Quantum Critical Scaling of Fidelity in 2D BCS-like models
Mariusz Adamski and Janusz Jedrzejewski
˛
Institute for Theoretical Physics, University of Wrocław
Results
Introduction
Using
ground-state fidelity
dν < 2
(A) Critical endpoint:
103
105
δ =10−6
101
10-1
c =0
c =1
c =2
c =0
c =1
c =2
102
∼ L2
∼ δ1
10-1
10-3
−lnF
Our objectives
verifying known scaling laws in small- and macroscopic systems in the
range of their validity (dν < 2)
comparing exponents of power-like behaviour with obtained earlier critical
indices beyond that range
L =103
−lnF
We are investigating
2D spinful fermion model
BCS-like interactions
Quantum critical points
10-5
Consistent10-4with
ν = 1/2
∼ L4
10-7
ξ
10-9 1
10
102
104
103
L
105
10-7 -9
10
The model
δ(ξ = L)
10-7
10-6
10-5
δ
108
2|J|
t †
†
H =−
al,σ al+ei,σ + h.c. − µal,σ al,σ +
2
l,σ,i
X
J
† †
† †
−
al,↑al+ei,↓ − al,↓al+ei,↑ + h.c.
2
105
see ref. [1]
1
2
where σ ∈ {↑, ↓}, i ∈ {1, 2} and ei are unit vectors along the axes.
k,i
∼ L2
∼ L ≈ 2.307
∼ L4
10-4 1
10
Pairing interaction
εk = − cos k1 − cos k2 + µ.
ξdiag
104
103
L
√ 1 + J2
≈ 2
,
|µJ|
∼ δ2
10-1
105
∼ δ ≈ 1.146
δ(ξdiag = L)
δ(ξoffdiag = L)
10-4 -9
10
10-8
10-7
10-6
√ √
2 4 1 + J2
ξoffdiag ≈ p
2 |µ| sin
Fails
10-5
δ
10-4
105
101
105
L =103
c =0
c =1
c =2
∼ L2
−lnF
−lnF
ξ ≈ 106
∼ L4
10-5
Wrongly suggests
10-4
ν = 1/2
10-7
Phase diagram
10-9 1
10
102
J
L
10-8
10-7
δ(ξ = L)
10-6
10-5
δ
10-4
p
2 − |µ|
≈
|µJ|
see ref. [1]
L =103
c =0
c =1
c =2
105
L2
∼ L ≈ 2.285
−lnF
Wrongly suggests
ν=1
10-4
∼ L2
102
νdiag = 1/2
νoffdiag = 1/2
104
103
L
∼ δ1
10-3
Wrongly suggests
ν = 1/2
10-7
105
10-11 -9
10
√
ξdiag
Critical lines
c =0
c =1
c =2
101
10-1
µ
10-2
dν > 2
−lnF
(A)
10-3
109
102
10-7 1
10
νdiag = 1
νoffdiag = 1
ξoffdiag
δ =10−4
2
(D)
√ p
2 4 − µ2
≈
,
2 |µJ|
10-7 -9
10
108
105
(C)
105
∼ δ2
(D) Multicritical point:
νdiag = 1
νoffdiag = 1/2
νdiag = 2
νoffdiag = 3/2
104
103
ξdiag
(B)
∼ δ1
10-1
10-3
[1]
c =0
c =1
c =2
102
10-1
i
10-2
dν = 2
δ =10−6
103
10-3
see ref. [1]
1
2 arctan |J|
(C) Across J = 0 critical line, µ = 1:
Dispersion relation after Bogoliubov transformation
v
2
u
X
u
t
2
Ek = ε k + J
cos ki
Multicritical point
∼ δ ≈ 0.858
102
Correctly identifies
νoffdiag = 1/2
102
ξdiag
c =0
c =1
c =2
105
ξoffdiag
Momentum-space Hamiltonian
X †
X
†
†
εkckσ ck,σ − J
cos ki ck,↑c−k,↓ + h.c.
H =
k,σ Kinetic term
c =0
c =1
c =2
−lnF
10-1
dνoffdiag < 2
L =104
102
l,i
10-2
108
δ =10−3
10-3
−lnF
X
10-4
2
(B) Across µ = 0 critical line, J = 1:
Real-space Hamiltonian
−2
10-8
√
ξdiag ≈ ξoffdiag ≈
∼ δ2
2
≈
,
|µJ|
10-8
√
ξoffdiag ≈ p
∼ δ2
10-7
δ
2
10-6
10-5
see ref. [1]
|µ||J|
Observations
As long as the condition dν < 2 holds, the scaling laws are satisfied,
correctly reproducing critical indices
If dν ≥ 2, the exponents characterising power-like behaviour of − ln Fe do
not match critical indices obtained from correlation functions
Fidelity
Definition
Conclusions
− + Fe(λ, δ) = hg(λ − eδ)|g(λ + eδ)i ≡ g g Small system (L ξ ) [2, 3, 4]
Macroscopic system (L ξ )[2, 3, 5]
− ln Fe(λc + cδe, δ) ∼ L2/ν δ 2
− ln Fe(λc + cδe, δ) ∼ Ldδ dν Ae(c)
In both cases
dν < 2
In our case
− ln Fe = −
X
Typically, in systems with dimension d ≥ 2, the ground-state fidelity does not
allow for determining critical indices.
ln f (k)
k
− +
− +
2
εk εk + J J cos k
1
f (k) =
1+
2
Ek−Ek+
Scaling function
References
[1] Mariusz Adamski, Janusz Jedrzejewski,
˛
and Taras
Krokhmalskii.
Quantum phase transitions and ground-state
correlations in BCS-like models
[arXiv:1311.1080].
[3] Marek M. Rams and Bogdan Damski.
Quantum fidelity in the thermodynamic limit
Phys. Rev. Lett. 106, 055701 (2011).
[4] A. Fabricio Albuquerque, Fabien Alet, Clément Sire,
and Sylvain Capponi.
[2] Marek M. Rams and Bogdan Damski.
Quantum critical scaling of fidelity susceptibility
Scaling of ground-state fidelity in the thermodynamic
Phys. Rev. B 81, 064418 (2010).
limit: XY model and beyond
[5] Lorenzo Campos Venuti and Paolo Zanardi.
Phys. Rev. A 84, 032324 (2011).
Quantum critical scaling of the geometric tensors
Phys. Rev. Lett. 99, 095701 (2007).
50th KARPACZ WINTER SCHOOL OF THEORETICAL PHYSICS , 2-9 MARCH 2014, KARPACZ, POLAND
Mail:
[email protected]
© Copyright 2024 ExpyDoc
