
Hamazaki, Kawabata, Kura & Ueda, PRR 2, 023286 (2020)

Summary

arXiv:2005.00604

・We develop a scaling theory of localization in non-Hermitian systems.
・Non-Hermiticity introduces a new scale and breaks down theone-parameter scaling: two-parameter scaling.
・Threefold universality based on reciprocity.
(a) 10

64

2004

6

disorder strength

10localization(Hermitian)

21delocalization(non-Hermitian)

8class AII†

8

localization length8

localization length(Hermitian)(b) 10=0.4 (left)
=0.4 (right)
=1.0 (left)
=1.0 (right)

10

64

200.0

class A0.5

1.0

non-Hermiticity

1.5

2.0