Dislocation-Mediated Quantum Melting Aron Beekman

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Dislocation-Mediated Quantum Melting

Aron Beekman

Department of Physics & Research and Education Center for Natural Sciences Keio University (Hiyoshi, Yokohama)

aron@phys-h.keio.ac.jp

“Chiral Matter and Topology” workshop, NTU, December 7^th 2018

supported by:

• MEXT-Supported Program for the Strategic Research Foundation at Private Universities

“Topological Science” (grant no. S1511006)

• JSPS Kakenhi Grant-in-Aid for Early-Career Scientists (grant no. 18K13502)

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Dislocation-Mediated Quantum Melting

Aron Beekman

supported by:

• JSPS Kakenhi Grant-in-Aid for Early-Career Scientists (grant no. 18K13502)

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Dislocation-Mediated Quantum Melting

Aron Beekman

supported by:

• JSPS Kakenhi Grant-in-Aid for Early-Career Scientists (grant no. 18K13502) Nematic

Isotropic Smectic

Crystal

Figure 3 Schematic view of the local stripe order in the various phases discussed in the text. Here, we have assumed that the stripes maintain their integrity throughout, although in reality they must certainly become less and less well def ned as the system becomes increasingly quantum, until eventually they are not the correct variables for describing the important correlations in the system.

Heavy lines represent liquid-like stripes, along which the electrons can f ow, whereas the f lled circles representpinned, density-wave order along the stripes.

The stripes are shown executing more or less harmonic oscillations in the smectic phase. Two dislocations, which play an essential role in the smectic-to- nematic phase transition, are shown in the view of the nematic phase.

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Dislocations

Topological defect associated with translational order

Topological charge: Burgers vector B^a

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Outline

• Classical dislocations

• restricted motion

• interdependence with disclinations

• Dislocation condensation = quantum melting

• duality

• deconfinement of disclinations

• Recent developments

• critical properties of dislocation condensation

• relation to fractons

• superfluids without U (1) breaking

Aron Beekman Dislocation-Mediated Quantum Melting 2 / 21

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Dislocation motion

(a) initial dislocation (b) glide motion (c) climb motion

climb motion involves the addition/removal of (interstitial) particles and is suppressed ↔ particle number conservation

Glide constraint:

“dislocations can only move in the direction of their Burgers vector”

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Dislocations and disclinations

(d) dislocation (e) disclination (f) Volterra construction

• dislocation : Burgers vector B^a, torsion

• disclination : Frank scalar Ω, curvature

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Interdependence of dislocations and disclinations

(g) atoms (h) disclination (i) stack of dislocations

(j) disclination pair (k) two disclination pairs

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Defect-mediated melting

Nobel Prize in Physics 2016 citation :

‘‘for theoretical discoveries of topological phase transitions ...”

Berezinshkii–Kosterlitz–Thouless melting

Berezinskii 1970-71; Kosterlitz Thouless 1972-73

• in 2D, no true long-range order

• higher dimensions: order–disorder defect-unbinding phase transition

• 2+1D superfluid–Bose-Mott insulator quantum phase transition

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Defect-mediated melting

Nobel Prize in Physics 2016 citation :

‘‘for theoretical discoveries of topological phase transitions ...”

Berezinshkii–Kosterlitz–Thouless melting

• in 2D, no true long-range order

• higher dimensions: order–disorder defect-unbinding phase transition

• 2+1D superfluid–Bose-Mott insulator quantum phase transition

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Going quantum

• D-dim. quantum field theory ↔ D + 1-dim. statistical physics, e.g. 2D superfluid–insulator QPT is in the 3D XY universality class.

• Time axis is the additional dimension. Statistical physics of worldlines.

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Two-dimensional classical melting

• Unbinding of dislocations = loss of translational order

• Two types of topological defects

(a) dislocation – translational (b) disclination – rotational

Nelson Halperin 1978-79; Young 1979

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• Why is the ordinary solid-to-liquid transition first order?

• Simultaneous unbinding

Kleinert 1983

• Towards quantum melting, zero-temperature phase transition

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Kleinert 1983

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Kleinert 1983

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Two-dimensional quantum melting

(a) 2D bound pairs (b) 2D unbound

(c) 2+1D bound loops (d) 2D unbound worldlines

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Two-dimensional quantum melting

(a) 2D bound pairs (b) 2D unbound

(c) 2+1D bound loops (d) 2D unbound worldlines

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2+1D dislocation-mediated quantum melting

• ‘Statistical physics’/quantum partition sum of dislocation worldlines

• Role of inverse temperature is played by temporal correlations

• 2D quantum corresponds to 3D classical

• Proliferation of dislocation lines 3D classical

Kleinert 1980s

• Time direction manifestly different from space directions

• Condensation of dislocations = proliferation of worldlines 2+1D quantum

Zaanen Nussinov Mukhin 2004; Cvetkovic Zaanen 2006; AJB et al. 2017

Essence: the dislocation condensate is decribed by a collective complex field Ψ^a(x), a = x, y, |Ψ^a|²∼ density of ‘worldline tangle’ (1)

Ginzburg–Landau-type action LGL= X

a=x,y

1

2αa|Ψ^a|²+1 4βa|Ψ^a|⁴

+1

2γ|Ψ^x|²|Ψ^y|² (2)

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2+1D dislocation-mediated quantum melting

• ‘Statistical physics’/quantum partition sum of dislocation worldlines

• Role of inverse temperature is played by temporal correlations

• 2D quantum corresponds to 3D classical

• Proliferation of dislocation lines 3D classical

Kleinert 1980s

• Time direction manifestly different from space directions

• Condensation of dislocations = proliferation of worldlines 2+1D quantum

Zaanen Nussinov Mukhin 2004; Cvetkovic Zaanen 2006; AJB et al. 2017

Essence: the dislocation condensate is decribed by a collective complex field Ψ^a(x), a = x, y, |Ψ^a|²∼ density of ‘worldline tangle’ (1)

Ginzburg–Landau-type action LGL= X

a=x,y

1

2αa|Ψ^a|²+1 4βa|Ψ^a|⁴

+1

2γ|Ψ^x|²|Ψ^y|² (2)

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Dual gauge field theory of defect-mediated melting

• Duality mapping, analogous to vortex–boson / Abelian-Higgs duality

• Phonons are gauge bosons orstress photons

• Dislocations areshear stress charges

• A solid is astress vacuumorCoulomb gas of stress charges

• An hexatic is astress superconductor

• Dual Meissner effect: shear stress is expelled from the liquid crystal

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Dual stress effective action

Classical stress energy Esolid= ¹₂σ^am C_mnab⁻¹

| {z }

elastic moduli

σ^bn

Quantum stress Lagrangian Lsolid=_2ρ¹(στ^a)²+¹₂σ^amC_mnab⁻¹ σ^bn

Lsolid= 1

2(µκλ∂κb^a_λ)C_µνab⁻¹ (νρσ∂ρb^a_σ), LGL= X

a=x,y

1

2αa|Ψ^a|²+1 4βa|Ψ^a|⁴

+1

2γ|Ψ^x|²|Ψ^y|², Lcoupling= 1

2 X

a=x,y

|(∂µ− ib^a_µ− iλτ µa)Ψ^a|².

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| {z }

elastic moduli

σ^bn

Stress is conserved ∂τστâ+ ∂mσmâ = ∂µσµâ= 0 Dual stress gauge field σ_µâ= µνλ∂νbâ_λ, a = x, y

Lsolid= 1

2(µκλ∂κb^aλ)C_µνab⁻¹ (νρσ∂ρb^aσ), LGL= X

a=x,y

1

2αa|Ψ^a|²+1 4βa|Ψ^a|⁴

+1

2 X

a=x,y

|(∂µ− ib^aµ− iλτ µa)Ψ^a|².

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| {z }

elastic moduli

σ^bn

Lsolid= 1

2(µκλ∂κb^aλ)C_µνab⁻¹ (νρσ∂ρb^aσ),

LGL= X

a=x,y

1

2αa|Ψ^a|²+1 4βa|Ψ^a|⁴

+1

2 X

a=x,y

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| {z }

elastic moduli

σ^bn

Lsolid= 1

a=x,y

1

2αa|Ψ^a|²+1 4βa|Ψ^a|⁴

+1

2 X

a=x,y

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Main results

1 Phonons are gauge bosons

2 The disordered solid is a stress superconductor

3 The disordered solid is a real superfluid(longitudinal response)

4 Rotational Goldstone mode deconfines in qu. hexatic(transverse response) 5 Transverse phonon becomes gapped shear mode in quantum hexatic 6 The gapped shear mode is detectable by finite-momentum spectroscopy

AJB et al. Phys. Rep. 683, 1 (2017)

stress vacuum stress superconductor

dislocations unbound dislocations bound torque vacuum torque conﬁned torque superconductor

disclinations bound disclinations conﬁned disclinations unbound

solid superﬂuid quantum hexatic

ρ

longitudinal transverse

phonon phase mode

phase mode

rotational NG mode phonon response

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Disclination deconfinement

• Displacement field u^a(x)

• Rotation field ω^ab= ∂au^b(x) − ∂bu^a(x)

solid hexatic

Lagrangian stress uâ(∂_t²+ ∇²)uâ uâ(∂_t²+ ∇²+ |Ψ|²)uâ rotation ωâb∇²(∂t²+ ∇²)ωâb . . . + ωâb|Ψ|²(∂²t + ∇²)ωâb propagator stress _ω₂_+q¹ ₂ _ω₂_+q₂¹_+|Ψ|₂

rotation _q2(ω²¹+q²)

. . . +

_ω^|Ψ|2+q²² static limit stress ¹

q²

1 q²+|Ψ|² rotation _q¹₄

. . . +

^|Ψ|_q2²

For the same reason, rotational Nambu–Goldstone modes are absent in solid, but present in quantum hexatic.

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Helium monolayer experiments

Helium monolayers on ZYX exfoliated graphite S. Nakamura et al. PRB 94, 180501(R) (2016)

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• anomaly in specific heat : BKT-like defect-unbinding transition

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• three separate peaks

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Critical properties of solid-to-hexatic quantum melting

Effective field theory (Ginzburg–Landau) Lsolid= 1

a=x,y

1

2αa|Ψ^a|²+1 4βa|Ψ^a|⁴

+1

2 X

a=x,y

|(∂µ− ib^aµ− iλτ µa)Ψ^a|². Simplified to

Lsolid=1

2(∇ × b^x)²+1

2(∇ × b^y)², LGL=1

2m²(|Ψ^x|²+ |Ψ^y|²) +1

4λ(|Ψ^x|⁴+ |Ψ^y|⁴) +1

2g|Ψ^x|²|Ψ^y|², Lcoupling =1

2|(∇ − ieb^x)Ψ^x|²+1

2|(∇ − ieb^x)Ψ^y|²

• Onlym² andλ²: O(2) Wilson-Fisher theory

• withe: charged O(2), Abelian-Higgs model

• withg: O(2) × O(2), two-component BEC

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Critical properties of solid-to-hexatic quantum melting in d = 3

Abelian-Higgs d = 3

shown with FRG down to N = 2:

G. Fejos & T. Hatsuda

PRD 93, 121702 (2016) 96, 056018 (2017)

Two-component BEC d = 3

ε-expansion Ceccarelli et al.

PRA 92, 024513 (2016) 93, 033647 (2017)

Work in progress (with Gergely Fejos):

• FRG for charged O(2) × O(2) in d = 3, charged fixed points

• Influence of stress gauge field dynamics

• Influence of glide constraint

• Quantum critical exponent for specific heat

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Relation to fracton physics

Fractons: objects/particles with spatially restricted dynamics

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Superfluid sound

“Superfluid Goldstone mode arises only when U (1) particle conservation symmetry is broken, i.e. when the glide constraint is relaxed.”

Pretko & Radzihovsky arXiv:1808.05616 Kumar & Potter arXiv:1808.05621

Redundant Goldstone modes: Superfluid breaks boosts and particle conservation. Solid breaks boosts, translations and rotations.

Nicolis, Brauner, Watanabe, Hidaka, . . .

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Superfluid sound

“Superfluid Goldstone mode arises only when U (1) particle conservation symmetry is broken, i.e. when the glide constraint is relaxed.”

Pretko & Radzihovsky arXiv:1808.05616 Kumar & Potter arXiv:1808.05621

Redundant Goldstone modes: Superfluid breaks boosts and particle conservation. Solid breaks boosts, translations and rotations.

Nicolis, Brauner, Watanabe, Hidaka, . . .

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Summary

• Condensation of topological defects as symmetry-restoring phase transition

• Topological defects in solids study case for restricted mobility

• dislocations: glide constraint

• disclination: confinement

• Possibly novel critical behaviour

• Nature of superfluid sound

Collaborators:

Jan Zaanen Leiden Robert-Jan Slager Dresden Jaakko Nissinen Aalto Vladimir Cvetkovic

Kai Wu Zohar Nussinov St. Louis

Ke Liu Munich Gergely Fejos Keio U

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Assumptions and limitations

• Zero temperature

• Ginzburg–Landau → only near the phase transition

• London limit, phase fluctuations only

• Maximal crystalline correlations (collective physics only)

• No interstitials

• No disclinations

• Bosons onlybut 4-He and 3-He experiments similar

• Isotropic solid only

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3D quantum liquid crystals

t x y

µ

dislocation worldline

t x y

ν µ

dislocation worldsheet

• phonons are now two-form gauge fields bµν

• quantum versions of columnar, smectic and nematic liquid crystals

Goldstone modes phonons

rotational

solid columnar smectic nematic

2+1D 2/0 – 1/0 0/1

3+1D 3/0 2/0 1/1 0/3

AJB, J. Nissinen, K. Wu, J. Zaanen, Physical Review B 96, 165115 (2017)