Numerical Python - Cornell Laboratory of Atomic and Solid State
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...
Strain and symmetry breaking
Brad Ramshaw Laboratory for Atomic and Solid State Physics,
Cornell University
Symmetry breaking and order parameters
• Disordered state preserves symmetry. • Different symmetry-breaking ground states • Need more information in the ordered state to describe
thermodynamics.
Probing order parameters with fields
• Field has the same symmetry as the order parameter. • Susceptibility diverges at Tc.
χ
Probing order parameters with fields
• What about the AFM? • “Wrong” field – need Pi,Pi – neutron.
Iron pnictide nematicity
Jiang et al., J. Phys. C. M. (2009), J.H. Chu et al., Science (2012)
+nema'c
Compression strain fields
• Compression strains – change volume.
𝑥
𝜖↓𝑥𝑥
𝜖↓𝑥𝑥 = 𝑑(𝑥− 𝑥↓0 )/𝑑𝑥
Compression waves
• Compression strains supported in solids, liquids, and gasses.
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
Shear waves
• Solids support shear and compression. • Liquids support only compression.
Strain in lower symmetry systems
M. Rotter et al., PRL (2008)
• 1 compression • 1 shear
• 1 compression • 2 shear
• 2 compression • 3 shear
Resonant Ultrasound Spectroscopy
Resonant Ultrasound Spectroscopy
Drive at frequency ω Detect transmitted power
Resonant Ultrasound Spectroscopy
Resonant Ultrasound Spectroscopy
Temperature dependence of RUS on PuCoGa5
J. L. Sarrao et al., Nature (2002)
Temperature dependence of RUS on YBa2Cu3O7
A.Shekhter et al. Nature (2013)
T
Temperature dependence of RUS on URu2Si2
Complex spectra from complex strains
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
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
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?
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 )
Representations of D4
• There are minimal, or “irreducible” representations.
Ψ=+1 Ψ=−1
1 2
34
1 2
34
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
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
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
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
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
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
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
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
Reducible representation
?
= +
E B1
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!
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
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
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
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
Strains of different symmetry
Compressional
+
+
+
=
62more
Shear
Symmetry-resolved temperature dependence in PuCoGa5
B. J. Ramshaw et al., PNAS (2015)
A1g
B2g B1g Eg
Symmetry-resolved temperature dependence in URu2Si2
Universal features at Tc
URu2Si2 PuCoGa5
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
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
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
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
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
Jumps show up in other thermodynamic parameters
K.A. Modic et al., Nat.Comm. (2018)
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
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
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
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 )
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…
Valence transition in YbInCu4
B. Kindler et al., PRB (1994), B. J. Ramshaw et al., PNAS (2015)
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)
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)
Summary
Singlet order parameter
Physics
URu2Si2
PuCoGa5
• A1g order parameter • Preserves time reversal • Q=0 • Valence fluctuation driven
superconductivty