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Fractional fRG Fractionalization in spin systems An fRG perspective Dietrich Roscher with Michael M. Scherer, Nico Gneist, Simon Trebst, Sebastian Diehl arXiv:1905.01060 Institute for Theoretical Physics / Universitt zu Kln Cold Quantum

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SLIDE 1

Fractional fRG

Fractionalization in spin systems

An fRG perspective

Dietrich Roscher

with Michael M. Scherer, Nico Gneist, Simon Trebst, Sebastian Diehl

arXiv:1905.01060

Institute for Theoretical Physics / Universität zu Köln

Cold Quantum Coffee, Heidelberg May 7th, 2019

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SLIDE 2

Fractional fRG

High energy... “fractionalization”

ATLAS Experiment c 2016 CERN

Shattering bound states by brute force

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SLIDE 3

Fractional fRG

Fractionalization in solids

[R. Willet, J.P. Eisenstein, H.L. Störmer, D.C. Tsui, A.C. Gossard, J.H. English, ’87]

Low-energy Collective Effect

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SLIDE 4

Fractional fRG

Fractionalization in solids

[R. Willet, J.P. Eisenstein, H.L. Störmer, D.C. Tsui, A.C. Gossard, J.H. English, ’87]

Low-energy Collective Effect

Seems perfectly suited for field theory & (f)RG Extremely rich/confusing field: anyons, majoranas, gauge fields... Actual physical observables/interpretation?

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SLIDE 5

Fractional fRG

Spin systems & Spin liquids

Heisenberg model: H =

  • i,j

JijSµ

i · Sµ j

Magnetic phases (SU(2) symmetry breaking):

  • i

Sµ

i = 0

[F.L. Buessen, S. Trebst, ’16]

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SLIDE 6

Fractional fRG

Spin systems & Spin liquids

Heisenberg model: H =

  • i,j

JijSµ

i · Sµ j

Magnetic phases (SU(2) symmetry breaking):

  • i

Sµ

i = 0

[F.L. Buessen, S. Trebst, ’16]

Spin liquids are...

[L. Savary, L. Balents ’16] :

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SLIDE 7

Fractional fRG

Spin systems & Spin liquids

Heisenberg model: H =

  • i,j

JijSµ

i · Sµ j

Magnetic phases (SU(2) symmetry breaking):

  • i

Sµ

i = 0

[F.L. Buessen, S. Trebst, ’16]

Spin liquids are...

[L. Savary, L. Balents ’16] : ...are fancy and...

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SLIDE 8

Fractional fRG

Spin systems & Spin liquids

Heisenberg model: H =

  • i,j

JijSµ

i · Sµ j

Magnetic phases (SU(2) symmetry breaking):

  • i

Sµ

i = 0

[F.L. Buessen, S. Trebst, ’16]

Spin liquids are...

[L. Savary, L. Balents ’16] : ...are fancy and... hot and...

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SLIDE 9

Fractional fRG

Spin systems & Spin liquids

Heisenberg model: H =

  • i,j

JijSµ

i · Sµ j

Magnetic phases (SU(2) symmetry breaking):

  • i

Sµ

i = 0

[F.L. Buessen, S. Trebst, ’16]

Spin liquids are...

[L. Savary, L. Balents ’16] : ...are fancy and... hot and... and...

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SLIDE 10

Fractional fRG

Spin systems & Spin liquids

Heisenberg model: H =

  • i,j

JijSµ

i · Sµ j

Magnetic phases (SU(2) symmetry breaking):

  • i

Sµ

i = 0

[F.L. Buessen, S. Trebst, ’16]

Spin liquids are...

[L. Savary, L. Balents ’16] : ...are fancy and... hot and... and... I DUNNO!!!

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SLIDE 11

Fractional fRG

Spin systems & Spin liquids

Heisenberg model: H =

  • i,j

JijSµ

i · Sµ j

Magnetic phases (SU(2) symmetry breaking):

  • i

Sµ

i = 0

[F.L. Buessen, S. Trebst, ’16]

Spin liquids are...

[L. Savary, L. Balents ’16] : ...are fancy and... hot and... and... I DUNNO!!!

Spin systems with non-magnetic but non-trivial ground states Long-range entanglement Topological order Fractionalization

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SLIDE 12

Fractional fRG

Pseudofermion representation

Spin decomposition [A.A. Abrikosov, ’65]: Sµ

i ≡ 1

2f †

iασµ αβfiβ

GIVEN: “Microscopic” Spin model: HUV = J

  • i,j

Sµ

i · Sµ j ≃ −J

2

  • i,j

f †

iαfjαf † jβfiβ

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SLIDE 13

Fractional fRG

Pseudofermion representation

Spin decomposition [A.A. Abrikosov, ’65]: Sµ

i ≡ 1

2f †

iασµ αβfiβ

GIVEN: “Microscopic” Spin model: HUV = J

  • i,j

Sµ

i · Sµ j ≃ −J

2

  • i,j

f †

iαfjαf † jβfiβ

WANTED: Low-energy Spin liquid model [X.-G. Wen, ’02] HIR ∼

  • i,j
  • Qijf †

iαfjα + ∆ijǫαβf † iαf † jβ + h.c. + . . .

  • 246 different classes (symmetries of Q, ∆)

Fractionalization, “Topological order” Ad hoc postulated...

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SLIDE 14

Fractional fRG

Pseudofermion functional RG

∂tΓk = 1 2STr

  • ∂tRk

Γ(2)

k

+ Rk

  • [C. Wetterich, ’93]

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ

 

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SLIDE 15

Fractional fRG

Pseudofermion functional RG

∂tΓk = 1 2STr

  • ∂tRk

Γ(2)

k

+ Rk

  • [C. Wetterich, ’93]

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ

  Let’s consider: SU(2) → SU(N) “spins”

  • n a 2D square lattice

with antiferromagnetic coupling

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SLIDE 16

Fractional fRG

Pseudofermion functional RG

∂tΓk = 1 2STr

  • ∂tRk

Γ(2)

k

+ Rk

  • [C. Wetterich, ’93]

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ

  Let’s consider: SU(2) → SU(N) “spins”

  • n a 2D square lattice

with antiferromagnetic coupling Result: Jk → ∞

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SLIDE 17

Fractional fRG

Suppose we would bosonize...

Jk 2 f †

iαfjαf † jβfiβ

  • mQQ†

ijQij + Qijf † αjfαi RG

  • mQ,kQ†

ijQij + Un>2 k

  • Q†Q

n

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SLIDE 18

Fractional fRG

Suppose we would bosonize...

Jk 2 f †

iαfjαf † jβfiβ

  • mQQ†

ijQij + Qijf † αjfαi RG

  • mQ,kQ†

ijQij + Un>2 k

  • Q†Q

n Second order (well...) phase transition (mQ ∼ J−1

k ):

Jk → ∞ designates onset of “some kind” of order

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SLIDE 19

Fractional fRG

Suppose we would bosonize...

Jk 2 f †

iαfjαf † jβfiβ

  • mQQ†

ijQij + Qijf † αjfαi RG

  • mQ,kQ†

ijQij + Un>2 k

  • Q†Q

n Second order (well...) phase transition (mQ ∼ J−1

k ):

Jk → ∞ designates onset of “some kind” of order Drawbacks of bosonization: Bias by choice of channel and/or massive cost Fierz ambiguity Spatially inhomogeneous phases?

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SLIDE 20

Fractional fRG

Infinitesimal explicit symmetry breaking

[M. Salmhofer, C. Honerkamp, W. Metzner, O. Lauscher, ’04]

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ+

  • i,j

Qij,kf †

iαfjα + ...

  Minimal bias as Q∞ → 0 New vertices (Fierz-completeness!) Exact for SU(N → ∞)

[DR, F.L. Buessen, M.M. Scherer, S. Trebst, S. Diehl, ’18]

0.05 0.1 0.15 0.2 0.25 0.1 0.2 0.3 0.4 0.5

  • rder parameter Q

temperature T/J Q∞ = 10−2 Q∞ = 10−3 Q∞ = 10−5 Qmf(T)

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SLIDE 21

Fractional fRG

Infinitesimal explicit symmetry breaking

[M. Salmhofer, C. Honerkamp, W. Metzner, O. Lauscher, ’04]

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ+

  • i,j

Qij,kf †

iαfjα + ...

  Minimal bias as Q∞ → 0 New vertices (Fierz-completeness!) Exact for SU(N → ∞)

[DR, F.L. Buessen, M.M. Scherer, S. Trebst, S. Diehl, ’18]

0.05 0.1 0.15 0.2 0.25 0.1 0.2 0.3 0.4 0.5

  • rder parameter Q

temperature T/J Q∞ = 10−2 Q∞ = 10−3 Q∞ = 10−5 Qmf(T)

Wait a minute... symmetry breaking??

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SLIDE 22

Fractional fRG

Symmetries... what’s real?

HUV = J

  • i,j

Sµ

i · Sµ j ,

Sµ

i ≡ 1

2f †

iασµ αβfiβ

The obvious: Global SU(N): better not be broken for spin liquid

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SLIDE 23

Fractional fRG

Symmetries... what’s real?

HUV = J

  • i,j

Sµ

i · Sµ j ,

Sµ

i ≡ 1

2f †

iασµ αβfiβ

The obvious: Global SU(N): better not be broken for spin liquid DONE

0.05 0.1 0.15 0.2 0.25 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 converged order parameter |QΛ→0| magnetic susceptibility χmag temperature T/J1 |QMF| |QFRG| χmag,MF χmag,FRG

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SLIDE 24

Fractional fRG

Symmetries... what’s real?

HUV = J

  • i,j

Sµ

i · Sµ j ,

Sµ

i ≡ 1

2f †

iασµ αβfiβ

The obvious: Global SU(N): better not be broken for spin liquid DONE Translation invariance: Wen’s classification, but...

0.05 0.1 0.15 0.2 0.25 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 converged order parameter |QΛ→0| magnetic susceptibility χmag temperature T/J1 |QMF| |QFRG| χmag,MF χmag,FRG

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SLIDE 25

Fractional fRG

Symmetries... what’s real?

HUV = J

  • i,j

Sµ

i · Sµ j ,

Sµ

i ≡ 1

2f †

iασµ αβfiβ

The obvious: Global SU(N): better not be broken for spin liquid DONE Translation invariance: Wen’s classification, but...

0.05 0.1 0.15 0.2 0.25 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 converged order parameter |QΛ→0| magnetic susceptibility χmag temperature T/J1 |QMF| |QFRG| χmag,MF χmag,FRG

The not-so-obvious: Local U(1): broken by Qij ∼ f †

iαfjα

Artificial symmetry breaking? Actually, that’s not even all...

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SLIDE 26

Fractional fRG

Artificial & Local

Pseudofermion Spin operator: Sµ

i = f † iασµ αβfiα

Reformulate (for N = 2): ψi ≡

  • fi↑

fi↓ f †

i↑

−f †

i↓

  • Heisenberg model:

H = J

  • i,j

Sµ

i · Sµ j = J

16

  • i,j

Tr

  • ψ†

i ψi σµ,T

· Tr

  • ψ†

j ψj σµ,T

...invariant under ψi → hiψi with hi ∈ SU(2).

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SLIDE 27

Fractional fRG

Hilbert spaces & Constraints

Spin operator: {| ↑, | ↓}

?

← → Pseudofermions: {|0, 0, |0, 1, |1, 0, |1, 1}

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SLIDE 28

Fractional fRG

Hilbert spaces & Constraints

Spin operator: {| ↑, | ↓}

?

← → Pseudofermions: {|0, 0, |0, 1, |1, 0, |1, 1} To be enforced: f †

iαfiα = 1

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SLIDE 29

Fractional fRG

Hilbert spaces & Constraints

Spin operator: {| ↑, | ↓}

?

← → Pseudofermions: {|0, 0, |0, 1, |1, 0, |1, 1} To be enforced: f †

iαfiα = 1 ⇐

⇒ 1

2Tr

  • ψ†

i σzψi

  • = 0

No local SU(2) invariance!

U(1) still valid!

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SLIDE 30

Fractional fRG

Hilbert spaces & Constraints

Spin operator: {| ↑, | ↓}

?

← → Pseudofermions: {|0, 0, |0, 1, |1, 0, |1, 1} To be enforced: f †

iαfiα = 1 ⇐

⇒ 1

2Tr

  • ψ†

i σzψi

  • = 0

No local SU(2) invariance!

U(1) still valid!

However, gauging [I. Affleck, Z. Zou, T. Hsu, P.W. Anderson ’88] L = 1 2

  • i

Tr

  • ψ†

i (i∂τ)ψi

  • − H :

(i∂t) → (i∂t) + Aµσµ enforces constraint and restores full local SU(2). Emergent gauge fields in spin liquids!

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SLIDE 31

Fractional fRG

Implementing the constraint

Issues of the SU(N) gauge theory: Overcompleteness f †

αfα = 1

f †

↑ f † ↓ = 0

f↑ f↓ = 0

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SLIDE 32

Fractional fRG

Implementing the constraint

Issues of the SU(N) gauge theory: Overcompleteness Non-abelian gauge theory... ...on a lattice - monopoles?

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SLIDE 33

Fractional fRG

Implementing the constraint

Issues of the SU(N) gauge theory: Overcompleteness Non-abelian gauge theory... ...on a lattice - monopoles? Reliable truncation scheme?

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SLIDE 34

Fractional fRG

Implementing the constraint

Issues of the SU(N) gauge theory: Overcompleteness Non-abelian gauge theory... ...on a lattice - monopoles? Reliable truncation scheme? Gauge invariant regularization/mWTI?

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SLIDE 35

Fractional fRG

Implementing the constraint

Alternatively: add µPF = i

2πT

[V.N. Popov, S.A. Fedotov ’88]

No local SU(N) to begin with Simple implementation Straightforward truncation Physical interpretation!?

0.2 0.4 0.6 0.8 1 1.2 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 divergence scale Λdiv temperature T/J1 µΛ = 0 µΛ = µPF Without Katanin With Katanin

slide-36
SLIDE 36

Fractional fRG

Symmetries... what’s real?

HUV = J

  • i,j

Sµ

i · Sµ j ,

Sµ

i ≡ 1

2f †

iασµ αβfiβ

The obvious: Global SU(N): better not be broken for spin liquid DONE Translation invariance: Wen’s classification, but...

0.05 0.1 0.15 0.2 0.25 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 converged order parameter |QΛ→0| magnetic susceptibility χmag temperature T/J1 |QMF| |QFRG| χmag,MF χmag,FRG

The not-so-obvious: Local U(1): broken by Qij ∼ f †

iαfjα

Artificial symmetry breaking? Actually, that’s not even all... DONE

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SLIDE 37

Fractional fRG

Spatial Inhomogeneity - Staggered Flux Spin Liquid

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ+

  • i,j

Qij,kf †

iαfjα + ...

  Spin liquid states at large N: Naïve implementation: homogeneous “BZA” state

slide-38
SLIDE 38

Fractional fRG

Spatial Inhomogeneity - Staggered Flux Spin Liquid

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ+

  • i,j

Qij,kf †

iαfjα + ...

  Spin liquid states at large N: Naïve implementation: homogeneous “BZA” state

slide-39
SLIDE 39

Fractional fRG

Spatial Inhomogeneity - Staggered Flux Spin Liquid

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ+

  • i,j

Qij,kf †

iαfjα + ...

  Spin liquid states at large N: Naïve implementation: homogeneous “BZA” state Expected result: inhomogeneous “flux state”

[e.g. D. Arovas, A. Auerbach, ’88]

slide-40
SLIDE 40

Fractional fRG

Spatial Inhomogeneity - Staggered Flux Spin Liquid

Γk =

  • τ

 

i

f †

iα(i∂τ)fiα − Jk

2

  • i,j

f †

iαfjαf † jβfiβ+

  • i,j

Qij,kf †

iαfjα + ...

  Spin liquid states at large N: Naïve implementation: homogeneous “BZA” state Expected result: inhomogeneous “flux state”

[e.g. D. Arovas, A. Auerbach, ’88]

How to include spatially structured order parameters? Brute force is not and option!

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SLIDE 41

Fractional fRG

Spatial Inhomogeneity - Staggered Flux Spin Liquid

Solution: Introduce of M artificial sublattices fi → Ψ = (f A

i , f B i , f C i , ...)

f †

iαfjαf † jβfiβ −

→ (Ψ†

αηACΨα)(Ψ† βηCAΨβ) + ...

Finite-ranged OPs addressed by M-dimensional matrices Formally: mapping part of the translation group to an SU(M)

slide-42
SLIDE 42

Fractional fRG

Spatial Inhomogeneity - Staggered Flux Spin Liquid

Solution: Introduce of M artificial sublattices fi → Ψ = (f A

i , f B i , f C i , ...)

f †

iαfjαf † jβfiβ −

→ (Ψ†

αηACΨα)(Ψ† βηCAΨβ) + ...

Finite-ranged OPs addressed by M-dimensional matrices Formally: mapping part of the translation group to an SU(M) Advantage: straightforward treatment within fRG, “minimal bias” Cost: 3 → 24 flow equations for 4-sublattice Heisenberg model

slide-43
SLIDE 43

Fractional fRG

Spatial Inhomogeneity - Staggered Flux Spin Liquid

Solution: Introduce of M artificial sublattices fi → Ψ = (f A

i , f B i , f C i , ...)

f †

iαfjαf † jβfiβ −

→ (Ψ†

αηACΨα)(Ψ† βηCAΨβ) + ...

Finite-ranged OPs addressed by M-dimensional matrices Formally: mapping part of the translation group to an SU(M) Advantage: straightforward treatment within fRG, “minimal bias” Cost: 3 → 24 flow equations for 4-sublattice Heisenberg model Bonus: Group theory analysis of symmetry breaking patterns

slide-44
SLIDE 44

Fractional fRG

Symmetries... what’s real?

HUV = J

  • i,j

Sµ

i · Sµ j ,

Sµ

i ≡ 1

2f †

iασµ αβfiβ

The obvious: Global SU(N): better not be broken for spin liquid DONE Translation invariance: Wen’s classification DONE

0.05 0.1 0.15 0.2 0.25 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 converged order parameter |QΛ→0| magnetic susceptibility χmag temperature T/J1 |QMF| |QFRG| χmag,MF χmag,FRG

The not-so-obvious: Local U(1): broken by Qij ∼ f †

iαfjα DONE

Artificial symmetry breaking? Actually, that’s not even all... DONE

slide-45
SLIDE 45

Fractional fRG

Real world

  • well...

challenge: J1 − J2 Heisenberg model

arXiv:1905.01060

SU(N): Competition of SL ordering patterns

slide-46
SLIDE 46

Fractional fRG

Real world

  • well...

challenge: J1 − J2 Heisenberg model

arXiv:1905.01060

0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.2 0.9 0.95 1 1.05 1.1 1.15 1.2

  • rder parameters |Q(n)|

coupling ratio g Q(1..4) Q(5..6)

SU(N): Competition of SL ordering patterns

slide-47
SLIDE 47

Fractional fRG

Real world

  • well...

challenge: J1 − J2 Heisenberg model

arXiv:1905.01060

SU(N): Competition of SL ordering patterns SU(2): Localization of magnetic phases consistent with literature

slide-48
SLIDE 48

Fractional fRG

Real world

  • well...

challenge: J1 − J2 Heisenberg model

arXiv:1905.01060

SU(N): Competition of SL ordering patterns SU(2): Localization of magnetic phases consistent with literature Non-magnetic regime is not a spin liquid of the type presented here fRG currently not sensitive to 4/6/8... fermion order parameters

slide-49
SLIDE 49

Fractional fRG

Conclusions & Outlook

Achievements of spin liquid fRG: Systematic emergence of spin liquids from microscopic spin models Clear picture of underlying ordering mechanisms Successful proof of principle Systematic inclusion of the fermion number constraint Spatially structured order parameters Exclusion of bilinear spin liquids for the J1 − J2 Heisenberg model

slide-50
SLIDE 50

Fractional fRG

Conclusions & Outlook

Achievements of spin liquid fRG: Systematic emergence of spin liquids from microscopic spin models Clear picture of underlying ordering mechanisms Successful proof of principle Systematic inclusion of the fermion number constraint Spatially structured order parameters Exclusion of bilinear spin liquids for the J1 − J2 Heisenberg model Future prospects: Reproduce further exact solutions (Kitaev model) Analyze more complicated geometries Systematic scan of candidate materials

slide-51
SLIDE 51

Fractional fRG

Conclusions & Outlook

Achievements of spin liquid fRG: Systematic emergence of spin liquids from microscopic spin models Clear picture of underlying ordering mechanisms Successful proof of principle Systematic inclusion of the fermion number constraint Spatially structured order parameters Exclusion of bilinear spin liquids for the J1 − J2 Heisenberg model Future prospects: Reproduce further exact solutions (Kitaev model) Analyze more complicated geometries Systematic scan of candidate materials

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