Real-space renormalization group and entanglement Zoltn Zimbors - - PowerPoint PPT Presentation

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Real-space renormalization group and entanglement

Zoltán Zimborás

Department of Theoretical Physics, Wigner Research Centre for Physics

Non-Perturbative Methods in Quantum Field Theory 20 September 2017

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

An easy start: Ma-Dasgupta-Hu-Fisher RG

• Let us consider a disordered Heisenberg XXZ chain

H =

• i

Ji(Sx

i Sx i+1 + Sy i Sy i+1 + ∆iSz i Sz i+1) .

• The strong disorder renormalization group (Ma-Dasgupta-Hu, Fisher)

˜ J = Ji−1Ji+1 (1 + ∆i)ji , ˜ ∆ = (1 + ∆i) 2 ∆i−1∆i+1 .

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

The approximate ground-state and its entanglement

• Continuing this procedure, we end up with a singlet distribution as an

approximate ground state - random singlet phase, infinite disorder fixpoint SL = ln 2 3 log L .

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

The approximate ground-state and its entanglement

• Continuing this procedure, we end up with a singlet distribution as an

approximate ground state - random singlet phase, infinite disorder fixpoint SL = ln 2 3 log L .

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

The transverse-field Ising model

• The disordered transverse field Ising model

H =

• i

JiSx

i Sx i+1 + hiSz i .

• Two types of RG steps

SL = ln 2 6 log L .

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Translation-invariant Quantum Critical Systems in 1+1 dimensions

• The entanglement entropy asymptotics of gapless models

( C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424-443 (1994); Vidal et al. PRL 90, 227902 (2003);

• P. Calabrese and J. Cardy, JSTAT 06002 (2004); J. Eisert et al, Rev. Mod. Phys. 82, 277 (2010).)

SL = c 3 log L + k

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Translation-invariant Quantum Critical Systems in 1+1 dimensions

• The entanglement entropy asymptotics of gapless models

( C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424-443 (1994); Vidal et al. PRL 90, 227902 (2003);

• P. Calabrese and J. Cardy, JSTAT 06002 (2004); J. Eisert et al, Rev. Mod. Phys. 82, 277 (2010).)

SL = c 3 log L + k

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Quasi-periodic models: the Professor-Student sequence

• Consider the substitution rule: L → LS; S → L.

L LS LSL LSLLS LSLLSLSL LSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLS LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSL LSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLSLLSLLSLSLLSLLSLSLLSLSLLS

• Professors and students reproduce themselves as rabbits! Actually, the

above sequence was originally constructed as a simple model for rabbit reproduction in 1202 by Leonardo of Pisa, also known as Fibonacci.

6 / 14

Real-space renormali- zation group and entangle- ment Zoltán Zimborás

The Fibonacci XXZ chain

• The universality class of the model is not the infinite disorder fixpoint.

This quasiperiodic modulation is only marginal. SL = ln 2 3 ln 5 log L .

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Kadanoff blocking RG in translation-invariant models

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Kadanoff blocking RG

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Entanglement renormalization

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Entanglement renormalization

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Multiscale Entanglement Renormalization Ansatz

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

Entanglement Properties of MERA

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Real-space renormali- zation group and entangle- ment Zoltán Zimborás

An interesting connection to AdS/CFT

Ryu-Takanayagi formula

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