Disorder-free localization in lattice gauge
theories
Markus Heyl

Center for Electronic Correlations and Magnetism
University of Augsburg

17.11.22

Workshop “Novel Quantum States in Condensed Matter 2022”

Work together withICTP & SISSA: M. Dalmonte, M. Brenes, A. Scardicchio, F. Surace, R. Fazio, S. Notarnicola
MPI-PKS: P. Karpov, R. Moessner, N. Chakraborty, G.-Y. Zhu, R. Verdel, A. Russomanno, Y.-P. Huang
Cologne: M. Schmitt

Gauge theories in physics

Wikipedia

Markus HeylBalents Nature ‘10

Disorder-free localization in gauge theories

Satzinger et al. Science ‘212

Dynamics in gauge theories

Mildenberger et al. arXiv:2203.08905

Quantum simulators
Particle collisions
Martinez, MH et al. Nature ‘16

CERN

Markus Heyl

Disorder-free localization in gauge theories

3

Goal:eChallenge
xplor: calce dynami
Dynamics in gauge theories
cal p

Mildenberger et al. arXiv:2203.08905

ulatedrincipl es a

Quantum simulators
Particle collisions

ynamics theoretind dycallynamic al univer
salitiesMartinez, MH et al. Nature ‘16

CERN

Markus Heyl

Disorder-free localization in gauge theories

4

New kinds of order

Temperature T

Conventional matter

0Markus Heyl

Ordered phases throughcoolingmagnetparamagnet

gc

Controlparameter gDisorder-free localization in gauge theories

5New kinds of order
Matter with gauge constraints

Ordered phases throughcoolingmagnet0paramagnet

gc

Controlparameter ggauge glassquasi-long range orderstrong multipartite entanglement
subdiffusion

Temperature T
Energy density

Temperature T

Conventional matterparamagnet

New types of phases
through “heating”

0gc

Controlparameter g

Not happening (generically) in thermodynamic systems → requires new mechanism
Markus Heyl

Disorder-free localization in gauge theories

6

Outline

Quantum thermalization

Disorder-free localization (DFL)

Breaking thermalization through gauge
invariance

DFL in 2D lattice gauge theories
Quantum link model (quantum spin ice,quantum dimer model)

New types of order

Gauge time crystals
Subdiffusion in the Kitaev honeycomb

Markus Heyl

Disorder-free localization in gauge theories

7

Quantum thermalization
Isolated quantum magnettime?

Realistic (homogeneous) systems thermalize
Ensemble equivalence: long-time steady state ↔ thermal Gibbs statesubsystems thermalize(remainder acting as an effective bath)Markus Heyl

Disorder-free localization in gauge theories

8Breaking ergodicity
No thermalization: avoid equipartitionenergy E

Occupation
probability

Occupation
probability

equal a priori
probabilities

energy E

New quantum states beyond thermodynamic constraints
Markus Heyl

Disorder-free localization in gauge theories

9Breaking ergodicity

Integrability
Fine-tuning● Not robust→ no stablenonthermal states●Polkovnikov et al. RMP ‘11

Markus Heyl

Disorder(Many-body)
Localization● Robust in 1D● But unstable beyond1D!
●Local constraints
Fractons● Quantum many-bodyscars● Gauge invariance→ Disorder-freelocalization●Vosk & Altman, Nandkishore &Huse, Annu. Rev. ‘15

Disorder-free localization in gauge theories

Smith et al. PRL‘17
Brenes, MH et al. PRL ‘18

10

Breaking ergodicity

Integrability
Fine-tuning● Not robust→ no stablenonthermal states●Polkovnikov et al. RMP ‘11

Markus HeylRobusterg→ od i c
Dis ity
ord breer- akifre nge l inoc a n
aliz y dati ime
DisorderLocal constraints
on nsi(Many-body)
Localization● Robust in 1D● But unstable beyond
1D!
●onFractons● Quantum many-bodyscars● Gauge invariance→ Disorder-freelocalization●Vosk & Altman, Nandkishore &Huse, Annu. Rev. ‘15

Disorder-free localization in gauge theories

Smith et al. PRL‘17
Brenes, MH et al. PRL ‘18

11

Disorder-free localization

Gauge invariance & conservation laws
Lattice gauge theory(LGT)
Local gauge symmetry → Generators:

Noether theorem: local symmetry → local conservation law
Extensive # local conserved operators/quantities(almost like integrable systems, but not equal to the # DOF)

LGTs have builtin constrained dynamics
Markus Heyl

Disorder-free localization in gauge theories

13

Fragmentation & Superselection sectors
Abelian LGT → Generators:

Hamiltonian eigenstates can be labeled by the eigenvalues of the generators
Superselection sector

Hamiltonian decomposes intodisconnected blocks(fragmentation builtin automatically)‘static background charge’
Markus Heyl

Disorder-free localization in gauge theories

14

Background charges & dynamics
Dynamics for superposition states:
Diagonalobservable(gauge invariant)“disorder average” even though the state might be homogeneous

Disorder-free localization from strong random background charges
Smith et al. PRL ‘17 & Brenes, MH et al. PRL ‘18

Markus Heyl

Disorder-free localization in gauge theories

15

DFL Mechanisms
Localization throughinterference

Localization through kinetic
constraints

Disorder landscape generated by
the background charges

Fragmentation in real & Hilbert
space→ Anderson or many-body
localization→ Quantum/classical percolation
problem(disorder often spatially correlated)→ this talkMarkus Heyl

Disorder-free localization in gauge theories

16

DFL in interacting 2D LGTs

U(1) quantum link model (QLM)

Potential energy
counting # flippableplaquettes

Correlated spin flip on
full plaquette

Local symmetry:

Quantum spin ice (Gn=0) → Emergent strong-coupling QED, quantum dimer model (G n=(-1)n), ...
Markus Heyl

Disorder-free localization in gauge theories

18Nonequilibrium dynamics
Initial condition:→ superposition over all superselection sectors

No propagation of quantum correlations
Is this just slow dynamics or reallynonergodic behavior?
Markus Heyl

Karpov, MH et al. PRL ‘21

Solution via classical networks

Disorder-free localization in gauge theories

Verdel, MH et al. PRB ‘21

19

Background charges & kinetic constraints
All 4 spins will be frozen foreverAt least 2 neighboring plaquettes
unflippable

Key question: kinetics local or can excitations still
propagate?
Markus Heyl

Disorder-free localization in gauge theories

20

Mapping onto percolation problem
Typical superselection
sectorKinetics & clusters

Percolation of clusters

fully-flippable sector

Karpov, MH et al. PRL ‘21

No percolation → Small clusters → Ergodicity breaking
Markus Heyl

Disorder-free localization in gauge theories(no background charges)

21

Signatures in quantum dynamics

locEnergy density in column

Create an energy
inhomogeneity in column

ergodic

Markus Heyl

Disorder-free localization in gauge theories

22

Localization protected quantum order
Gauge time crystal

Quasi long-range orderRussomanno, MH et al. PRR ‘20

Markus Heyl

Zhu & MH PRR ‘21

Disorder-free localization in gauge theories

23

Summary & outlook●●

Disorder-free localization as a new mechanism for ergodicity
breaking
Robust ergodicity breaking even in 2D (provided gauge
invariance is preserved)●

What new types of nonequilibrium phases are possible?

●

What about 3D? Can there still be a nonergodic phase?

●

What happens in the presence of matter in 2D?

●

Implications on high-temperature spectral functions?

Quantum localization transition
Chakraborty, MH et al. ‘22

Level-spacing statistics for individual clusters

clustersize

Clusters ergodic
Universality class of 2D site percolation
Markus Heyl

Quantum thermalization transition= classical percolation transition

Disorder-free localization in gauge theories

25

Signatures in quantum dynamics

locEnergy density in column

Create an energy
inhomogeneity in column

ergodic

Markus Heyl

Disorder-free localization in gauge theories

26

Method: classical networks
Challenge: no efficient compression of quantum states available 2D# amplitudes exponential in system size

Markus Heyl

Many-body
Disorder-freelocalization
localization
dynamicsinfrom
gauge
gauge
theories
invariance27
27

Method: classical networks
Challenge: no efficient compression of quantum states available 2D# amplitudes exponential in system size

Classical networks: “Don’t store. Generate on the fly.”
MachineSample using MC techniques
Schmitt & MH SciPost ‘18Markus Heyl

Many-body
Disorder-free
localization
localization
dynamicsinfrom
gauge
gauge
theories
invariance28
28

Method: classical networks
Classical network
Schmitt & MH SciPost ‘18

Effective classical Hamiltonian●Structure obtained from cumulant
expansion(around a classical limit)

Further variationally optimized● “Simple artificial neural network”●

Markus Heyl

Many-body
Disorder-free
localization
localization
dynamicsinfrom
gauge
gauge
theories
invariance29
29

Method: classical networks
Classical network
Schmitt & MH SciPost ‘18

Dynamics in zero charge
superselection sector(worst case for method)

Effective classical Hamiltonian●Structure obtained from cumulant
expansion(around a classical limit)

Further variationally optimized● “Simple artificial neural network”●Markus Heyl

Many-body
Disorder-free
localization
localization
dynamicsinfrom
gauge
gauge
theories
invariance3030