The Schwinger model in the canonical formulation Urs Wenger Albert - PowerPoint PPT Presentation

The Schwinger model in the canonical formulation Urs Wenger Albert Einstein Center for Fundamental Physics University of Bern in collaboration with Patrick B¨ uhlmann XQCD 19, 26 June 2019, Tsukuba/Tokyo

Motivation for the canonical formulation ▸ Consider the grand-canonical partition function at finite µ : Z GC ( µ ) = Tr [ e −H( µ )/ T ] = Tr ∏ T t ( µ ) t ▸ The sign problem at finite density is a manifestation of huge cancellations between different states: ▸ all states are present for any µ and T ▸ some states need to cancel out at different µ and T ▸ In the canonical formulation: Z C ( N f ) = Tr N f [ e −H/ T ] = Tr ∏ T ( N f ) t t ▸ dimension of Fock space tremendously reduced ▸ less cancellations necessary: ( N Q ) = 0 for N Q ≠ 0 mod N c ▸ e.g. Z QCD C

Motivation for the canonical formulation ▸ Consider the grand-canonical partition function at finite µ : Z GC ( µ ) = Tr [ e −H( µ )/ T ] = Tr ∏ T t ( µ ) t ▸ The sign problem at finite density is a manifestation of huge cancellations between different states: ▸ all states are present for any µ and T ▸ some states need to cancel out at different µ and T ▸ In the canonical formulation: Z C ( N f ) = Tr N f [ e −H/ T ] = Tr ∏ T ( N f ) t t ▸ dimension of Fock space tremendously reduced ▸ less cancellations necessary: ( N Q ) = 0 for N Q ≠ 0 ▸ e.g. Z U ( 1 ) C

Motivation for the canonical formulation ▸ Consider the grand-canonical partition function at finite µ : Z GC ( µ ) = Tr [ e −H( µ )/ T ] = Tr ∏ T t ( µ ) t ▸ The sign problem at finite density is a manifestation of huge cancellations between different states: ▸ all states are present for any µ and T ▸ some states need to cancel out at different µ and T ▸ In the canonical formulation: Z C ( N f ) = Tr N f [ e −H/ T ] = Tr ∏ T ( N f ) t t ▸ dimension of Fock space tremendously reduced ▸ less cancellations necessary: ▸ e.g. ”Silver Blaze” phenomenon realised automatically

Motivation for canonical formulation of QCD Canonical transfer matrices can be obtained explicitly! ▸ based on the dimensional reduction of the QCD fermion determinant [Alexandru, Wenger ’10; Nagata, Nakamura ’10]

Motivation for canonical formulation of QCD Canonical transfer matrices can be obtained explicitly! ▸ based on the dimensional reduction of the QCD fermion determinant [Alexandru, Wenger ’10; Nagata, Nakamura ’10] Outline: ▸ Overview ▸ Definition of the transfer matrices in canonical formulation ▸ Relation to fermion loop and worldline formulations ▸ Hubbard model and Super Yang-Mills QM ▸ Schwinger model

Overview ▸ Identification of transfer matrices: ▸ Dimensional reduction in QCD [Alexandru, UW ’10] ▸ SUSY QM and SUSY Yang-Mills QM [Baumgartner, Steinhauer, UW ’12-’15] ▸ solution of the sign problem ▸ connection with fermion loop formulation ▸ QCD in the heavy-dense limit ▸ absence of the sign problem at strong coupling ▸ solution of the sign problem in the 3-state Potts model [Alexandru, Bergner, Schaich, UW ’18] ▸ Hubbard model [Burri, UW ’19] ▸ HS field can be integrated out analytically ▸ N f = 1 , 2 Schwinger model [B¨ uhlmann, UW ’19]

General construction ▸ For a generic Hamiltonian H with µ ≡ { µ σ } one has Tr [ e −H( µ )/ T ] Z GC ( µ ) = e − ∑ σ N σ µ σ / T ⋅ Z C ({ N σ }) = ∑ { N σ } where Z C ({ N σ }) = Tr ∏ t T ({ N σ }) . t ▸ Trotter decomposition and coherent state representation yields Z GC ( µ ) = ∫ D φ e − S b [ φ ] ∫ D ψ † D ψ e − S [ ψ † ,ψ,φ ; µ ] with Euclidean action S b and fermion matrix M S [ ψ † ,ψ,φ ; µ ] = ∑ ψ † σ M [ φ ; µ ] ψ σ . σ

Fermion matrix and dimensional reduction ▸ The fermion matrix M [ φ ; µ σ ] has the generic structure ± e µ σ C N t − 1 ⎛ ⎞ e − µ σ C ′ 0 B 0 ... ⎜ ⎟ 0 ⎜ ⎟ e µ σ C 0 e − µ σ C ′ B 1 0 ⎜ ⎟ ⋱ ⋮ 1 ⎜ ⎟ e µ σ C 1 M = 0 B 2 ⎜ ⎟ ⋮ ⋱ ⋱ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ e − µ σ C ′ B N t − 2 ⎝ ⎠ ± e − µ σ C ′ N t − 2 e µ σ C N t − 2 0 B N t − 1 N t − 1 for which the determinant can be reduced to det M [ φ ; µ σ ] = ∏ B t ⋅ det ( 1 ∓ e N t µ σ T [ φ ]) det ˜ t where T [ φ ] = T N t − 1 ⋅ ... ⋅ T 0 . ▸ M [ φ ; µ σ ] is ( L s ⋅ N t ) × ( L s ⋅ N t ) , while T [ φ ] is L s × L s .

Fermion matrix and canonical determinants ▸ Fugacity expansion det M [ φ ; µ σ ] = ∑ e − N σ µ σ / T ⋅ det N σ M [ φ ] N σ yields the canonical determinants det N σ M [ φ ] = ∑ det T / J [ φ ] = Tr [∏ T ( N σ ) ] . J / t t J where det T / J is the principal minor of order N σ . J / ▸ States are labeled by index sets J ⊂ { 1 ,..., L s } , ∣ J ∣ = N σ ▸ number of states grows exponentially with L s at half-filling N states = ( L s N σ ) = N principal minors ▸ sum can be evaluated stochastically with MC

Transfer matrices ▸ Use Cauchy-Binet formula K = ∑ det ( A ⋅ B ) / I / det A / I / J ⋅ det B / J / K J to factorize into product of transfer matrices ▸ Transfer matrices in sector N σ are hence given by (T ( N σ ) ) IK = det ˜ B t ⋅ det [T t ] / I / K t with Tr [∏ t T ( N σ ) ] = (T ( N σ ) N t − 1 ) IJ ⋅ (T ( N σ ) N t − 2 ) JK ⋅ ... ⋅ (T ( N σ ) ) LI . t 0 ▸ Finally, we have Z C ({ N σ }) = ∫ D φ e − S b [ φ ] ∏ t ) ( ∏ det [T σ t ] / B t ⋅ ∑ ∏ det ˜ J σ t − 1 / J σ t { J σ t } t σ where ∣ J σ t ∣ = N σ and J σ N t = J σ 0 .

Example: Hubbard model ▸ Consider the Hamiltonian for the Hubbard model H( µ ) = − c y ,σ + ∑ µ σ N x ,σ + U ∑ ∑ c † t σ ˆ x ,σ ˆ N x , ↑ N x , ↓ x ,σ x ⟨ x , y ⟩ ,σ with particle number N x ,σ = ˆ c † x ,σ ˆ c x ,σ . ▸ After Trotter decomposition and Hubbard-Stratonovich transformation we have Z GC ( µ ) = ∫ D ψ † D ψ D φρ [ φ ] e − ∑ σ S [ ψ † σ ,ψ σ ,φ ; µ σ ] with S [ ψ † σ ,ψ σ ,φ ; µ σ ] = ψ † σ M [ φ ; µ σ ] ψ σ , and hence = ∫ D φρ [ φ ] ∏ det M [ φ ; µ σ ] . σ

Example: Hubbard model ▸ The fermion matrix has the structure ± e µ σ C ( φ N t − 1 ) ⎛ ⎞ B 0 ... − e µ σ C ( φ 0 ) ⎜ ⎟ M [ φ ; µ σ ] = ⎜ ⎟ B ... 0 ⎜ ⎟ ⋮ ⋱ ⋱ ⋮ ⎝ ⎠ − e µ σ C ( φ N t − 2 ) 0 ... B for which the determinant can be reduced to det M [ φ ; µ σ ] = det B N t ⋅ det ( 1 ∓ e N t µ σ T [ φ ]) where T [ φ ] = B − 1 C ( φ N t − 1 ) ⋅ ... ⋅ B − 1 C ( φ 0 ) . ▸ Fugacity expansion yields the canonical determinants det M N σ [ φ ] = ∑ det T / J [ φ ] = Tr [∏ T ( N σ ) ] . J / t J t where det T / J is the principal minor of order N σ . J /

Example: Hubbard model ▸ Transfer matrices are hence given by (T t ) IK = det B ⋅ det [ B − 1 ⋅ C ( φ t ) ] I / / K = det B ⋅ det ( B − 1 ) / J ⋅ det C ( φ t ) / I / J / K ▸ Moreover, using the complementary cofactor we get I = ( − 1 ) p ( I , J ) det B IJ det B ⋅ det ( B − 1 ) / J / where p ( I , J ) = ∑ i ( I i + J i ) and HS field can be integrated out, K = δ JK ∏ det C ( φ t ) / φ x , t � ⇒ ∏ w x , t ≡ W ({ J σ t }) . J / x x ∉ J ▸ Finally, only sum over discrete index sets is left: t ) W ({ J σ Z C ({ N σ }) = ∑ ( ∏ t }) , ∣ J σ t ∣ = N σ ∏ det B J σ t − 1 J σ t σ { J σ t }

Example: Hubbard model t ) W ({ J σ Z C ({ N σ }) = ∑ (∏ t }) ∏ det B J σ t − 1 J σ t { J σ t } σ index sets J t : { 3,6 } { 4,5 } { 4,5 } { 2,7 } { 2,7 } { 3,7 }

Example: Hubbard model ▸ In d = 1 dimension the ’fermion bags’ det B IJ can be calculated analytically: and one can prove that det B IJ ≥ 0 for open b.c. ⇒ there is no sign problem ▸ For periodic b.c. there is no sign problem either, because ( L s → ∞ ) = Z obc ( L s → ∞ ) Z pbc C C

Example: Hubbard model ▸ Since our formulation is factorized in time, we have Z C ( L t ) E 0 = lim Z C ( L t + 1 ) = ⟨∏ ( det B J σ t − 1 J σ t + 1 ) t })⟩ 1 t + 1 W ({ J σ det B J σ t − 1 J σ t det B J σ t J σ L t → ∞ σ Z C ( L t + 1 ) γ =0.025, G=0.13 0.8 0.7 0.6 E 0 /L s Ls=4, N/L s =1 Ls=6, N/L s =1 0.5 Ls=8, N/L s =1 Ls=4, N/L s =1/2 Ls=8, N/L s =1/2 0.4 0.3 0 20 40 L t

Grand canonical gauge theories ▸ Consider gauge theory, e.g. Schwinger model or QCD: Z GC ( µ ) = ∫ D U D ψ D ψ e − S g [ U ]− S f [ ψ,ψ, U ; µ ] where S g [ U ] = β ∑ [ 1 − 1 2 ( U P + U † P )] , P S f [ ψ,ψ, U ; µ ] = ψ M [ U ; µ ] ψ . ▸ for QCD: d = 4 , U ∈ SU ( N c ) ▸ for the Schwinger model: d = 2 , U ∈ U ( 1 ) ▸ Integrating out the Grassmann fields for N f flavours yields Z GC ( µ ) = ∫ D U e − S g [ U ] ( det M [ U ; µ ]) N f .

Dimensional reduction of gauge theories ▸ Consider the Wilson fermion matrix for a single quark with chemical potential µ : ± P − A − ⎛ ⎞ P + A + B 0 ⎜ ⎟ 0 L t − 1 ⎜ ⎟ P − A − P + A + B 1 ⎜ ⎟ ⋱ 0 1 ⎜ ⎟ M ± ( µ ) = P − A − B 2 ⎜ ⎟ ⋱ ⋱ 1 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ P + A + ⎝ ⎠ ± P + A + L t − 2 P − B L t − 1 L t − 1 ▸ B t are (spatial) Wilson Dirac operators on time-slice t , 2 ( I ∓ Γ 4 ) , ▸ Dirac projectors P ± = 1 ▸ temporal hoppings are t = e + µ ⋅ I d × d ⊗ U t = ( A − t ) − 1 A + ▸ all blocks are ( d ⋅ N c ⋅ L 3 s ) -matrices s × d ⋅ N c ⋅ L 3