Fractionalization in spin systems An fRG perspective Dietrich - PowerPoint PPT Presentation

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

Fractional fRG High energy... “fractionalization” ATLAS Experiment c � 2016 CERN Shattering bound states by brute force

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

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?

Fractional fRG Spin systems & Spin liquids Heisenberg model: � J ij S µ i · S µ H = j � i , j � Magnetic phases (SU(2) symmetry breaking): � � S µ i � � = 0 i [F.L. Buessen, S. Trebst, ’16]

Fractional fRG Spin systems & Spin liquids Heisenberg model: � J ij S µ i · S µ H = j � i , j � Magnetic phases (SU(2) symmetry breaking): � � S µ i � � = 0 i [F.L. Buessen, S. Trebst, ’16] Spin liquids are... [L. Savary, L. Balents ’16] :

Fractional fRG Spin systems & Spin liquids Heisenberg model: � J ij S µ i · S µ H = j � i , j � Magnetic phases (SU(2) symmetry breaking): � � S µ i � � = 0 i [F.L. Buessen, S. Trebst, ’16] Spin liquids are... [L. Savary, L. Balents ’16] : ...are fancy and...

Fractional fRG Spin systems & Spin liquids Heisenberg model: � J ij S µ i · S µ H = j � i , j � Magnetic phases (SU(2) symmetry breaking): � � S µ i � � = 0 i [F.L. Buessen, S. Trebst, ’16] Spin liquids are... [L. Savary, L. Balents ’16] : ...are fancy and... hot and...

Fractional fRG Spin systems & Spin liquids Heisenberg model: � J ij S µ i · S µ H = j � i , j � Magnetic phases (SU(2) symmetry breaking): � � S µ i � � = 0 i [F.L. Buessen, S. Trebst, ’16] Spin liquids are... [L. Savary, L. Balents ’16] : ...are fancy and... hot and... and...

Fractional fRG Spin systems & Spin liquids Heisenberg model: � J ij S µ i · S µ H = j � i , j � Magnetic phases (SU(2) symmetry breaking): � � S µ i � � = 0 i [F.L. Buessen, S. Trebst, ’16] Spin liquids are... [L. Savary, L. Balents ’16] : ...are fancy and... hot and... and... I DUNNO!!!

Fractional fRG Spin systems & Spin liquids Heisenberg model: � J ij S µ i · S µ H = j � i , j � Magnetic phases (SU(2) symmetry breaking): � � S µ i � � = 0 i [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

Fractional fRG Pseudofermion representation Spin decomposition [A.A. Abrikosov, ’65] : i ≡ 1 S µ 2 f † i α σ µ αβ f i β GIVEN: “Microscopic” Spin model: j ≃ − J � � H UV = J S µ i · S µ f † i α f j α f † j β f i β 2 � i , j � � i , j �

Fractional fRG Pseudofermion representation Spin decomposition [A.A. Abrikosov, ’65] : i ≡ 1 S µ 2 f † i α σ µ αβ f i β GIVEN: “Microscopic” Spin model: j ≃ − J � � H UV = J S µ i · S µ f † i α f j α f † j β f i β 2 � i , j � � i , j � WANTED: Low-energy Spin liquid model [X.-G. Wen, ’02] � � � H IR ∼ Q ij f † i α f j α + ∆ ij ǫ αβ f † i α f † j β + h . c . + . . . � i , j � 246 different classes (symmetries of Q , ∆) Fractionalization, “Topological order” Ad hoc postulated...

Fractional fRG Pseudofermion functional RG � � ∂ t Γ k = 1 ∂ t R k 2 STr Γ (2) + R k k [C. Wetterich, ’93]   � i α ( i ∂ τ ) f i α − J k � � f † f † i α f j α f † Γ k = j β f i β  2 τ i � i , j �

Fractional fRG Pseudofermion functional RG � � ∂ t Γ k = 1 ∂ t R k 2 STr Γ (2) + R k k [C. Wetterich, ’93]   � i α ( i ∂ τ ) f i α − J k � � f † f † i α f j α f † Γ k = j β f i β  2 τ i � i , j � Let’s consider: SU(2) → SU( N ) “spins” on a 2D square lattice with antiferromagnetic coupling

Fractional fRG Pseudofermion functional RG � � ∂ t Γ k = 1 ∂ t R k 2 STr Γ (2) + R k k [C. Wetterich, ’93]   � i α ( i ∂ τ ) f i α − J k � � f † f † i α f j α f † Γ k = j β f i β  2 τ i � i , j � Let’s consider: SU(2) → SU( N ) “spins” on a 2D square lattice with antiferromagnetic coupling Result: J k → ∞

Fractional fRG Suppose we would bosonize... J k �� � n � RG 2 f † i α f j α f † � m Q Q † ij Q ij + Q ij f † � m Q , k Q † ij Q ij + U n > 2 Q † Q j β f i β � α j f α i � � � k

Fractional fRG Suppose we would bosonize... J k �� � n � RG 2 f † i α f j α f † � m Q Q † ij Q ij + Q ij f † � m Q , k Q † ij Q ij + U n > 2 Q † Q j β f i β � α j f α i � � � k Second order (well...) phase transition ( m Q ∼ J − 1 k ): J k → ∞ designates onset of “some kind” of order

Fractional fRG Suppose we would bosonize... J k �� � n � RG 2 f † i α f j α f † � m Q Q † ij Q ij + Q ij f † � m Q , k Q † ij Q ij + U n > 2 Q † Q j β f i β � α j f α i � � � k Second order (well...) phase transition ( m Q ∼ J − 1 k ): J k → ∞ designates onset of “some kind” of order Drawbacks of bosonization: Bias by choice of channel and/or massive cost Fierz ambiguity Spatially inhomogeneous phases?

Fractional fRG Infinitesimal explicit symmetry breaking [M. Salmhofer, C. Honerkamp, W. Metzner, O. Lauscher, ’04]   � i α ( i ∂ τ ) f i α − J k � � � f † f † i α f j α f † Q ij , k f † Γ k = j β f i β + i α f j α + ...  2 τ i � i , j � � i , j � 0 . 25 Q ∞ = 10 − 2 Q ∞ = 10 − 3 Q ∞ = 10 − 5 Minimal bias as Q ∞ → 0 0 . 2 Q mf ( T ) order parameter Q New vertices (Fierz-completeness!) 0 . 15 0 . 1 Exact for SU( N → ∞ ) 0 . 05 [DR, F.L. Buessen, M.M. Scherer, S. Trebst, S. Diehl, ’18] 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 temperature T / J

Fractional fRG Infinitesimal explicit symmetry breaking [M. Salmhofer, C. Honerkamp, W. Metzner, O. Lauscher, ’04]   � i α ( i ∂ τ ) f i α − J k � � � f † f † i α f j α f † Q ij , k f † Γ k = j β f i β + i α f j α + ...  2 τ i � i , j � � i , j � 0 . 25 Q ∞ = 10 − 2 Q ∞ = 10 − 3 Q ∞ = 10 − 5 Minimal bias as Q ∞ → 0 0 . 2 Q mf ( T ) order parameter Q New vertices (Fierz-completeness!) 0 . 15 0 . 1 Exact for SU( N → ∞ ) 0 . 05 [DR, F.L. Buessen, M.M. Scherer, S. Trebst, S. Diehl, ’18] 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 temperature T / J Wait a minute... symmetry breaking??

Fractional fRG Symmetries... what’s real? i ≡ 1 � H UV = J 2 f † S µ i · S µ S µ i α σ µ j , αβ f i β � i , j � The obvious: Global SU(N) : better not be broken for spin liquid

Fractional fRG Symmetries... what’s real? i ≡ 1 � H UV = J 2 f † S µ i · S µ S µ i α σ µ j , αβ f i β � i , j � converged order parameter | Q Λ → 0 | 0 . 25 1 0 . 9 magnetic susceptibility χ mag The obvious: 0 . 2 0 . 8 0 . 7 0 . 15 0 . 6 Global SU(N) : better not be 0 . 5 broken for spin liquid DONE 0 . 1 0 . 4 | Q MF | | Q FRG | 0 . 3 χ mag , MF 0 . 05 0 . 2 χ mag , FRG 0 . 1 0 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 temperature T / J 1

Fractional fRG Symmetries... what’s real? i ≡ 1 � H UV = J 2 f † S µ i · S µ S µ i α σ µ j , αβ f i β � i , j � converged order parameter | Q Λ → 0 | 0 . 25 1 0 . 9 magnetic susceptibility χ mag The obvious: 0 . 2 0 . 8 0 . 7 0 . 15 0 . 6 Global SU(N) : better not be 0 . 5 broken for spin liquid DONE 0 . 1 0 . 4 | Q MF | | Q FRG | 0 . 3 χ mag , MF Translation invariance : 0 . 05 0 . 2 χ mag , FRG 0 . 1 Wen’s classification, but... 0 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 temperature T / J 1

Fractional fRG Symmetries... what’s real? i ≡ 1 � H UV = J 2 f † S µ i · S µ S µ i α σ µ j , αβ f i β � i , j � converged order parameter | Q Λ → 0 | 0 . 25 1 0 . 9 magnetic susceptibility χ mag The obvious: 0 . 2 0 . 8 0 . 7 0 . 15 0 . 6 Global SU(N) : better not be 0 . 5 broken for spin liquid DONE 0 . 1 0 . 4 | Q MF | | Q FRG | 0 . 3 χ mag , MF Translation invariance : 0 . 05 0 . 2 χ mag , FRG 0 . 1 Wen’s classification, but... 0 0 0 0 . 1 0 . 2 0 . 3 0 . 4 0 . 5 temperature T / J 1 The not-so-obvious: Local U(1) : broken by Q ij ∼ � f † i α f j α � Artificial symmetry breaking? Actually, that’s not even all...

Fractional fRG Artificial & Local Pseudofermion Spin operator: S µ i = f † i α σ µ αβ f i α � � f i ↑ f i ↓ Reformulate (for N = 2): ψ i ≡ f † − f † i ↑ i ↓ Heisenberg model: � i ψ i σ µ, T � � j ψ j σ µ, T � j = J � � S µ i · S µ ψ † ψ † H = J · Tr Tr 16 � i , j � � i , j � ...invariant under ψ i → h i ψ i with h i ∈ SU(2).