Download Quantum Phase Transitions - The Quantum

General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Phase Transitions - The
Quantum-Classical Mapping
Daniel Lechner
University of Vienna
13.6.2014
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
1
General Aspects
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
2
Quantum Ising Chain in a Transverse Field
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
3
References and Backup Slides
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
1. General Aspects
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
Quantum Fluctuations and Quantum Critical Point
Quantum phase transition: non-analyticity in the ground state
energy
QPTs not driven by thermal fluctuations
Quantum fluctuations = Heisenberg uncertainity fluctuations
For T 6= 0: Quantum fluctuations always overpowered by
thermal fluctuations (close to critical point)
Consequence
Quantum phase transitions can only take place at T = 0!
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
Quantum Critical Point
Hamiltonian often of the form:
Ĥ = Ĥ0 + g Ĥ1
Tuning of g across a critical value g c at T = 0 induces a
quantum phase transition
Quantum critical point:
g = gc
Daniel Lechner
at
Tc = 0
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
A Quantum Phase Diagram
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
Quantum-Classical Mapping
Exact spectrum Ĥ(g) |ψi = (g) |ψi almost never accessible
Especially powerful for investigating QPTs: QC-Mapping
D-dim. quantum model usually maps onto (D+1)-dim.
classical model
(imaginary) time → additional spatial direction
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
Strategy of QC-Mapping
1
Map to classical model
2
Solve classical model
3
Find the classical critical point(s)
4
Identify quantum critical point(s) (QCP) via map
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Quantum Fluctuations and Quantum Critical Point
A Quantum Phase Diagram
Quantum-Classical Mapping
2. Quantum Ising Chain in a Transverse Field
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Hamiltonian
Quantum Ising chain in a transverse field
Ĥ = −J
N
P
i=1
z σ̂ z − h
σ̂i+1
i
Note: HN = H1 ⊗ H1 ⊗ ... ⊗ H1 =
N
P
i=1
N
N
σ̂ix
H1
i=1
Ferromagnetic part:
Ĥ0 = −J
N
P
i=1
z σ̂ z
σ̂i+1
i
Paramagnetic part:
Ĥ1 = −h
N
P
i=1
Daniel Lechner
σ̂ix
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Limiting Cases - h = 0
h = 0 → ferromagnetic part
Ĥ → Ĥ0 = −J
N
P
i=1
z σ̂ z
σ̂i+1
i
Ground state:
|{S0z }i ≡
N
Q
i=1
|↑ii
Daniel Lechner
or
|{S0z }i ≡
N
Q
i=1
|↓ii
Quantum Phase Transitions - The Quantum-Classical Mapping
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian:
H ≡ h{S z }| Ĥ |{S z }i = −J
N
P
i=1
z Sz
Si+1
i
Partition function:
Z→
P
βJ
e
N
P
i=1
z Sz
Si+1
i
=
{Siz }
= (2 cosh(βJ))N + (2 sinh(βJ))N
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Limiting Cases - J = 0
J = 0 → paramagnetic part
Ĥ → Ĥ1 = −h
N
P
i=1
σ̂ix
Ground state:
|{S0x }i ≡
N
Q
i=1
|→ii
where
Daniel Lechner
|→i =
√1
2
(|↑i + |↓i)
Quantum Phase Transitions - The Quantum-Classical Mapping
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian:
H ≡ h{S x }| Ĥ |{S x }i = −h
N
P
i=1
Six
Partition function:
Z→
P
N
P
βh
e
i=1
Six
= (2 cosh(βh))N
{Six }
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Quantum-to-Classical Mapping
General situation: J 6= 0, h 6= 0
Goal: Partition function
h
i
h
i
Z = T r e−β Ĥ = T r e−β Ĥ0 −β Ĥ1
i.e.
Z=
P
N
N
P
P
z σ̂ z +βh
βJ
σ̂i+1
σ̂ix
i
z
i=1
h{S }| e i=1
|{S z }i
{S z }
Problem:
[Ĥ0 , Ĥ1 ] 6= 0
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Trotter Decomposition
Trotter´s theorem
e−β(A+B) = s − lim
L→∞
β
β
e− L A e− L B
L
for bounded-from-below, self adjoint operators A,B.
Exponential inside the trace:
L L
β
β
e−β Ĥ0 −β Ĥ1 ≈ e− L Ĥ0 e− L Ĥ1
≡ e−δτ Ĥ0 e−δτ Ĥ1
δτ =
β
L
Note: ≈ becomes = in the limit L → ∞
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Applying Trotters formula to the partition function:
P
h{S z }| e−β Ĥ1 −β Ĥ0 |{S z }i =
Z=
{S z }
=
P
h{S z }|
{S z }
L Q
e−δτ Ĥ1 e−δτ Ĥ0 |{S z }i
l=1
Insert complete sets
"
#
QN
P
P
z
z
1 = i=1
|Si i hSi | ≡
|{S z }i h{S z }|
Siz =±1
Daniel Lechner
{S z }
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Hence:
P QL Z=
l=1
z }
{Si,l
z } e−δτ Ĥ0 e−δτ Ĥ1 |{S z }i
{Sl+1
l
Ĥ0 acts on eigenstates:
z −δτ Ĥ −δτ Ĥ
1e
0 |{S z }i =
{Sl+1 } e
l
δτ J
=e
N
P
i=1
z
z
Si+1,l
Si,l
Daniel Lechner
z −δτ Ĥ
1 |{S z }i
{Sl+1 } e
l
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
The remaining matrix element is:
E
N D
z −δτ Ĥ
Q
δτ hσ̂ix z
z
1 |{S z }i =
{Sl+1 } e
S
e
Si,l
l
i,l+1
i=1
Now use the identity
E
D
δτ hσ̂ix z
γS z
Sz
z
Si,l+1
e
Si,l = Λe i,l+1 i,l
where:
γ = − 12 ln [tanh (δτ h)],
Daniel Lechner
Λ2 = sinh (δτ h) cosh (δτ h)
Quantum Phase Transitions - The Quantum-Classical Mapping
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Matrix element:
=
z } e−δτ Ĥ1 e−δτ Ĥ0 |{S z }i =
{Sl+1
l
ΛN e
N
P
δτ J
i=1
z
z +γ
Si+1,l
Si,l
N
P
i=1
z Sz
Si,l
i,l+1
Substituting leads to the
Partition function of the QIC
Z=
ΛN L
δτ J
P
e
N P
L
P
i=1 l=1
z
z +γ
Si,l
Si+1,l
N P
L
P
i=1 l=1
z Sz
Si,l
i,l+1
z }
{Si,l
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Comparison with Classical 2D Model
Partition function of the QIC
Z=
ΛN L
δτ J
P
e
L
N P
P
i=1 l=1
z
z +γ
Si+1,l
Si,l
L
N P
P
i=1 l=1
z Sz
Si,l
i,l+1
z }
{Si,l
Partition function of the classical 2D Ising model
β cl J x
P
Z=
e
N
Px N
Py
Si+1,j Si,j +β cl J y
i=1 j=1
N
Px N
Py
Si,j Si,j+1
i=1 j=1
{Si,j }
Identify:
δτ J = β cl J x
γ = β cl J y
Daniel Lechner
N = Nx
L = Ny
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Comparison with Classical 2D Model
Classical model shows PT in the thermodynamic limit
(Nx → ∞, Ny → ∞)
Kramers-Wannier duality:
sinh 2J x βccl sinh 2J y βccl = 1
Quantum case:
sinh (2δτ Jc ) sinh (2γc ) = 1
Remember: δτ =
β
L
!
L→∞
=⇒
Daniel Lechner
β → ∞ = βc
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Quantum Critical Point
Criticality condition:
sinh(2δτ Jc )
sinh(2δτ hc )
=1
ı.e.
hc = Jc
It is more conventional to set h = gJ and keep J fixed:
Critical point of the QIC
gc = 1
at
Daniel Lechner
Tc =
1
βc
=0
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Summarizing
Starting point: Hamiltonian with non-commuting parts Ĥ0
and Ĥ1
Trace evaluated in product states |{S z }i
Trotter decompositon of the density matrix
Evaluation of the matrix elements (trivial part with σ̂ z ,
non-trivial part with σ̂ x )
Result: partition fct. of 2D classical Ising model
Kramers-Wannier duality → critical point
Map → quantum critcal point
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Phase Diagram of QIC
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Hamiltonian
Limiting Cases
Quantum-to-Classical Mapping
Phase Diagram
Schematic Phase Diagram of 2D Quantum Ising Model
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
References
Sachdev, Subir (1999) Quantum Phase Transitions.
Cambrigde University Press
Batrouni C.G.. and Scalettar, R.T.. Quantum Phase
Transitions. Oxford University Press
Vojta, M. (2002) Physik Journal, 1, Nr. 3
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
3. Backup Slides
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Operator Convergence An → A
Norm convergence: A = lim An
n→∞
⇔ lim kAn − Ak = 0,
n→∞
kAk = sup
where
ψ∈D(A)
kAψk
kψk
Strong convergence: A = s − lim An
n→∞
⇔ lim k(An − A)ψk = 0
n→∞
Daniel Lechner
∀ψ ∈ D(A) ⊆ H
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Dual Operator Set
Pauli matrices obey:
i
h
i h
i h
σ̂ix , σ̂jy = σ̂iy , σ̂jz = σ̂iz , σ̂jx = 0
(i 6= j)
{σ̂ix , σ̂iy } = {σ̂iy , σ̂iz } = {σ̂iz , σ̂ix } = 0
(i = j)
(σ̂ix )2
=
2
(σ̂iy )
=
(σ̂iz )2
=1
New set of operators that obey the same algebra:
Q x
z ,
τ̂ix = σ̂iz σ̂i+1
τ̂iz =
σ̂k
k≤i
Subsitution
−J
N
P
i=1
(σ̂ix , σ̂iz )
→
z σ̂ z − h
σ̂i+1
i
Self-duality for Jc = hc
(τ̂iz , τ̂ix ):
N
P
i=1
σ̂ix → −h
N
P
i=1
z τ̂ z − J
τ̂i+1
i
N
P
i=1
τ̂ix
(h = gJ ⇒ gc = 1)
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Correlation Function
Dynamic Correlation Function:
Using:
S (~x, ~y , t) = h0| O† (~x, 0) O (~y , t) |0i =
P
O (~x, t) = e−iĤt O (~x, 0) eiĤt ,
|mi hm|, Fourier
m
transforming and going to imaginary time: t → −iτ
2
P
S(~k, τ ) = e−(Em −E0 )τ h0| O(~k, 0) |0i ≈
m
2
2
≈ h0| O(~k, 0) |0i + e−(E1 −E0 )τ h0| O(~k, 0) |0i
Comparison with e−τ /ξτ shows that time-correlation length ξτ
diverges for vanishing ground-state energy gap
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Exact Spectrum
It can be shown:
HI =
P
k
k γk† γk − 1/2
(k) = 2J 1 + g 2 − 2g cos(k)
1/2
From this follows:
(0) = 2J |g − 1| = 2J |g − gc | ⇒ gc = 1
Correlation length:
ξτ ∼ |g − gc |−1 ∼ ∆−1
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Mean Field Analysis
General d-dimensional Quantum Ising Hamiltonian:
Ĥ = −J
P
hi,ji
hi, ji
...
σ̂iz σ̂jz − gJ
N
P
i=1
σ̂ix
sum over nearest neighbours
Use Mean field approximation
σ̂iz = mz + δσ̂iz ,
where
mz = hσ̂iz i =
h
i
T r σ̂iz e−β Ĥ
T r[e−β Ĥ ]
Mean field Hamiltonian (γ = 2d):
ĤM F = γJ mz2N − γJ
Daniel Lechner
N
P
i=1
(mz σ̂iz + gσ̂ix )
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Partition function:
iN
h
p
2
ZM F = e−βN Jγmz /2 2 cosh βγJ m2z + g 2
1
Landau function LM F = − βN
ln(ZM F ):
h
i
p
2
1
z
2 + g2
LM F = γJm
−
ln
2
cosh
βγJ
m
z
2
β
Self-consistency condition
p
p
!
∂LM F !
2 + g2 =
2 + g2
=
0
⇒
m
tanh
βγJ
m
z
z
∂mz
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Important: trivial solution mz = 0 was discarded!
To obtain the critcal coupling, one now takes the limit
mz → 0:
|gc | = tanh (βc γJ |gc |)
Criticality condition
Tc
γJ
=
Daniel Lechner
|gc |
artanh(|gc |)
Quantum Phase Transitions - The Quantum-Classical Mapping
General Aspects
Quantum Ising Chain in a Transverse Field
References and Backup Slides
Mean Field Quantum Phase Diagram
Daniel Lechner
Quantum Phase Transitions - The Quantum-Classical Mapping