Quantum Order of Fermions : Broken Matsubara Time Translations and Quantum Order Fingerprints S.I. Mukhin

1. Quantum Order of Fermions : Broken Matsubara Time Translations and Quantum Order Fingerprints S.I. Mukhin Theoretical Physics & Quantum Technologies Department, Moscow Institute for Steel & Alloys, Moscow, Russia Serguey Brazovski Jan Zaanen

2. QUANTUM ORDER vsCLASSICAL ORDER CLASSICAL CONDENSATES describe CLASSICAL BROKEN SYMMETRY STATES (examples) • BROKEN SPACE TRANSLATIONS withCHARGE DENSITY WAVE Thermodynamic expectation value : CDW makes Hamiltonian quadratic: • BROKEN SPIN SYMMETRY withSPIN DENSITY WAVE Thermodynamic expectation value : SDW makes Hamiltonian quadratic: Thermodynamic expectation value : CLASSICAL ORDER PARAMETERS:

3. QUANTUM ORDER vsCLASSICAL ORDER CLASSICAL CONDENSATES describe CLASSICAL BROKEN SYMMETRY STATES (examples) • BROKEN SPACE TRANSLATIONS withCHARGE DENSITY WAVE Thermodynamic expectation value : CDW makes Hamiltonian quadratic: • BROKEN SPIN SYMMETRY withSPIN DENSITY WAVE Thermodynamic expectation value : SDW makes Hamiltonian quadratic: • BROKEN GUAGE SYMMETRY with SUPERCONDUCTING ORDER Thermodynamic expectation value : CLASSICAL ORDER PARAMETERS:

4. QUANTUM ORDER vsCLASSICAL ORDER CLASSICAL CONDENSATES describe CLASSICAL BROKEN SYMMETRY STATES (examples) • BROKEN SPACE TRANSLATIONS withCHARGE DENSITY WAVE Thermodynamic expectation value : CDW makes Hamiltonian quadratic: • BROKEN SPIN SYMMETRY withSPIN DENSITY WAVE Thermodynamic expectation value : SDW makes Hamiltonian quadratic: • BROKEN GUAGE SYMMETRY with SUPERCONDUCTING ORDER Thermodynamic expectation value : SC makes Hamiltonian quadratic: CLASSICAL ORDER PARAMETERS:

5. QUANTUM ORDER vsCLASSICAL ORDER CLASSICAL ROUTE OF MANY-BODY PHYSICS Hamiltonian is quadratic form of fermionic operators under theCLASSICAL ORDER PARAMETER(S): e.g. for Hubbard t-U-V model Free energy F is minimized with respect to the CLASSICAL ORDER PARAMETER(S) and a phase diagram of the system is found:

6. What is Stratonovich transformation ? A toy example: A sophisticated example: How to linearize exponential of non-commuting operators?

7. General Hubbard-Stratonovich transformation

8. QUANTUM ORDER HS-field If there exists a saddle point Hubbard-Stratonovich field, that dominates the path-integral, then it is the ‘QUANTUM ORDER’ of the problem

9. QUANTUM ORDER vsCLASSICAL ORDER SYMMETRY BREAKINGQUANTUM CONDENSATES OF HUBBARD-STRATONOVICH FIELDS Example3D+1 EUCLIDIAN ACTION OF FERMIONS WITH BROKENMATSUBARA TIME TRANSLATIONS:

10. Self-consistency equation for HS field that breaks Matsubara axis translations : as compared with Classical Order self-consistent algebraic equation :

11. HS field: exact solution, that breaks Matsubara axis translations S.I. Mukhin, J. Supercond. & Novel Magn, v. 24, 1165-71 (2011) S(τ)=k τQ-1sn{τ/τQ, k}; sn is Jacobi snoidal elliptic function,

12. HS QUANTUM ORDER at DIFFERENT TEMPERATURES IS COMMENSURATE WITH EUCLIDIAN 3D+1 SLAB nT=const

13. So, why quantum orders are so rare ? Or why we do not see them ? The (first) self-consistent solution HS breaking Matsubara axis translations is found for the Hubbard model with ‘spoiled’ nesting at the bare 2D Fermi surface S.I. Mukhin , J. Supercond. & Novel Magn, v. 24, 1165-71 (2011). • QUANTUM ORDER PARAMETER (QOP) – CONDENSED (DYNAMIC) HUBBARD-STRATONOVICH FIELD: • QOP GREEN’S FUNCTION HAS ONLY 2nd-ORDER POLES – QOP IS DIRECTLY ‘INVISIBLE’ (‘DARK MATTER’ of QUANTUM-CONDENSED BOSE-PAIRS) • THE “FINGERPRINTS” OF QOP in FERMIONIC SYSTEM: PSEUDO-GAP, ‘LIGHT FERMIONS’, COMMENSURATION JUMPS OF QOP (MATSUBARA) PERIODICITY WHEN T-> 0, etc. • EFFECTIVE EUCLIDIAN ACTION OF QOP and its GOLDSTONE MODES: • PERIODIC SOLUTIONS of the Schrödinger Equation with Weierstrass • periodic potential

14. FREE ENERGIES WITH HS QOP versusFREE ENERGIES WITH CLASSICAL ORDER PARAMETER COP: HOW QUANTUM ‘FIGHTS’ CLASSICAL

15. So, why quantum orders are so rare ? Or why we do not see them ? CALCULATED PHASE DIAGRAM

16. DISORDERING PARAMETER that FAVORS QUANTUM ORDER Definition of the NESTING in ANY-D case : Complete nesting condition: “ANTI-NESTING” PARAMETER : QUANTUM ORDER DOMAIN:

17. So, why quantum orders are so rare ? Or why we do not see them ? GREEN’S FUNCTION of the HS FIELD ( QOP) Usual COP -> Bragg peaks: What “Bragg peaks” are predicted for QOP ?

18. So, why quantum orders are so rare ? Or why we do not see them ? Definition of the averaging <…> on the mean-field level : ANALYTICAL EXPRESSION for THE QUANTUM ORDER PARAMETER (HS): nesting wave-vector Q; The ENVELOPE FUNCTION CAN BE EXPRESSED AS :

19. So, why quantum orders are so rare ? Or why we do not see them ? THE QOP GREEN’S FUNCTION - ANALYTIC SOLUTION THE ANALYTIC CONTINUATION TO THE REAL FREQUENCES AXIS: So, NO BRAGG PEAKS come from QOP!

20. So, why quantum orders are so rare ? Or why we do not see them ? QUNTUM ORDER PARAMETER IS DIRECTLY “INVISIBLE” ! (HIDDEN ORDER) Do we have “dark matter here” ??? SCATTERING CROSS-SECTION OF THE ORDER PARAMETER FIELD (see Abrikosov,Gor’kov,Dzyaloshiskii) BUT EXCHANGE OF ENERGY e.g. of NEUTRONS WITH HS IS ZERO (!) : - Fourier component of the external ‘force’ acting on the HS QOP

21. THE “FINGERPRINTS” OF QOP in the FERMI-SYSTEM Fermionic Greens function in the system with broken Matsubara time translations is found analytically (S.I. Mukhin, T.R. Galimzyanov, 2011, in preparation) Timur (outside the Department) To find measurable predictions one has to make analytical continuation from Matsubara to real time and derive :

22. THE “FINGERPRINTS” OF QOP in the FERMI-SYSTEM S.I. Mukhin, T.R. Galimzyanov, 2011, in preparation

23. THE “FAITH” OF FERMIONS UNDER QOP Strongly nonlinear HS PSEUDO GAP! Single harmonic HS NO PG, but SIDE-BANDS! S.I. Mukhin, T.R. Galimzyanov, 2011, in preparation

24. THE “FAITH” OF FERMIONS UNDER QOP at pF cut along the Strongly nonlinear HS Single harmonic HS S.I. Mukhin, T.R. Galimzyanov, 2011, in preparation

25. EFFECTIVE EUCLIDIAN ACTION OF QOP HAS INFINITE NUMBER OF DEGREES OF FREEDOM : HAMILTONIAN DYNAMICS OF “ANGELS” for nonlinear Schrödingerequation

26. EFFECTIVE EUCLIDIAN ACTION OF QOP HAS INFINITE NUMBER OF DEGREES OF FREEDOM : HAMILTONIAN DYNAMICS OF “ANGELS” “Holographyic principle” for the HS – QOP: HS-QOP that minimizes the lowest order Euclidian action also minimizes the full Euclidian action, , but with renormalized amplitude and period along the Matsubara’s time axis. QUESTION : is it Hamiltonian dynamics, since Lagrangian contains higher time-derivatives than 1 ???

27. EFFECTIVE EUCLIDIAN ACTION OF QOP HAS INFINITE NUMBER OF DEGREES OF FREEDOM : HAMILTONIAN DYNAMICS OF “ANGELS” ANSWER : YES, Euclidian action S of HS-field describes Hamiltonian dynamics, but with an infinite number of degrees of freedom (‘angels’) according to the rule : For any finite m>1 and m-th order Lagrangian The following canonical coordinates and momenta are defined: With the corresponding HS-’coordinates’ ‘Hamiltonian’ НQOP :

28. SUMMARY The (first) self-consistent solution HS breaking Matsubara axis translations is found for the Hubbard model with ‘spoiled’ nesting of the bare 2D Fermi surface S.I. Mukhin , J. Supercond. & Novel Magn, v. 24, 1165-71 (2011). • QUANTUM ORDER PARAMETER (QOP) – CONDENSED (DYNAMIC) HUBBARD-STRATONOVICH FIELD: • QOP GREEN’S FUNCTION HAS ONLY 2nd-ORDER POLES – QOP IS DIRECTLY ‘INVISIBLE’ (‘DARK MATTER’ of QUANTUM-CONDENSED BOSE-PAIRS) • THE “FINGERPRINTS” OF QOP IN FERMI-SYSTEM: PSEUDO-GAP, ‘LIGHT FERMIONS’, COMMENSURATION JUMPS OF QOP MATSUBARA • TIME- PERIODICITY WHEN T-> 0, etc. • EFFECTIVE EUCLIDIAN ACTION OF QOP HAS INFINITE NUMBER OF • DEGREES OF FREEDOM : HAMILTONIAN DYNAMICS OF “ANGELS” • d) THE GOLDSTONE MODES of QOP ARE GAPPED and EQUAL to DISCRETE Matsubara TIME-PERIODIC EIGENMODES of a HAMILTONIAN with WEIERSTRASS POTENTIAL (S.I. Mukhin 2011 , in preparation )

29. THANK YOU

30. QUANTUM ORDERED STATE as BROKEN “TIME”-INVARIANCE STATE OF MANY-BODY SYSTEM (“time” is Matsubara’s imaginary time) Appendix Partition function in broken “time”-invariance state ( i.e. with “time”-dependent Hubbard-Stratanovich field ) : - Hubbard-Stratanovich field action Definition of Floquet index :

31. Self-consistency condition in broken “time”-invariance (Quantum Ordered) state Appendix and in the explicit form : The miracle of the exact self-consistent solution with Jacobi elliptic functions:

32. The workings of the e-h symmetry break : (Horovitz, Gutfreund, Weger PRB (1975) ) Appendix

33. CLACULATED EUCLIDIAN ACTION (Free energy) OF THE COP and QOP STATES Appendix

34. Appendix

35. Appendix

36. Appendix Introduction of the Hubbard-Stratonovich fields in the Hubbard U-g Hubbard Hamiltonian Matveenko JETP Lett. (2003)

37. Appendix (anti)periodic conditions, with the temperature T defining the period b=T-1 Abrikosov, Gor’kov, Dzyaloshinski (1963) Dashen, Hasslacher, Neveu, PRD (1975) Floquet equation : with wave-functions of a “particle” in the HS fields M(t), D(t) as Matsubara’s- time-periodic potentials:

38. The self-consistent solutions of the periodic Bloch equations are obtained using wellknown solutions for the SSH solitonic lattices (TTF-TCNQ theory): Appendix Brazovskii, Gordyunin, Kirova JETP Lett. (1980) Machida Fujita, PRB (1984)