Full counting statistics and Edgeworth series for Matrix Product State Full counting statistics and Edgeworth series for Matrix Product State Yifei Shi, Israel Klich 21 November
Full counting statistics and Edgeworth series for Matrix Product State 1 Matrix Product State 2 Generalization of MPS to Tensor Network states 3 Full counting statistics and matrix product states 4 Edgeworth series
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State The AKLT state, where everything started < i , j > � S i · � S j + ∆( � S i · � S j ) 2 Spin-1 : H = � ∆ = 1 3 , Ground state, AKLT state, Affleck, Kennedy, Lieb and Tasaki
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State The AKLT state, where everything started < i , j > � S i · � S j + ∆( � S i · � S j ) 2 Spin-1 : H = � ∆ = 1 3 , Ground state, AKLT state, Affleck, Kennedy, Lieb and Tasaki Think of Spin-1 as 2 spin- 1 2 Also a simple example of topological state Gappless edge modes, string order parameter, fractional charge
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State AKLT state, the explicit form Use the physical picture to write the state: � � 0 � 1 � � 0 � 0 1 1 A + = = 0 0 − 1 0 0 0 � � 0 � − 1 � √ 0 A 0 = 1 � 0 � 1 1 2 √ = 1 1 0 − 1 0 0 √ 2 2 � � 0 � 0 � 0 � � 0 1 0 A − = = 0 1 − 1 0 − 1 0 � � A s 1 A s 2 ... A s n ) | s 1 s 2 ... s n � | AKLT � ∝ Tr( { s i } i
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Generalization of AKLT state to Matrix Product State Physical intuition Physical Space: d Auxiliary Space: DxD Neighboring auxiliary spins in a maximumly entangled state: � D i =1 | i �| i � ✗✔ ✗✔ � i | i �| i � A s i , j ✉ ✉ ✉ ✉ ✖✕ ✖✕ Use matrix A s k i , j to project from D × D dimensional auxiliary space to d dimensional physical space
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Generalization of AKLT state to Matrix Product State Mathematical forms � � Tr( A s 1 [1] A s 2 [2] ... A s N | Ψ M � = [ N ] ) | s 1 , s 2 , ..., s N � (PBC) { s i } i � � A s 1 [1] A s 2 [2] ... A s N | Ψ M � = [ N ] | s 1 , s 2 , ..., s N � (OBC) i { s i } A s i 1 and A s i N 1 × D and D × 1 dimensional Variational ansatz with D × D × d × N numbers. D ∼ e N , every state is exactly a MPS D = 1 meanfield limit D finite, only access a very small portion of Hilbert space!
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Schmidt decomposition Divide system into subsystem A( L sites) B( N − L sites) d L d N − L � � | ψ � = C ij | i � A ⊗ | j � B i =1 j =1 Single value decomposition: C = UDV , U and V unitary, D diagonal with semipositive elements called Schmidt coefficients λ α d L d N − L χ � � � | ψ � = λ α ( U i α | i � A ) ⊗ ( V α j | j � B ) α =1 i =1 j =1 χ � λ α | φ [ A ] α � ⊗ | φ [ B ] = α � α =1 � φ [ A ] β | φ [ A ] α � = δ αβ and � φ [ B ] β | φ [ B ] α � = δ αβ
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Schmidt decomposition Reduced density matrix and entanglement χ � λ 2 α | φ [ A ] α �� φ [ A ] ρ A = α | α =1 χ � λ 2 α | φ [ B ] α �� φ [ B ] ρ B = α | α =1 χ � λ 2 α log( λ α ) 2 S [ A , B ] = − α =1 rank of ρ A , B = rank of D
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Schmidt decomposition and Canonical form Gauge freedom, A [ i ] → X − 1 A [ i ] X i +1 , state invariant. i Divide system into two parts, D � | φ left α � ⊗ | φ right | Ψ M � = � α α =1 φ left � A s 1 [1] A s 2 [2] ... A s i = [ i ] | s 1 , s 2 , ..., s i � α { s 1 ,... s i } � A s i +1 [ i +1] A s i +2 φ right [ i +2] ... A s i = [ N ] | s i +1 , s i +2 , ..., s N � α { s i +1 ,... s N }
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Schmidt decomposition and Canonical form Gauge freedom, A [ i ] → X − 1 A [ i ] X i +1 , state invariant. i Divide system into two parts, D � | φ left α � ⊗ | φ right | Ψ M � = � α α =1 φ left � A s 1 [1] A s 2 [2] ... A s i = [ i ] | s 1 , s 2 , ..., s i � α { s 1 ,... s i } � A s i +1 [ i +1] A s i +2 φ right [ i +2] ... A s i = [ N ] | s i +1 , s i +2 , ..., s N � α { s i +1 ,... s N } Canonical Form: one particular gauge choice, so that the above equation is the Schmidt composition
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Properties of MPS ρ L Reduce density matrix of L neighboring spins rank of ρ L ≤ D ( D 2 for PBC)
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Properties of MPS ρ L Reduce density matrix of L neighboring spins rank of ρ L ≤ D ( D 2 for PBC) Tranlational invariant case: | Ψ M � = � { s i } Tr( � i A s i ) |{ s i }� Normalization: | Ψ M | 2 = Tr[( E A ) N ] i A s i ⊗ ¯ E A = � A s i Operator expectation value: � ˆ O i � = Tr(( E A ) N − 1 E O A ) O i ˆ ′ � ˆ j � = Tr( E O A ( E A ) j − i − 1 E O A ( E A ) N − j + i − 2 ) ′ O i , j O i , j A s i ⊗ ¯ E O A = � A s j
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Two point function Consider two point function: C ( r ) = � ˆ O 0 ˆ O r � − � O � 2 | Ψ M | 2 = Tr[( E A ) N ] → λ N M � ˆ O i � → ( l | E O A | r ) λ M largest eigenvalue, set to 1, ( l | and | r ) eigenvectors ( l | r ) = 1 Second largest eigenvalue λ 2 < 1, ξ = log(1 /λ 2 ): C ( r ) ∝ exp ( − l /ξ ) Second largest eigenvalue = 1: C ( r ) → const
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State Properties of MPS O r � − � O � 2 either decays Two point function: � ˆ O 0 ˆ exponentially or stays as constant MPS cannot describe critical system! In practice, one will have an effective correlation length ξ ∝ log D Matrix Product State represent ground state faithfully F. Verstraete and J. I. Cirac, Phys. Rev. B 73, 094423 (2006)
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State MPS and DMRG Density Matrix Renormalization Group Calculate ground state energy to almost machine precision Variational method with MPS ansatz Fix all A s i except on one site, minimize the energy, and move to the nex site • ... • • • • • • • • ... • system σ l +1 σ l +2 environment
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State DMRG and truncation Truncation is essential
Full counting statistics and Edgeworth series for Matrix Product State Matrix Product State DMRG and truncation Truncation is essential Truncate Only keep D states 1D gapped system, the eigenvalues of ρ L decay exponentially* Not true in 2D! Most error comes from truncation * U. Schollwck RevModPhys.77.259
Full counting statistics and Edgeworth series for Matrix Product State Generalization of MPS to Tensor Network states Graphic representation s k i A j A s k i , j : Ψ M : A 1 A 2 A 3 ... A i A n Calculate normalization: ¯ ¯ ¯ ¯ ¯ ... A 1 A 2 A 3 A i A n ... A 1 A 2 A 3 A i A n
Full counting statistics and Edgeworth series for Matrix Product State Generalization of MPS to Tensor Network states Tensor Network State Create any Tensor Network State Entanglement is build in!
Full counting statistics and Edgeworth series for Matrix Product State Generalization of MPS to Tensor Network states Tensor Network State Create any Tensor Network State Entanglement is build in! S ∼ # of legs cut by the partition
Full counting statistics and Edgeworth series for Matrix Product State Generalization of MPS to Tensor Network states PEPS Projected Entanglement Pair State Straight forward generalization to 2D Can also describe critical system Impossible to compute anything! (normalization, operator expectation value...) F. Verstraete, J.I. Cirac, V. Murg, arXiv:0907.2796
Full counting statistics and Edgeworth series for Matrix Product State Generalization of MPS to Tensor Network states MERA Multi-scale entanglement renormalization ansatz Designed to describe Critical system, for block of L sites, S E ∼ log( L ) Found its role in AdS/CFT G.Vidal, Phys.Rev.Lett.101.110501
Full counting statistics and Edgeworth series for Matrix Product State Full counting statistics and matrix product states Full counting statistics FCS generating function: n P n e i λ n χ ( λ ) = � κ n ( i λ ) n log χ ( λ ) = � n n ! κ cumulants, κ 1 = � n � , κ 2 = � n 2 � − � n � 2 ... It characterizes quantum noise and fluctuation*. Related to entanglement entropy † *L. S. Levitov and G. B. Lesovik, JETP Lett. 58, 230 (1993) † I. Klich and L. Levitov, Phys. Rev. Lett. 102, 100502 (2009)
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