MERA for Spin Chains with Critical Lines

Transcription

1 MERA for Spin Chains with Critical Lines Jacob C. Bridgeman Aroon O Brien Stephen D. Bartlett Andrew C. Doherty ARC Centre for Engineered Quantum Systems, The University of Sydney, Australia January 16, 2013

2 Overview Developing numerical methods to study 1D critical systems Variational algorithm to optimise a MERA description of the ground state

3 Overview Developing numerical methods to study 1D critical systems Variational algorithm to optimise a MERA description of the ground state Ashkin-Teller model Believed to be described by c = 1 CFT with continuously varying critical indices

4 Overview Developing numerical methods to study 1D critical systems Variational algorithm to optimise a MERA description of the ground state Ashkin-Teller model Believed to be described by c = 1 CFT with continuously varying critical indices Extract physical information about the model of interest Output of our algorithm is consistent with a conformal field theory conjectured to describe the thermodynamic limit of the spin models examined

5 x µ = Tensor Network Formalism

6 Tensor Network Formalism x µ = x µ =

7 Tensor Network Formalism x µ = x µ = x µ x µ =

8 Tensor Network Formalism x µ = x µ = u γδ αβ = x µ x µ =

9 Tensor Network Formalism x µ = x µ = u γδ αβ = x µ x µ = w δ αβγ =

10 Multiscale Entanglement Renormalization Ansatz G. Vidal, Physical Review Letters 99, (2007).

11 Multiscale Entanglement Renormalization Ansatz χ u χ l G. Evenbly et al, Physical Review B 82, (2010).

12 Algorithm G. Evenbly and G. Vidal, Physical Review B 79, (2009). G. Evenbly and G. Vidal, (2011), arxiv: v1 [quant-ph]. R. N. C. Pfeifer, Simulation of Anyons Using Symmetric Tensor Network Algorithms, PhD Thesis,The University of Queensland, G. Evenbly, Foundations and Applications of Entanglement Renormalization, PhD Thesis, The University of Queensland, S. Singh, R. Pfeifer, and G. Vidal, Physical Review A 82, (2010).

13 Scaling of Algorithm 10 3 χ l = χ u = χ/5 Time (Seconds) Fit: aχ b Nonsymmetric Nonsymmetric with projector Z 2 Z 2 with projector 10 5 Z 2 Z 2 Z 2 Z 2 with projector χ 3.46 GHz Dual Core, 192 Gb RAM

14 Scaling of Algorithm 10 3 χ l = χ u = χ/5 Time (Seconds) Fit: aχ b Nonsymmetric χ Z 2 Z 2 Z 2 Z 2 with projector 3.46 GHz Dual Core, 192 Gb RAM

15 Scaling of Algorithm Time (Seconds) χ l = χ u = χ/5 Fit: aχ b χ Nonsymmetric Z 2 Z GHz Dual Core, 192 Gb RAM

16 Ashkin-Teller Model H = N (Z j + Z j ) j=1 N 1 β (X j X j+1 + X j X j+1 ) j=1 J. Ashkin and E. Teller, Physical Review 64, 178 (1943). J. Sólyom, Physical Review B 24, 230 (1981).

17 H AT = N (Z j + Z j + λz j Z j ) j=1 Ashkin-Teller Model N 1 β (X j X j+1 + X j X j+1 + λx j X j X j+1 X j+1 ) j=1 J. Ashkin and E. Teller, Physical Review 64, 178 (1943). J. Sólyom, Physical Review B 24, 230 (1981).

18 H AT = S 1 = N (Z j + Z j + λz j Z j ) j=1 Ashkin-Teller Model N 1 β (X j X j+1 + X j X j+1 + λx j X j X j+1 X j+1 ) j=1 N Z j S 2 = j=1 N j=1 Z j J. Ashkin and E. Teller, Physical Review 64, 178 (1943). J. Sólyom, Physical Review B 24, 230 (1981).

19 Ashkin-Teller Phase Diagram M. Yamanaka, Y. Hatsugai, and M. Kohmoto, Physical Review B 50, 559 (1994).

20 Ashkin-Teller Phase Diagram M. Yamanaka, Y. Hatsugai, and M. Kohmoto, Physical Review B 50, 559 (1994).

21 Ashkin-Teller Ground State Energy Ground state energy per site λ MERA Exact χ l = 12, χ u = 8

22 Ashkin-Teller Ground State Energy Ground state energy per site λ MERA Exact χ l = 12, χ u = 8

23 Ashkin-Teller Ground State Energy Ground state energy per site λ MERA Exact Error in GSE ( 10 4 ) λ χ l = 12, χ u = 8

24 Conformal Data Thermodynamic limit of critical spin chain described by a conformal field theory Central charge c conformal exponents h + h = OPE coefficients

25 Conformal Data Thermodynamic limit of critical spin chain described by a conformal field theory Central charge c conformal exponents h + h = OPE coefficients Ashkin-Teller (on our line) thought to be described by orbifold boson CFT c = 1 continuously varying exponents

26 Conformal Data Thermodynamic limit of critical spin chain described by a conformal field theory Central charge c conformal exponents h + h = OPE coefficients φ = λ φ Ashkin-Teller (on our line) thought to be described by orbifold boson CFT c = 1 continuously varying exponents

27 Ashkin-Teller Scaling Dimensions λ = 0 χ l = 28 = χ/5, χ u = φ 9/8 1/ φ 9/8 1/ φ 5/4 1/4 2 4 φ MERA PrimarySDs Descendant SDs

28 Ashkin-Teller Scaling Dimensions λ = 0 χ l = 28 = χ/5, χ u = φ 9/8 1/ φ 9/8 1/ φ 5/4 1/4 2 4 φ MERA PrimarySDs Descendant SDs λ = 2/2 χ l = 36 = χ/5, χ u = 20 3/2 2 1/ φ 9/8 1/ φ 9/8 1/ φ 3/2 2 1/ φ

29 Ashkin-Teller Central Charge Central Charge MERA Exact R 2 AT = π 2 cos 1 ( λ) R 2 AT χ l = 12, χ u = 8

30 Ashkin-Teller Continuously Varying Exponents MERA obcft MERA obcft / /2 1 2 R 2 AT R 2 AT χ l = 12, χ u = 8

31 Ashkin-Teller Continuously Varying Exponents MERA obcft MERA obcft / /2 1 2 R 2 AT R 2 AT χ l = 12, χ u = 8

32 Ashkin-Teller Nonlocal/Twisted R 2 AT R 2 AT χ l = 12, χ u = 8

33 Ashkin-Teller Nonlocal/Twisted R 2 AT R 2 AT χ l = 12, χ u = 8

34 Ashkin-Teller Ising Line M. Yamanaka, Y. Hatsugai, and M. Kohmoto, Physical Review B 50, 559 (1994).

35 Ashkin-Teller Ising Line M. Yamanaka, Y. Hatsugai, and M. Kohmoto, Physical Review B 50, 559 (1994).

36 Ashkin-Teller Ising Line 2 9/ λ 1/8 χ l = 16 = χ/4, χ u = λ

37 V. Alba, L. Tagliacozzo, P. Calabrese, J. Stat. Mech P06012 (2011) c = 1 CFTs

38 Conclusions Independently developed code to optimize a MERA description of ground state Incorporated Abelian symmetries present in the models

39 Conclusions Independently developed code to optimize a MERA description of ground state Incorporated Abelian symmetries present in the models Obtained conformal data for Ashkin-Teller consistent with the orbifold boson CFT

40 Conclusions Independently developed code to optimize a MERA description of ground state Incorporated Abelian symmetries present in the models Obtained conformal data for Ashkin-Teller consistent with the orbifold boson CFT Obtained conformal data for perturbed cluster state consistent with the free boson CFT

41 Conclusions Independently developed code to optimize a MERA description of ground state Incorporated Abelian symmetries present in the models Obtained conformal data for Ashkin-Teller consistent with the orbifold boson CFT Obtained conformal data for perturbed cluster state consistent with the free boson CFT Demonstrated a critical line which does not have continuously varying critical indices

42 V. Alba, L. Tagliacozzo, P. Calabrese, J. Stat. Mech P06012 (2011) c = 1 CFTs

43 S 1 Boson CFT

44 Perturbed Cluster State Ground State Energy Ground state energy per site λ MERA Exact Error in GSE ( 10 5 ) λ χ l = χ u = χ/4 = 20

45 Perturbed Cluster State Central Charge Central Charge MERA Exact R 2 pcl = 2 π (π cos 1 (λ)) R 2 pcl χ l = χ u = χ/4 = 20

46 pcl Continuously Varying Exponents MERA S 1 boson MERA S 1 boson / /2 1 2 R 2 pcl R 2 pcl χ l = χ u = χ/4 = 20

47 pcl Continuously Varying Exponents MERA S 1 boson MERA S 1 boson / /2 1 2 R 2 pcl R 2 pcl χ l = χ u = χ/4 = 20

48 Avoided Crossing MERA S 1 boson R 2 pcl = 2 π (π cos 1 (λ)) 1 R 2 pcl χ l = χ u = χ/4 = 20