exotic phases and quantum phase transitions in model systems
TRANSCRIPT
![Page 1: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/1.jpg)
Exotic phases and quantum phasetransitions in model systems
Subir SachdevHarvard University
![Page 2: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/2.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 3: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/3.jpg)
Outline1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
![Page 4: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/4.jpg)
Ground state has long-range Néel order
Square lattice antiferromagnet
H =∑
〈ij〉
Jij!Si · !Sj
Order parameter is a single vector field !ϕ = ηi!Si
ηi = ±1 on two sublattices〈!ϕ〉 #= 0 in Neel state.
![Page 5: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/5.jpg)
Square lattice antiferromagnet
H =∑
〈ij〉
Jij!Si · !Sj
J
J/λ
Weaken some bonds to induce spin entanglement in a new quantum phase
![Page 6: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/6.jpg)
λλc
M. Matsumoto, C. Yasuda, S. Todo, and H. Takayama, Phys. Rev.B 65, 014407 (2002).
Quantum critical point with non-local entanglement in spin wavefunction
![Page 7: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/7.jpg)
λλcCFT3O(3) order parameter !ϕ
S =∫
d2rdτ[(∂τϕ)2 + c2(∇r $ϕ)2 + s$ϕ2 + u
($ϕ2
)2]
![Page 8: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/8.jpg)
S. Wenzel and W. Janke, arXiv:0808.1418M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997)
Quantum Monte Carlo - critical exponents
![Page 9: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/9.jpg)
Quantum Monte Carlo - critical exponents
Field-theoretic RG of CFT3E. Vicari et al.
S. Wenzel and W. Janke, arXiv:0808.1418M. Troyer, M. Imada, and K. Ueda, J. Phys. Soc. Japan (1997)
![Page 10: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/10.jpg)
TlCuCl3
![Page 11: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/11.jpg)
Phase diagram as a function of the ratio ofexchange interactions, λ
λλc
Pressure in TlCuCl3
![Page 12: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/12.jpg)
Christian Ruegg, Bruce Normand, Masashige Matsumoto, Albert Furrer, Desmond McMorrow, Karl Kramer, Hans–Ulrich Gudel, Severian Gvasaliya,
Hannu Mutka, and Martin Boehm, Phys. Rev. Lett. 100, 205701 (2008)
TlCuCl3 with varying pressure
Observation of 3 → 2 low energy modes, emergence of new longi-tudinal mode (the “Higgs boson”) in Neel phase, and vanishing ofNeel temperature at quantum critical point
![Page 13: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/13.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 14: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/14.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 15: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/15.jpg)
Triangular lattice antiferromagnet
H = J∑
〈ij〉
!Si · !Sj
Nearest-neighbor model has non-collinear Neel order
![Page 16: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/16.jpg)
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Triangular lattice antiferromagnet
![Page 17: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/17.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
![Page 18: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/18.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
![Page 19: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/19.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
![Page 20: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/20.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
![Page 21: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/21.jpg)
P. Fazekas and P. W. Anderson, Philos. Mag. 30, 23 (1974).
Spin liquid obtained in a generalized spin model with S=1/2 per unit cell
=
Triangular lattice antiferromagnet
![Page 22: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/22.jpg)
First approach
Look for spin liquids across continuous (or weakly first-order)
quantum transitions from antiferromagnetically ordered states
D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988).
![Page 23: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/23.jpg)
Second approach
Look for spin liquids across continuous (or weakly first-order) quantum transitions to an insulator
from a metal
S. Burdin, D.R. Grempel, and A. Georges, Phys. Rev. B 66, 045111 (2002).T. Senthil, M. Vojta and S. Sachdev, Phys. Rev. B 69, 035111 (2004).
S. Florens and A. Georges, Phys. Rev. B 70, 035114 (2004).T. Senthil, Phys. Rev. B 78 045109 (2008).
![Page 24: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/24.jpg)
First approach
Look for spin liquids across continuous (or weakly first-order)
quantum transitions from antiferromagnetically ordered states
D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988).
![Page 25: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/25.jpg)
ssc
non-collinear Neel state
N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)X.-G. Wen, Phys. Rev. B 44, 2664 (1991)
Z2 spin liquidwith paired spinonsand a vison excitation
![Page 26: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/26.jpg)
Excitations of the Z2 Spin liquid
=A spinon
![Page 27: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/27.jpg)
Excitations of the Z2 Spin liquid
=A spinon
![Page 28: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/28.jpg)
Excitations of the Z2 Spin liquid
=A spinon
![Page 29: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/29.jpg)
Excitations of the Z2 Spin liquid
=A spinon
![Page 30: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/30.jpg)
• A characteristic property of a Z2 spin liquidis the presence of a spinon pair condensate
• A vison is an Abrikosov vortex in the paircondensate of spinons
• Visons are are the dark matter of spin liq-uids: they likely carry most of the energy,but are very hard to detect because they donot carry charge or spin.
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
![Page 31: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/31.jpg)
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
![Page 32: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/32.jpg)
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
![Page 33: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/33.jpg)
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
![Page 34: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/34.jpg)
=
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
![Page 35: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/35.jpg)
=
-1
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
![Page 36: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/36.jpg)
=
-1
-1
Excitations of the Z2 Spin liquid
A vison
N. Read and B. Chakraborty, Phys. Rev. B 40, 7133 (1989)N. Read and S. Sachdev, Phys. Rev. Lett. 66, 1773 (1991)
![Page 37: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/37.jpg)
Solvable models of Z2 spin liquids
Z2 gauge theory - T. Senthil and M.P.A. Fisher, Phys. Rev. B 63, 134521 (2001).
Quantum dimer models - D. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 2376 (1988); R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001).
Integrable Z2 gauge theory - A. Y. Kitaev, Annals of Physics 303, 2 (2003).
Majorana fermion models - X.-G. Wen, Phys. Rev. Lett. 90, 016803 (2003).
![Page 38: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/38.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 39: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/39.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 40: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/40.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 41: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/41.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 42: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/42.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 43: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/43.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 44: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/44.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 45: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/45.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 46: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/46.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 47: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/47.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 48: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/48.jpg)
A Simple Toy Model (A. Kitaev, 1997)
Spins Sα living on the links of a square lattice:
− Hence, Fp's and Ai's form a set of conserved quantities.
![Page 49: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/49.jpg)
Properties of the Ground State
Ground state: all Ai=1, Fp=1 Pictorial representation: color each link with an up-spin. Ai=1 : closed loops. Fp=1 : every plaquette is an equal-amplitude superposition of
inverse images.
The GS wavefunction takes the same value on configurationsconnected by these operations. It does not depend on the geometry of the configurations, only on their topology.
![Page 50: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/50.jpg)
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition. Charges and vortices interact via topological Aharonov-Bohm
interactions.
![Page 51: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/51.jpg)
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line (a “spinon”) “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition. Charges and vortices interact via topological Aharonov-Bohm
interactions.
![Page 52: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/52.jpg)
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line (a “spinon”) “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition (a “vison”). Charges and vortices interact via topological Aharonov-Bohm
interactions.
![Page 53: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/53.jpg)
Properties of Excitations
“Electric” particle, or Ai = –1 – endpoint of a line (a “spinon”) “Magnetic particle”, or vortex: Fp= –1 – a “flip” of this plaquette
changes the sign of a given term in the superposition (a “vison”). Charges and vortices interact via topological Aharonov-Bohm
interactions.
Spinons and visons are mutual semions
![Page 54: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/54.jpg)
Beyond Z2 spin liquids
String nets - M. A. Levin and X.-G. Wen, Phys. Rev. B 71, 045110 (2005).
Doubled SU(2)k Chern-Simons theory - M. Freedman, C. Nayak, K. Shtengel, K. Walker, and Z. Wang, Annals of Physics 310, 428 (2004).
Spin model on the honeycomb lattice - A. Y. Kitaev, Annals of Physics 321, 2 (2006).
Quantum spin liquids with anyonic excitations
![Page 55: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/55.jpg)
Kitaev Model
![Page 56: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/56.jpg)
Kitaev Model
Z2 spin liquids
![Page 57: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/57.jpg)
Kitaev Model
Non-abelian anyons in a magnetic field
![Page 58: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/58.jpg)
ssc
non-collinear Neel state
Z2 spin liquidwith paired spinonsand a vison excitation
Quantum “disordering” magnetic order
![Page 59: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/59.jpg)
Spin liquid with a “photon”, whichis unstable to the appearance ofvalence bond solid (VBS) order
ssc
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989).D. P. Arovas and A. Auerbach, Phys. Rev. B 38, 316 (1988).
collinear Neel state
Quantum “disordering” magnetic order
![Page 60: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/60.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 61: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/61.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 62: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/62.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 63: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/63.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 64: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/64.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 65: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/65.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 66: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/66.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 67: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/67.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 68: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/68.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 69: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/69.jpg)
Ψvbs(i) =∑
〈ij〉
!Si · !Sjei arctan(rj−ri)
Order parameter of VBS state
![Page 70: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/70.jpg)
N. Read and S. Sachdev, Phys. Rev. Lett. 62, 1694 (1989)O.I. Motrunich and A. Vishwanath, Phys. Rev. B 70, 075104 (2004).
T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
• Near the Neel-VBS transition, the (nearly)gapless photon can be identified with theGoldstone mode associated with an emer-gent circular symmetry
Ψvbs → Ψvbseiθ.
![Page 71: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/71.jpg)
Quantum Monte Carlo simulations display convincing evidence for a transition from a
Neel state at small Qto a
VBS state at large Q
A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007).R.G. Melko and R.K. Kaul, Phys. Rev. Lett. 100, 017203 (2008).F.-J. Jiang, M. Nyfeler, S. Chandrasekharan, and U.-J. Wiese, arXiv:0710.3926
![Page 72: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/72.jpg)
Distribution of VBS order Ψvbs at large Q
Emergent circular symmetry is
evidence for U(1) photon and
topological order
A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007).
Re[Ψvbs]
Im[Ψvbs]
![Page 73: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/73.jpg)
Many other models display VBS order: e.g. the planar pyrochlore lattice
C.H. Chung, J.B. Marston, and S. Sachdev, Phys. Rev. B 64, 134407 (2001)J.-B. Fouet, M. Mambrini, P. Sindzingre, and C. Lhuillier, Phys. Rev. B 67, 054411 (2003)R. Moessner, O. Tchernyshyov, and S. L. Sondhi, J. Stat. Phys. 116, 755 (2004).O.A. Starykh, A. Furusaki, and L. Balents, Phys. Rev. B 72, 094416 (2005)M. Raczkowski and D. Poilblanc, arXiv:0810.0738
![Page 74: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/74.jpg)
Many other models display VBS order: e.g. the planar pyrochlore lattice
C.H. Chung, J.B. Marston, and S. Sachdev, Phys. Rev. B 64, 134407 (2001)J.-B. Fouet, M. Mambrini, P. Sindzingre, and C. Lhuillier, Phys. Rev. B 67, 054411 (2003)R. Moessner, O. Tchernyshyov, and S. L. Sondhi, J. Stat. Phys. 116, 755 (2004).O.A. Starykh, A. Furusaki, and L. Balents, Phys. Rev. B 72, 094416 (2005)M. Raczkowski and D. Poilblanc, arXiv:0810.0738
![Page 75: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/75.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 76: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/76.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 77: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/77.jpg)
ssc
Neel state〈zα〉 #= 0 〈zα〉 = 0
(a)
(b)
(a)
(b)or
Neel-VBS quantum transition
VBS state with a nearly gapless photon
Critical theory for photons and deconfined spinons:
Sz =∫
d2rdτ
[|(∂µ−iAµ)zα|2+s|zα|2+u(|zα|2)2+
12e2
0
(εµνλ∂νAλ)2]
O.I. Motrunich and A. Vishwanath, Phys. Rev. B 70, 075104 (2004).T. Senthil, A. Vishwanath, L. Balents, S. Sachdev and M.P.A. Fisher, Science 303, 1490 (2004).
![Page 78: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/78.jpg)
ssc
Neel state〈zα〉 #= 0 〈zα〉 = 0
(a)
(b)
(a)
(b)or
Neel-VBS quantum transition
VBS state with a nearly gapless photon
A.W. Sandvik, Phys. Rev. Lett. 98, 2272020 (2007).R.G. Melko and R.K. Kaul, Phys. Rev. Lett. 100, 017203 (2008).A. B. Kuklov, M. Matsumoto, N. V. Prokof ’ev, B. V. Svistunov, and M. Troyer, Phys. Rev. Lett. 101, 050405 (2008)F.-J. Jiang, M. Nyfeler, S. Chandrasekharan, and U.-J. Wiese, arXiv:0710.3926
Monte Carlo studies support CP1 field theory and presence of emer-gent photon. At system sizes larger than L ∼ 50, and most studiesshow weakly first-order behavior at larger scales.
![Page 79: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/79.jpg)
Second approach
Look for spin liquids across continuous (or weakly first-order) quantum transitions to an insulator
from a metal
S. Burdin, D.R. Grempel, and A. Georges, Phys. Rev. B 66, 045111 (2002).T. Senthil, M. Vojta and S. Sachdev, Phys. Rev. B 69, 035111 (2004).
S. Florens and A. Georges, Phys. Rev. B 70, 035114 (2004).T. Senthil, Phys. Rev. B 78 045109 (2008).
![Page 80: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/80.jpg)
Mott transition on the honeycomb latticet
M. Hermele, Phys. Rev. B 76, 035125 (2007).
![Page 81: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/81.jpg)
Mott transition on the honeycomb latticet
M. Hermele, Phys. Rev. B 76, 035125 (2007).
Turn up interactions to transform semi-metal into an insulator with a charge gap
![Page 82: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/82.jpg)
Mott transition on the honeycomb lattice
M. Hermele, Phys. Rev. B 76, 035125 (2007).
Possible intermediate critical (or algebraic) spin liquid (ASL) phase with neutral fermionic Dirac spinons, which are
remnants of the electrons of the semi-metal
![Page 83: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/83.jpg)
M. Hermele, Phys. Rev. B 76, 035125 (2007).
Possible intermediate critical (or algebraic) spin liquid (ASL) phase with neutral fermionic Dirac spinons, which are
remnants of the electrons of the semi-metal
SASL =∫
d2rdτ
[Ψγµ(∂µ − iAµ))Ψ
]
Fermionic critical spin liquids
![Page 84: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/84.jpg)
Fermionic critical spin liquids
Similar states have been proposed for the square and kagome lattices
SASL =∫
d2rdτ
[Ψγµ(∂µ − iAµ))Ψ
]
I. Affleck and J. B. Marston, Phys. Rev. B 37, 3774-3777 (1988). W. Rantner and X.-G. Wen, Phys. Rev. Lett. 86, 3871-3874 (2001). M. Hermele, T. Senthil, and M.P.A. Fisher, Phys. Rev. B 72 104404 (2005). Y. Ran, M. Hermele, P.A. Lee, and X.-G. Wen, Phys. Rev. Lett. 98, 117205 (2007).
![Page 85: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/85.jpg)
Fermionic critical spin liquids
Evidence critical spin liquid is stable for SU(4) but not SU(2)
![Page 86: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/86.jpg)
Hubbard model on triangular lattice
Phase diagram at Half filling?
U/t0
Free Fermi gas,un-nested
Fermi liquid
Heisenberg n.n. AFM3-sublattice Neel state
Weak coupling Strong coupling
metal insulator?????
J=t2/U
O.I. Motrunich, Phys. Rev. B 72, 045105 (2005). D.N. Sheng, O.I. Motrunich, S. Trebst, E. Gull, and M.P.A. Fisher, Phys. Rev. B 78, 054520 (2008)
![Page 87: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/87.jpg)
“Weak” Mott insulator - Ring exchange
Insulator --> effective spin model
Ring exchange:(mimics charge fluctuations)
U/t0
metal insulator???
Mott insulator with small charge gap
O.I. Motrunich, Phys. Rev. B 72, 045105 (2005). D.N. Sheng, O.I. Motrunich, S. Trebst, E. Gull, and M.P.A. Fisher, Phys. Rev. B 78, 054520 (2008)
![Page 88: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/88.jpg)
Is a Spinon Fermi sea actually the ground state of Triangular ring model (or Hubbard model)?
Variational Monte Carlo analysis suggests it might be for J4/J2 >0.3
O.I. Motrunich, Phys. Rev. B 72, 045105 (2005). D.N. Sheng, O.I. Motrunich, S. Trebst, E. Gull, and M.P.A. Fisher, Phys. Rev. B 78, 054520 (2008)
![Page 89: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/89.jpg)
Quasi-1d route to “Spin-Metals”
ky
kx
Neel or Critical Spin liquid
Triangular strips:
Few gapless 1d modes Fingerprint of 2d singular surface - many gapless 1d modes, of order N
New spin liquid phases on quasi-1d strips, each a descendent of a 2d spin-metal
Spin-Metal
Q-Q
D.N. Sheng, O.I. Motrunich, S. Trebst, E. Gull, and M.P.A. Fisher, Phys. Rev. B 78, 054520 (2008)
![Page 90: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/90.jpg)
Singularities in momentum space locate the “Bose” surface (points in 1d)
J1
J2
D. N. Sheng, O. I. Motrunich, S. Trebst, E. Gull, and M. P. A. Fisher, Phys. Rev. B 78, 054520 (2008)
![Page 91: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/91.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 92: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/92.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 93: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/93.jpg)
G. Misguich, C. Lhuillier, B. Bernu, and C. Waldtmann, Phys. Rev. B 60, 1064 (1999).W. LiMing, G. Misguich, P. Sindzingre, and C. Lhuillier, Phys. Rev. B 62, 6372,6376 (2000).
Triangular lattice antiferromagnet: exact diagonalization
![Page 94: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/94.jpg)
G. Misguich, C. Lhuillier, B. Bernu, and C. Waldtmann, Phys. Rev. B 60, 1064 (1999).W. LiMing, G. Misguich, P. Sindzingre, and C. Lhuillier, Phys. Rev. B 62, 6372,6376 (2000).
Triangular lattice antiferromagnet: exact diagonalization
Z2 spin liquid
![Page 95: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/95.jpg)
G. Misguich, C. Lhuillier, B. Bernu, and C. Waldtmann, Phys. Rev. B 60, 1064 (1999).W. LiMing, G. Misguich, P. Sindzingre, and C. Lhuillier, Phys. Rev. B 62, 6372,6376 (2000).
Triangular lattice antiferromagnet: exact diagonalization
Z2 spin liquidor
spinon Fermi surface
or both ?
Z2 spin liquid
![Page 96: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/96.jpg)
Q2D organics κ-(ET)2X; spin-1/2 on triangular lattice
dimer model
ET layer
X layer
Kino & Fukuyama
t’/t = 0.5 ~ 1.1
Triangular latticeHalf-filled band
![Page 97: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/97.jpg)
Heat capacity measurements
S. Yamashita, et al., Nature Physics 4, 459 - 462 (2008)
Evidence for Gapless spinon?
A. P. Ramirez, Nature Physics 4, 442 (2008)
![Page 98: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/98.jpg)
Thermal Conductivity below 300 mK•Convex, non-T^3 dependence in κ•Magnetic fields enhance κ
κ/T vs T2 Plot
γ = 0Note: phonon contribution has no effect on this conclusion.
M. Yamashita, H. Nakata, Y. Kasa-hara, S. Fujimoto, T. Shibauchi,Y. Matsuda, T. Sasaki, N. Yoneyama,and N. Kobayashi, preprint and25th International Conference onLow Temperature Physics, SaM3-2, Amsterdam, August 9, 2008.
![Page 99: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/99.jpg)
Half-filled band Mott insulator with spin S = 1/2
Triangular lattice of [Pd(dmit)2]2 frustrated quantum spin system
X[Pd(dmit)2]2 Pd SC
X Pd(dmit)2
t’t
t
![Page 100: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/100.jpg)
Magnetic Criticality
t’/t
T N (K
)
Magnetic order
Quantum critical
Quantum disordered
Me4P
Me4As
EtMe3As
Et2Me2As Me4Sb
Et2Me2P
EtMe3Sb
EtMe3Pt’/t = 1.05
Y. Shimizu et al. J.Phys. Conden. Matter, in press.
X[Pd(dmit)2]2Et2Me2Sb (CO)
![Page 101: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/101.jpg)
Pressure-temperature phase diagram of ETMe3P[Pd(dmit)2]2
M. Tamura, A. Nakao and R. Kato, J. Phys. Soc. Japan 75, 093701 (2006)Y. Shimizu, H. Akimoto, H. Tsujii, A. Tajima, and R. Kato, Phys. Rev. Lett. 99, 256403 (2007)
Observation of a valence bond solid (VBS)
Spin gap ~ 40 K J ~ 250 K
![Page 102: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/102.jpg)
Recent DMRG
• Large systems (up to 108 spins)
• Spin Gap 0.055(5)• Singlet gap Tiny• E=-0.433
Kagome antiferromagnet
H.C. Jiang, Z.Y. Weng, and D.N. Sheng, Phys. Rev. Lett. 101, 117203 (2008)
![Page 103: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/103.jpg)
Proposed VBS ground state
3rd Order: Bind 3Es into H—maximize H
4th Order: Honeycomb Lattice
Leftover: Pinwheels
24*2^(N/36) Low energy states
J. Marston and C. Zeng, J. Appl. Phys. 69 5962 (1991)P. Nikolic and T. Senthil, Phys. Rev. B 68, 214415 (2003)
R. R. P. Singh and D.A. Huse, Phys. Rev. B 76, 180407(R) (2007)
Kagome antiferromagnets
![Page 104: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/104.jpg)
Series show excellent ConvergenceOrder & Honeycomb & Stripe VBC & 36-site PBC \\ 0 & -0.375 & -0.375 & -0.375 \\ 1 & -0.375 & -0.375 & -0.375 \\ 2 & -0.421875 & -0.421875 & -0.421875 \\ 3 & -0.42578125 & -0.42578125 & -0.42578125 \\ 4 & -0.431559245 & -0.43101671 & -0.43400065 \\ 5 & -0.432088216 & -0.43153212 & -0.43624539 \\
Ground State Energy per site
Estimated H-VBC energy: -0.433(1) (ED, DMRG)
36-site PBC: Energy=-0.43837653
Variational state of Ran et al -0.429
Kagome antiferromagnet
R. R. P. Singh and D.A. Huse, Phys. Rev. B 76, 180407(R) (2007)
![Page 105: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/105.jpg)
Rb2Cu3SnF12
Spin gap ~ 20 K J ~ 200 K
Distorted kagome(or VBS order in kagome spin-phonon model ?)
Kagome antiferromagnet
K. Morita, M. Yano. T. Ono, H. Tanaka, K. Fujii, H. Uekusa, Y. Narumi, and K. Kindo, J. Phys. Soc. Japan 77, 043707 (2008)
![Page 106: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/106.jpg)
• New material: Herbertsmithite
ZnCu_3(OH)_6Cl_2 Cu atoms carry spin-
half Kagome-layers of Cu Separated by layers of
Zn
Kagome antiferromagnet
![Page 107: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/107.jpg)
Two factors complicate interpretation of experiments on herbertsmithite
• Substitutional Impurities (Zn for Cu)
Causes extra spins and dilution
• Anisotropy Dzyaloshinski-Moria
Interaction can cause weak ferromagnetism
Dz and Dp
Kagome antiferromagnet
![Page 108: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/108.jpg)
Lecheminant et al. PRB 1997Waldtman et al. Here is a clear separation of energy scales
Collapse for N infinity : onset of a broken symmetry state
Energy levels of
S2
E/NJ
N
Sz2/2IN
Signature of a broken-symmetry state : Signature of a broken-symmetry state :
by exact diagonalization ofN=21, 24, 27, 30, 36 sites
z
Q=0
D=0D=0 D/J=0.1D/J=0.1
Dezij.
Is there a critical D/JD/J ? (!Sz !" >h )
O. Cepas, C.M. Fong, P.W. Leung, and C. Lhuillier, arXiv:0806.0393
![Page 109: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/109.jpg)
MOMENT-FREEPHASE(S)
pressure?
ZnCu3(OH)6Cl2
D = 0.08J (from ESR)Zorko et al. PRL 2008
Quantum critical point at DQuantum critical point at Dc c !! 0.1J 0.1J
m2AF (T=0)T
???
D/JVBC, spin liquid?Singh and Huse 2007Budnik and Auerbach 2004Syromyatnikov and Maleyev 2004Nikolic and Senthil 2003Marston and Zeng 1991 algebraic spin liquidHastings 2000; Ran et al. 2007;Ryu et al. 2007; Hermele et al. 2008
Power law behaviour?• Algebraic spin liquid• Disorder : Rozenberg and Chitra, PRB 2008• or proximity to this quantum critical point ???
NÉEL PHASE
Olivier Cépas, C.M. Fong, P.W. Leung, and C.Lhuillier, cond-mat/0806.0393v2
QUANTUMCRITICAL?
Chakravarty, Halperin, and Nelson 1989 ;Sachdev’s Quantum Phase Transitions(Cambridge University Press)
vs. SWT : Dc = 0.03J Elhajal et al. 2002
ASL : Dc = 0 Hermele et al. 2008
![Page 110: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/110.jpg)
Three-dimensional S=1/2 Frustrated Magnet has a Hyper-Kagome sublattice of Ir ions
Pyrochlore Hyper-Kagome
3/4 Ir, 1/4 Na
Ir Na
Ir 4+ (5d ) 5
All Ir-Ir bonds are equivalent
carries “S=1/2” moment ?
Y. Okamoto, M. Nohara, H. Agura-Katrori, and H. Takagi, PRL 99, 137207 (2007)
![Page 111: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/111.jpg)
Inverse Spin Susceptibility; Strong Spin Frustration
No magnetic ordering down to
Curie-Weiss fit
Large Window of Cooperative Paramagnet
Y. Okamoto, M. Nohara, H. Agura-Katrori, and H. Takagi, PRL 99, 137207 (2007)
Strong Frustration - Macroscopic degeneracy of classical ground states
![Page 112: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/112.jpg)
Theory of Spin Liquid with Gapless Fermionic Spinons
J = 304K obtained from ED and Gutzwiller factor g=3
(can only fix the overall scale, but it is a zero-parameter fit !)
C/T looks linear for 5K < T < 20KM. J. Lawler, A. Paramekanti, Y. B. Kim, L. Balents, arXiv:0806.4395Y. Zhou, P.A. Lee, T.-K. Ng, F.-C. Zhang, arXiv:0806.3323
![Page 113: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/113.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 114: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/114.jpg)
1. Landau-Ginzburg criticality Coupled-dimer antiferromagnets
2. Quantum “disordering” magnetic order Z2 spin liquids and valence bond solids
3. Critical spin liquids Deconfined criticality; fermionic spinons near the Mott transition
4. Triangular, kagome, and hyperkagome lattices Connections to experiments
[[[ 5. Correlated boson modelSupersolids and stripes ]]]
Outline
![Page 115: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/115.jpg)
D. Heidarian and K. Damle, Phys. Rev. Lett. 95, 127206 (2005)
R. G. Melko, A. Paramekanti, A. A. Burkov, A. Vishwanath, D. N. Sheng and L. Balents, Phys. Rev. Lett. 95, 127207 (2005)
S. Wessel and M. Troyer, Phys. Rev. Lett. 95, 127205 (2005)
Bosons with repulsive interactions on the triangular lattice near half-filling
![Page 116: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/116.jpg)
R. G. Melko, A. Del Maestro, and A. A. Burkov, Phys. Rev. B 74, 214517 (2006)
Bosons with (longer range) repulsive interactions on the triangular lattice near half-filling
Striped supersolid with very small anisotropy in superfluid stiffness
![Page 117: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/117.jpg)
Open questions: phase transitions with Fermi surfaces/points
Slave-particle gauge theories from conventional phases with Fermi surfaces/points lead to exotic phases with “ghosts” of Fermi surfaces/points
Is it possible to have direct transitions between conventional Fermi liquid phases but with distinct topologies of Fermi surfaces/points ?
![Page 118: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/118.jpg)
Increasing AFM order
Large Fermi surface metal
AFM Metal
A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997)
Spin density wave theory for Fermi surface evolution in the hole-doped cuprates
![Page 119: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/119.jpg)
Increasing AFM order
Large Fermi surface metal
AFM Metal
A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997)
Spin density wave theory for Fermi surface evolution in the hole-doped cuprates
Hole pocket
![Page 120: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/120.jpg)
Increasing AFM order
Large Fermi surface metal
AFM Metal
A. V. Chubukov and D. K. Morr, Physics Reports 288, 355 (1997)
Spin density wave theory for Fermi surface evolution in the hole-doped cuprates
Electron pocket (seen in high magnetic fields ?)
![Page 121: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/121.jpg)
Increasing AFM order
Large Fermi surface metal
AFM Metal
Direct transition between small and large Fermi surfaces ?
![Page 122: Exotic phases and quantum phase transitions in model systems](https://reader031.vdocument.in/reader031/viewer/2022012509/6186ac75cebcb90a745c98b9/html5/thumbnails/122.jpg)
Direct transition between small and large Fermi surfaces ?
Is there a critical theory for such a transition ?
Does such a critical point/phase control strange metal behavior in the cuprates ?
Beyond slave particle gauge theories: help from the AdS/CFT correspondence ? (Sung-Sik Lee, arXiv:0809.3402)