Real-space renormali- zation group and entangle- ment Real-space renormalization group and entanglement Zoltán Zimborás Zoltán Zimborás Department of Theoretical Physics, Wigner Research Centre for Physics Non-Perturbative Methods in Quantum Field Theory 20 September 2017 1 / 14
An easy start: Ma-Dasgupta-Hu-Fisher RG Real-space renormali- zation group and entangle- • Let us consider a disordered Heisenberg XXZ chain ment Zoltán � J i ( S x i S x i + 1 + S y i S y i + 1 + ∆ i S z i S z H = i + 1 ) . Zimborás i • The strong disorder renormalization group (Ma-Dasgupta-Hu, Fisher) J = J i − 1 J i + 1 ∆ = ( 1 + ∆ i ) ˜ ˜ ∆ i − 1 ∆ i + 1 . ( 1 + ∆ i ) j i , 2 2 / 14
The approximate ground-state and its entanglement Real-space renormali- zation group and entangle- ment • Continuing this procedure, we end up with a singlet distribution as an Zoltán Zimborás approximate ground state - random singlet phase, infinite disorder fixpoint S L = ln 2 3 log L . 3 / 14
The approximate ground-state and its entanglement Real-space renormali- zation group and entangle- ment • Continuing this procedure, we end up with a singlet distribution as an Zoltán Zimborás approximate ground state - random singlet phase, infinite disorder fixpoint S L = ln 2 3 log L . 3 / 14
The transverse-field Ising model • The disordered transverse field Ising model Real-space renormali- zation � J i S x i S x i + 1 + h i S z H = group and i . entangle- i ment Zoltán • Two types of RG steps Zimborás S L = ln 2 6 log L . 4 / 14
Translation-invariant Quantum Critical Systems in 1+1 dimensions Real-space renormali- zation group and entangle- ment • The entanglement entropy asymptotics of gapless models Zoltán ( C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424-443 (1994); Vidal et al. PRL 90, 227902 (2003); Zimborás P. Calabrese and J. Cardy, JSTAT 06002 (2004); J. Eisert et al, Rev. Mod. Phys. 82, 277 (2010).) S L = c 3 log L + k 5 / 14
Translation-invariant Quantum Critical Systems in 1+1 dimensions Real-space renormali- zation group and entangle- ment • The entanglement entropy asymptotics of gapless models Zoltán ( C. Holzhey, F. Larsen, F. Wilczek, Nucl. Phys. B 424-443 (1994); Vidal et al. PRL 90, 227902 (2003); Zimborás P. Calabrese and J. Cardy, JSTAT 06002 (2004); J. Eisert et al, Rev. Mod. Phys. 82, 277 (2010).) S L = c 3 log L + k 5 / 14
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
Quasi-periodic models: the Professor-Student sequence Real-space renormali- zation group and entangle- • Consider the substitution rule: L → LS ; S → L . ment L Zoltán LS Zimborás 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
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