quantum criticality beyond the landau-ginzburg-wilson …qpt.physics.harvard.edu/talks/sces.pdf ·...
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Quantum criticality beyond the Landau-Ginzburg-Wilson paradigm
Leon Balents (UCSB)Lorenz Bartosch (Yale/Frankfurt)
Anton Burkov (UCSB)Matthew Fisher (UCSB)
Subir Sachdev (Yale)Krishnendu Sengupta (Yale)
T. Senthil (MIT)Ashvin Vishwanath (MIT)Matthias Vojta (Karlsruhe)
Phys. Rev. Lett. 90, 216403 (2003).Science 303, 1490 (2004).
cond-mat/0408xxxTalk online: Google Sachdev
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( )( )
cos .
sin .
jj
j
K r
K r
=
+
1
2
S N
N
Collinear spins: 0Non-collinear spins: 0
× =× ≠
1 2
1 2
N NN N
0j =S
T=0SDW
Pressure,carrier concentration,….
Quantum critical point
States on both sides of critical point could be either (A) Insulators
(B) Metals(C) Superconductors
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SDWs in Mott insulators
2 4La CuO
( )4 / 3, 4 / 3K π π=Non-collinear spins
( ),K π π=Collinear spins
“Disorder” the spins by enhancing quantum fluctuations in a variety of ways…..
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OutlineOutlineA. “Dimerized” Mott insulators
Landau-Ginzburg-Wilson (LGW) theory.
B. Kondo lattice models “Large” Fermi surfaces and the LGW SDW paramagnon theory.
C. Fractionalized Fermi liquidsSpin liquids and Fermi volume changing transitions with a topological order parameter.
D. Multiple order parameters LGW forbidden transitions
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(A) Magnetic quantum phase transitions in “dimerized” Mott insulators
Landau-Ginzburg-Wilson (LGW) theory:Second-order phase transitions described by
fluctuations of an order parameterassociated with a broken symmetry
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Coupled Dimer Antiferromagnet
jiij
ij SSJH ⋅= ∑><
10 ≤≤ λ
JλJM. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, 10801-10809 (1989).N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994).J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999).M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002).
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close to 0λ Weakly coupled dimers
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close to 0λ Weakly coupled dimers
0, 0iS ϕ= =Paramagnetic ground state
( )↓↑−↑↓=2
1
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close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
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close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
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close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
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close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
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close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon
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close to 0λ Weakly coupled dimers
( )↓↑−↑↓=2
1
Excitation: S=1 triplon(exciton, spin collective mode)
Energy dispersion away fromantiferromagnetic wavevector
2 2 2 2
2x x y y
p
c p c pε
+= ∆ +
∆spin gap∆→
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Coupled Dimer Antiferromagnet
jiij
ij SSJH ⋅= ∑><
10 ≤≤ λ
JλJM. P. Gelfand, R. R. P. Singh, and D. A. Huse, Phys. Rev. B 40, 10801-10809 (1989).N. Katoh and M. Imada, J. Phys. Soc. Jpn. 63, 4529 (1994).J. Tworzydlo, O. Y. Osman, C. N. A. van Duin, J. Zaanen, Phys. Rev. B 59, 115 (1999).M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev. B 65, 014407 (2002).
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close to 1λ Weakly dimerized square lattice
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close to 1λ Weakly dimerized square lattice
Excitations: 2 spin waves (magnons)
2 2 2 2p x x y yc p c pε = +
Ground state has long-range spin density wave (Néel) order at wavevector K= (π,π)
0 ϕ ≠
spin density wave order parameter: ; 1 on two sublatticesii i
SS
ϕ η η= = ±
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λc = 0.52337(3)M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama,
Phys. Rev. B 65, 014407 (2002)T=0
λ 1 cλ Pressure in TlCuCl3
Quantum paramagnetNéel state
0ϕ ≠ 0ϕ =
The method of bond operators (S. Sachdev and R.N. Bhatt, Phys. Rev. B 41, 9323 (1990)) provides a quantitative description of spin excitations in TlCuCl3 across the quantum phase transition (M. Matsumoto, B. Normand, T.M. Rice, and M. Sigrist,
Phys. Rev. Lett. 89, 077203 (2002))
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LGW theory for quantum criticalitywrite down an effective action
for the antiferromagnetic order parameter by expanding in powers of and its spatial and temporal d
Landau-Ginzburg-Wilson theor
erivatives, while preserving
y:
all s
ϕϕ
ymmetries of the microscopic Hamiltonian
( ) ( ) ( )( ) ( )22 22 2 2 212 4!x c
ud xd cSϕ ττ ϕ ϕ λ λ ϕ ϕ⎡ ⎤= ∇ + ∂ + − +⎢ ⎥⎣ ⎦∫S. Chakravarty, B.I. Halperin, and D.R. Nelson, Phys. Rev. B 39, 2344 (1989)
~3∆
( )2 2
; 2 c
c pp cε λ λ= ∆ + ∆ = −∆
( )For oscillations of about 0 lead to the following structure in the dynamic structure factor ,
c
S pλ λ ϕ ϕ
ω< =
ω
( ),S p ω
( )( )Z pδ ω ε−
Three triplon continuumTriplon pole
Structure holds to all orders in u
A.V. Chubukov, S. Sachdev, and J.Ye, Phys.
Rev. B 49, 11919 (1994)
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(B) Kondo lattice models
“Large” Fermi surfaces and the Landau-Ginzburg-Wilson spin-density-wave
paramagnon theory
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Kondo lattice
+
Local moments fσ Conduction electrons cσ† †
'K ij i j K i i fi fi fji j i ij
H t c c J c c S J S Sσ σ σ σσ στ<
= + ⋅ + ⋅∑ ∑ ∑
At large JK , magnetic order is destroyed, and we obtain a non-magnetic Fermi liquid (FL) ground state
S. Doniach, Physica B 91, 231 (1977).
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Luttinger’s Fermi volume on a d-dimensional lattice for the FL phase
Let v0 be the volume of the unit cell of the ground state,nT be the total number density of electrons per volume v0.
(need not be an integer)
( )( )
( )
02 Volume enclosed by Fermi surface2
mod 2
d
T
v
n
π×
=
1c cT fn n n n= + = +
A “large” Fermi surface
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Argument for the Fermi surface volume of the FL phase
Single ion Kondo effect implies at low energiesKJ →∞
( )† † † † 0i i i ic f c f↑ ↓ ↓ ↑− † 0 , =1/2 holeif S↓
Fermi liquid of S=1/2 holes with hard-core repulsion
( )( ) ( )
Fermi surface volume density of holes mod 2
1 1 mod 2c cn n
= −
= − − = +
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Doniach’s T=0 phase diagram for the Kondo lattice
JK
“Heavy” Fermi liquid with moments Kondo screened by conduction electrons.
Fermi surface volume equals the Luttinger value.
Local moments choose some static spin
arrangement
SDW FL
JKc
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LGW theory for quantum critical point
( )( ) ( )( ) ( )2 22 2
1 ,42
dd
K Kcdd qd uS q q J J d rdϕ
ω ϕ ω ω τ ϕπ += + + − +∫ ∫
K
Write down effective action for SDW order parameter ϕ
fluctuations are damped by mixing with fermionic quasiparticles near the Fermi surface
ϕ
Fluctuations of about 0 the triplon is now a paramagno nϕ ϕ = ⇒J. Mathon, Proc. R. Soc. London A, 306, 355 (1968); T.V. Ramakrishnan, Phys. Rev. B 10, 4014 (1974); M. T. Beal-Monod and K. Maki, Phys. Rev. Lett. 34, 1461 (1975); J.A. Hertz, Phys. Rev. B 14, 1165 (1976).T. Moriya, Spin Fluctuations in Itinerant Electron Magnetism, Springer-Verlag, Berlin (1985); G. G. Lonzarich and L. Taillefer, J. Phys. C 18, 4339 (1985); A.J. Millis, Phys. Rev. B 48, 7183 (1993).
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Doniach’s T=0 phase diagram for the Kondo lattice
Local moments choose some static spin arrangement.
Near the quantum critical point, the Fermi surface is modified
from the “large Fermi surface” only by the appearance of “gaps” near the hot spots.
JK
“Heavy” Fermi liquid with moments Kondo screened by conduction electrons.
Fermi surface volume equals the Luttinger value.
SDW FL
JKc
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(C) Fractionalized Fermi liquids (FL*)
Spin liquids and Fermi volume changing transitions with a topological order parameter
Beyond LGW: quantum phases and phase transitions with emergent gauge excitations
and fractionalization
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Ground state has Neel order with 0ϕ ≠
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Destroy SDW order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Destroy SDW order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
2
A ground state with 0 and no broken lattice symmetries.
Such a state has emergent excitations described by a or U(1) gauge theory
spin liquid
Z
ϕ =
P. Fazekas and P.W. Anderson, Phil Mag 30, 23 (1974). N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991); X. G. Wen, Phys. Rev. B 44, 2664 (1991).
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Influence of conduction electrons
+
Local moments fσConduction electrons cσ
( ) ( )† †' ,ij i j K i i fi H fi fj
i j i i jH t c c J c c S J i j S Sσ σ σ σσ στ
< <
= + ⋅ + ⋅∑ ∑ ∑
Determine the ground state of the quantum antiferromagnet defined by JH, and then couple to conduction electrons by JK
Choose JH so that ground state of antiferromagnet is a Z2 or U(1) spin liquid
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Influence of conduction electrons
+
Conduction electrons cσLocal moments fσ
At JK= 0 the conduction electrons form a Fermi surface on their own with volume determined by nc.
Perturbation theory in JK is regular, and so this state will be stable for finite JK.So volume of Fermi surface is determined by
(nT -1)= nc(mod 2), and does not equal the Luttinger value.
The (U(1) or Z2) FL* state
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A new phase: FL*
( )( )
( )( )
02 Volume enclosed by Fermi surface2
1 mod 2
d
T
v
n
π×
= −
This phase preserves spin rotation invariance, and has a Fermi surface of sharp electron-like quasiparticles.
The state has “topological order” and associated neutral excitations. The topological order can be detected by the violation of Luttinger’s
Fermi surface volume. It can only appear in dimensions d > 1
Precursors: N. Andrei and P. Coleman, Phys. Rev. Lett. 62, 595 (1989).Yu. Kagan, K. A. Kikoin, and N. V. Prokof'ev, Physica B 182, 201 (1992).Q. Si, S. Rabello, K. Ingersent, and L. Smith, Nature 413, 804 (2001).S. Burdin, D. R. Grempel, and A. Georges, Phys. Rev. B 66, 045111 (2002).L. Balents and M. P. A. Fisher and C. Nayak, Phys. Rev. B 60, 1654, (1999); T. Senthil and M.P.A. Fisher, Phys. Rev. B 62, 7850 (2000).F. H. L. Essler and A. M. Tsvelik, Phys. Rev. B 65, 115117 (2002).
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Phase diagram
U(1) FL* FL
JKc JK
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Phase diagram
Fractionalized Fermi liquid with moments paired in a spin liquid. Fermi surface volume does not include
moments and is unequal to the Luttinger value.
U(1) FL* FL
JKc JK
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Phase diagram
Fractionalized Fermi liquid with moments paired in a spin liquid. Fermi surface volume does not include
moments and is unequal to the Luttinger value.
“Heavy” Fermi liquid with moments Kondo screened by conduction electrons.
Fermi surface volume equals the Luttinger value.
JK
U(1) FL* FL
JKc
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Phase diagram
Fractionalized Fermi liquid with moments paired in a spin liquid. Fermi surface volume does not include
moments and is unequal to the Luttinger value.
“Heavy” Fermi liquid with moments Kondo screened by conduction electrons.
Fermi surface volume equals the Luttinger value.
JK
U(1) FL* FL
JKc
Sharp transition at T=0 in compact U(1) gauge theory; compactness “irrelevant” at critical point
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Phase diagram
JK
FLU(1) FL*
JKc
Quantum Critical
No transition for T>0 in compact U(1) gauge theory; compactness essential for this feature
T
Sharp transition at T=0 in compact U(1) gauge theory; compactness “irrelevant” at critical point
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Phase diagram
FLU(1) FL*
JKc JK
Quantum Critical
• Specific heat ~ T ln T• Violation of Wiedemann-FranzT
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Phase diagram
FLU(1) FL*
JKc JK
Quantum Critical
( )Resistivity ~ 1/ ln 1/TT
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Phase diagram (after allowing for conventional magnetic order)
Topological and SDW order parameters suggest two Topological and SDW order parameters suggest two separate quantum critical pointsseparate quantum critical points
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(D) Multiple order parameters
Berry phases and the breakdown of the LGW paradigm
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Ground state has Neel order with 0ϕ ≠
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Destroy SDW order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Destroy SDW order by perturbations which preserve full square lattice symmetry e.g. second-neighbor or ring exchange.
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Possible paramagnetic ground state with 0ϕ =
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
VBSΨ
Possible paramagnetic ground state with 0ϕ =
VBS
VBS
Such a state breaks lattice symmetry and has 0, where is the
Ψ ≠
Ψ valence bond solid (VBS) order parameter
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Possible paramagnetic ground state with 0ϕ =
VBS
VBS
Such a state breaks lattice symmetry and has 0, where is the
Ψ ≠
Ψ valence bond solid (VBS) order parameter
VBSΨ
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
Possible paramagnetic ground state with 0ϕ =
VBS
VBS
Such a state breaks lattice symmetry and has 0, where is the
Ψ ≠
Ψ valence bond solid (VBS) order parameter
VBSΨ
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
VBSΨ
Possible paramagnetic ground state with 0ϕ =
VBS
VBS
Such a state breaks lattice symmetry and has 0, where is the
Ψ ≠
Ψ valence bond solid (VBS) order parameter
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Work in the regime with small JK, and consider destruction of magnetic order by frustrating
(RKKY) exchange interactions between f moments
VBSΨ
N. Read and S. Sachdev,
Phys. Rev. Lett. 62, 1694 (1989).
Possible paramagnetic ground state with 0ϕ =
VBS
VBS
Such a state breaks lattice symmetry and has 0, where is the
Ψ ≠
Ψ valence bond solid (VBS) order parameter
VBS order (and confinement) appear for collinear spins in d=2
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Naïve approach: add VBS order parameter to LGW theory “by hand”
First order
transitionVBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
g
gVBS
Coexistence0,
0
ϕ ≠
Ψ ≠VBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
gVBS
( topologically ordered)
" "
0, 0
Disordered
ϕ≠
= Ψ =VBS
Neel order0, 0ϕ ≠ Ψ =
VBS
VBS order0, 0ϕ = Ψ ≠
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Naïve approach: add VBS order parameter to LGW theory “by hand”
First order
transitionVBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
g
gVBS
Coexistence0,
0
ϕ ≠
Ψ ≠VBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
gVBS
( topologically ordered)
" "
0, 0
Disordered
ϕ≠
= Ψ =VBS
Neel order0, 0ϕ ≠ Ψ =
VBS
VBS order0, 0ϕ = Ψ ≠
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Superfluid-insulator transition of hard core bosons at f=1/2 (Neel-valence bond solid transition of S=1/2 AFM)
A. W. Sandvik, S. Daul, R. R. P. Singh, and D. J. Scalapino, Phys. Rev. Lett. 89, 247201 (2002)Large scale (> 8000 sites) numerical study of the destruction of superfluid (i.e.
magnetic Neel) order at half filling with full square lattice symmetry
VBS insulator
Superfluid(Neel) order
g=
( ) ( )i j i j i j k l i j k lij ijkl
H J S S S S K S S S S S S S S+ − − + + − + − − + − +
⊂
= + − +∑ ∑
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Boson-vortex duality
Quantum mechanics of two-
dimensional bosons: world
lines of bosons in spacetime
xy
τ
Express theory of S=1/2 AFM as a theory of Sz= -1 “spin down” bosons at filling f=1/2
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);
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Boson-vortex duality
Classical statistical mechanics of a “dual” three-dimensional
superconductor:vortices in a
“magnetic” field
xy
z
Express theory of S=1/2 AFM as a theory of Sz= -1 “spin down” bosons at filling f=1/2
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);
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Boson-vortex duality
Classical statistical mechanics of a “dual” three-dimensional
superconductor:vortices in a
“magnetic” field
xy
z
Strength of “magnetic” field = density of bosons = f flux quanta per plaquette
C. Dasgupta and B.I. Halperin, Phys. Rev. Lett. 47, 1556 (1981); D.R. Nelson, Phys. Rev. Lett. 60, 1973 (1988); M.P.A. Fisher and D.-H. Lee, Phys. Rev. B 39, 2756 (1989);
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Boson-vortex dualityStatistical mechanics of dual superconductor is invariant
under the square lattice space group:, : Translations by a lattice spacing in the , directions
: Rotation by 90 degrees.x yT T x y
R
2
1 1 1 4
Magnetic space group: ;
; ; 1
ifx y y x
y x x y
T T e T T
R T R T R T R T R
π
− − −
=
= = =
Strength of “magnetic” field = density of bosons = f flux quanta per plaquette
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Boson-vortex duality
At density = / ( , relatively prime integers) there are species of vortices, (with =1 ), associated with gauge-equivalent regions of the Brillouin zone
f p q p qq
ϕ …
Hofstäder spectrum of dual “superconducting” order
2
1 1 1 4
Magnetic space group: ;
; ; 1
ifx y y x
y x x y
T T e T T
R T R T R T R T R
π
− − −
=
= = =
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Boson-vortex duality
21
2
1
The vortices form a representation of the space group
: ; :
1 :
i fx y
qi mf
mm
q projective
T T e
R eq
π
π
ϕ ϕ ϕ ϕ
ϕ ϕ
+
=
→ →
→ ∑
Hofstäder spectrum of dual “superconducting” order
At density = / ( , relatively prime integers) there are species of vortices, (with =1 ), associated with gauge-equivalent regions of the Brillouin zone
f p q p qq
ϕ …
See also X.-G. Wen, Phys. Rev. B 65, 165113 (2002)
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Boson-vortex duality
The fields characterize Neel (superfluid) and VBS orderbothϕ
Neel (superfluid) order: 0ϕ =
( )
ˆ
* 2
1
VBS (charge) order:
Status of space group symmetry determined by 2density operators at wavevectors ,
: ;
:
qi mnf i mf
mn
i xx
n
y
mn
p m n
e
T
e
q
e T
π π
πρ
ρ ρ
ρ ϕ ϕ +=
=
→
= ∑i
Q
QQ Q
Q
( ) ( )
ˆ
:
i ye
R R
ρ ρ
ρ ρ
→
→
iQQ Q
Q Q
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Naïve approach: add VBS order parameter to LGW theory “by hand”
First order
transitionVBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
g
gVBS
Coexistence0,
0
ϕ ≠
Ψ ≠VBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
gVBS
( topologically ordered)
" "
0, 0
Disordered
ϕ≠
= Ψ =VBS
Neel order0, 0ϕ ≠ Ψ =
VBS
VBS order0, 0ϕ = Ψ ≠
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Predictions of extended “LGW” theory with projective symmetry
First order
transitionVBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
g
gVBS
Coexistence0,
0
ϕ ≠
Ψ ≠VBS
Neel order0, 0ϕ ≠ Ψ = VBS
VBS order0, 0ϕ = Ψ ≠
gVBS
Neel order0, 0ϕ ≠ Ψ =
VBS
VBS order0, 0ϕ = Ψ ≠
Second order
transition
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Phase diagram of S=1/2 square lattice antiferromagnet
g*
Neel order~ 0z zα αβ βϕ σ ≠
VBSVBS order 0, 1/ 2 spinons confined, 1 triplon excitations
S zS
α
Ψ ≠=
=
or
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Main lesson:
Novel second-order quantum critical point between phases with conventional order parameters.
A direct second-order transition between such phases is forbidden by symmetry in LGW theory.
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Key question for metallic systems:
Is a direct second-order quantum critical point possible between metallic states distinguished by
two conventional order parameters: (A) SDW order(B) Shape of Fermi surface ?
Such a transition is obtained if the FL* phase is unstable to confinement to a SDW state at low
energies.
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Phase diagram for the Kondo lattice ?
JK
Local moments choose some static spin arrangement.
The shape of the Fermi surface differs strongly from that of the
heavy Fermi liquid
“Heavy” Fermi liquid with moments Kondo screened by conduction electrons.
Fermi surface volume equals the Luttinger value.
SDW FL
JKc
See also Q. Si, S. Rabello, K. Ingersent, and J. L. Smith, Nature 413, 804 (2001); S. Paschen, T. Luehmann, C. Langhammer, O. Trovarelli, S. Wirth, C. Geibel, F. Steglich,
Acta Physica Polonica B 34, 359 (2003).
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Conclusions
Two possible routes to “exotic” quantum criticalityI. New FL* phase with a Fermi surface of electron-like
quasiparticles (whose volume violates the Luttinger theorem), topological order, emergent gauge excitations, and neutral fractionalized quasiparticles.
Novel Fermi-volume-changing quantum criticality in the transition between the FL and FL* phases (and associated SDW and SDW* phases).
Conclusions
Two possible routes to “exotic” quantum criticalityI. New FL* phase with a Fermi surface of electron-like
quasiparticles (whose volume violates the Luttinger theorem), topological order, emergent gauge excitations, and neutral fractionalized quasiparticles.
Novel Fermi-volume-changing quantum criticality in the transition between the FL and FL* phases (and associated SDW and SDW* phases).
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Conclusions
Two possible routes to “exotic” quantum criticalityII. Conventional FL and SDW phases (but with very different
shapes of Fermi surfaces) undergo a direct quantum phase transition.
Analogous quantum critical point found in a direct transition between Neel and VBS states in S=1/2 Mott insulators in two dimensions. Mapping to this scenario to metals we obtain the above scenario if the FL* phase is unstable to confinement to a SDW phase.
Conclusions
Two possible routes to “exotic” quantum criticalityII. Conventional FL and SDW phases (but with very different
shapes of Fermi surfaces) undergo a direct quantum phase transition.
Analogous quantum critical point found in a direct transition between Neel and VBS states in S=1/2 Mott insulators in two dimensions. Mapping to this scenario to metals we obtain the above scenario if the FL* phase is unstable to confinement to a SDW phase.