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Matrix Product States for frustrated spin chains, lattices with an - - PowerPoint PPT Presentation

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Matrix Product States for frustrated spin chains, lattices with an extended Hilbert space and constrained models in 1D Natalia Chepiga Swiss National Science Foundation University of Amsterdam, The Netherlands 23 July 2019 Natalia Chepiga

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Matrix Product States

for frustrated spin chains, lattices with an extended Hilbert space and constrained models in 1D

Natalia Chepiga

Swiss National Science Foundation University of Amsterdam, The Netherlands

23 July 2019

Natalia Chepiga (SNF, UvA) 23 July 2019 1 / 68

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Scope

Basics of DMRG Area law Graphical notations MPO construction Variational optimization (finite- and infinite-size DMRG) Abelian symmetry Spin-1 chain with three-site interactions Phase diagram Excitation spectrum and DMRG iterations. Conformal towers Comb tensor networks Tree tensor network DMRG investigation of a hard-boson model of Rydberg atoms Implementing local constraint into DMRG Floating phase versus chiral transition

Natalia Chepiga (SNF, UvA) 23 July 2019 2 / 68

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Area law

Why tensor networks work?

Natalia Chepiga (SNF, UvA) 23 July 2019 3 / 68

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Area law

Exponential growth of the Hilbert space dimH = dN Exact diagonalization is limited to small clusters.

Natalia Chepiga (SNF, UvA) 23 July 2019 3 / 68

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Area law

Exponential growth of the Hilbert space dimH = dN Exact diagonalization is limited to small clusters. Area law for the entanglement entropy

Full Hilbert space Low energy states (local H)

Ground states of local Hamiltonians are less entangled than a random state in the Hilbert space

Natalia Chepiga (SNF, UvA) 23 July 2019 3 / 68

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Area law

subsystem A environment B subsystem A environment B environment B

1D 2D

Entanglement entropy: SA = −tr(ρA log ρA) GS of local Hamiltonians Area law: SA(L) ∝ Ld−1 1D: SA(L) = const 2D: SA(L) ∝ L Random state Volume law: SA(L) ∝ Ld Critical state in 1D SA(L) ∝ log(L)

Natalia Chepiga (SNF, UvA) 23 July 2019 4 / 68

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Area law

Full Hilbert space Low energy states (local H)

Our goal: to diagonalize the Hamiltonian directly in the truncated basis Number of relevant states D ∝ exp(S) GS of local Hamiltonians Area law: SA(L) ∝ Ld−1 1D: SA(L) = const 2D: SA(L) ∝ L Random state Volume law: SA(L) ∝ Ld Critical state in 1D SA(L) ∝ log(L)

Natalia Chepiga (SNF, UvA) 23 July 2019 5 / 68

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Graphical notations

Number Vector Matrix rank-3 tensor (MPS) rank-4 tensor (MPO) rank-5 tensor (PEPS)

Natalia Chepiga (SNF, UvA) 23 July 2019 6 / 68

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Contraction

Summation over connected bonds In practice: reshape tensors into matrices and use optimized matrix multipliers Rank of the resulting tensor = number of open legs

Natalia Chepiga (SNF, UvA) 23 July 2019 7 / 68

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Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 12

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 13

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 14

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters! Exponential growth of complexity!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 15

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 16

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 17

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 18

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 19

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 20

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SLIDE 21

Contraction

Complexity

  • i∈ connected legs

Di ·

  • j∈ open legs

Dj The order of contraction matters! Complexity stays finite!

Natalia Chepiga (SNF, UvA) 23 July 2019 8 / 68

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SVD

singular values decomposition

Natalia Chepiga (SNF, UvA) 23 July 2019 9 / 68

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Singular Values Decomposition (SVD)

For any rectangular matrix Mi,j exists a decomposition M = Ui,kSk,kV †

k,j

such that: U †U = I S is a diagonal matrix with non-negative entries V †V = I

Natalia Chepiga (SNF, UvA) 23 July 2019 9 / 68

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Schmidt decomposition

Quantum state: |ψ =

  • i,j

Ψi,j|iA|jB, where |iA and |jB are orthonormal basis of subsystems A and B. Treat Ψi,j as a matrix and perform SVD Schmidt decomposition |ψ =

  • i,j
  • k

Ui,kSk,kV †

k,j|iA|jB

Natalia Chepiga (SNF, UvA) 23 July 2019 10 / 68

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Schmidt decomposition

Quantum state: |ψ =

  • i,j

Ψi,j|iA|jB, where |iA and |jB are orthonormal basis of subsystems A and B. Treat Ψi,j as a matrix and perform SVD Area law - D relevant states only |ψ =

  • i,j

D

  • k

Ui,kSk,kV †

k,j|iA|jB

Natalia Chepiga (SNF, UvA) 23 July 2019 11 / 68

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Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 27

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 28

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 29

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 30

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 31

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 32

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 33

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D D=const Area law:

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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SLIDE 34

Bring quantum states into MPS

reshape reshape reshape

SVD SVD

D D=const Area law:

Mixed-canonical form

B S A A

= =

Normalization:

Natalia Chepiga (SNF, UvA) 23 July 2019 12 / 68

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Normalization

The goal is to find |Ψ that minimizes the energy: E = Ψ| ˆ H|Ψ Ψ|Ψ If norm is fixed Ψ|Ψ = 1, it becomes E = Ψ| ˆ H|Ψ In variational optimization a generalized eigenvalue problem is reduced to a generalized eigvenvalue problem: ˆ Heff|ψ = E|ψ instead of ˆ Heff|ψ = E ˆ Neff|ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 13 / 68

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Variational optimization of the MPS

Fix all tensors but two Hamiltonian

Natalia Chepiga (SNF, UvA) 23 July 2019 14 / 68

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MPO

Full Hamiltonian as a product of local tensors

Natalia Chepiga (SNF, UvA) 23 July 2019 15 / 68

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MPO construction

For a given site j write all possible terms in the Hamiltonian: Transverse field Ising model: H = JSx

i Sx i+1 + hSz i

I...I hSz

j

I...I I...IJSx

j−1

Sx

j

I...I I...I JSx

j

Sx

j+1I...I

I...JSx

i Sx i+1...I

I I...I I...hSz

i ...I

I I...I I...I I I...JSx

i Sx i+1...I

I...I I I...hSz

i ...I

Natalia Chepiga (SNF, UvA) 23 July 2019 15 / 68

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MPO construction

For a given site j write all possible terms in the Hamiltonian: Transverse field Ising model: H = JSx

i Sx i+1 + hSz i

I...I hSz

j

I...I I...IJSx

j−1

Sx

j

I...I I...I JSx

j

Sx

j+1I...I

I...JSx

i Sx i+1...I

I I...I I...hSz

i ...I

I I...I I...I I I...JSx

i Sx i+1...I

I...I I I...hSz

i ...I

Natalia Chepiga (SNF, UvA) 23 July 2019 15 / 68

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MPO construction

For a given site j write all possible terms in the Hamiltonian: Transverse field Ising model: H = JSx

i Sx i+1 + hSz i

I...I hSz

j

I...I I...IJSx

j−1

Sx

j

I...I I...I JSx

j

Sx

j+1I...I

I...Full...I I I...I I...I I I...Full...I Five non-trivial entries in the MPO

Natalia Chepiga (SNF, UvA) 23 July 2019 15 / 68

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MPO construction

For a given site j write all possible terms in the Hamiltonian: Transverse field Ising model: H = JSx

i Sx i+1 + hSz i

I...I hSz

j

I...I I...IJSx

j−1

Sx

j

I...I I...I JSx

j

Sx

j+1I...I

I...Full...I I I...I I...I I I...Full...I Five non-trivial entries in the MPO Look at the left and right basis in which the MPO is going to be written

Natalia Chepiga (SNF, UvA) 23 July 2019 15 / 68

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MPO construction

For a given site j write all possible terms in the Hamiltonian: Five non-trivial entries in the MPO Look at the left and right basis in which the MPO is going to be written I...Full...I I...IJSx

j−1

I...I I...I Sx

j+1I...I

I...Full...I

Natalia Chepiga (SNF, UvA) 23 July 2019 16 / 68

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MPO construction

For a given site j write all possible terms in the Hamiltonian: Five non-trivial entries in the MPO Look at the left and right basis in which the MPO is going to be written Fill-in the matrix: I...Full...I I I...IJSx

j−1

Sx

j

I...I hSz

j

JSx

j

I I...I Sx

j+1I...I

I...Full...I

Natalia Chepiga (SNF, UvA) 23 July 2019 16 / 68

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MPO exercise:

Heisenberg nearest-neighbor: H = J

  • j

Sj · Sj+1 = J

  • j

1 2

  • S+

j S− j+1 + S− j S+ j+1

  • + Sz

j Sz j+1

(1) J1 − J2 model: HJ1−J2 = J1

  • j

Sj · Sj+1 + J2

  • j

Sj · Sj+2 (2) Three-site interaction: H = HJ1−J2 + J3

  • j

[(Sj · Sj+1)(Sj+1 · Sj+2) + h.c.] (3)

Natalia Chepiga (SNF, UvA) 23 July 2019 17 / 68

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MPO answers:

Heisenberg nearest-neighbor: H = J

  • j

1 2

  • S+

j S− j+1 + S− j S+ j+1

  • + Sz

j Sz j+1

(4) Hj =        I . . . S−

j

. . . S+

j

. . . Sz

j

. . . .

J 2 S+ j J 2 S− j

JSz

j

I       

Natalia Chepiga (SNF, UvA) 23 July 2019 18 / 68

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MPO answers:

J1 − J2 model: HJ1−J2 = J1

  • j

Sj · Sj+1 + J2

  • j

Sj · Sj+2 (5)

Hj =             I . . . . . . S−

j

. . . . . . S+

j

. . . . . . Sz

j

. . . . . . . I . . . . . . . I . . . . . . . I . . . .

J1 2 S+ j J1 2 S− j

J1Sz

j J2 2 S+ j J2 2 S− j

J2Sz

j

I            

Natalia Chepiga (SNF, UvA) 23 July 2019 19 / 68

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MPO answers:

Three-site interaction: H = HJ1−J2 + J3

  • j

[(Sj · Sj+1)(Sj+1 · Sj+2) + h.c.] (6) Hi =           

I . . . . . . S−

i

. . . . . . S+

i

. . . . . . Sz

i

. . . . . . .

J2 2 I + J3Q+−

J3Q−− J3Q−z . . . . J3Q++

J2 2 I + J3Q+−

J3Q+z . . . . J3Q+z J3Q−z J2I + J3Qzz . . . .

J1 2 S+ i J1 2 S− i

J1Sz

i

S+

i

S−

i

Sz

i

I

           , where Qαβ

i

= Sα

i Sβ i + Sβ i Sα i with Sα = {S+ √ 2, S− √ 2, Sz}

Natalia Chepiga (SNF, UvA) 23 July 2019 20 / 68

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SLIDE 48

DMRG / variational MPS

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Matrix Product Operators

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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DMRG sweep

Left-to-right: Right-to-left:

Group the legs and treat this rank-8 tensor as a matrix

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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SLIDE 53

DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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SLIDE 54

DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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SLIDE 55

DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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SLIDE 56

DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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SLIDE 57

DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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SLIDE 58

DMRG sweep

...

diag SVD diag SVD diag SVD diag SVD

Left-to-right: Right-to-left:

Natalia Chepiga (SNF, UvA) 23 July 2019 21 / 68

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SLIDE 59

Initial guess

Product state Random state Infinite-size DMRG

Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

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SLIDE 60

Infinite-size DMRG

...

diag SVD diag SVD

Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

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SLIDE 61

Infinite-size DMRG

...

diag SVD diag SVD

Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

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SLIDE 62

Infinite-size DMRG

...

diag SVD diag SVD

Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

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SLIDE 63

Infinite-size DMRG

...

diag SVD diag SVD

Natalia Chepiga (SNF, UvA) 23 July 2019 22 / 68

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SLIDE 64

Abelian symmetry

S A B A

Assign quantum numbers - labels to physical bonds of MPS Using fusion rules of the symmetry, find quantum numbers on auxiliary legs When local basis is sorted according to the quantum number of states, the MPS takes a block-diagonal form

Natalia Chepiga (SNF, UvA) 23 July 2019 23 / 68

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Abelian symmetry. Examples

A A A Natalia Chepiga (SNF, UvA) 23 July 2019 24 / 68

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SLIDE 66

Abelian symmetry. Examples

A A A Natalia Chepiga (SNF, UvA) 23 July 2019 24 / 68

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SLIDE 67

Abelian symmetry. Examples

A A A Natalia Chepiga (SNF, UvA) 23 July 2019 24 / 68

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SLIDE 68

Observables

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 69

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 70

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 71

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 72

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 73

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 74

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 75

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 76

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 77

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 25 / 68

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SLIDE 78

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ On-site measures Ψ|Oi|Ψ

  • r

Natalia Chepiga (SNF, UvA) 23 July 2019 26 / 68

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SLIDE 79

Observables

Nearest-neighbor correlations Ψ|Si · Si+1|Ψ On-site measures Ψ|Oi|Ψ

  • r

Long range correlations Ψ|Oi · Oj|Ψ

Natalia Chepiga (SNF, UvA) 23 July 2019 26 / 68

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SLIDE 80

Dimerization transition in frustrated spin-1 chain

in collaboration with Frederic Mila (EPFL) and Ian Affleck (UBC)

Natalia Chepiga (SNF, UvA) 23 July 2019 26 / 68

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SLIDE 81

Spin chains

Heisenberg Hamiltonian: H = J1

  • i

Si · Si+1 Spin-1/2 chain is critical Spin-1 chain:

finite bulk gap topologically non-trivial ground state spin-1/2 edge states

Haldane, Phys. Lett. A 93, 464 ’83 Affleck, Kennedy, Lieb, Tasaki, PRL 59, 799 ’87 Kennedy J. Phys: Cond. Mat 2, 5737 ’90 Natalia Chepiga (SNF, UvA) 23 July 2019 27 / 68

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SLIDE 82

Introduction

Add biquadratic interaction: H =

  • i

J1Si · Si+1 + Jb(Si · Si+1)2

Takhtajan-Babudjian

Critical spin-1 chains: WZW SU(2)2 SU(3)

Affleck, Nucl.Phys.B 265, 409 ’86 F´ ath, S´

  • lyom, PRB 44,11836 ’91

Schollw¨

  • ck, Jolicœur, Garel,

PRB 53, 3304 ’96 L¨ auchli, Schmid, Trebst, PRB 74, 144426 ’06 Natalia Chepiga (SNF, UvA) 23 July 2019 28 / 68

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SLIDE 83

The model

Hamiltonian: H =

  • i

(J1Si · Si+1 + J2Si−1 · Si+1) +

  • i

J3 [(Si−1 · Si)(Si · Si+1) + H.c.]

Three-site term: Appears in next-to-leading order in the strong coupling expansion of the two-band Hubbard model Induces spontaneous dimerization Reduces to next-nearest-neighbor interaction for spin-1/2

Natalia Chepiga (SNF, UvA) 23 July 2019 29 / 68

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SLIDE 84

Motivation

H =

  • i

(J1Si · Si+1 + J2Si−1 · Si+1) +

  • i

J3 [(Si−1 · Si)(Si · Si+1) + H.c.]

1 0.8 0.6 0.4 0.2

1st order

1st order transition between two topologically different phases

Kolezhuk, Roth, Scholw¨

  • ck,

PRL 77, 5142 ’96 Kolezhuk, Scholw¨

  • ck, PRB

65, 100401 ’01 Natalia Chepiga (SNF, UvA) 23 July 2019 30 / 68

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SLIDE 85

Motivation

H =

  • i

(J1Si · Si+1 + J2Si−1 · Si+1) +

  • i

J3 [(Si−1 · Si)(Si · Si+1) + H.c.]

1 0.8 0.6 0.4 0.2

  • r

0.05 0.1 0.15

+

1/6

1st order

Generalization of the Majumdar-Ghosh model: fully dimerized state is an exact ground state at J3/J1 = 1/6 Continuous WZW SU(2)k=2 transition at J3/J1 = 0.111

Michaud, Vernay, Manmana, Mila, PRL 108, 127202 ’12 Natalia Chepiga (SNF, UvA) 23 July 2019 31 / 68

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SLIDE 86

Motivation

H =

  • i

(J1Si · Si+1 + J2Si−1 · Si+1) +

  • i

J3 [(Si−1 · Si)(Si · Si+1) + H.c.]

1 0.8 0.6 0.4 0.2

  • r

0.05 0.1 0.15

+

1/6 E x a c t l y d i m e r i z e d l i n e

1st order

There is a line in J2 − J3 parameter space, at which the fully dimerized state is exact eigenstate First order phase transition has to appear between the Haldane and dimerized phases

Michaud, Vernay, Manmana, Mila, PRL 108, 127202 ’12 Wang, Furuya, Nakamura, Komakura, PRB 88, 224419’13 Natalia Chepiga (SNF, UvA) 23 July 2019 32 / 68

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SLIDE 87

Phase diagram

0.05 0.1 0.15 1 0.8 0.6 0.4 0.2

c=3/2

  • r

c=1/2

The transition between the Haldane and dimerized phases is continuous WZW SU(2)2 below and including at the end point The transition between the NNN-Haldane phase and the dimerized phase is in the Ising universality class Topological transition between the Haldane and NNN-Haldane phases is always first order

Natalia Chepiga (SNF, UvA) 23 July 2019 33 / 68

slide-88
SLIDE 88

Phase diagram. Field theory

0.05 0.1 0.15 1 0.8 0.6 0.4 0.2

c=3/2

  • r

c=1/2

Effective Hamiltonian: H = HWZW + λ1(trg)2 + λ2 JR · JL λ2 < 0 Continuous SU(2)2 λ2 = 0 End point λ2 > 0 First order Free boson and Ising fields: trg ∝ σ sin √πθ (trg)2 ∝ ǫ − C1 cos √ 4πθ

  • JL ·

JR ∝ ǫ cos √ 4πθ + C2∂xφL∂xφR

Second order transition between the Haldane and dimerized phases

  • ccurs simultaneously in Ising and boson sectors. Far from the

WZW critical end point the Ising and boson critical lines could split

Natalia Chepiga (SNF, UvA) 23 July 2019 34 / 68

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SLIDE 89

Dimerization

Local dimerization: D(j, N) = |Sj · Sj+1 − Sj−1 · Sj|

0.1 0.2 1 2 0.1 1 2 0.1 1 2

Natalia Chepiga (SNF, UvA) 23 July 2019 35 / 68

slide-90
SLIDE 90

Ising transition. Dimerization

Local dimerization: D(j, N) = |Sj · Sj+1 − Sj−1 · Sj| Finite-size scaling of the middle-chain dimerization in log-log scale The separatrix is associated with the phase transition The slope corresponds to the critical exponent Critical exponent in Ising chain: d = 1/8

0.057 0.0575 0.058 0.0585 0.059 ,

Natalia Chepiga (SNF, UvA) 23 July 2019 36 / 68

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SLIDE 91

Ising transition. Dimerization

Local dimerization: D(j, N) = |Sj · Sj+1 − Sj−1 · Sj| Open boundary favors dimerization In the transverse-field Ising chain it is equivalent to the applied boundary magnetic field At the critical point the magnetization decays away from the edges as σ(x) ∝ [N sin(πj/N)]−1/8

Fit DMRG 1.5 2 1 200 400 600 800

,800 ,

In spin-1 chain the dimerization decays away from the boundaries in the same way and with the same critical exponent

Natalia Chepiga (SNF, UvA) 23 July 2019 37 / 68

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SLIDE 92

Conformal towers = fingerprints of a critical theory Ising critical theory is minimal model of CFT It is described by a finite number of primary fields Combining boundary CFT with DMRG we can probe all of them numerically!

Natalia Chepiga (SNF, UvA) 23 July 2019 38 / 68

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SLIDE 93

Conformal towers from entanglement spectrum

Transverse field Ising model:

0.1 0.2 0.3

1/Ln(L)

1 1 1 1 1 1 1 2 2 2 2 3 3 degeneracy

  • bc: 0 + 1/2

1 1 1 2 2 3 1 1 1 1 2 2 3 1 1 1 2 2 3 4 1/2 1/16

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

Ising CFT c=1/2 0.1 0.2 0.3

1/Ln(L)

1/2 1/2+1 0+2 1/2+2 0+3 1/2+3 0+4 1/2+4 0+5 1/2+5 0+6 1/2+6 pbc: 0 + 1/2

primary f e lds +de sce ndants

A.L¨ auchli, arxiv:1303.0441

Natalia Chepiga (SNF, UvA) 23 July 2019 39 / 68

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SLIDE 94

Energy excitation spectrum with DMRG

Natalia Chepiga (SNF, UvA) 23 July 2019 40 / 68

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SLIDE 95

Excitation spectrum with DMRG/MPS

1 The excited state is the ’ground-state’ of the different symmetry

sector

Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

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SLIDE 96

Excitation spectrum with DMRG/MPS

1 The excited state is the ’ground-state’ of the different symmetry

sector

2 Conventional DMRG: Mixed states

The ground-state is spoilt Heavy memory usage

Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

slide-97
SLIDE 97

Excitation spectrum with DMRG/MPS

1 The excited state is the ’ground-state’ of the different symmetry

sector

2 Conventional DMRG: Mixed states

No longer variational Heavy memory usage

3 MPS: Construct the lowest-energy state orthogonal to the

previously constructed ones

Time consuming Accumulation of the error

Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

slide-98
SLIDE 98

Excitation spectrum with DMRG/MPS

1 The excited state is the ’ground-state’ of the different symmetry

sector

2 Conventional DMRG: Mixed states

No longer variational Heavy memory usage

3 MPS: Construct the lowest-energy state orthogonal to the

previously constructed ones

Time consuming Accumulation of the error

4 MPS: Domain wall or special tensor (see Laurens’ talk)

Translation invariant MPS

Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

slide-99
SLIDE 99

Excitation spectrum with DMRG/MPS

1 The excited state is the ’GS’ of the different symmetry sector 2 Conventional DMRG: Mixed states

No longer variational & Heavy memory usage

3 MPS: Construct the lowest-energy state orthogonal to the

previously constructed ones

Time consuming & Accumulation of the error

4 MPS: Domain wall or special tensor (see Laurens’ talk)

Translation invariant MPS

There is a cheaper option:

Sometimes it is sufficient to target multiple eigenstates of the effective Hamiltonian and keep track of the energies as a function of iterations

[NC, Mila, Phys.Rev.B 96, 054425 (2017)] Natalia Chepiga (SNF, UvA) 23 July 2019 41 / 68

slide-100
SLIDE 100

Effective Hamiltonian

Hamiltonian Approximate basis Truncation and rotation of the basis D D

The Hamiltonian is written in a truncated and rotated basis. This basis is selected for the ground state. Could this basis be suitable for other low-energy states?

Natalia Chepiga (SNF, UvA) 23 July 2019 42 / 68

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SLIDE 101

Trivial case - non-truncated MPS

When no truncation is imposed and all basis states are kept in MPS, the DMRG is equivalent to exact diagonalization and one can access the entire spectrum!

2 4 6 8 10 12 14

  • 6.4
  • 6
  • 5.6
  • 5.2
  • 4.8

2 4 6 8 10 12 14 0.4 0.8 1.2 1.6 Energy 1 N Iterations N/2 1 N Iterations N/2 Error

a) b)

Natalia Chepiga (SNF, UvA) 23 July 2019 43 / 68

slide-102
SLIDE 102

When does it work?

Local impurities Localized excitations MPS is the same except for a few sites

Impurity

Natalia Chepiga (SNF, UvA) 23 July 2019 44 / 68

slide-103
SLIDE 103

When does it work?

Edge states Edge spins are entangled through the entire network All edge states are in the basis Local impurities Localized excitations MPS is the same except for a few sites B

2 ⊗ ↑

  • ⊕

B

2 ⊗ ↓

  • Natalia Chepiga

(SNF, UvA) 23 July 2019 45 / 68

slide-104
SLIDE 104

Edge states in the Haldane chain

  • 140.5
  • 140
  • 139.5

b) Energy

  • 139
  • 138.5
  • 138

Energy a) Iterations c)

1000 1500

  • 20
  • 15
  • 10
  • 5
  • 138.940086+10-8

+

N=100 d)

1000 1500 20 15 10 5

  • 140.34157+10-8

+

Iterations N=101 N=101 N=100

200 400 600 800 1000 1200 1400 1600 200 400 600 800 1000 1200 1400 1600

N 1 1 N N 1 1 N N 1 1 N N 1 1 N 1

Natalia Chepiga (SNF, UvA) 23 July 2019 46 / 68

slide-105
SLIDE 105

When does it work?

Critical systems Divergent correlation length Slow decay of Schmidt values Special structure of spectrum Edge states Edge spins are entangled through the entire network All edge states are in the basis Local impurities Localized excitations MPS is the same except for a few sites

10 0 10 1 10 2 10 3 10 -15 10 -10 10 -5 10 0 Singular value Number of singular value

Gapped Critical

Natalia Chepiga (SNF, UvA) 23 July 2019 46 / 68

slide-106
SLIDE 106

Transverse field Ising model H =

  • i

JSx

i Sx i+1 + hSz i

Critical at h = J/2 Solved by Jordan-Wigner transformation Corresponds to the minimal model (4,3) in CFT

Natalia Chepiga (SNF, UvA) 23 July 2019 46 / 68

slide-107
SLIDE 107

Transverse field Ising model. Excitation spectrum

100 200 300 400 500 600 700 800

  • 31.75
  • 31.7
  • 31.65
  • 31.6

Energy Iterations

{ { { {

sweep 1 sweep 2 sweep 3 sweep 4

30 states within a single run! Flat modes signal convergence

NC, F. Mila, Phys. Rev. B 96, 054425’17 Natalia Chepiga (SNF, UvA) 23 July 2019 46 / 68

slide-108
SLIDE 108

Transverse field Ising model. Excitation spectrum

600 700

  • 31.75
  • 31.7
  • 31.65
  • 31.6

1 N 1 Iterations

710 730 750

5 8 9 12 13 16-18

710 730 750 710 730 750 710 730 750

Iterations Energy Critical

  • 39.6
  • 39.5
  • 39.4
  • 39.3
  • 39.2
  • 39.1
  • 27.1
  • 27.05
  • 27
  • 26.95
  • 26.9
  • 26.85
  • 26.8
  • 26.75
  • 26.7

600 700 600 700

1 N 1 1 N 1

two-fold degenerate ground-state

Iterations Iterations Gapped Energy

Remarkable accuracy for critical system Wrong spectrum for gapped system

[NC, F. Mila, Phys. Rev. B 96, 054425’17] Natalia Chepiga (SNF, UvA) 23 July 2019 47 / 68

slide-109
SLIDE 109

Finite-size scaling of the excitation energy

0.1 0.2 0.3 0.4 0.1 0.2 0.3 0.4 0.5 0.1 0.2 0.3 0.4 i)

j) k)

0.01 0.02 0.03 0.01 0.02 0.03 0.01 0.02 0.03 0.1 0.2 0.3 0.4 0.5

free

0.01 0.02 0.03

free l)

χI(q) = q−1/48 1 + q2 + q3 + 2q4 + 2q5 + 3q6 + 3q7 + 5q8 χǫ(q) = q1/2−1/48 1 + q + q2 + q3 + 2q4 + 2q5 + 3q6 + 4q7 + 5q8

BCFT prediction: Cardy, Nuc. Phys. B, 324 581-596’89 DMRG results: NC, Mila, Phys. Rev. B 96, 054425’17 Natalia Chepiga (SNF, UvA) 23 July 2019 48 / 68

slide-110
SLIDE 110

Ising transition in spin-1 chain

Natalia Chepiga (SNF, UvA) 23 July 2019 49 / 68

slide-111
SLIDE 111

Ising conformal towers in spin-1 chain

2 4 0.2 0.4 0.6 2 4 0.2 0.4 0.6 0.02 0.04 0.02 0.04

c) d)

Triplet Singlet

OBC-even OBC-odd

Singlet-triplet gap is open Critical scaling of the gap in the singlet sector N even I conformal tower N odd ǫ conformal tower

NC, Affleck, Mila, PRB 93, 241108’16 Natalia Chepiga (SNF, UvA) 23 July 2019 49 / 68

slide-112
SLIDE 112

WZW SU(2)2 end point

0.05 0.1 0.15 1 0.8 0.6 0.4 0.2

c=3/2

  • r

c=1/2

Natalia Chepiga (SNF, UvA) 23 July 2019 50 / 68

slide-113
SLIDE 113

WZW SU(2)2 end point

0.2 0.3 0.4 0.04 0.08 0.12 0.16 0.2

Associate the critical point with the separatrix in the log-log plot of the finite-size scaling of the dimerization The slope gives an apparent critical exponent. It is different from the WZW SU(2)2 due to logarithmic corrections At the end point the logarithmic corrections vanish and the critical exponent is d = 3/8

Natalia Chepiga (SNF, UvA) 23 July 2019 51 / 68

slide-114
SLIDE 114

WZW SU(2)2 end point

5 10 15 0.4 0.8 1.2

OBC-even

0.02 0.02 5 10 15 0.4 0.8 1.2

OBC-odd

4 8 −2 −1

OBC-even

4 8 1

OBC-odd

NC, Affleck, Mila, PRB 93, 241108’16 Natalia Chepiga (SNF, UvA) 23 July 2019 52 / 68

slide-115
SLIDE 115

WZW SU(2)2 end point

0.1 0.2 1.12 1.16 1.2 1.24 1.12 1.16 1.2 0.1 0.2

OBC-even OBC-odd The ’velocities’ extracted from the gap between the ground state and a few lowest excited states cross at the end point Away from this point the lines are divergent and the conformal towers are destroyed

Natalia Chepiga (SNF, UvA) 23 July 2019 53 / 68

slide-116
SLIDE 116

Ising vs WZW SU(2)2

0.05 0.1 0.15 1 0.8 0.6 0.4 0.2

c=3/2

  • r

c=1/2

Natalia Chepiga (SNF, UvA) 23 July 2019 54 / 68

slide-117
SLIDE 117

Ising vs WZW SU(2)2

The domain wall between the Haldane and dimerized phase carries spin-1/2 and corresponds to the magnetic WZW SU(2)2 transition The domain wall between the NNN-Haldane phased and the dimerized phase does not carry any spin and therefore the singlet-triplet gap remains open

Natalia Chepiga (SNF, UvA) 23 July 2019 55 / 68

slide-118
SLIDE 118

Solitons at the transition between the Haldane and dimerized phases

  • 0.2

0.2 0.4

  • 0.6
  • 0.4
  • 0.2

1 2 3

  • 0.2

0.2 0.4

  • 0.6
  • 0.4
  • 0.2

1 2 3 50 100

  • 0.2

0.2 0.4 50 100

  • 0.6
  • 0.4
  • 0.2

50 100 1 2 3

a) b) d) i) c) e) f) j) k)

NC, Affleck, Mila, PRB 94, 205112’16 Natalia Chepiga (SNF, UvA) 23 July 2019 56 / 68

slide-119
SLIDE 119

Outlook-1

In spin-1 chains the transition to spontaneously dimerized phase can be either continuous in the WZW SU(2)2 or in the Ising universality class, or first

  • rder

The choice between the Ising and WZW SU(2)2 transition depends on the nature of the domain walls between the corresponding phases Continuous WZW SU(2)2 critical line turn into a first order phase transition at the end point due to the presence of marginal operator Universality class can be deduced from the finite-size scaling of the energy spectrum

Natalia Chepiga (SNF, UvA) 23 July 2019 57 / 68

slide-120
SLIDE 120

Comb tensor networks

in collaboration with S.R.White

Natalia Chepiga (SNF, UvA) 23 July 2019 58 / 68

slide-121
SLIDE 121

Comb geometry

tooth backbone

Spin chains (teeth) coupled through one edge Highly decorated spin chain (backbone) One dimensional... in which direction?

Natalia Chepiga (SNF, UvA) 23 July 2019 58 / 68

slide-122
SLIDE 122

Comb geometry

tooth backbone

Y-DMRG: Guo, White, Phys. Rev. B 74, 060401 (2006) Fork tensor networks:

Holzner, Weichselbaum, von Delft, Phys. Rev. B 81, 125126 (2010); Bauernfeind, Zingl, Triebl, Aichhorn, Evertz, Phys. Rev. X 7, 031013 (2017)

Natalia Chepiga (SNF, UvA) 23 July 2019 59 / 68

slide-123
SLIDE 123

Comb geometry

tooth backbone

Natalia Chepiga (SNF, UvA) 23 July 2019 59 / 68

slide-124
SLIDE 124

Comb geometry

tooth backbone

Natalia Chepiga (SNF, UvA) 23 July 2019 59 / 68

slide-125
SLIDE 125

Generic comb

The goal is to split two channels of entanglement: along the backbone and within the tooth Finite-size clusters form local degrees of freedom Ad-lib complicated interactions within the clusters (DMRG-limited) The wave-function is expected to obey the area law

Natalia Chepiga (SNF, UvA) 23 July 2019 59 / 68

slide-126
SLIDE 126

A comb network. Mixed-canonical form

(a) (b)

Auxiliary backbone tensors: Each tensor is at most of rank 3 Split degrees of freedom on a backbone

NC, White, Phys. Rev. B 99, 235426 (2019)

Natalia Chepiga (SNF, UvA) 23 July 2019 60 / 68

slide-127
SLIDE 127

Variational optimization

(a)

=

Hamiltonian in terms of local tensors - PEPO

Natalia Chepiga (SNF, UvA) 23 July 2019 61 / 68

slide-128
SLIDE 128

Variational optimization

=

(a) (b) (c) (d)

Hamiltonian in terms of local tensors - PEPO Optimization within the tooth = DMRG

Natalia Chepiga (SNF, UvA) 23 July 2019 61 / 68

slide-129
SLIDE 129

Variational optimization

=

(a) (b) (c) (d)

Hamiltonian in terms of local tensors - PEPO Optimization within the tooth = DMRG Fully contracted tooth can be viewed as an MPO with fat physical bonds

Natalia Chepiga (SNF, UvA) 23 July 2019 61 / 68

slide-130
SLIDE 130

Variational optimization

=

(a) (b) (c) (d)

Optimization within the tooth = DMRG Fully contracted tooth can be viewed as an MPO with fat physical bonds Optimization of two backbone tensors = DMRG

Natalia Chepiga (SNF, UvA) 23 July 2019 61 / 68

slide-131
SLIDE 131

Variational optimization

=

(a) (b) (c) (d)

Optimization within the tooth = DMRG Optimization of two backbone tensors = DMRG Connect update = DMRG and involves three environments

Natalia Chepiga (SNF, UvA) 23 July 2019 61 / 68

slide-132
SLIDE 132

Complexity

=

(a) (b) (c) (d)

Complexity (χ ≈ ζ ≈ λ ≈ D) Backbone update: D5 Connect update: D4 Tooth update: D3 For AKLT-like states (finite ξ, ζ) the complexity is λ3

Natalia Chepiga (SNF, UvA) 23 July 2019 61 / 68

slide-133
SLIDE 133

DMRG versus comb. Schmidt values

50 100 150 10 -6 10 -4 10 -2 10 0 200 400 600 10 -6 10 -4 10 -2 10 0 50 100 150 10 -6 10 -4 10 -2 10 0 500 1000 10 -6 10 -4 10 -2 10 0

singular values singular values states states Comb DMRG (a) (b) (d) (c)

Heisenberg spin-1/2 Backbone cut is the same for the comb and for the DMRG DMRG: the largest bond dimension is inside the tooth Comb: the bond dimension decreases upon approaching the tip of the tooth

NC, White, Phys. Rev. B 99, 235426 (2019)

Natalia Chepiga (SNF, UvA) 23 July 2019 62 / 68

slide-134
SLIDE 134

DMRG versus comb. Complexity

0.05 0.1 0.15 100 200 300 400 500 600 0.05 0.1 0.15 10 9 10 10 10 11 10 12 0.05 0.1 200 400 600 800 1000 1200 0.05 0.1 10 9 10 10 10 11 10 12 10 13

Bond dimension Bond dimension Complexity Complexity

Comb DMRG

(a) (b) (c) (d)

DMRG, mid-tooth DMRG/Comb, backbone Comb, connect Comb, mid-tooth

Natalia Chepiga (SNF, UvA) 23 July 2019 63 / 68

slide-135
SLIDE 135

Spin-1 Heisenberg comb H = Jbb

N−1

  • i=1

Si,1 · Si+1,1 + Jt

N

  • i=1

L−1

  • j=1

Si,j · Si,j+1,

tooth backbone

Natalia Chepiga (SNF, UvA) 23 July 2019 64 / 68

slide-136
SLIDE 136

Spin-1 Heisenberg comb H = Jbb

N−1

  • i=1

Si,1 · Si+1,1 + Jt

N

  • i=1

L−1

  • j=1

Si,j · Si,j+1,

(a)

Natalia Chepiga (SNF, UvA) 23 July 2019 65 / 68

slide-137
SLIDE 137

Emergent spin-1/2 chain

20 40 60 80 100 0.1 0.2 0.3 0.4 10 -1 10 0 0.1 0.2 0.3 0.4

(a) (b)

CFT prediction for WZW SU(2)1: d = 1/2 and c = 1

Natalia Chepiga (SNF, UvA) 23 July 2019 66 / 68

slide-138
SLIDE 138

Spin-1 comb. Correlations

5 10 15 20 4 8 12 16 20

Natalia Chepiga (SNF, UvA) 23 July 2019 66 / 68

slide-139
SLIDE 139

Spin-1 comb

8 9 10 11 12 13 1 2 3 4 8 9 10 11 12 13 1 2 3 4

(a) (b)

Natalia Chepiga (SNF, UvA) 23 July 2019 66 / 68

slide-140
SLIDE 140

Higher-order edge states

Tooth with odd number of sites Edge states of each tooth couple to a triplet

Natalia Chepiga (SNF, UvA) 23 July 2019 67 / 68

slide-141
SLIDE 141

Higher-order edge states

Tooth with odd number of sites Edge states of each tooth couple to a triplet Effective spin-1 chain - Haldane state → Edge states

Natalia Chepiga (SNF, UvA) 23 July 2019 67 / 68

slide-142
SLIDE 142

Higher-order edge states

Jbb = Jt

NC, White, Phys. Rev. B 99, 235426 (2019)

Natalia Chepiga (SNF, UvA) 23 July 2019 67 / 68

slide-143
SLIDE 143

Outlook-2

Comb lattice - quasi-one-dimensional system Exotic critical behavior induced by the backbone interaction Flexible and powerful algorithm ...and many geometries to play with

Natalia Chepiga (SNF, UvA) 23 July 2019 68 / 68