Integrable states, exact overlaps, and the Boundary Yang-Baxter relation Balázs Pozsgay „Premium Postdoctoral Program” Hungarian Academy of Sciences and BME Statistical Field Theory Research group, Budapest University of Technology and Economics RAQIS, Annecy, 13. September 2018
Work with Lorenzo Piroli (SISSA) and Eric Vernier (Oxford)
Outline Non-equilibrium dynamics: | Ψ( t ) � = e − iHt | Ψ 0 � | Ψ 0 � → → �O ( t ) � =? Overlaps: � Ψ 0 |{ λ } N � Integrable initial states | Ψ 0 � Relation to boundary integrability Exact solutions for quantum quenches (root densities ρ ( λ ) )
Outline Non-equilibrium dynamics: | Ψ( t ) � = e − iHt | Ψ 0 � | Ψ 0 � → → �O ( t ) � =? Overlaps: � Ψ 0 |{ λ } N � Integrable initial states | Ψ 0 � Relation to boundary integrability Exact solutions for quantum quenches (root densities ρ ( λ ) )
Outline Non-equilibrium dynamics: | Ψ( t ) � = e − iHt | Ψ 0 � | Ψ 0 � → → �O ( t ) � =? Overlaps: � Ψ 0 |{ λ } N � Integrable initial states | Ψ 0 � Relation to boundary integrability Exact solutions for quantum quenches (root densities ρ ( λ ) )
Outline Non-equilibrium dynamics: | Ψ( t ) � = e − iHt | Ψ 0 � | Ψ 0 � → → �O ( t ) � =? Overlaps: � Ψ 0 |{ λ } N � Integrable initial states | Ψ 0 � Relation to boundary integrability Exact solutions for quantum quenches (root densities ρ ( λ ) )
Outline Non-equilibrium dynamics: | Ψ( t ) � = e − iHt | Ψ 0 � | Ψ 0 � → → �O ( t ) � =? Overlaps: � Ψ 0 |{ λ } N � Integrable initial states | Ψ 0 � Relation to boundary integrability Exact solutions for quantum quenches (root densities ρ ( λ ) )
Integrable initial states Overlaps display the pair structure, are non-zero only if { λ + , − λ + } N / 2 { λ + , − λ + } ( N − 1 ) / 2 ∪ { 0 } { λ } N = or They take a factorized form. In simple cases N / 2 |� Ψ 0 |{ λ } N �| 2 j ) × det G + � v ( λ + �{ λ } N |{ λ } N � = det G − j = 1 More generally some linear combination of these, with different v -functions
Integrable initial states Overlaps display the pair structure, are non-zero only if { λ + , − λ + } N / 2 { λ + , − λ + } ( N − 1 ) / 2 ∪ { 0 } { λ } N = or They take a factorized form. In simple cases N / 2 |� Ψ 0 |{ λ } N �| 2 j ) × det G + � v ( λ + �{ λ } N |{ λ } N � = det G − j = 1 More generally some linear combination of these, with different v -functions
Integrable initial states In spin-1/2 XXZ: all two site states | ψ � ∈ C 2 ⊗ C 2 | Ψ � = ⊗ L / 2 j = 1 | ψ � , In higher spin and higher rank models: A subset of two-site states Matrix Product States (from AdS/CFT) ω ( i ) , i = 1 , . . . , N N � ω ( i 1 ) ω ( i 2 ) . . . ω ( i L ) � � | MPS � = | i 1 , i 2 , . . . , i L � tr 0 i 1 ,..., i L = 1 M de Leeuw, C. Kristjansen, K. Zarembo, I. Buhl-Mortensen, S. Mori, G. Linardopoulos, 2015-2018 Overlaps calculated in SO ( 6 ) -symmetric spin chain
Integrable initial states In spin-1/2 XXZ: all two site states | ψ � ∈ C 2 ⊗ C 2 | Ψ � = ⊗ L / 2 j = 1 | ψ � , In higher spin and higher rank models: A subset of two-site states Matrix Product States (from AdS/CFT) ω ( i ) , i = 1 , . . . , N N � ω ( i 1 ) ω ( i 2 ) . . . ω ( i L ) � � | MPS � = | i 1 , i 2 , . . . , i L � tr 0 i 1 ,..., i L = 1 M de Leeuw, C. Kristjansen, K. Zarembo, I. Buhl-Mortensen, S. Mori, G. Linardopoulos, 2015-2018 Overlaps calculated in SO ( 6 ) -symmetric spin chain
Integrable initial states In spin-1/2 XXZ: all two site states | ψ � ∈ C 2 ⊗ C 2 | Ψ � = ⊗ L / 2 j = 1 | ψ � , In higher spin and higher rank models: A subset of two-site states Matrix Product States (from AdS/CFT) ω ( i ) , i = 1 , . . . , N N � ω ( i 1 ) ω ( i 2 ) . . . ω ( i L ) � � | MPS � = | i 1 , i 2 , . . . , i L � tr 0 i 1 ,..., i L = 1 M de Leeuw, C. Kristjansen, K. Zarembo, I. Buhl-Mortensen, S. Mori, G. Linardopoulos, 2015-2018 Overlaps calculated in SO ( 6 ) -symmetric spin chain
Integrable initial states S. Ghoshal and A. B. Zamolodchikov, Int. J. Mod. Phys., A9 3841, hep-th/9306002 ξ n � Ψ 0 | . . . ξ 2 ξ 1 1 L-1 L 2 3 . . . 4 . . . | Ψ 0 � n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � j = 1
Integrable initial states S. Ghoshal and A. B. Zamolodchikov, Int. J. Mod. Phys., A9 3841, hep-th/9306002 ξ n � Ψ 0 | . . . ξ 2 ξ 1 1 L-1 L 2 3 . . . 4 . . . | Ψ 0 � n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � j = 1
K + K − K + K − K + K − K + K − � Ψ 0 | | Ψ 0 � K + K − = K + K − K + K − . . . . . . ξ n ξ 2 ξ 1 ξ n ξ 2 ξ 1 For two site product states in XXZ: boundary transfer matrix τ ( u ) = Tr 0 { K + ( u ) T ( u ) K − ( u ) T − 1 ( − u ) } BP, J. Stat. Mech. (2013) P10028
Integrable MPS N � ω ( i 1 ) ω ( i 2 ) . . . ω ( i L ) � � | MPS � = | i 1 , i 2 , . . . , i L � tr 0 i 1 ,..., i L = 1 i 1 i 2 i 3 i 4 ω ω ω ω Let’s baxterize them! Looking for matrices ψ ab ( u ) with a , b = 1 . . . N a b u − u ψ ( u ) Initial condition: = ψ ab ( 0 ) = ω ( a ) ω ( b ) ω ω ψ ( 0 )
Integrable MPS N � ω ( i 1 ) ω ( i 2 ) . . . ω ( i L ) � � | MPS � = | i 1 , i 2 , . . . , i L � tr 0 i 1 ,..., i L = 1 i 1 i 2 i 3 i 4 ω ω ω ω Let’s baxterize them! Looking for matrices ψ ab ( u ) with a , b = 1 . . . N a b u − u ψ ( u ) Initial condition: = ψ ab ( 0 ) = ω ( a ) ω ( b ) ω ω ψ ( 0 )
Boundary Yang-Baxter relation: Relation to (operator valued) K -matrices K ( u ) ∈ End ( V ) K ( u ) b a = ψ ab ( u ) Twisted BYB relation in End ( V ⊗ V ) K 2 ( v ) R t 1 21 ( − u − v ) K 1 ( u ) R 12 ( u − v ) = R 21 ( u − v ) K 1 ( u ) R t 1 12 ( − u − v ) K 2 ( v ) Soliton non-preserving boundary conditions Extended Twisted Yangian Integrability properties follow
Boundary Yang-Baxter relation: Relation to (operator valued) K -matrices K ( u ) ∈ End ( V ) K ( u ) b a = ψ ab ( u ) Twisted BYB relation in End ( V ⊗ V ) K 2 ( v ) R t 1 21 ( − u − v ) K 1 ( u ) R 12 ( u − v ) = R 21 ( u − v ) K 1 ( u ) R t 1 12 ( − u − v ) K 2 ( v ) Soliton non-preserving boundary conditions Extended Twisted Yangian Integrability properties follow
Boundary Yang-Baxter relation: Relation to (operator valued) K -matrices K ( u ) ∈ End ( V ) K ( u ) b a = ψ ab ( u ) Twisted BYB relation in End ( V ⊗ V ) K 2 ( v ) R t 1 21 ( − u − v ) K 1 ( u ) R 12 ( u − v ) = R 21 ( u − v ) K 1 ( u ) R t 1 12 ( − u − v ) K 2 ( v ) Soliton non-preserving boundary conditions Extended Twisted Yangian Integrability properties follow
Boundary Yang-Baxter relation: Relation to (operator valued) K -matrices K ( u ) ∈ End ( V ) K ( u ) b a = ψ ab ( u ) Twisted BYB relation in End ( V ⊗ V ) K 2 ( v ) R t 1 21 ( − u − v ) K 1 ( u ) R 12 ( u − v ) = R 21 ( u − v ) K 1 ( u ) R t 1 12 ( − u − v ) K 2 ( v ) Soliton non-preserving boundary conditions Extended Twisted Yangian Integrability properties follow
. ξ 4 n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � ξ 3 j = 1 ξ 2 = Tr ( T QTM ( 0 )) L / 2 ξ 1 TBA side: Quench Action method e − E ( λ ) / T v ( λ + ) ∼ Data: ρ ( h ) x ( λ ) ρ x ( λ ) = e ε x ( λ ) = Y x ( λ ) Y -system relations follow from factorized overlaps QTM side: Fusion of double row transfer matrices Data: T x ( λ ) , Y x ( λ )
. ξ 4 n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � ξ 3 j = 1 ξ 2 = Tr ( T QTM ( 0 )) L / 2 ξ 1 TBA side: Quench Action method e − E ( λ ) / T v ( λ + ) ∼ Data: ρ ( h ) x ( λ ) ρ x ( λ ) = e ε x ( λ ) = Y x ( λ ) Y -system relations follow from factorized overlaps QTM side: Fusion of double row transfer matrices Data: T x ( λ ) , Y x ( λ )
. ξ 4 n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � ξ 3 j = 1 ξ 2 = Tr ( T QTM ( 0 )) L / 2 ξ 1 TBA side: Quench Action method e − E ( λ ) / T v ( λ + ) ∼ Data: ρ ( h ) x ( λ ) ρ x ( λ ) = e ε x ( λ ) = Y x ( λ ) Y -system relations follow from factorized overlaps QTM side: Fusion of double row transfer matrices Data: T x ( λ ) , Y x ( λ )
. ξ 4 n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � ξ 3 j = 1 ξ 2 = Tr ( T QTM ( 0 )) L / 2 ξ 1 TBA side: Quench Action method e − E ( λ ) / T v ( λ + ) ∼ Data: ρ ( h ) x ( λ ) ρ x ( λ ) = e ε x ( λ ) = Y x ( λ ) Y -system relations follow from factorized overlaps QTM side: Fusion of double row transfer matrices Data: T x ( λ ) , Y x ( λ )
. ξ 4 n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � ξ 3 j = 1 ξ 2 = Tr ( T QTM ( 0 )) L / 2 ξ 1 TBA side: Quench Action method e − E ( λ ) / T v ( λ + ) ∼ Data: ρ ( h ) x ( λ ) ρ x ( λ ) = e ε x ( λ ) = Y x ( λ ) Y -system relations follow from factorized overlaps QTM side: Fusion of double row transfer matrices Data: T x ( λ ) , Y x ( λ )
. ξ 4 n � Z = � Ψ 0 | T ( ξ j ) | Ψ 0 � ξ 3 j = 1 ξ 2 = Tr ( T QTM ( 0 )) L / 2 ξ 1 TBA side: Quench Action method e − E ( λ ) / T v ( λ + ) ∼ Data: ρ ( h ) x ( λ ) ρ x ( λ ) = e ε x ( λ ) = Y x ( λ ) Y -system relations follow from factorized overlaps QTM side: Fusion of double row transfer matrices Data: T x ( λ ) , Y x ( λ )
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