Gauge invariance and topological order in quantum many-particle systems

1. 2005 년 10월 28일 @ 한국고등과학원 Gauge invariance and topological order in quantum many-particle systems 오시가와 마사기 (Masaki Oshikawa) 동경공대 (Tokyo Institute of Technology)

2. Commensurability and Luttinger’s theorem implications of (fractional) particle density (“old” stuffs)  Ground-state degeneracy and topological order what is the topological order and when do we find it? (more recent developments) 

3. Quantum phases and transitions (at T=0) Phase I gap Phase II critical point (gapless) Typical example: Ising model with a transverse field in d-dim. (equivalent to classical Ising in (d+1)-dim.) ordered phase disordered phase

4. Renormalization Group Critical point = gapless RG fixed point There is always a relevant perturbation! We have to fine-tune the coupling to achieve the criticality

5. However ……. there are many gapless systems in cond-mat physics, without any apparent fine-tuning! solids, metals, etc. …… Why is the gapless phase “protected”? Nambu-Goldstone theorem: gapless excitations exist if a continuous symmetry is spontaneously broken explains gapless phonons in solids but what about metals?? Let’s seek a new mechanism……

6. Magnetization process of an antiferromagnet (at T=0 ) classical picture H

7. magnetization curve m saturation H

8. Magnetization process in quantum antiferromagnets Long history of study Exact magnetization curve for S=1/2 Heisenberg antiferromagnetic chain (Bethe Ansatz exact solution) Quantitave difference from classical case No qualitative difference??

9. New feature in the quantum case Shiramura et al. (1998) [H. Tanaka group, Tokyo Inst. Tech.]

10. 10 difficult to understand in classical picture! T=0 m magnetization plateau H Quantization condition for a plateau: (M.O.-Yamanaka-Affleck ’97) n : # of spins per unit cell of the groundstate S : spin quantum number

11. Understanding the quantum magnetization process At T=0, the system should be in the ground state magnetization curve = magnetization of the ground state for the Hamiltonian (which depends on the magnetic field)

12. Hamiltonian: Exchange interaction (typical example) Magnetic field (Zeeman term) Let us assume that the interaction is invariant under the rotation about z-axis (direction of the applied field)

13. We can choose simultaneous eigenstates of and They are also always eigenstates of no change in the eigenstates even if the magnetic field is changed! how does the ground-state magnetization increase by the magnetic field?

14. E lowest energy state with gap  lowest energy state with H g.s. magnetization = M M+1 plateau of width  !

15. For any (finite size) quantum magnet (with the axial symmetry) the magnetization curve at T=0 consists of plateaus and steps! In the thermodynamic limit (infinite system size) “gapless” (! 0 above the ground state) : smooth magnetization curve “gapful” ( remains finite above the g.s.): plateau

16. m gapless T=0 gapful! H gapful phases are rather “special”! when can the quantum magnet be gapful?

17. Quantum magnet as a many-particle system e.g. consider S=1/2 occupied by a particle “down” “up” empty site particle creation op. annihilation op. particle hopping interaction

18. When can the quantum many-particle system on a lattice be gapful? usually, particles can move around, giving gapless (arbitrary low-energy) excitations A finite excitation gap may appear if the particles are “locked” by the lattice to form a stable ground state. (particles are then mobilized only by giving an energy larger than the gap.)

19. To have the particles “locked”, the density of the particles must be commensurate with the lattice. 1 particle/ unit cell (= 2 sites) add extra particles (“doping”) mobile carriers

20. 20 commensurate density # of particles/unit cell of the g.s. particle density (# of particle/site) # of sites/ unit cell of the g.s. particles may be “locked” to form an insulator, with a finite gap (possibly with SSB of translation symmetry ---- will come back on this later) incommensurate density particles are mobile, forming a conductor with gapless excitations

21. Finite-temperature transition near the plateau magnetization//H vs. T m MFT T

22. Magnon BEC picture Tsuneto-Murao 1971 ...........Nikuni et al. 2000 vacuum singlet on dimer (lowest) triplet on dimer magnon (boson) Dispersion: (near the bottom) magnetic field chemical potential magnon BEC ordering transition

23. Consequences of the BEC picture condensed magnons Nikuni, MO, Oosawa, Tanaka 2000 Quantum spin system in a field = “particles” with a tunable chemical potential

24. Back to the quantization….. e.g. consider S=1/2 occupied by a particle “down” “up” empty site commensurability condition

25. Is it really true? physical properties of the system (such as magnetization curve): generally depends on Hamiltonian ground state in strongly interacting system: very complicated! why would the commensurability condition be valid in strongly interacting systems??

26. d=1 A generalization of Lieb-Schultz-Mattis argument (1961) shows There are q degenerate groundstates if  = p/q and if the system has a gap (M.O.-Yamanaka-Affleck, 1997) d¸ 2 Topological argument (with assumptions) (M.O. 2000) Relation to Drude/Kohn argument (M.O. 2003) Rigorous proofs (Hastings 2004, 2005)

27. Insulator vs. conductor Linear response theory Drude weight D=0 : insulator D>0 : conductor (Kohn, 1963)

28. Real-time formulation of D initial condition: ground state at t=0 taking t!1, T !1 (as long as the linear response theory is valid)

29. circumference: E  uniform electric field cf. Laughlin (1981)

30. 30 energy gain

31. (unit flux quantum) choose and take the limit Hamiltonian at t=T with the unit flux quantum is equivalent to that at t=0 with =0 (no Aharonov-Bohm effect) Does the groundstate go back to the groundstate? If so, the energy gain =0 thus the system is an insulator

32. No change in the momentum?! As long as we choose constant-A gauge, Hamiltonian is translational invariant. Momentum is gauge-dependent!! large gauge transf.

33. has same momentum and To compare the momentum, we compare and lattice translation operator cross section Total momentum change (after large gauge tr.) (Lieb-Schultz-Mattis, 1961)

34. Momentum Px is defined modulo 2 The final state must be different from the initial state (g.s.) if Z (for appropriate C) In order to have an insulator for an incommensurate particle density  Z, one must have low-energy state with the extra momentum (M.O. 2003) 2dim.: 1 dim. 3 dim. and higher: no constraint from D=0

35. Application to gapless system Consider a system of electrons (fermions) non-interacting electrons = free Fermi gas Fermi sea

36. Landau’s Fermi liquid theory Interacting electrons: what happens?? elementary excitation: “quasiparticles” collective excitation in terms of electrons but behaves like free fermions “Fermi sea” of quasiparticles What is the volue of the “Fermi sea”? Luttinger’s theorem: VF is not renormalized by interactions

37. In some cases, the original proof by Luttinger does not apply, or is questionable…. eg. one dimensional systems systems involving localized spins (Kondo lattice) non-Fermi liquids Alternative approach?

38. adiabatically insert unit flux quantum (again!) E  cf. Laughlin (1981)

39. calculate the momentum change due to the flux insertion • by Fermi liquid theory (or any effective theory) (ii) using the large gauge transformation

40. Applications electrons coupled to localized spins (Kondo lattice) localized spins do contribute to Fermi sea volume! (if low-energy excitations are exhausted by Fermi liquid) “Fractionalized Fermi liquid” a phase that has similar low-energy excitations as the Fermi liquid but violates Luttinger’s theorem (with fractionalized spin exc.) (Senthil-Sachdev-Vojta, 2003)

41. [From http://sachdev.physics.harvard.edu/] M. Oshikawa, Phys. Rev. Lett.84, 3370 (2000).

42. Effect of flux-piercing on a topologically ordered quantum paramagnet Ly N. E. Bonesteel, Phys. Rev. B 40, 8954 (1989). G. Misguich, C. Lhuillier, M. Mambrini, and P. Sindzingre, Eur. Phys. J. B 26, 167 (2002). vison Lx-1 Lx 2 Lx-2 1 3 [From http://sachdev.physics.harvard.edu/]

43. Flux piercing argument in Kondo lattice A Fractionalized Fermi liquid. cond-mat/0209144 Shift in momentum is carried by nT electrons, where nT = nf+ nc In topologically ordered, state, momentum associated with nf=1 electron is absorbed by creation of vison. The remaining momentum is absorbed by Fermi surface quasiparticles, which enclose a volume associated with ncelectrons. [From http://sachdev.physics.harvard.edu/]

44. “Bose volume” The present argument actually applies to system of boson as well. The momentum change due to applied electric field is “quantized”! The corresponding “Luttinger’s theorem” gives a quantization of magnus force in lattice bose systems at T=0 (Vishwanath and Paramekanti, 2004)

45. Summary Quantum many-particle systems on a periodic lattice : # of particles / unit cell Topological restrictions: If the system is gapless the “Fermi(Bose) volume” is quantized -- “Luttinger’s theorem” If the system is gapful for  Z there must be q-foldgroundstate degeneracy

46. magnetization plateau gauge invariance and QHE (Laughlin, 1981) Haldane conjecture (1983) topological quantization conductor or insulator? (Kohn, 1963) Lieb-Schultz-Mattis theorem (1961) Luttinger’s theorem (1960)

47. Topological restrictions: If the system is gapless the “Fermi(Bose) volume” is quantized -- “Luttinger’s theorem” If the system is gapful there must be groundstate degeneracy what does this mean? “Usually” it is a consequence of Spontaneous Symmetry Breaking characterized by a local order parameter e.g. Neel order

48. Topological degeneracy There is also an “unusual” possibility that the groundstate degeneracy is due to a “topological order” Characteristics of the topological degeneracy (i) Degeneracy (# of g.s.) depending on the topology of the system (sphere, torus….) well known for Fractional Quantum Hall Liquids [ cannot be understood with the ordinary SSB] (ii) Absence of the local order parameter

49. Topological degeneracy ground-state degeneracy N depends on topology of the system g=0 g=1 g=2 not a consequence of a ordinary SSB….. a signature of a topological order! degenerate g.s.: indistinguishable by any local operator

50. Quantum many-particle systems on a periodic lattice : # of particles / unit cell Topological restrictions: If the system is gapful for  Z there must be some kind of order, either the standard SSB with a local order parameter or a topological order