correlations and order parameters in infinite matrix
TRANSCRIPT
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Correlations and order parameters in infinite matrixproduct states
Ian McCulloch
University of QueenslandSchool of Mathematics and Physics
April 22, 2021
Ian McCulloch (UQ) iMPS April 22, 2021 1 / 31
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Outline
1 Approaches to detecting quantum criticality
2 Cumulants
3 ExamplesIsing modelPotts model
4 Generalising beyond local order parametersstring order parametersSPT and time reversal
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Approaches to detecting criticality
Bipartite fluctuationsBipartite entropyEntanglement spectrumBipartite fluctuations in quantum numbers (eg J. Stat. Mech. (2014) P10005from Karyn le Her’s group; Kjall, Phys. Rev. B 87, 235106 (2013))
Transfer matrix spectrumEigenvalues λi
λ0 = 1 by construction (normalization condition)Spectrum of correlation lengths ξi = − 1
lnλi
Or consider εi = − lnλi = 1/ξi
– Behaves like an energy scaleChoice of which quantity to use as the scaling variable
Scaling with respect to the bond dimensionScaling with respect to the correlation length – diverges at critical pointScaling with respect to δ = ε2 − ε1 (or some other combination of εi)
Order parameter and order parameter fluctuations (higher moments)Lots of history behind finite-size scaling, can be modified forfinite-entanglement scalingJ. Pillay and IPM, Phys. Rev. B 100, 235140 (2019)
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=5
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=6
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=13
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=16
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=20
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=25
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=30
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0.98 0.99 1 1.01 1.02lambda
0
0.1
0.2
0.3
0.4
0.5
1/l
ength
Inverse correlation length spectrum m=40
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Critical scaling exampleTwo-species bose gas with linear tunneling Ω, from F. Zhan et al, Phys. Rev. A 90, 023630 2014
1
10
100
50 100 200
Ω = 0.2148Ω = 0.215Ω = 0.2152Ω = 0.2154Ω = 0.2156Ω = 0.2158
ξ
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Higher moments
It is straightforward to evaluate a local order parameter, eg
M =∑
i
Mi
The first moment of this operator gives the order parameter,
〈M〉 = m1(L)
It is also useful to calculate higher moments, eg
〈M2〉 = m2(L)
or generally
〈Mk〉 = mk(L)
These are polynomial functions in the system size L.
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Cumulant expansions
Express the moments mi in terms of the cumulants per site κj,
m1(L) = κ1Lm2(L) = κ2
1L2 + κ2Lm3(L) = κ3
1L3 + 3κ1κ2L2 + κ3Lm4(L) = κ4
1L4 + 6κ21κ2L3 + (3κ2
2 + 4κ1κ3)L2 + κ4L
κ1 is the order parameter itselfκ2 is the variance (related to the susceptibility)κ3 is the skewnessκ4 is the kurtosis
The cumulants per site κk are well-defined for a translationally invariantiMPSThe moments (and hence cumulants) can be obtained directly as apolynomial-valued expectation value (arXiv:1008.4667)
(I don’t know of a good way to calculate the cumulants per site for a finitesystem!)
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The cumulants have power-law scaling at a critical point, κi ∝ mαi
Ising model example
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Scaling functionsFor finite systems, scale with respect to system size LControl parameter t ≡ λ−λc
λch (external field)
Critical exponentsExponent relationν ξ ∝ |t|−νβ 〈M〉 ∝ (−t)β
γ∗ σ2 = 〈M2〉 − 〈M〉2 ∝ |t|−γ∗
Scaling relation 2β + γ∗ = ν
Note: no fluctuation-dissipation theorem: γ∗ 6= γ is different to dM/dH.In d = 2 classical systems, 2β + γ = νd
Finite-size scaling functions
ξ = L X (t L1/ν)〈M〉 = L−β/νM(t L1/ν)
σ2 = Lγ∗/ν G(t L1/ν)
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Finite entanglement
Effective ’system size’ L→ mκ
New scaling functions that scale with t mκ/ν .
Finite-entanglement scaling functions
ξ = mκ X (t mκ/ν)〈M〉 = m−βκ/νM(t mκ/ν)σ2 = mγκ/ν G(t mκ/ν)
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Cumulant exponent relationFollowing V. Privman and M.E. Fisher, Phys. Rev. B 30, 322 (1984)
Singular part of the free energy:
f ' L−d Y(
C1tL1/ν ,C2hL(β+γ)/ν)
Finite-entanglement version:
f ' m−κd Y(
C1tmκ/ν ,C2hm(β+γ)κ/ν)
Order parameter:
κ1 = −∂f∂h
= C2m−βκ/νY ′(
C1tmκ/ν ,C2hm(β+γ)κ/ν)
Higher cumulants:
κn = −∂nf∂hn = Cn
2m((n−2)β+(n−1)γ)κ/νY(n)(
C1tmκ/ν ,C2hm(β+γ)κ/ν)
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Cumulant exponent relation
This gives relation for all of the exponents (recall κn ∼ mαn )
αn = [(n− 2)β + (n− 1)γ∗]κ
ν
α1 = −βκν
α2 =γ∗κ
ν
This also works for α0 = −(2β + γ)κ/ν = −κwhich is the correlation length exponent!Hence include κ0 ≡ ξ as the 0th order cumulant ξ ∼ mκ → κ0 ∼ m−α0
Cumulant relationαn = (n− 1)α2 + (2− n)α1
= nα1 + (1− n)α0
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Ising model example(from J. Pillay, IPM, Phys. Rev. B 100, 235140 (2019))
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Ising model cumulant scaling functions
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Ising model correlation length scaling function
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Binder cumulant scaling
For finite systems, the Binder cumulant of the order parameter cancels theleading-order finite size effects
UL = 1− 〈m4〉3〈m2〉2
0
0.25
0.5
0.75
1
0.18 0.2 0.22 0.24
L = 50L = 100L = 150L = 200iDMRGfinite size scaling
|〈∆NL/2〉|
Ω
0
0.2
0.4
0.6
0.8
0.18 0.2 0.22 0.24
L = 50L = 100L = 150L = 200
UL
Ω
0.4
0.425
0.215 0.216
The 2-component Bose-Hubbard model, with a linear coupling betweencomponents, has an Ising-like transition from immersible (small Ω) to miscible(large Ω).
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Binder Cumulant for iMPS
Naively taking the limit L→∞ for the Binder cumulant doesn’t produceanything useful:
if the order parameter κ1 6= 0,
UL = 1− 〈m4〉L
3〈m2〉2L→ 2
3
if κ1 = 0, then m4(L) = 3k22L2 + k4L
Hence
UL = 1− 3k22L2 + k4L3k2
2L2 → 0
Finally, a step function that detects whether the order parameter isnon-zero
2019 approach: Evaluate the moment polynomial using L ∝ correlation lengthBetter approach: Find a combination of cumulants such that the bonddimension cancels
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Products of cumulants κa00 κ
a11 κ
a22 . . . scales as ma0α0+a1α1+a2α2+...
Find a combination such that a0α0 + a1α1 + a2α2 + . . . = 0This quantity is independent of basis size at the critical pointSimplest case:
C2 =κ2
κ21κ0
=σ2
〈m〉2ξ
Closely related to the 2nd order cumulant 〈m2〉
〈m〉2
Higher order cumulants are possible,
C4 =κ4
κ22κ0
(Closely related to the Binder cumulant)
C(3)4 =
κ4κ21
κ32
(Avoids using the correlation length κ0)In cases I’ve tried so far, C2 works best.
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Ising model example
0.985 0.99 0.995 1 1.005λ
0.25
0.5
1
2
4
κ2/κ
1
2κ
0
m=3m=4m=5m=6m=7m=8m=9m=10m=11m=12
Transverse Field Ising Model 2nd order cumulant
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Ising model example
10 20 30m
0.5
1
1.5
2
C2
λ=0.999λ=1λ=1.001
Ising model C2 sensitivity to λ
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Quantum 3-state Potts model example
0.996 0.998 1 1.002 1.004g
0.4
0.6
0.8
1
1.2
1.4
C2
m=10m=15m=20m=25m=30
Quantum Potts model
2nd order cumulant
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Quantum 3-state Potts model example
10 15 20 25 30 35 40Bond dimension
0.1
1
10
g=1
g=0.999
g=1.001
Linear fit, slope=0.00075
Quantum 3-state Potts model2nd order cumulant
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Critical exponents
Once we know the critical point, there are several approaches to determiningthe exponents
Direct fit as a function of gScaling function collapse – choice of scaling parameter
bond dimension scaling – J. Pillay, IPM, Phys. Rev. B 100, 235140 (2019)δ = εi − εj scaling – Vanhecke, Haegeman, et al, Phys. Rev. Lett. 123,250604 (2019)
Correlation length scaling at the critical pointThe exponents of the order parameter and 2nd cumulant are
κ1 ∼ m−βκ/ν
κ2 ∼ mγκ/ν
And recall κ0 ∼ mκ. So at the critical point, κ1 = κ−β/ν0 and κ2 = κ
γ/ν0 .
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Quantum 3-state Potts model example
10 100
κ0 (= ξ)
0.4241
0.46651
0.51316
0.56447
0.62092
κ1
Fitted exponent = -0.1342 (exact value -0.1333...)
Quantum Potts model fit for β/ν
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Alternative scaling variable – κ2/κ21
10 100
κ2
/ κ1
2
0.4241
0.46651
0.51316
0.56447
0.62092
κ1
Fitted exponent = -0.12981 (exact value -0.1333...)
Quantum Potts model fit for β/ν
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Quantum 3-state Potts model example – exponent γ
10 100
κ0 (= ξ)
1
10
100
κ2
Fitted exponent = 0.75561 (exact value 0.73333...)
Quantum Potts model fit for γ/ν
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Generalising beyond local order parametersThe approach of calculating moments of the order parameter doesn’trequire that the order parameter is strictly local.Example: Mott insulator at integer filling n = 1 particles everywhere, withonly short-range fluctuations. String order parameter:
O2P = lim
|j−i|→∞〈Πj
k=i(−1)nk−1〉
We can write this as a correlation function of ‘kink operators’,
pi = Πk<i (−1)nk−1
This turns the string order into a 2-point correlation function:
O2P = lim
|j−i|→∞〈 pi pj 〉
Or as an order parameter:
P =∑
i
pi MPO: P =
((−1)n−1 I
0 I
)Then O2
p = 1L2 〈P2〉
(see J. Pillay and IPM, Phys. Rev. B 100, 235140 (2019) for examples)Ian McCulloch (UQ) iMPS April 22, 2021 27 / 31
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SPT transitions
Can use conventional string order parameter 〈Sz(0) Πx−1j=1 (−1)Sz(x) Sz(x)〉
Not universal – can deform state, string order parameter arbitrarily smallProper way to understand SPT: symmetry fractionalization
figure shamelessly stolen from Pollman and Turner, PRB 86, 125441 (2012)
Dihedral: UxUy = ±UyUx
Time reversal: UτUτ∗ = ±1Space refection: URUR∗ = ±1
Requires entanglement spectrum spectroscopy – not observable from thephysical degrees of freedom
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Kink operator for time reversalTime reversal operator τ = UK, τsατ−1 = −sα
K = complex conjugationU = some basis-dependent unitaryNormal basis: U = exp iπSy
For an MPS, we can treat this as a local actionτ(As) =
∑t〈 s |U | t 〉A∗t
Kink operator for time evolution:
pτ (x) =∏j<x
τ(j)
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Kink operator for time reversal
String correlator 〈pτ (0)pτ (x)〉
Order parameter: Pτ =∑
x pτ (x) has MPO form Pτ =
[eiπSy
K I0 I
]Trivial phase: symmetric under time reversal, 〈pτ (0)pτ(x)〉 6= 0SPT phase: antisymmetric under time reversal, 〈pτ (0)pτ(x)〉 = 0
Can calculate 〈P2τ 〉 straightforwardly, and higher moments, 〈P4
τ 〉, etc
In principle the same scheme applies to spatial reflection too – the MPOcontains the R operator that transposes an A-matrix. But no examples yet!
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Conclusions
MPO techniques for higher momentsBond-dimension scaling is inaccurate – explore methods that areinsensitive to convergence of the MPSHigher cumulants are effective for calculating critical phenomenaSecond order cumulant κ2/κ
21κ0 is very effective for locating critical points
with minimal effort.Generalising order parameters to strings, time reversal, . . .Spatial reflection can be considered in a similar way
Future directions:2D – some (confusing) results already for 2D cylinders (S.N. Saadatmandand IPM, Phys. Rev. B 96, 075117 (2017))
Field-induced fluctuations
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