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Opal Wilcox
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1 = x 3 4

2 Application of DMRG to 2D systems L y L x Unit cell Periodic boundary conditions for both x and y directions k y = 2n/L y = X n / l 2 Initial basis states (Landau gauge) (r) = exp [ i X n y (x X n ) 2 XN l 2l ] H ( x X n ) 2 2 N l One particle states are uniquely specified by X n and N H N : Hermite polynomials X n : guiding center N : Landau level index Mapping on to effective D lattice model D lattice model Xn

(Ne=) DMRG m= -3.23934 m=2-3.239686 m=3-3.23998 m=4-3.

3 Density matrix renormalization Ground state ij i > j > i j Density matrix ii' =j ij * i'j Norm Tr Ground state energy (Ne=) DMRG m= m= m= m= N=2 =/2 Exact Eigenvalues of N= =/3 Ne=8 M = i

4 Ground state of quantum Hall systems Type-II stripe Stripe II FQHE N= WC Fermi liquid N= Stripe II Stripe I Laughlin state N=2 WC Bubble Stripe I Wigner crystal Bubble Type-I stripe

5 Fractional quantum Hall effect R. Willett et al (987) Hall resistance Longitudinal resistance

6 N=2 Landau level Stripe state =3/7 E G 3-stripes 2-stripes (b) 4-stripes (a) (c) L x /L y (a) L x /L y =.3 (b) L x /L y =.8 (c) L x /L y =2.3 x x x y y y

Ground state of N=2 Landau level..2.3.4.

7 Ground state of N=2 Landau level N Wigner crystal Two-electron bubble state Type I stripe state Shibata and Yoshioka: Phys. Rev. Lett (2)

Stripe and bubble states 3 g(,y) 2 stripe I Wigner crystal /7 /6 /5 =2/9 /3 /4 3/ 4/ 2/5 /2 3/7 2-electron bubble 2 4 6 8 y 9/2 Lilly et al.

8 Stripe and bubble states 3 g(,y) 2 stripe I Wigner crystal /7 /6 /5 =2/9 /3 /4 3/ 4/ 2/5 /2 3/7 2-electron bubble y 9/2 Lilly et al. (999) Two-electron bubble state Type I stripe state DMRG WC 2-electron bubble stripe I y Hartree-Fock x electron bubble

Effective interaction in higher Landau levels

4 8 r N= Veff (xx) x 4 Veff (x) + Veff (xx) Rc :

9 Effective interaction in higher Landau levels Pure Coulomb Potential energy generated by the two electrons separated by x x = 5 Veff N= N= N= Veff (x) + Veff (xx) Veff (x) N=2 N=3 2Rc N= r N= Veff (xx) x 4 Veff (x) + Veff (xx) Rc : cyclotron radius e 2Rc e Veff (x) Rc e Veff (xx) x x = 5 4 e

independent of S2 - Not correlated - ( > > ) SS2 = > S ( > > ) S2

10 Entanglement entropy Wave function S=/2 2-Spin system S S2 S = or S2 = or SS2 S > S2 > Reduced density matrix Entanglement entropy S is independent of S2 - Not correlated - ( > > ) SS2 = > S ( > > ) S2 disentangled S depends on S2 - correlated - ( > > ) Singlet state entangled

11 Scaling of entanglement entropy D system Short range correlation : L D-XXZ model ( L >> correlation length ) 2 gapless Power law correlation : (D critical system ) gapfull L

Topological order in 2D Non-trivial universal correction Boundary term

12 Scaling of entanglement entropy Short range correlation ( L >> ) entanglement D-dimensional system Area law L boundary size (length) Topological order in 2D Non-trivial universal correction Boundary term Topological term Fractional quantum Hall state = /m Laughlin state : D = m /2

13 Torus geometry Density matrix eigenvalues i Fractional quantum Hall state = /3 i. Ne=8 Ne= Ne=2 torus. Ne : electron number Lx Ly. boundary i

14 Entanglement entropy Torus geometry Fractional quantum Hall state = /3 Pair correlation functions =/3 4 Ne = 8 ~6 Ly / l = 7.4 Ly / l = 6.2 x/l Ly / l = 5. 3 Ly / l = 3.7 ( boundary length ) Ly / l = torus Lx Ly boundary Lx / l

15 Entanglement entropy 6 Entanglement entropy =/3 fractional quantum Hall state boundary term topological term sphere 4 2 topological entropy sphere N= 6 22 N= torus spherical : ln m /2 torus : 2 ln m /2 sphere ln 3 /2 ~.55 torus 2ln 3 /2 ~ =/m topological term L: boundary length torus L / l : boundary length L: boundary length

6 4 2 topological entropy 2 Entanglement entropy Fractional quantum Hall state topological term spherical : ln m /2 torus : 2 ln m /2 ( =/m ) Entanglement entropy boundary term =/5 sphere 36

16 6 4 2 topological entropy 2 Entanglement entropy Fractional quantum Hall state topological term spherical : ln m /2 torus : 2 ln m /2 ( =/m ) Entanglement entropy boundary term =/5 sphere N=8 =/3 torus =/3 sphere 2 24 sphere ln 3/2 ~.55 ln 5 /2 ~ L / l : boundary length topological term torus 2ln 3/2 ~. sphere L: boundary length torus L: boundary length

17 f ( x) 2 sin x L f ( x) sin 2 x L x L A. Gendiar, R. Krcmar, and T. Nishino, Prog. Theor. Phys. 22, 953 (29); 23, 393 (2).

18 H ( )c Original k k k / 2 L k c k L 2 f ( x) sin x L f ( x) sin 2 kx / 2 coskx 2 exp 2 4 k 2 L ikx exp ikx H Deform k k k ( k ( ( ) c k k 2 k k 2 k c k ) c ) c k k k c c k k k H Deform T. Hikihara and T. Nishino, Phys. Rev. B 83, 644(R) (2) H. Katsura, J. Phys. A 44, 252 (2)

19 Naokazu Shibata and Daijiro Yoshioka: Phys. Rev. Lett (2) Shibata and Hotta: Phys. Rev. B (2) Hotta and Shibata: Phys. Rev. B 86 48(R) (22)

20 E E - F x L x L

21 S=/2 Heisenberg spin ladder.578. E/L J / J =. H =. M S=/2 spin ladder J J J J / J = /L H E/L.59 J / J =. H =.8 M /L 3.3 H =.6 x J / J =. 6

22 Nishimoto, Shibata and Hotta: Nature Communications (23)

25 .52.5 = Ne=49 Ne=48 2 x 3 4 7/9 M 5/9 3/9 /9 2 3 H

Sine square deformation(ssd) and Mobius quantization of 2D CFTs

riken_ssd_2017 arxiv:1603.09543/ptep(2016) 063A02 Sine square deformation(ssd) and Mobius quantization of 2D CFTs Niigata University, Kouichi Okunishi Related works: Kastura, Ishibashi Tada, Wen Ryu Ludwig