Laboratory for Atomic and Solid State Physics, Cornell University · 2019-01-14 · Brad Ramshaw...
Transcript of Laboratory for Atomic and Solid State Physics, Cornell University · 2019-01-14 · Brad Ramshaw...
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Strain and symmetry breaking
Brad Ramshaw Laboratory for Atomic and Solid State Physics,
Cornell University
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Symmetry breaking and order parameters
• Disordered state preserves symmetry. • Different symmetry-breaking ground states • Need more information in the ordered state to describe
thermodynamics.
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Probing order parameters with fields
• Field has the same symmetry as the order parameter. • Susceptibility diverges at Tc.
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χ
Probing order parameters with fields
• What about the AFM? • “Wrong” field – need Pi,Pi – neutron.
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Iron pnictide nematicity
Jiang et al., J. Phys. C. M. (2009), J.H. Chu et al., Science (2012)
+nema'c
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Compression strain fields
• Compression strains – change volume.
𝑥
𝜖↓𝑥𝑥
𝜖↓𝑥𝑥 = 𝑑(𝑥− 𝑥↓0 )/𝑑𝑥
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Compression waves
• Compression strains supported in solids, liquids, and gasses.
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Shear strain fields
𝝐=(█𝜖↓𝑥𝑥 &𝜖↓𝑥𝑦 &𝜖↓𝑥𝑧 @𝜖↓𝑦𝑥 &𝜖↓𝑦𝑦 &𝜖↓𝑦𝑧 @𝜖↓𝑧𝑥 &𝜖↓𝑧𝑥 &𝜖↓𝑧𝑧 )
𝑥
𝑥
𝑦
𝜖↓𝑥𝑥
𝜖↓𝑥𝑦
𝜖↓𝑥𝑦 = 𝑑(𝑥− 𝑥↓0 )/𝑑𝑦 + 𝑑(𝑦− 𝑦↓0 )/𝑑𝑥
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Shear waves
• Solids support shear and compression. • Liquids support only compression.
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Strain in lower symmetry systems
M. Rotter et al., PRL (2008)
• 1 compression • 1 shear
• 1 compression • 2 shear
• 2 compression • 3 shear
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Resonant Ultrasound Spectroscopy
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Resonant Ultrasound Spectroscopy
Drive at frequency ω Detect transmitted power
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Resonant Ultrasound Spectroscopy
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Resonant Ultrasound Spectroscopy
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Temperature dependence of RUS on PuCoGa5
J. L. Sarrao et al., Nature (2002)
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Temperature dependence of RUS on YBa2Cu3O7
A.Shekhter et al. Nature (2013)
T
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Temperature dependence of RUS on URu2Si2
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Complex spectra from complex strains
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Symmetry to the rescue!
𝑖ℏ 𝜕/𝜕𝑡 Ψ(𝒓,𝑡)=− ℏ↑2 /2𝑚 𝛻↑2 Ψ(𝒓,𝑡)+V(r,t)Ψ(𝒓,𝑡)
?
SO(3)
D4h
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Square (D4)
1 2
34
1 2
34
1 2
34
4 1
23
1 2
34
2 3
41
1 2
34
3 4
12
1 2
34
2 1
43
1 2
34
4 3
21
1 2
34
3 2
14
1 2
34
1 4
32
x1 x1 x2
I S4+ S2 S4
-
S4
R100 R010
R100
R110 R1-10
R110
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Representations of D4
1 2
34
Ψ=(██2@3 @█2@1 )
• Square is a “representation” of D4 that preserves all symmetries of the group.
• What if we want to break symmetry? More complicated things on a square?
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Representations of D4
• There are minimal, or “irreducible” representations.
1 2
34
1 2
34
Ψ=(██4@3 @█2@1 ) Ψ=(██2@1 @█4@3 )
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Representations of D4
• There are minimal, or “irreducible” representations.
Ψ=+1 Ψ=−1
1 2
34
1 2
34
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Representations of D4
Ψ=+1
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
Ψ=−1
χ 1 -1 1 1 -1
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Representations of D4
Ψ=+1 Ψ=−1
x2-y2
B11 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
χ 1 -1 1 1 -1
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Representations of D4
xy
B2
Ψ=+1 Ψ=−1
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
χ 1 -1 1 -1 1
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Representations of D4
Ψ=+1
A1
x2+y2z2
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
χ 1 1 1 1 1
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Representations of D4
A2
xvy-yvx=Jz
Ψ=+1 Ψ=−1
z
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
χ 1 1 1 -1 -1
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Representations of D4
E
Ψ=+1 Ψ=+1
Ψ=−1 Ψ=−1
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
χ
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Representations of D4
E
Ψ=(█+1@0 )
Ψ=(█−1@0 )
Ψ=(█0@+1 )
Ψ=(█0@−1 )
(█1&0@0&1 )
(█0&1@1&0 )
(█−1&0@0&−1 )
(█1&0@0&−1 )
(█0&−1@1&0 )
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
χ
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Representations of D4
E
Ψ=(█+1@0 )
(xz,yz)(x,y)(Jx,Jy)Ψ=(█−1
@0 )
Ψ=(█0@+1 )
Ψ=(█0@−1 )
tr(█1&0@0&1 )=2
tr(█0&1@1&0 )=0
tr(█−1&0@0&−1 )=-2
tr(█1&0@0&−1 )=0
tr(█0&−1@1&0 )=0
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
χ
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Reducible representation
?
= +
E B1
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Character table of D4
1 2
34
1 2
34
1 2
34
1 2
34
1 2
34x1 x2
I S2 S4 R100 R110
A1 1 1 1 1 11 1 1 -1 -1A21 -1 1 1 -1B11 -1 1 -1 1B22 0 -2 0 0ETableis5x5,notacoincidence!
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Representations in 3D – a simple example
𝐸 =(█𝐸↓𝑥 @𝐸↓𝑦 @𝐸↓𝑧 )
ℓ=1
𝐸=(█𝐸↓𝑥 @𝐸↓𝑦 )⊕ 𝐸↓𝑧 𝐸↓𝑢
𝐴↓2𝑢
SO(3)
D4h
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Representations in 3D – a simple example
𝐵 =(█𝐵↓𝑥 @𝐵↓𝑦 @𝐵↓𝑧 )
𝐵=(█𝐵↓𝑥 @𝐵↓𝑦 )⊕ 𝐵↓𝑧 𝐸↓𝑔
𝐴↓2𝑔
D4h
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Strain in 3D
𝝐=(█𝜖↓𝑥𝑥 &𝜖↓𝑥𝑦 &𝜖↓𝑥𝑧 @𝜖↓𝑦𝑥 &𝜖↓𝑦𝑦 &𝜖↓𝑦𝑧 @𝜖↓𝑧𝑥 &𝜖↓𝑧𝑥 &𝜖↓𝑧𝑧 )
𝝐=(𝜖↓𝑥𝑥 + 𝜖↓𝑦𝑦 )⊕ 𝜖↓𝑧𝑧 ⊕(𝜖↓𝑥𝑥 − 𝜖↓𝑦𝑦 )⊕ 𝜖↓𝑥𝑦 ⊕{𝜖↓𝑥𝑧 , 𝜖↓𝑦𝑧 }𝐴↓1𝑔
𝐵↓1𝑔
SO(3)𝝐= ( 𝜖↓𝑥𝑥 + 𝜖↓𝑦𝑦 + 𝜖↓𝑧𝑧 )⊕ {█𝜖↓𝑥𝑦 @𝜖↓𝑥𝑧 @█𝜖↓𝑦𝑧 @𝜖↓𝑥𝑥 − 𝜖↓𝑦𝑦 @𝜖↓𝑥𝑥 − 𝜖↓𝑧𝑧 }
ℓ=0
𝐴↓1𝑔
𝐵↓2𝑔
𝐸↓𝑔
ℓ=2
D4h
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Basis of strains
Compressional
Shear
(c11+c12)/2 c33
(c11-c12)/2 c66 c44
(𝜖↓𝑥𝑥 + 𝜖↓𝑦𝑦 )
{𝜖↓𝑥𝑧 , 𝜖↓𝑦𝑧 }
𝜖↓𝑧𝑧
(𝜖↓𝑥𝑥 − 𝜖↓𝑦𝑦 )
𝜖↓𝑥𝑦
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Strains of different symmetry
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Compressional
+
+
+
=
62more
Shear
Symmetry-resolved temperature dependence in PuCoGa5
B. J. Ramshaw et al., PNAS (2015)
A1g
B2g B1g Eg
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Symmetry-resolved temperature dependence in URu2Si2
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Universal features at Tc
URu2Si2 PuCoGa5
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How do order parameters and strains couple?
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 𝑐↓𝑘 = 𝜕↑2 ℱ/𝜕𝜖↓𝑘↑2
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2
• What are we measuring? Elastic moduli – curvature of free energy with respect to strain.
• How does free energy transform?
+𝛽𝜖↑𝜈 𝜂↑𝜌 + 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4
𝐴↓1𝑔
𝐴↓1𝑔
𝐴↓1𝑔
𝐴↓1𝑔
𝐴↓1𝑔
𝐴↓1𝑔
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How do order parameters and strains couple?
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 𝑐↓𝑘 = 𝜕↑2 ℱ/𝜕𝜖↓𝑘↑2
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 +𝛽𝜖↑𝜈 𝜂↑𝜌
+ 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4
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How do order parameters and strains couple?
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 +𝛽𝜖↑𝜈 𝜂↑𝜌
+ 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4
• Order parameters also break other symmetries – time reversal, gauge, translation – that prevent linear coupling.
• Lowest-order term is often linear in A1g strain, quadratic in OP.
𝛽𝜖↓𝐴↓1𝑔 𝜂↑2 𝛽𝜖↓𝐵↓1𝑔 𝜂↑2
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How do order parameters and strains couple?
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 +𝛽𝜖↓𝐴↓1𝑔 𝜂↑2
+ 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4
𝜕ℱ/𝜕𝜂 =𝛼(𝑇− 𝑇↓𝑣 )𝜂+𝛾𝜂↑3 +2𝛽𝜖↓𝐴↓1𝑔 𝜂=0
𝜂 =0
𝜂 =√𝛼(𝑇↓𝑣 −𝑇)−2𝛽𝜖↓𝐴↓1𝑔 /𝛾
𝑇> 𝑇↓𝑣
𝑇< 𝑇↓𝑣
𝑐↓𝐴↓1𝑔 (𝑇)= 𝜕↑2 ℱ/𝜕𝜖↓𝐴↓1𝑔 ↑2 = 𝑐↓𝐴↓1𝑔 𝑇> 𝑇↓𝑣
= 𝑐↓𝐴↓1𝑔 −2𝛽↑2 /𝛾 𝑇< 𝑇↓𝑣
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Universal features at Tc
URu2Si2 PuCoGa5
𝑐↓𝐴↓1𝑔 (𝑇)= 𝜕↑2 ℱ/𝜕𝜖↓𝐴↓1𝑔 ↑2 = 𝑐↓𝐴↓1𝑔 𝑇> 𝑇↓𝑣
= 𝑐↓𝐴↓1𝑔 −2𝛽↑2 /𝛾 𝑇< 𝑇↓𝑣
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Jumps show up in other thermodynamic parameters
K.A. Modic et al., Nat.Comm. (2018)
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Unique couplings for two-component order parameters
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 +𝛽𝜖↑𝜈 𝜂↑𝜌
+ 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4
𝜂↓𝐸 = {𝜂↓𝑥 , 𝜂↓𝑦 } ∴ 𝜂↓𝐸↑2 =( 𝜂↓𝑥↑2 + 𝜂↓𝑦↑2 )⊕( 𝜂↓𝑥↑2 − 𝜂↓𝑦↑2 )
⊕ 𝜂↓𝑥 𝜂↓𝑦 E
A↓1𝑔
B↓1𝑔
B↓2𝑔
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Unique couplings for two-component order parameters
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 + 𝛽↓1 𝜖↓𝐵↓1𝑔 𝜂↓𝐸↑2 + 𝛽↓2 𝜖↓𝐵↓2𝑔 𝜂↓𝐸↑2 + 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4 +
• Same mathematics now predicts jumps in cA1g, cB1g, and cB2g.
𝛽𝜖↓𝐴↓1𝑔 𝜂↓𝐸↑2
URu2Si2
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Linear coupling
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 𝑐↓𝑘 = 𝜕↑2 ℱ/𝜕𝜖↓𝑘↑2
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 +𝛽𝜖↑𝜈 𝜂↑𝜌
+ 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4
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Linear coupling
ℱ=∑𝑘↑▒1/2 𝑐↓𝑘 𝜖↓𝑘↑2 +𝛽𝜖↓𝑘 𝜂↓𝑘
+ 1/2 𝛼(𝑇− 𝑇↓𝑣 )𝜂↑2 + 1/4 𝛾𝜂↑4
𝜕ℱ/𝜕𝜂 =𝛼(𝑇− 𝑇↓𝑣 )𝜂+𝛽𝜖↓𝑘 =0+𝛽𝜖↓𝑘 =0
𝜂 =−𝛽𝜖↓𝑘 /𝛼(𝑇− 𝑇↓𝑣 )
𝑐↓𝑘 (𝑇)= 𝜕↑2 ℱ/𝜕𝜖↓𝑘↑2 = 𝑐↓𝑘 − 𝛽↑2 /𝛼(𝑇− 𝑇↓𝑣 )
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Soft A1g moduli in PuCoGa5
B. J. Ramshaw et al., PNAS (2015)
𝑐↓𝑘 (𝑇)= 𝑐↓𝑘 − 𝛽↑2 /𝛼(𝑇− 𝑇↓𝑣 )
𝑇↓𝑣 ≈9𝐾Whatdoeslinearcouplingsay?• Q=0• Non-magne'c• C4preserving• etc…
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Valence transition in YbInCu4
B. Kindler et al., PRB (1994), B. J. Ramshaw et al., PNAS (2015)
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Mixed plutonium valence in PuCoGa5
J. L. Smith et al., JLCM (1983), J. M. Lawrence et al., Rep. Prog. Phys. (1981), C. H. Booth et al., JESRP (2014)
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Valence fluctuations as a mechanism for high-Tc
H. Q. Yuan et al., Science (2003), K. Miyake, J. Phys. C. M. (2008), E.D. Bauer et al., J.Physics C. M. (2012)
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Summary
Singlet order parameter
Physics
URu2Si2
PuCoGa5
• A1g order parameter • Preserves time reversal • Q=0 • Valence fluctuation driven
superconductivty