1. Anomaly Realization in Condensed Matter
Eun-Gook Moon
RIKEN. DEC. 6, 2016

2. Yesterday and today…
We heard beautiful talks about anomalies in WSM / DSM.
(CME, CVE, gravitational anomaly....)
(almost all) WSM / DSM are made of s and p orbitals
~ weakly correlated systems
~ almost free theory is applicable (ARPES)
In this talk, I focus on another realization of anomalies.
What happens if strongly correlated systems have anomalies?

3. In condensed matter, more interests in anomalies!
At PRB
In condensed matter, more interests in anomalies!

4. In condensed matter, more interests in anomalies!
At PRB
In condensed matter, more interests in anomalies!

5. Outline Competing Orders Landau-Ginzburg-Wilson description
Exotic class of competing orders (beyond LGW)
Anomalies’ lesson in competing orders
Conclusion
Let us review conventional LGW theory before

6. Concepts Anomaly Competing Orders Symmetry / Topology
Quantum Criticality
Massless

7. Phase Diagrams in Strongly Correlated Systems
Nandi et.al., PRL 2010
L. Taillefer, 2010
Moll et.al., NC 2015
Kurosaki et.al., PRL 2005
Witczak-Krempa et.al., 2014
Muhlbauer et.al., Science 2009

8. Phase Diagrams in Strongly Correlated Systems
Nandi et.al., PRL 2010
L. Taillefer, 2010
Moll et.al., NC 2015
Kurosaki et.al., PRL 2005
Witczak-Krempa et.al., 2014
Muhlbauer et.al., Science 2009
Various Colors : Various Phases

9. Observations
Various phases (Superconductors, Magnetism, Weyl semi-metals,…)
Criteria of phases : Symmetry / Topology (ex : order parameters)
SC : U(1) local symmetry + Higgs mechanism
SDW : SU(2) spin rotation symmetry

10. Observations
Various phases (Superconductors, Magnetism, Weyl semi-metals,…)
Criteria of phases : Symmetry / Topology (ex : order parameters)
SC : U(1) local symmetry + Higgs mechanism
SDW : SU(2) spin rotation symmetry
Overlapped colors / Non-monotonic phase boundaries
Interplay between order parameters (Competition / Attraction)
Competing orders

11. Competing orders + anomalies ???
Observations
Various phases (Superconductors, Magnetism, Weyl semi-metals,…)
Criteria of phases : Symmetry / Topology (ex : order parameters)
SC : U(1) local symmetry + Higgs mechanism
SDW : SU(2) spin rotation symmetry
Overlapped colors / Non-monotonic phase boundaries
Interplay between order parameters (Competition / Attraction)
Competing orders + anomalies ???

12. Outline Competing Orders Landau-Ginzburg-Wilson description
Exotic class of competing orders (beyond LGW)
Anomalies’ lesson in competing orders
Conclusion
Let us review conventional LGW theory before

13. Theories of competing orders
1. Landau-Ginzburg-Wilson
2. Deconfined Quantum Criticality
3. Non-linear sigma model with Wess-Zumino-Witten term
4. Non-linear sigma model with theta term
5. …

14. Theories of competing orders
1. Landau-Ginzburg-Wilson
2. Deconfined Quantum Criticality
3. Non-linear sigma model with Wess-Zumino-Witten term
4. Non-linear sigma model with theta term
5. …
(ex : 1+1D spin chain)

15. Theories of competing orders
1. Landau-Ginzburg-Wilson
2. Deconfined Quantum Criticality
3. Non-linear sigma model with Wess-Zumino-Witten term
4. Non-linear sigma model with theta term
5. …
(spin liquid physics : spin fractionalization)
(ex : 1+1D spin chain)

16. Theories of competing orders
1. Landau-Ginzburg-Wilson
2. Deconfined Quantum Criticality
3. Non-linear sigma model with Wess-Zumino-Witten term
4. Non-linear sigma model with theta term
5. …
(spin liquid physics : spin fractionalization)
(ex : Witten, Non-abelian bosonization in D=2)
(ex : 1+1D spin chain)

17. Theories of competing orders
1. Landau-Ginzburg-Wilson
2. Deconfined Quantum Criticality
3. Non-linear sigma model with Wess-Zumino-Witten term
4. Non-linear sigma model with theta term
5. …
(spin liquid physics : spin fractionalization)
(ex : 1+1D spin chain)

18. Landau-Ginzburg-Wilson
Two order parameters
Vector reps of

19. Landau-Ginzburg-Wilson
Two order parameters
Vector reps of
O(3) ~ magnetism (Neel)

20. Landau-Ginzburg-Wilson
Two order parameters
Vector reps of

21. Landau-Ginzburg-Wilson
Two order parameters
Vector reps of
Four phases
(massive excitation)
(Nambu-Goldstone)
(Nambu-Goldstone)
(Nambu-Goldstone)

22. Landau-Ginzburg-Wilson
Two order parameters
Vector reps of

23. Landau-Ginzburg-Wilson
Near the multi-critical point,

24. Landau-Ginzburg-Wilson
Near the multi-critical point,
define super-spin

25. Landau-Ginzburg-Wilson
Near the multi-critical point,
define super-spin
Enjoys O(N+M) symmetry
Equivalent to O(N+M) NLSM

26. Landau-Ginzburg-Wilson
Near the multi-critical point,
define super-spin
+ anisotropy terms Ex :
S : all excitations are massive

27. Outline Competing Orders Landau-Ginzburg-Wilson description
Exotic class of competing orders (beyond LGW)
Anomalies’ lesson in competing orders
Conclusion
Let us review conventional LGW theory before

28. Deconfined Quantum Criticality
Singh (2010)
Direct Phase Transition without fine-tuning
between magnetism (Neel) and charge density wave (VBS)

29. Deconfined Quantum Criticality
Singh (2010)
Direct Phase Transition without fine-tuning
between magnetism (Neel) and charge density wave (VBS)
In LGW, this is impossible! → New competing order physics
Spin liquid physics (fractionalization) : ex) electrical insulator but thermal metal
Not spin excitation ( ) but spinon (z)!

30. Deconfined Quantum Criticality
Singh (2010)
Direct Phase Transition without fine-tuning
between magnetism (Neel) and charge density wave (VBS)
In LGW, this is impossible! → New competing order physics
Critical theory for competing orders
CP(1) with non-compact U(1) gauge in (2+1)D

31. Deconfined Quantum Criticality
Another perspective :
O(3) sym. broken
O(2) sym. broken
O(5) symmetric?
O(3) ~ magnetism (Neel)
O(2) ~ VBS
NLSM ~ LGW.
→ “superspin” → O(5) NLSM ? NO!

32. Deconfined Quantum Criticality
Another perspective :
O(3) sym. broken
O(2) sym. broken
O(5) symmetric?
O(3) ~ magnetism (Neel)
O(2) ~ VBS
→ “superspin” → O(5) NLSM –WZW??
O(5) NLSM-WZW ~ spin liquid physics in 3D
Senthil and Fisher (2006)

33. Deconfined Quantum Criticality
In 2+1D,
Senthil and Fisher : dQC ~ spin liquid physics ~ O(5) NLSM-WZW
Many subsequent works!
Grover and Senthil (2009), EGM (2012),…..
In 3+1D,
Hosur, Ryu, and Vishivanath studied O(6) NLSM–WZW.
O(3) ~ magnetism (Neel)
O(3) ~ VBS
Hosur, Ryu, and Vishwanath (2010)
But, no concrete results in O(N+M) NLSM-WZW in D>2.

34. Novel class of competing orders?
NLSM in D space-time dimensions
ex :
k : integer
WZW on one higher dimensional manifold
Imaginary WZW term (phase in path integral)

35. Novel class of competing orders?
NLSM in D space-time dimensions
Same universality class as LGW’s? or different one?
If different, how different?

36. Novel class of competing orders?
NLSM in D space-time dimensions
D+1 derivatives
two derivatives
Same universality class as LGW’s? or different one?
If different, how different?
Two opposite arguments
- Same! WZW terms has many derivatives in D>2.
- Different! WZW term is topological.

37. Novel class of competing orders
In D=2, Witten showed NLSM-WZW is different from NLSM.

38. Novel class of competing orders
In D=2, Witten showed NLSM-WZW is different from NLSM.
k=0 : NLSM,
ground state : massive
k=1,2,.. : NLSM-WZW
ground state : massless (free fermion)
Witten (1984)
Non-abelian bosonization
cf. no spontaneous symmetry breaking : Mermin-Wagner theoerem

39. Novel class of competing orders
two derivatives
D+1 derivatives
What about D>2? Very Difficult!
Spontaneous symmetry breaking
No 𝜺 / large N expansion due to WZW term
No perturbative calculation
Non-perturbative calculation is necessary

40. Outline Competing Orders Landau-Ginzburg-Wilson description
Exotic class of competing orders (beyond LGW)
Anomalies’ lesson in competing orders
Conclusion
Let us review conventional LGW theory before

41. Anomaly
Anomaly
: quantum mechanical fluctuations violate a conservation “law”.
Continuous Symmetry → Conservation law (Noether’s theorem)
Anomaly → Conservation law is “spoiled”
Pure quantum effects
no classical analogue
appears in path-integral (or loop calculation)

42. Anomaly
Non-perturbative nature : tool for strong coupling physics.

43. Anomaly Non-perturbative nature : tool for strong coupling physics.
Example : U(1) Chiral anomaly
Weyl semi-metal
Classically,
Quantum Mechanically,

44. Anomaly Example : Quantum Hall states Example : Topological Insulator
Son, Wiegmann, Fradkin, Ryu, Hughes,…
Example : Topological Insulator
Ryu et. al (2012)
Anomalies in quantum phase transitions ?

45. Anomaly How to find anomaly?
Gauging symmetries and doing gauge-transformation
Zero variation : no anomaly
Non-zero variation : anomaly

46. Anomaly How to find anomaly?
Gauging symmetries and doing gauge-transformation
Zero variation : no anomaly
Non-zero variation : anomaly
Ex ) chiral fermions with (classically) conserved currents

47. Anomaly How to find anomaly?
Gauging symmetries and doing gauge-transformation
Zero variation : no anomaly
Non-zero variation : anomaly

48. Anomaly How to find anomaly?
Gauging symmetries and doing gauge-transformation
Zero variation : no anomaly
Non-zero variation : anomaly

49. Gauging NLSM + WZW 1. Gauge SO(D+2) symmetry of NLSM +WZW in D=2, 4
Lie algebra of SO(D+2)
Local transformation
2. Introduce gauge potential and covariant derivative
3. Gauge transform
Hull and Spence (1991)

50. Gauging NLSM + WZW In D=2, WZW term in a differential form notation is
(variation is two dimensional)
Introduce minimal coupling
Its variation is not two dimensional (EOM is not two dimensional).
Must be appended to make EOM two-dimensional

51. Gauging NLSM + WZW In D=2, we find the total action is
Anomaly is found by gauge transformation

52. Gauging NLSM + WZW
In D=2, anomaly equation is

53. Gauging NLSM + WZW In D=2, anomaly equation is Gauge potential
Anomaly coefficient
Gauge transformation

54. Gauging NLSM + WZW In D=2, anomaly equation is
Gauge potential
Anomaly coefficient
Gauge transformation
In D=4, similar calculation gives

55. Anomaly of NLSM + WZW In 2D, 4D , anomaly coefficients are
Non-zero (anomalies) :
k is non-zero,
all directions in SO(D+2) group are used.
Implication
Anomaly is from WZW term. (interaction strength independent)
Full symmetry cannot be gauged (anomaly).
Subgroup can be gauged (SO(2) or SO(3)).
2D results are consistent with Witten’s results.

56. Anomaly matching ‘t Hooft anomaly matching
Anomaly at UV fixed point and IR fixed point should be matched.
Local deformation of theories do not change anomaly.
Roughly speaking, anomaly is conserved.
Powerful tool to investigate low energy physics.
Implication of anomaly
Existence of the massless degrees of freedom
(Singularity in correlation functions of currents, Coleman and Grossman 1982)

57. Anomaly of NLSM + WZW In 4D, without anomaly (NLSM, LGW)
NG modes
critical modes
massive
In 4D, with anomaly (NLSM-WZW)
NG modes
critical modes
massless
(anomaly enforced)

58. Anomaly of NLSM + WZW
In 4D, depending on microscopic details, RG flows are generic.
We do not know how many first / second order phase transitions.
NG modes
??? ??
If there is anomaly, then S should contain massless excitation!

59. Anomaly of NLSM + WZW
In 4D, depending on microscopic details, RG flows are generic.
We do not know how many first / second order phase transitions.
NG modes
??? ??
If there is anomaly, then S should contain massless excitation!
Do we have a concrete model for any RG flows?

60. Minimal model Integrating out fermions induces the WZW term.
Abanov and Wiegmann (2000)
Symmetry group :
Low energy degrees of freedom of the minimal model
N+M : Goldstones
QC : Goldstones + Fermions
S : Fermions

61. Minimal model + Anisotropy
Anisotropy operators :
N+M, N, M : Goldstones
QC : Goldstones + Fermions
S : Fermions
S is stable!
No deconfined QCP!

62. Non-Minimal models (a) Anisotropy is marginal
S is stable with one sign and unstable with an opposite sign.
X : either a symmetry broken state or a CFT with the subgroup
RG flow is similar to 2D results (dimerization).
(b) Anisotropy is relevant
Candidate for deconfined QCP if two opposite flows break different symmetries.
Must be strongly coupled CFT.
No U(1) spin liquid.

63. Statements 1. O(6) NLSM-WZW describes competing physics beyond-LGW .
2. O(6) NLSM-WZW describes a different class from one of CP(1) in 4D.
(The former has anomaly but the latter does not)
(Qualitatively different from 3D)
3. Deconfined QC requires relevant operators
which requires strongly interacting fixed points for S.
4. If a system with Weyl points (Nd2Ir2O7?) become
a symmetric paramagnetic insulator, then it should be a gapless spin-liquid.

64. Conclusions 1. Anomaly is cool.
2. Not only in transports of WSM/DSM but also in competing order physics!
3. Realization :
Weyl semi-metals in pyrochlore (8 Weyl points )
→ charge gapped : anomaly enforced gapless spin liquid!
(thermal conductivity!)

65. Thank you for your attention!