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

Transcription

1 riken_ssd_2017 arxiv: /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

2 Sine square deformation (SSD) SSD : smooth cutoff to suppress the boundary scattering e.g. spin chains Gendiar, Krcmar and Nishino, Prog. Theor. Phys. 122, 953(2009) Hikihara and Nishino, Phys. Rev. B 83, (2011) sin + ) sin coupling amplitude 1 = SSD system(open B.C.) uniform system with periodic B.C. The groundstate wavefunctions are identical to each other!

3 Correlation functions of the XXZ chain(massless regime) Open uniform / SSD Periodic Hikihara and Nishino, Phys. Rev. B 83, (2011) The overlap of the wavefunctions between SSD and PBC systems is 1, within numerical accuracy!

4 SSD the exact equivalence between SSD and PBC systems for the gapless groundstate Hikihara and Nishino, Phys. Rev. B 83, (2011) exact example: XY model, free fermion(lattice) free fermion(non rela) Katsura, J. Phys. A: Math. theror. 44, (2011) Maruyama, Katsura and Hikihara, Phys. Rev. B 84, (2011) Okunishi, Katsura, J.Phys.A:Math.Teore. 48 (2015) applications grand canonical approach for magnetization curves Shibata and Hotta, Phys. Rev.B 84, (2011) Hotta Nishimoto and Shibata, Phys. Rev. B 87, (2013) CFT string theory/cft H. Katsura, J. Phys. A:Math. Theore. 45, (2012) Tada, arxiv: [hep th] Ishibashi and Tada, arxiv: [hep th] Ishibashi and Tada, arxive: [hep th]

5 Lattice free fermion Maruyama, Katsura and Hikihara, Phys. Rev. B 84, (2011) sin ) sin 1 2 Fourier transform. ) 2cos uniform / spatial deformation Nearest neighbor hopping in the momentum space! can annihilates the Fermi sea, if the chemical potential is appropriately chosen

6 excitations drastically reduce the finite size effect 1/L^2 dependence of low energy excitations Note Non relativistic free fermion: SUSY quantum mechanics exact 1/L^2 dependence of excitations (Almost) continuous curve Hotta Nishimoto and Shibata, Phys. Rev. B 87, (2013) application: grandcanonical approach Okunishi, Katsura, J.Phys.A:Math.Teore. 48 (2015)

7 CFT Hamiltonian H. Katsura J. Phys. A: Math. Thoer. 45, (2012) Hamiltonian for the cylinder 2 2 / / 2 log SSD Hamiltonian )] SL(2,C) invariance of CFT 0 0 The SSD vacuum is equivalent to that of the uniform system,

8 SSD/CFT dipolar quantization Ishibashi and Tada, arxiv: [hep th] Ishibashi and Tada, arxive: [hep th] Regarding as a Hamiltonian of a CFT, ) Quantization of the CFT, starting with the classical Virasoro algebra Quantization on the dipolar coordinate (singularity at z=1) infinite circumference limit continuous Virasoroalgebra

9 Parameterization: uniform and SSD cosh 2 sinh 2 2 θ= 0 : Uniform : SSD The CFT vacuum is always the same for any real SL(2,R) invariance of CFT Lorentz transformation If we construct a CFT for the Hamiltonian of, what happens? interpolating between radial quantization(uniform system) and dipolar quantization(ssd system).

10 Classical Virasoro classical Virasoro generators new classical Virasoro generators Mobius transformation coordinate classical Virasoro algebra

11 Mobius coordinate define complex cooridinate with : tanh : 1/tanh radial quantization ( 0) : 0, : dipolar quantization ( ) : 1, essential singularity constant contours in z plane time source and sink approach to each other as θ increases

12 Virasoro charges integration path : constant τ contour Relation with the conventional Virasoro generators SL(2) subalgebra

13 For general n(>1), we have series expansion form : F: Gauss s hypergeometric function, tanh satisfies the Virasoro algebra commutator contour integral

14 Continuum limit (connection to the dipolar quantization) Lorentz transformation contains a diversive factor of scale normalized generator θ= : dipolar quantization overall scale factor The spectrum of becomes continuous : κ / The spatial coordinate on the constant τ contours continuous Virasoro algebra in the limit with

15 Primary fields primary field of the scaling dimension h primary state for can be related with through translation operator c.f., This primary state is normalizable for However, primary fields at the SSD/dipolar point( ) is still unknown

16 conformal mapping approach We can obtain without passing through analysis of the Mobius coordinate. Conformal mapping of SL(2,R) the same as Mobius quantization However, the Mobius quantization is essential to reveal the continuum limit of the Virasoro algebra.

17 simple example: lattice free fermion θ= 0 : Uniform : SSD Fermi surface single particle spectrum SSD ( 1 parabola dispersion uniform( 0 cos dispersion The linear dispersion around the Fermi surface is invariant High energy states changes into the parabola dispersion due to the SSD effect. quantum index

18 role of (the same Hamiltonian except for the overall scale) single particle spectrum SSD ( parabola dispersion θ= 0 : Uniform : SSD Fermi surface uniform( 0 cos dispersion The linear dispersion is compressed around the Ferimi surface. Parabolic dispersion at the SSD point quantum index

19 summary We analyzed the SSD problem in terms of 2D CFT The Mobius coordinate of SL(2,R) plays an essential role. For a finite, a primary state is well defined. In the limit, we have the continuous Virasoro algebra corresponding to the dipolar quantization.

20 prospective The entanglement Hamiltonian/modular Hamiltonian Radial quantization angular quantization Rindler Baxter s CTM DMRG half infinite cut [0: ] If asystem is integrable, its entanglement Hamiltonian is equivalent to log of Baxter s CTM. H. Itoyama and H. B. Thacker, Nucl.Phys. B320 (1989) 541 Mobius quantization the entanglement Hamiltonian can be defined for the finite length(= 2/sinh(2θ) ) interval. Wen, Ryu, Ludiwg, arxiv:

21 SSD in the continuous space SSD : sin uniform : 1 with periodic boundary Ψ,,, Ψ,,, system length = L Single particle problem uniform : SSD : sin plane waves cos(k) dispersion localized shape???? SUSY quantum mechanics

22 the exact groundstate correspondence N free fermions in the uniform 1D ring of length L. Slater determinant of plane waves below the Fermi energy Φ,,, det N free fermions in the SSD system of length L. Slater determinant of the SSD basis below the Fermi(zero) energy Ψ,,, det If the chemical potential is tuned so that the highest energy level corresponds to zero energy, Ψ,,, Φ,,, equivalence of the linear spaces between uniform and SSD systems

23 inverse problem sin sin for a given chemical potential solve the eigenvalue problem of for a given energy eigenvalue evaluate the potential coupling The inverse problem of the 1/sin 2 potential problem parametarization 1 1 sin potential problem Integrable = SUSY quantum mechanics + shape invariance

24 SUSY quantum mechanics intertwiner super potential ln sin, super charge shape invariance recursion relation for, partner Hamiltonian 1 wavefunction th excited state for x x x 0 with x sin energy eigenvalue inverse problem x 1

25 wavefunctions potential problem : column SSD inverse problem : row 0 : Neumann 1 : Dirichret boundary connectivity of the wavefunction : P/AP 1: smooth boundary 0 0 0

26 N body groundstate: uniform: ~cos ~sin 0,1, 2 1 1, 2 SSD:, 1,2, two zero energy states unoccupied /Neumann b.c. occupied / Dirichlet b.c.

27 proof : case the equivalence of the linear vector space between uniform and SSD systems SSD x~ direct calculation of polynomial of trigonometric function cos 2 subtract ~cos 2 sin2 with,,, 1

28 spectrum 101, 0; Lattice fitting Continuum consistent around Fermi point(zero energy) finite size spectrum with 1 exact(no higher order correction)! cf. universal 1/ of CFT DOS consistent with Hotta, Nishimoto and Shibata, Phys. Rev. B 87, (2013)

29 wavefunctions 1 2

30 zero energy wavefunction( ) 101 & 50 blue : ) with red : sin ) cos plane wave; OK! remark for 0, is not normalizable because of Neumann boundary., a certain cutoff at the boundary is required

31 summary J. Phys. A: Math. Theor. 48 (2015) (arxiv: ) We defined the SSD problem in the 1D continuous space The single particle state of the SSD is formulated as the inverse problem of the 1/sin^2 potential problem The SUSY quantum mechanics with the shape invariance plays a significant role to solve the inverse problem The comparison between the Lattice and continuum SSD The exact 1/ dependence of the excitation & / behavior of DOS. 1st quantization approach Field theoretical treatment