Download S. Mukhin

Quantum Order of Fermions :
Broken Matsubara Time
Translations and Quantum Order
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S.I. Mukhin
Theoretical Physics & Quantum Technologies Department,
Moscow Institute for Steel & Alloys, Moscow, Russia
Serguey Brazovski
Jan Zaanen
QUANTUM ORDER vs CLASSICAL ORDER
CLASSICAL CONDENSATES describe CLASSICAL BROKEN SYMMETRY STATES
(examples)
•BROKEN SPACE TRANSLATIONS with CHARGE DENSITY WAVE
Thermodynamic expectation value :
CDW makes Hamiltonian quadratic:
n̂i,  n̂i,  (ri )
U n̂i,n̂i,  i ci,† ci,
•BROKEN SPIN SYMMETRY with SPIN DENSITY WAVE
Thermodynamic expectation value :
SDW makes Hamiltonian quadratic:
 (ri )
Sz (ri )
s  n̂i,  n̂i,  Sz (ri )
U n̂i,n̂i,  Sz (ri ) ci,† ci,
 sc (i, j;  )
Thermodynamic expectation value :
CLASSICAL ORDER PARAMETERS:
 c j, ci,   sc (i, j;  )
(ri ), Sz (ri ), sc (i, j;  )
QUANTUM ORDER vs CLASSICAL ORDER
CLASSICAL CONDENSATES describe CLASSICAL BROKEN SYMMETRY STATES
(examples)
•BROKEN SPACE TRANSLATIONS with CHARGE DENSITY WAVE
Thermodynamic expectation value :
CDW makes Hamiltonian quadratic:
n̂i,  n̂i,  (ri )
n̂i,n̂i,  i ci,† ci,
•BROKEN SPIN SYMMETRY with SPIN DENSITY WAVE
Thermodynamic expectation value :
SDW makes Hamiltonian quadratic:
Sz (ri )
s  n̂i,  n̂i,  Sz (ri )
n̂i,n̂i,  Sz (ri ) ci,† ci,
•BROKEN GUAGE SYMMETRY with SUPERCONDUCTING ORDER
Thermodynamic expectation value :
CLASSICAL ORDER PARAMETERS:
 (ri )
 sc (i, j;  )
 c j, ci,   sc (i, j;  )
(ri ), Sz (ri ), sc (i, j;  )
QUANTUM ORDER vs CLASSICAL ORDER
CLASSICAL CONDENSATES describe CLASSICAL BROKEN SYMMETRY STATES
(examples)
•BROKEN SPACE TRANSLATIONS with CHARGE DENSITY WAVE
Thermodynamic expectation value :
CDW makes Hamiltonian quadratic:
n̂i,  n̂i,  (ri )
n̂i,n̂i,  i ci,† ci,
•BROKEN SPIN SYMMETRY with SPIN DENSITY WAVE
Thermodynamic expectation value :
SDW makes Hamiltonian quadratic:
Sz (ri )
s  n̂i,  n̂i,  Sz (ri )
n̂i,n̂i,  Sz (ri ) ci,† ci,
•BROKEN GUAGE SYMMETRY with SUPERCONDUCTING ORDER
Thermodynamic expectation value :
SC makes Hamiltonian quadratic:
CLASSICAL ORDER PARAMETERS:
 (ri )
 sc (i, j;  )
 c j, ci,   sc (i, j;  )
c j, ci, ci,† c†j,   sc (i, j;  )ci,† c†j,
(ri ), Sz (ri ), sc (i, j;  )
QUANTUM ORDER vs CLASSICAL ORDER
CLASSICAL ROUTE OF MANY-BODY PHYSICS
 Ĥ 
Z  Tr exp 
  partition function; F  kBT ln Z  free energy

 kBT 

Hamiltonian is quadratic form of fermionic operators under the CLASSICAL
ORDER PARAMETER(S):
H mf  t
V
c
i, j , 

i, j , 
†
i,  j, 
c

 U   c c  Sz (ri ) c c
i
†
i i,  i, 

†
i,  i, 

 sc (i, j;  )ci,† c†j,   h.c.    n̂i,
i.
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:
F  i ,Sz ,  sc  0
What is Stratonovich transformation ?
A toy example:

 Ĥ 
Z  Tr exp 
  partition function;
 kBT 


2


y
1/ 2
exp gA  A   
  exp  g  exp 2yAdy


quadratic in A  y  Stratonovich 'field ' linear in A 
A sophisticated example:
 1 
1

;    0,
  Matsubara (imaginary) time
kBT
 kBT 
How to linearize exponential of
non-commuting operators?
  †

exp   g     d 
 0

†  ,     non-commuting quantum operators;
General Hubbard-Stratonovich transformation
  †

exp   g     d  
 0


'quadratic' in  


linear in  



2

 







†
  D  D  exp   d
T
exp
 
  d Â      h.c.
g 
 0
 0






    Hubbard  Stratonovich fields, that depend on Matsubara' s time 
QUANTUM ORDER HS-field
2

 








†
D


D


exp

d

T
exp
d

Â




     h.c.





0

g 
 0


2
 
 

0   
†
 exp   d
 T exp   d Â  0    h.c. 
g 
 0

 0








0    QUANTUM ORDER HS field, that depends on Matsubara time 
If there exists a saddle point Hubbard-Stratonovich field, that dominates
the path-integral, then it is the ‘QUANTUM ORDER’ of the problem
QUANTUM ORDER vs CLASSICAL ORDER
SYMMETRY BREAKING QUANTUM CONDENSATES
OF
HUBBARD-STRATONOVICH FIELDS
Example
3D+1 EUCLIDIAN ACTION OF FERMIONS WITH BROKEN
MATSUBARA TIME TRANSLATIONS:
rr
iQr
HS field : Sz ( ,r)  S( )e
rr
iQr
 S ( )e

Sz ( ,ri )  QUANTUM ORDER PARAMETER :
H HS  t

i, j , 
ci,† c j,   Sz ( ,ri )  ci,† ci, ;
i, 
Sz ( , ri )
1/T
0

1 
Sz   
, r   Sz ( , ri )- periodicity condition for HS fields
kBT  i

Self-consistency equation for HS field that breaks Matsubara axis
translations :


S( )  M  sn   Q  is Matsubara  time  dependent,
hence, the self  consistency equation is a functional equation!
r
Sz ( , r )
r
r
 n 
 F  Sz ( , ri )  0 
  tanh    n ( , r ){ M Ĥ HS } n ( , r )
 2
U
n
as compared with Classical Order self-consistent algebraic equation :

{Úqr2  M 2 }1/2  t qr 
M
M
   tanh
 r2
2 1/2
r
U
2T
q 
 {Úq  M }
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,
 Q  n kBT; n  1,2....
HS QUANTUM ORDER at DIFFERENT TEMPERATURES
IS COMMENSURATE WITH EUCLIDIAN 3D+1 SLAB
 Q  is 'quantum lattice' constant along the Matsubara axis
nT=const
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) HUBBARDSTRATONOVICH FIELD:
a) QOP GREEN’S FUNCTION HAS ONLY 2nd-ORDER POLES – QOP IS DIRECTLY
‘INVISIBLE’ (‘DARK MATTER’ of QUANTUM-CONDENSED BOSE-PAIRS)
a)THE “FINGERPRINTS” OF QOP in FERMIONIC SYSTEM: PSEUDO-GAP, ‘LIGHT
FERMIONS’, COMMENSURATION JUMPS OF QOP (MATSUBARA) PERIODICITY
WHEN T-> 0, etc.
c)EFFECTIVE EUCLIDIAN ACTION OF QOP and its GOLDSTONE MODES:
PERIODIC SOLUTIONS of the Schrödinger Equation with Weierstrass
periodic potential
FREE ENERGIES WITH HS QOP versus FREE ENERGIES WITH
CLASSICAL ORDER PARAMETER COP: HOW QUANTUM ‘FIGHTS’ CLASSICAL
CALCULATED PHASE DIAGRAM
DISORDERING PARAMETER that FAVORS QUANTUM ORDER
Definition of the NESTING in ANY-D case :
Complete nesting condition:
“ANTI-NESTING” PARAMETER :
QUANTUM ORDER DOMAIN:
tq  0
tq  0
So, why quantum orders are so rare ? Or why we do not see them ?
GREEN’S FUNCTION of the HS FIELD ( QOP)


r
r
r
r
K  T  ( 1 , r1 ) ( 1 , r1 )  ( 2 , r2 )  ( 2 , r2 )  
r
r
2
 Z
1 Sz ( 1 , r1 )Sz ( 2 , r2 )

r
r
2
 Sz ( 1 , r1 ) Sz ( 2 , r2 ) Z
U
r
r
r
Usual COP -> Bragg peaks: K q,        q  K n

n
What “Bragg peaks” are predicted for QOP ?

So, why quantum orders are so rare ? Or why we do not see them ?
Definition of the averaging <…> on the mean-field level :
r r r
cos(Q·(r1  r2 )) 
r r
K( 1   2 , r1  r2 ) 
Sz ( 1   0 )Sz ( 2   0 )d 0
2

0
U
ANALYTICAL EXPRESSION for THE QUANTUM ORDER PARAMETER (HS):
rr
iQr
Sz ( ,r)  S( )e
rr
iQr
 S ( )( )e
nesting wave-vector Q;
The ENVELOPE FUNCTION CAN BE EXPRESSED AS :
sin( m )
;
 (2m  1)q 
m 0
sinh 


2
 m  2 nT (2m  1); q   K(k  ) / K(k)

S( )  4 nT 
So, why quantum orders are so rare ? Or why we do not see them ?
THE QOP GREEN’S FUNCTION - ANALYTIC SOLUTION
rr
(4 nT ) cos( m )cos(Q·r )
r
K( , r )  
2
2  (2m  1)q 
m 0
2U sinh 


2

2
THE ANALYTIC CONTINUATION TO THE REAL FREQUENCES AXIS:


k
Tn
1
 kBTn
R
B
K    

2  
2 m  %T2 4   m 
2sin 2 (  i )T2 4 
T2  K  (KnkBT ); %   i
So, NO BRAGG PEAKS come from QOP!
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 (!) :

d
2
R
Q  i 
K   f    0
2

f   - Fourier component of the external ‘force’ acting on the HS QOP
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)
L̂Ĝ   (   '),
   S  
p

L̂  
p
  S  

 G11 G12
Ĝ    
 G21 G22




  Matsubara  time  dependent Green's function

Timur
(outside the Department)
To find measurable predictions one has to make analytical continuation from
Matsubara to real time and derive :
 Im GR ( , p)  DOS
THE “FINGERPRINTS” OF QOP in the FERMI-SYSTEM
 F11  F12
G

0
0
F11  F12





m
m


1
1
1



 B

F11 
  Bm (a)
2  m
2
2 
2 (e  e2 )(e  e3 ) (2 1 )  m
m
i n    i
m
i n    i
m

2 1
2 1 
2



1
 
1
1


F12 

B

B
(a)
 m
 m
2
2

2 (e  e2 )(e  e3 ) (2 1 )2  m
i n    i
m m
i n    i
m

2 1
2 1 
1
Bm  

2
a 
sh 2   m 3   i
2 1
2 1 

2



1
 
1
1


G11  F11  F12 
B
(a)

B
 2m
 2 m 1
2
2

(e  e2 )(e  e3 ) (2 1 )2  m
i n    i
2m m
i n    i
(2m  1) 


2 1
2 1
2




1

1
1
  B2 m

G22  F11  F12  
  B2 m 1 (a)
2
2
(e  e2 )(e  e3 ) (2 1 )2  m
m
i n    i
2m
i n    i
(2m  1) 


2 1
2 1
2
S.I. Mukhin, T.R. Galimzyanov, 2011, in preparation
THE “FAITH”
OF FERMIONS
UNDER QOP
Im GR ( , p)  DOS
 Strongly nonlinear
HS 
 PSEUDO GAP!
Single harmonic
HS  

NO PG, but
SIDE-BANDS!
S.I. Mukhin, T.R.
Galimzyanov, 2011,
in preparation
THE “FAITH” OF FERMIONS UNDER QOP
Im GR ( , p)  DOS : at pF cut along the   axis
Strongly nonlinear HS
Single harmonic HS
S.I. Mukhin, T.R. Galimzyanov, 2011, in preparation
 
 
EFFECTIVE EUCLIDIAN ACTION OF QOP HAS INFINITE NUMBER OF DEGREES
OF FREEDOM : HAMILTONIAN DYNAMICS OF “ANGELS”
L1 ( )  L0 ( )  L1 ( )  Lsingle-PG ( )
C1 4
C1  &2
1 &&2
 2
2
Lsingle-PG ( )   ( )   ( )  C2  ( )   5 ( )    ( )   ( )

2
2
2
&  (n1)
Ln1  Ln ,...,  4 n 2 ; ,...,
6




S    Lsingle-PG ( )   Ln ( )  d  Euclidian action of the HS(QOP) field,
n1

0 
after fermions are integrated out exactly,
L is expressed via an infinite sum of
1/T
so  called 'auxiliary integrals of motion' Ln
of Lax LA  pair inverse scattering theory
for nonlinear Schrödinger equation
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
1/T
minimizes the full Euclidian action, S 

0


L
(

)

L
(

)
 single-PG
 n  d ,
n 1


but with renormalized amplitude and period along the
Matsubara’s time axis.
1/T
S0 
L
single-PG
( )d ;
 S0  0  0;

 0    M  sn   Q

0
QUESTION : is it Hamiltonian dynamics, since Lagrangian contains higher
(l 1)
time-derivatives    than 1 ???
Lsingle-PG ( )   6 ( ) 
C1 4
C 
1 &&2

 ( )  C2  2 ( )   5 2 ( )  1  &2 ( )  
( )

2
2
2
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 :
m
For any finite m>1 and m-th order Lagrangian
L(m)   Ln
s 0
The following canonical coordinates and momenta are defined:
qi   (i 1) ;
 d s L(m) 
s
pi    s
1
 
(i  s) 

s  0  d 
m i
With the corresponding HS-’coordinates’ ‘Hamiltonian’ НQOP :
m
H
(m)
QOP
  pi q&i  L(m)
i 1
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) HUBBARDSTRATONOVICH FIELD:
a) QOP GREEN’S FUNCTION HAS ONLY 2nd-ORDER POLES – QOP IS DIRECTLY
‘INVISIBLE’ (‘DARK MATTER’ of QUANTUM-CONDENSED BOSE-PAIRS)
a)THE “FINGERPRINTS” OF QOP IN FERMI-SYSTEM: PSEUDO-GAP, ‘LIGHT
FERMIONS’, COMMENSURATION JUMPS OF QOP MATSUBARA
TIME- PERIODICITY WHEN T-> 0, etc.
c)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 )
THANK YOU
QUANTUM ORDERED STATE as BROKEN “TIME”-INVARIANCE STATE
OF MANY-BODY SYSTEM
Appendix
(“time” is Matsubara’s imaginary time)
Partition function in broken “time”-invariance state ( i.e. with
r
“time”-dependent Hubbard-Stratanovich field M ( , r ) ) :
- Hubbard-Stratanovich field action
Definition of Floquet index  n :

k 
Ek
T
when M( ,r)  M(r)
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:
The workings of the e-h symmetry break :  t
(Horovitz, Gutfreund, Weger PRB (1975) )
Appendix

Tc E F
c  tc

W
CLACULATED EUCLIDIAN ACTION (Free energy) OF THE COP and QOP STATES
Appendix
Appendix
Appendix
Appendix
Introduction of the Hubbard-Stratonovich fields in the Hubbard
U-g Hubbard Hamiltonian
Matveenko JETP Lett. (2003)
Appendix
(anti)periodic conditions, with the temperature T defining the period =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,
as Matsubara’s- time-periodic potentials:
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)