Spectral functions from the time dependent density
matrix renormalization group
• DMRG
– Low entanglement approach
– Curent capabilities
– tDMRG
• How to use tDMRG for spectral functions
• S=1 Heisenberg chain, S=1/2 XXZ chain
collaborators:
Adrian Feiguin, Ian Affleck, Rodrigo
Pereira, Sasha Chernyshev
|ψ! = Statistical
ψij |i!|j! Mechani
Reference:
Lectures R.P. Feynman,
ij
Lectures
OriginsLetof|i!DMRG--RG,
Stat
Mech
be the states of the
block
(the system), a
The density matrix is
the|i!states
of the
rest
ofofthe
lattice
(the
restsystem
of the u
!
Let
be the
states
the
block
(the
• RG: throw
away
unimportant
states,
effective
H
in
∗
!
ρ
=
ψ
ψi! j
ii
ij
If
ψ
is
a
state
of
the
entire
lattice,
states
the rest
of the lattice
(the
rest
of
t
truncatedthe
basis
(Kenof
Wilson,
“NRG”)
j
!
ψ is a state
of the |ψ!
entire
lattice,
• StatisticalIfMechanics
Viewpoint
(Feynman
SM
lectures)
=
ψsystem,
|i!|j!
ij
If operator A acts only on the
!
! ij
! iij
"A!|ψ!
= = Aii! ρiψ
=|i!|j!
TrρA
Rest of the
Universe: |j>
The density matrix is
System |i>
ii!
ij
!
∗
Let
ρ
have
eigenstates
|v
!
and
eigenvalues wα
α
!
ρ
=
ψ
"
ii
ij ψi! j
The density
matrix
is
( α wα = 1). Then
j
!
m
!
∗
!
ρ
=
ψ
"A!iion
= thewsystem,
ij ψi! j
α "α|A|α!
If operator A acts only
! α=1j
"A! =
Aii! ρi! i = TrρA
If for
a particular
α, wαprobability
≈ 0,the
we system,
make no error in "A
If operator
A acts
on
• Key idea: throw
away
eigenstates
withonly
small
ii!
|vα !. matrix
One can
also
make no error i
!show wegroup
• Algorithm based ondiscard
this: density
renormalization
Let
eigenstates
|vαA!ii!and
eigenvalues
w
!i =
"A!
=
ρ
TrρA
i
"If ρthehave
(DMRG, srw(1992))
rest of the universe is regarded as a “heat b
1). Then β toii!which the system is weakl
( inverse
α wα =temperature
!
DMRG as a low entanglement approximation
• Vidal,Verstraete, Cirac: DMRG and QI entangled.
• Entanglement: Which is more entangled?
– 1) |↑↑> + |↓↓> or
– 2) |↑↑> + |↓↓> + |↑↓> + |↓↑> ??
• Answer: 1) is perfectly entangled. 2) is unentangled:
– (|↑> + |↓>) (|↑> + |↓>)
• To measure entanglement, must change to the Schmidt
basis where the Ψ is diagonal: Ψ= U D V
• Density matrix eigenvalues are square of D !
– ρ=U D2 U✝
• Now QI is supplying many ideas to DMRG!
DMRG Algorithm
• Finite system method:
Extensions
–
2D
and
Fermion
S
• Wavefunction = matrix product state(Ostlund & Rommer,1995)
s1 s 2
Tr{A1 A2
ψ(s1 , s2 , . . .) =
• 2D: map onto chain
. .(Noack,
.} White, Scalapino, 1994)
– Accuracy falls of exp’ly in width
system block
environment block
• convergence depends strongly on wi
Current capabilities
S=1/2, S=1 chains
Near exact
Very oversimplified
graph
Effort
Dynamics
(resolution)
Triangular
Heisenberg
Static
1D Dimensionality
(Ly)
2D
Third axis (not shown):
complexity of model
• Computational scaling to 2D
– TV-scan 1D DMRG cpu ~ Lx Ly m3, m ~ exp(a Ly)
– Projected entangled pair states (PEPS, tensor prod states):
• cpu ~ Lx Ly m10, m ~ constant ~ 10 (x 1000’s of iterations)
Heisenberg square lattice
0.45
•
•
•
•
Tilted lattice has smaller DMRG errors for its width
For this cluster obtain M = 0.3052(4) in center
“Exact” 2D M = 0.307 (Sandvik, QMC)
Standard PBC cluster, this size, M = 0.34
Heisenberg square lattice
Ly=5
Ly=6
Ly=7
Ly=8
0.308
Sandvik QMC
(error bars)
0.306
0.304
New finite size
scaling approach
using cylindrical
BCs
1.85
1.9
1.95
2
Lx/Ly
See White and Chernyshev, PRL 99, 127004 (2007).
Triangular Heisenberg, pinned cylindrial BCs
0.4
Triangular Lattice
17.3 x 9
lattice
<S z >
0.35
Pinning
fields
• Only one sublattice pinned, other two rotate in a cone
• Other two have z component -M/2
• Here only have L y= 3, 6, 9, ...
Triangular lattice, Scaled Data
0.35
Ly= 3
M
0.3
Ly= 6
Ly= 9
0.25
0.2
0.15
0.5
1
1.5
!
2
2.5
3
Results are consistent with, and as good as best GFMC and series
expansions.
See White and Chernyshev, PRL 99, 127004 (2007).
Back to 1D: Time Evolution (Vidal,...)
Suzuki Trotter decomposition:
exp(-iHτ) ≈ exp(-iH12τ) exp(-iH34τ) ... exp(-iH23τ) ...
Key
idea: adapt basis to each instant of time
Each sweep = one time step
Fourth order breakup--negligible time step errors
Growth of entanglement with t: stop after moderate time
Calculation of Spectral functions
• Start with standard ground state DMRG, get φ
• Apply operator to center site
|ψ(t = 0)! = S0+ |φ0 !
• Time evolve:
|ψ(t)! = e−i(H−E0 )t |ψ(0)!
• Measure time dependent correlation function
G(x, t) = !φ0 |Sx− |ψ(t)" = !φ0 |Sx− (t)S0+ (0)|φ0 "
• Fourier transform with x=0 to get N(ω) or in x and t
to get S(k,ω)
– But what about finite size effects, finite time, broadening,
etc??
Finite size effects: gapped systems
S=1 Heis chain
Real and Imag parts
t=14
0.4
z
<S (x)>
G(x,t=10)
0.2
-0.2
t=2
0.5
t=0
-0.4
0
50
100
x
0
15
-50
0
x
For t < L/(2 v), finite size effects are negligible.
50
Finite size effects: gapless systems
For t < L/(2 v), finite size
effects are negligible in the
imaginary part. Fourier
transform can be done just on
the imaginary part (then throw
away negative freq part).
S=1/2 Heisenberg, L=400
0.02
Real
Imag
0
-0.01
-0.02
S=1/2 Heisenberg model
0.2
L=100, Re
L=100, Im
L=400, Re
L=400, Im
L=800, Re
L=800, Im
-0.03
0.1
-0.04
-200
-100
0
x
100
200
G(k=π/2,t)
G(x,t=10)
0.01
0
-0.1
-0.2
0
5
10
t
15
20
Errors in time evolution
S=1 chain
4
!tot
|"S(x=0,t)|
|"S(x=1,t)|
5
3
Error x 10
Conclusion: We can obtain
very accurate data representing
the thermodynamic limit up to
t=20-40. Beyond that the
numerical work grows rapidly.
2
Note: one run gives all k and ω
(but need to worry about
broadening in ω).
1
0
0
5
t
10
Errors vary as total summed
truncation error. Here a
specified truncation error was
specified and m slowly grew.
Extrapolation to large time: linear prediction
S=1 chain
1
0.8
Windows
for time
FT
0.6
0.4
0
-3
-2
-1
0
1
t
2
N(!)
0.2
3
2
tmax = 20
Extrapolated for t > 10
DMRG
0.1
zz
2nd order
4th order
cos window
1
0.2
Re S (x=0,t)
3
yi =
0
n
!
0
dj yi−j
0
2
!
4
j=1
-0.1
0
5
10
t
15
20
2
See Numerical Recipes
Parameters dj determined from correlation
functions of available data.
6
Extrapolation in time: fitting singularities
• The large time behavior is usually very simple,
determined by a few singularities in the spectrum often
giving oscillating power law decay.
• We can fit the larger time data to asymptotic forms
and extend the data to very large times.
• Example: N(ω) for S=1 chain: inverse square root
singularities come from top and bottom of magnon
band
Fitting edge singularities
The following identity fits lower edge singularities:
!
∞
−∞
dωe−iωt θ(ω − b)e−a(ω−b) (ω − b)g = Γ(1 + g)e−ibt (a + it)−1−g
Only the r.h.s. is used: the fit is used to extend the data, then we FT
Simulated annealing fit:
1
0.77848e−0.41043it (0.5513 + it)−1/2 +
0.29358e−2.7249it (−3.2282 − it)−1/2
G(x=0,t)
0.5
Deviations are ~10-4
Value for Haldane gap is correct to 4 places!
0
-0.5
0
5
10
15
S=1, Re, Im, fit
20
N(ω) for S=1 chain, asymptotic fit
10
6
10-20
5-15
3 magnon
5
4
"/#
N(!)
2 magnon
5
3
2
1
0
0
0
1
2
!
3
4
5
0
kc
Detailed results for S=1: see White & Affleck,
PRB, 2008
0.5
k/!
1
S=1/2 XXZ model (Pereira,White, Affleck, Nov. PRL)
• Physics very nicely described by Alex Kamenev yesterday.
• Singularities in spectrum: look at p-h excitations in spinless
fermion model from the Jordan Wigner transfromation
– p, h both near kF: low energy, LL description. In N(ω),
describes lower edge singularity at ω=0
– p and/or h away from kF: need to go beyond usual treatment
• New treatment: p/h away from kF have few decay modes:
treat as mobile impurities in LL (Balents, Pustilnik, Khodas, Kamenev, Glazman)
– Bethe ansatz to exactly determine the key couplings
– Similar approach to Cheianov and Pustilnik, small disagreement
• Result: exact exponents for all singularities in N(ω)
and S(k,ω)
D
M
R
G
e−iW t
e−i2W t
B3
B4
G(t) ∼ B1 η + B2 η2 + σ + 2
t
t
t
t
x=0
S=1/2 Chain, XXZ model
0.4
Jz=0.125
Jz=0.25
Jz=0.375
Jz=0.5
N(!)
0.3
0.2
0.1
0
0
1
!
2
How accurate are the spectra?
0.4
Jz = 0.5
L=200
L=400
N(ω)
0.3
0.2
0.1
0
0
0.5
1
ω
1.5
2
2.5
S(k,ω)
S(q="/2,!)
3
# = $0.25
#=0
# = 0.125
# = 0.25
# = 0.5
2
1
0
0.8
1
1.2
!
1.4
1.6
1.8
Conclusions
• DMRG can do big enough systems for some models to
extrapolate order parameters to the thermodynamic
limit
• tDMRG allows precise determination of spectral
functions for 1D systems
• New results for singularity exponents in the XXZ
model