Recent Surprises in the Simulation of Quantum Phase T ransitions - - PowerPoint PPT Presentation
Recent Surprises in the Simulation of Quantum Phase T ransitions Lei W ang, ETH Zrich Cologne 2015.04 Quantum Phase T ransitions H ( ) = H 0 + H 1 Quantum Temperature Where is the QCP ? Critical What are the phases ?
Lei W ang, ETH Zürich Cologne 2015.04
ˆ H(λ) = ˆ H0 + λ ˆ H1
Temperature
λc
Disordered Ordered Quantum Critical
Where is the QCP ? What are the phases ? What is the universality class ? What are the experimental signatures ?
ˆ H(λ) = ˆ H0 + λ ˆ H1
Temperature
λc
Disordered Ordered Quantum Critical
Where is the QCP ? What are the phases ? What is the universality class ? What are the experimental signatures ?
exact diagonalization quantum Monte Carlo dynamical mean field theories tensor network states
Algorithms for quantum many body systems
ˆ H(λ) = ˆ H0 + λ ˆ H1
Temperature
λc
Disordered Ordered Quantum Critical
Where is the QCP ? What are the phases ? What is the universality class ? What are the experimental signatures ?
exact diagonalization quantum Monte Carlo dynamical mean field theories tensor network states
Algorithms for quantum many body systems
Buffon 1777
The first recorded Monte Carlo simulation
Statistical Mechanics: Algorithms and Computations W erner Krauth
`
d {
{
hNhitsi = 2 ⇡ ` d
THE 0 R Y
0 F
T RAe KEF FEe T SIN R A D I 0 L Y SIS 0 F
W ATE R
1087 instead, only water molecules with different amounts of excitation energy. These may follow any of three paths: (a) The excitation energy is lost without dissociation into radicals (by collision, or possibly radiation, as in aromatic hydrocarbons). (b) The molecules dissociate, but the resulting radi- cals recombine without escaping from the liquid cage. (c) The molecules dissociate and escape from the
more than a few molecular diameters through the dense medium before being thermalized.
In accordance with the notation introduced by
Burton, Magee, and Samuel,22 the molecules following
22 Burton, Magee, and Samuel, J. Chern. Phys. 20, 760 (1952).
THE JOURNAL OF CHEMICAL PHYSICS
paths (a) and (b) can be designated H 20* and those following path (c) can be designated H 20t. It seems reasonable to assume for the purpose of these calcula- tions that the ionized H 20 molecules will become the H 20t molecules, but this is not likely to be a complete correspondence. In conclusion we would like to emphasize that the qualitative result of this section is not critically de- pendent on the exact values of the physical parameters
treatment must be wave mechanical; therefore the result of this section cannot be taken as an a priori theoretical prediction. The success of the radical diffu- sion model given above lends some plausibility to the
crude calculation. Further work is clearly needed.
VOLUME 21, NUMBER 6
JUNE, 1953
Equation of State Calculations by Fast Computing Machines
NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER,
Los Alamos Scientific Laboratory, Los Alamos, New Mexico
AND
EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois
(Received March 6, 1953) A general method, suitable for fast computing machines, for investigatiflg such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion.
HE purpose of this paper is to describe a general
method, suitable for fast electronic computing machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules. Classical statistics is assumed,
field of a molecule is assumed spherically symmetric. These are the usual assumptions made in theories of
is not restricted to any range of temperature or density. This paper will also present results of a preliminary two- dimensional calculation for the rigid-sphere system. Work on the two-dimensional case with a Lennard- Jones potential is in progress and will be reported in a later paper. Also, the problem in three dimensions is being investigated.
*
Now at the Radiation Laboratory of the University of Cali- fornia, Livermore, California.
POTENTIAL BETWEEN THE PARTICLES
In order to reduce the problem to a feasible size for
numerical work, we can, of course, consider only a finite number of particles. This number N may be as high as several hundred. Our system consists of a squaret con- taining N particles. In order to minimize the surface effects we suppose the complete substance to be periodic, consisting of many such squares, each square contain- ing N particles in the same configuration. Thus we define dAB, the minimum distance between particles A and B, as the shortest distance between A and any of the particles B, of which there is one in each of the squares which comprise the complete substance. If we have a potential which falls off rapidly with distance, there will be at most one of the distances AB which can make a substantial contribution; hence we need consider only the minimum distance dAB.
t We will use
two-dimensional nomenclature here since it is easier to visualize. The extension to three dimensions is obvious.
THE 0 R Y
0 F
T RAe KEF FEe T SIN R A D I 0 L Y SIS 0 F
W ATE R
1087 instead, only water molecules with different amounts of excitation energy. These may follow any of three paths: (a) The excitation energy is lost without dissociation into radicals (by collision, or possibly radiation, as in aromatic hydrocarbons). (b) The molecules dissociate, but the resulting radi- cals recombine without escaping from the liquid cage. (c) The molecules dissociate and escape from the
more than a few molecular diameters through the dense medium before being thermalized.
In accordance with the notation introduced by
Burton, Magee, and Samuel,22 the molecules following
22 Burton, Magee, and Samuel, J. Chern. Phys. 20, 760 (1952).
THE JOURNAL OF CHEMICAL PHYSICS
paths (a) and (b) can be designated H 20* and those following path (c) can be designated H 20t. It seems reasonable to assume for the purpose of these calcula- tions that the ionized H 20 molecules will become the H 20t molecules, but this is not likely to be a complete correspondence. In conclusion we would like to emphasize that the qualitative result of this section is not critically de- pendent on the exact values of the physical parameters
treatment must be wave mechanical; therefore the result of this section cannot be taken as an a priori theoretical prediction. The success of the radical diffu- sion model given above lends some plausibility to the
crude calculation. Further work is clearly needed.
VOLUME 21, NUMBER 6
JUNE, 1953
Equation of State Calculations by Fast Computing Machines
NICHOLAS METROPOLIS, ARIANNA W. ROSENBLUTH, MARSHALL N. ROSENBLUTH, AND AUGUSTA H. TELLER,
Los Alamos Scientific Laboratory, Los Alamos, New Mexico
AND
EDWARD TELLER, * Department of Physics, University of Chicago, Chicago, Illinois
(Received March 6, 1953) A general method, suitable for fast computing machines, for investigatiflg such properties as equations of state for substances consisting of interacting individual molecules is described. The method consists of a modified Monte Carlo integration over configuration space. Results for the two-dimensional rigid-sphere system have been obtained on the Los Alamos MANIAC and are presented here. These results are compared to the free volume equation of state and to a four-term virial coefficient expansion.
HE purpose of this paper is to describe a general
method, suitable for fast electronic computing machines, of calculating the properties of any substance which may be considered as composed of interacting individual molecules. Classical statistics is assumed,
field of a molecule is assumed spherically symmetric. These are the usual assumptions made in theories of
is not restricted to any range of temperature or density. This paper will also present results of a preliminary two- dimensional calculation for the rigid-sphere system. Work on the two-dimensional case with a Lennard- Jones potential is in progress and will be reported in a later paper. Also, the problem in three dimensions is being investigated.
*
Now at the Radiation Laboratory of the University of Cali- fornia, Livermore, California.
POTENTIAL BETWEEN THE PARTICLES
In order to reduce the problem to a feasible size for
numerical work, we can, of course, consider only a finite number of particles. This number N may be as high as several hundred. Our system consists of a squaret con- taining N particles. In order to minimize the surface effects we suppose the complete substance to be periodic, consisting of many such squares, each square contain- ing N particles in the same configuration. Thus we define dAB, the minimum distance between particles A and B, as the shortest distance between A and any of the particles B, of which there is one in each of the squares which comprise the complete substance. If we have a potential which falls off rapidly with distance, there will be at most one of the distances AB which can make a substantial contribution; hence we need consider only the minimum distance dAB.
t We will use
two-dimensional nomenclature here since it is easier to visualize. The extension to three dimensions is obvious.
Z = Tr ⇣ e−β ˆ
H⌘
Z = Tr ⇣ e− β
M ˆ
H . . . e− β
M ˆ
H⌘
space imaginary-time
β/M
β
imaginary-time axis
ˆ H = ˆ H0 + λ ˆ H1
Z =
∞
X
k=0
λk Z β dτ1 . . . Z β
τk−1
dτk× Tr h (−1)ke−(β−τk) ˆ
H0 ˆ
H1 . . . ˆ H1e−τ1 ˆ
H0i
Traditional approach Modern approach
Beard and Wiese, 1996 Prokof’ev et al, 1996
. Sandvik et al, PRB, 43, 5950 (1991)
. Prokof’ev et al, JETP , 87, 310 (1998) Gull et al, RMP , 83, 349 (2011)
bosons World-line Approach
Time Space
quantum spins Stochastic Series Expansion fermions Determinantal Methods
. Sandvik et al, PRB, 43, 5950 (1991)
. Prokof’ev et al, JETP , 87, 310 (1998) Gull et al, RMP , 83, 349 (2011)
bosons World-line Approach quantum spins Stochastic Series Expansion fermions Determinantal Methods
β
Sign problem What about a negative probability ? There has always been surprise… General solution implies P=NP But, do we need a general solution ?
T royer and Wiese, 2005
Berg et al, Science, 2012
Huffman and Chandrasekharan, PRB, 2014
“designer” Hamiltonian new solution to the sign problem
Maps to an XXZ model Orders at infinitesimal V due to Fermi surface nesting
Vc/t = 2
ˆ H = −t X
hi,ji
⇣ ˆ c†
i ˆ
cj + ˆ c†
jˆ
ci ⌘ + V X
hi,ji
✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆
Maps to an XXZ model Orders at infinitesimal V due to Fermi surface nesting
Vc/t = 2
ˆ H = −t X
hi,ji
⇣ ˆ c†
i ˆ
cj + ˆ c†
jˆ
ci ⌘ + V X
hi,ji
✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆
Ising
V/t T/t
CDW Semi- Metal
Maps to an XXZ model Orders at infinitesimal V due to Fermi surface nesting
Critical point ? Universality class ? C
Vc/t = 2
ˆ H = −t X
hi,ji
⇣ ˆ c†
i ˆ
cj + ˆ c†
jˆ
ci ⌘ + V X
hi,ji
✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆
Ising
V/t T/t
CDW Semi- Metal
w(C) = det M
No sign problem model at But, h
w(C) = det M↑ × det M↓ = | det M↑|2 ≥ 0
e.g. spinless t-V model Meron cluster approach, Chandrasekharan and Wiese, PRL, 1999
Wu et al, PRB, 2005
Kramers pairs
Gubernatis et al, PRB, 1985 Scalapino et al, PRB, 1984
up to 8*8 square lattice and T≥ solves sign problem only for V ≥ 2t
For real skew-symmetric
Huffman and Chandrasekharan, PRB, 2014
det M = (pf M)2 ≥ 0
SU(3) split Dirac cone strain
spinless fermions
LW and T royer, PRL 2014 LW , Corboz, T royer, NJP 2014 (IOPselect) LW , Iazzi, Corboz, T royer, 1501.00986, PRB in press (Editors' Suggestion)
M T = −M
Appears naturalmy in modern CT-QMC
±1 for A(B) sublattice
up to 450 sites
2L2 M2 = 1 N 2 X
i,j
ηiηj ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆
M4 = 1 N 4 X
i,j,k,l
ηiηjηkηl ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆ ✓ ˆ nk − 1 2 ◆ ✓ ˆ nl − 1 2 ◆
±1 for A(B) sublattice
up to 450 sites
2L2
Scalings ansatz close to the QCP
M2 = L−z−ηF[L1/ν(V − Vc), Lz/β]
M4 = L−2z−2ηG[L1/ν(V − Vc), Lz/β]
M2 = 1 N 2 X
i,j
ηiηj ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆
M4 = 1 N 4 X
i,j,k,l
ηiηjηkηl ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆ ✓ ˆ nk − 1 2 ◆ ✓ ˆ nl − 1 2 ◆
z = 1
relativistic invariance
±1 for A(B) sublattice
up to 450 sites
2L2
Scalings ansatz close to the QCP
M2 = L−z−ηF[L1/ν(V − Vc), Lz/β]
M4 = L−2z−2ηG[L1/ν(V − Vc), Lz/β]
M2 = 1 N 2 X
i,j
ηiηj ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆
M4 = 1 N 4 X
i,j,k,l
ηiηjηkηl ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆ ✓ ˆ nk − 1 2 ◆ ✓ ˆ nl − 1 2 ◆
z = 1
relativistic invariance
±1 for A(B) sublattice
up to 450 sites
2L2
Scalings ansatz close to the QCP
M2 = L−z−ηF[L1/ν(V − Vc), Lz/β]
M4 = L−2z−2ηG[L1/ν(V − Vc), Lz/β]
M2 = 1 N 2 X
i,j
ηiηj ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆
M4 = 1 N 4 X
i,j,k,l
ηiηjηkηl ⌧✓ ˆ ni − 1 2 ◆ ✓ ˆ nj − 1 2 ◆ ✓ ˆ nk − 1 2 ◆ ✓ ˆ nl − 1 2 ◆
R = M4 (M2)2
LW , Corboz, T royer, NJP 16, 103008 (2014)
Vc/t = 1.356(1) ν = 0.80(3) η = 0.302(7)
M2Lz+η = F(L1/ν(V − Vc))
M4L2z+2η = G(L1/ν(V − Vc))
* Errorbars
χ2 + 1
LW , Corboz, T royer, NJP 16, 103008 (2014)
ε
Rosenstein et al, PLB, 1993
ν = 0.797
functional renormalization group Rosa et al, PRL,2001 Höfling et al, PRB, 2002
η = 0.502 ν = 0.80(3) η = 0.302(7)
Honeycomb
ν = 0.738 ∼ 0.927 η = 0.525 ∼ 0.635
* Field theory calculations are based on 2-flavors of 2-component Dirac fermions with the
also features two Dirac points
Vc/t = 1.304(2) ν = 0.80(6) η = 0.318(8)
π
Philippe Corboz Amsterdam
Vc = 1.36(3)
∼ (V − Vc)
˜ β 0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V OPCDW D=4 D=5 D=7 D=9 extrapolated CTQMC
Philippe Corboz Amsterdam
Vc = 1.36(3)
∼ (V − Vc)
˜ β
CTQMC
˜ β = ν 2(z + η) = 0.52(3)
0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 V OPCDW D=4 D=5 D=7 D=9 extrapolated CTQMC
L Majorana
LW , Iazzi, Corboz, T royer, 1501.00986 Li, Jiang and Y ao, 1408.2269 Li, Jiang and Y ao, 1411.7383
1.32 1.33 1.34 1.35 1.36 1.37 1.38 1.8 2.0 2.2 2.4 2.6
Binder ratio V
L=9 L=12 L=15 L=18 L=21 1.30 1.35 1.40 1.45 1.50 0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 ∆CDW V
Consistent with finite-T results
η = 0.45(2) ν = 0.77(3) Vc/t = 1.355(1)
Iazzi, T royer, 1411.0683
Compare Make sure to compare with the Larger systems
Vc/t = 1.356(1) ν = 0.80(3) η = 0.302(7)
CTQMC
The new solution
There are still some discrepancies, to resolve them
Critical point ? Universality class ? C
ˆ H(λ) = ˆ H0 + λ ˆ H1
Temperature
λc
Phase 2 Phase 1 Quantum Critical
= 1 − F 2 ✏2 + . . . F(, ✏) = |hΨ()|Ψ( + ✏)i|
Fidelity Susceptibility Fidelity
Y
Campos V enuti et al, 2007
A general indicator of quantum phase transitions
No need for local order parameter e.g. Kitaev model, Abasto et al 2008, Y ang et al 2008
ˆ H(λ) = ˆ H0 + λ ˆ H1
= 1 − F 2 ✏2 + . . . F(, ✏) = |hΨ()|Ψ( + ✏)i|
Fidelity Susceptibility Fidelity Fulfills scaling law around QCP Gu et al 2009,
Albuquerque et al 2010
Y
Campos V enuti et al, 2007
A general indicator of quantum phase transitions
No need for local order parameter e.g. Kitaev model, Abasto et al 2008, Y ang et al 2008
ˆ H(λ) = ˆ H0 + λ ˆ H1
Fulfills scaling law around QCP Gu et al 2009,
Albuquerque et al 2010
Y
Campos V enuti et al, 2007
χF = hkLkRi hkLi hkRi 2λ2
β
LW , Liu, Imriška, Ma and T royer, 1502.06969
χF = hkLkRi hkLi hkRi 2λ2
β
LW , Liu, Imriška, Ma and T royer, 1502.06969
χF = hkLkRi hkLi hkRi 2λ2
β
kL = 2 kR = 4
LW , Liu, Imriška, Ma and T royer, 1502.06969
χF = hkLkRi hkLi hkRi 2λ2 kL kR kL kR
Stochastic Series Expansion
(quantum spins)
Determinantal Methods
(fermions)
Worldline Algorithms
(bosons)
Time Space
kL kR
LW , Liu, Imriška, Ma and T royer, 1502.06969
Mott Insulator Superfluid Divergence of fidelity susceptibility correctly single out the quantum critical point
ˆ H = U 2 X
i
ˆ ni (ˆ ni − 1) − λ X
hi,ji
⇣ ˆ b†
iˆ
bj + ˆ b†
jˆ
bi ⌘
Mott Insulator Superfluid Divergence of fidelity susceptibility correctly single out the quantum critical point
1.0 0.8 0.6 0.4 0.2 0.06 0.05 0.04 0.03 0.02 0.01 0.40 0.38 0.36 0.34 0.0594 0.0588µ/U
J/U
Capogrosso-Sansone, et al., PRA, 77, 015602 (2008)
ˆ H = U 2 X
i
ˆ ni (ˆ ni − 1) − λ X
hi,ji
⇣ ˆ b†
iˆ
bj + ˆ b†
jˆ
bi ⌘
SM SL AFMI 0.1 0.2 0.1 0.2 0.3 0.4 0.5 2 2.5 3 3.5 4 4.5 5 5.5 6 ms U/t ms
sp(K)/t
Δ
s (×6)
Δ
sp/t,
Δ
Γ
K K′ M a1 b1 b2 a2
kx ky y
Δ s/t x
Meng et al, Nature 2010 Sorella et al, Sci.Rep 2012
Toldin et al, PRB 2015
A hotly debated problem in recent years
ˆ H = −t X
hi,ji
X
σ={",#}
⇣ ˆ c†
iσˆ
cjσ + ˆ c†
jσˆ
ciσ ⌘ + λ X
i
✓ ˆ ni" − 1 2 ◆ ✓ ˆ ni# − 1 2 ◆
Suggesting a single transition, i.e. no intermediate phase
Calculated using LCT-INT
χF = hkLkRi hkLi hkRi 2λ2
β
Z =
∞
X
k=0
λk Z β dτ1 . . . Z β
τk−1
dτk× Tr h (−1)ke−(β−τk) ˆ
H0 ˆ
H1 . . . ˆ H1e−τ1 ˆ
H0i
fugacity Quantum Phase T ransition Classical Particle Condensation
uval, 1969
Maps the Kondo model to a classical Coulomb gas
Fidelity Susceptibility: A general purpose indicator of quantum phase transition
Mauro Iazzi Philippe Corboz Ping Nang Ma Matthias Troyer Jakub Imriška Ye-Hua Liu
Fidelity Susceptibility: A general purpose indicator of quantum phase transition
Mauro Iazzi Philippe Corboz Ping Nang Ma Matthias Troyer Jakub Imriška Ye-Hua Liu