強相関電子系/スピン系の汎用シミュレータ: ALPS
藤堂眞治 (東大院理/物性研/ALPS Collaboration)
Why quantum lattice models?

QMC challenges for spin/bosonic lattice models

The ALPS/looper LibraryMulti-Cluster Quantum Monte Carlo Library

QLM: quantum spin models,bosonic Hubbard models, etc• Effects of strong correlations in
multi-degree-of-freedom systems• Universality• depends on few parameters:
dimensionality, symmetry oforder parameter, etc• various types of long-range order• quantum disordered phase(quantum liquids, spin gap phases)

cold atoms on optical lattice• phase transitions and criticalphenomena• quantum critical point• Powerful simulation algorithms• quantum Monte Carlo methods
exact diagonalization, DMRG, etc• Criticality with large (logarithmic) corrections to scaling (KosterlitzThouless transition, phase transition in upper critical dimension, etc)
• Supersolid: co-existence of diagonal and off-diagonal order infrustrated spin/bosonic lattice models• Deconfined criticality: direct continuous quantum phase transition
between long-range ordered phase with incompatible symmetries (orweak 1st order phase transition?)• Long-range interacting system (magnetic dipole, RKKY, etc): change ofeffective spatial dimension, exotic boundary effects, etc• Strongly anisotropic systems (layered magnets, etc)
• ...

⇒ large-scale parallel simulations on the K Computer

quantum critical point

The ALPS ProjectMethod• path-integral and SSE representations• multi-cluster loop algorithm• hybrid (MPI + OpenMP) parallelization
for K Computer

Generic QMC Library• successor of Looper 2(Fortran 90)
Based on ALPS framework• implemented in C++• ALPS/Lattice Library• ALPS/Model Library, etcModels• generic XXZ Hamiltonians• generic lattices• any spin size S

http://wistaria.comp-phys.org/alps-looper/

Spin Ladder Material Na2Fe2(C2O4)3(H2O)2

Orbital Ordering in eg Orbital Systems

Algorithms and Libraries for Physics Simulations

C693.5 GHz

010acc847.0 GHz(c) 730.5 GHzFeba

Susceptibility (emu/mol)b20655.7 GHz

H//a584.8 GHz b

T=1.3 K

30

40

c50

applications

MCdomain-specificlibraries

lattice model observables scheduler

numerics

QMCrandom ublas

EDDMRG

DMFT

iterative eigenvalue solver

Boost librarygeneric C++
C / Fortran

graph
BLAS

serialization XML/XSLTLAPACKMPI

HDF5

Infrastructure for International Collaboration
World-wide Workshops• Developers Workshop• Users Workshop, Summar School

Web Page (Wiki)• http://alps.comp-phys.org

x0.00

Magnetization (μB /ion)

424°

H⊥ a
020 H = 100
40 Oe 60
Temperature (K)

80

1.5
10.5

T x polarized(xy planes)

T z polarized

experiment
100

"0"

#

van Rynbach et al (2010)

!

0!

0.2

2

#!

!

!

0.4!

0.6

0.8!

1Jx / Jz

1.2

1.4!

1.6

1.8!

24!

y2.2 Experimental methods
3354
The magnetic
susceptibilities of a single crystal of SIO
H⊥ awere measured at 100 Oe with a superconducting quantum
1400
2H⊥ a
2 interference
device (SQUID) magnetometer (Quantum21200
Design MPMS-XL7). High-ﬁeld
magnetizations in pulsed
1experiment T=1.3 K
1magneticﬁelds
up
toabout
53
T were measured utilizing a1000
T = 1.3 K
3non-destructive
pulse magnet, and the magnetizations were
R=0.1, J1/kB=26 K
1 0800detected by an induction methodH//ausing a standard pick-up
01020304050 R=0.3,60 J1/kB=21 K
coilsystem.
Magneticfield(T)600
T =1.3 K
R=0.5, J1/kB=17 K
ESR measurements on a single crystal of SIO in pulsed
0Fig. 3. Magnetization
curves of Na
400
strongsignalcarried out using our
2 Fe2 (C 2 O4 )3 (H 2 O)2 at 1.3 K for H k amagnetic ﬁelds up to about 53
T were010203040andH?
a.
weak signal
pulse ﬁeld ESR apparatus equippedwith a non-destructive<LATTICES>
200calculation
Magnetic Field (T)<LATTICE name="square lattice" dimension="2">pulse magnet, a far-infrared laser, and several kinds of Gunn
<PARAMETER name="a" default="1"/>Oscillators for the frequency range from about 50 GHz
0Fig.
6.
(Coloronline)(a)
QMC
calculations of magnetic
susceptibility for a</VECTOR></BASIS><BASIS><VECTOR>a0</VECTOR><VECTOR>015,16)
0 to about51015 We 20
2530 ESR
35 measurements
Note that the maximum
number of retained
bases
es.
1.6 THz.
also performed
diﬀerent
ratiosofR.
Theillustrationshowstherelation
betweena-axisthe</LATTICE>
Magnetic
Field (T) ﬁelds up to 14 T utilizing am is up to 50, with which
ﬃcientsuconvergenceof
theof SIO in static
magnetic
name="simple2d"dimension="2">and the principal axes.<UNITCELL(b) DMRG calculationsof magnetization
process
LATTICE = "square lattice"magnetization
curve
can<VERTEX/>
be conﬁrmed.
superconducting magnet(Oxford Instruments) and a vector
fordiﬀ
erent
ratiosofR.
Theillustrationshows
therelation
betweenthe Fe
MODEL= "spin"
Fig. 5. (Color online)
Frequency-ﬁeldof the resonanceﬁelds
network
analyzer Lwithdiagram
someextension(ABmm).
Allthe<EDGE>= 16 and open triangles ladderand the principle axes. Thez-axis is nearly perpendicular to the
3. Results
k a. Closedof Na2Fe2(C 2O4)experiments3(H 2 O)2 for Hwere<SOURCEvertex="1"
offset="00"/> andcarried circles
out1 at KYOKUGEN in Osakarung
Jxy=oftheladder.
In
bothﬁgures,theopen diamonds,
open circles,
denote strong andweaksignals,
respectively.
Thesolidlines
indicate
the<TARGET
vertex="1"
offset="01"/>
Figure2 shows
the thetemperatureof theUniversity.
Jz= 2opensquares
represent
calculateddependences
results
R¼
for0:1, 0.3,
and 0.5,
resonance modes calculated fromSWEEPS
the energy
diagramshown
inFig.
8.
</EDGE>= 10000
magnetic susceptibilities for
H k a ( k ) and H ? a ( ? ).
respectively.
<EDGE>
THERMALIZATION= 1000
They show a broad peak at<SOURCE
about 25 vertex="1"and 20 K, respectively.
2.3 Calculation methods
offset="0 0"/>times larger
than the peak
The peak k is about three <TARGETThe quantum Monte Carlo (QMC) method used in this?,vertex="1"
offset="10"/>{ T = 0.1 }calculation is codedon
thebasisofacontinuous-time
loop
which
indicates
alargeanisotropy.
Inaddition,
although</EDGE>?
X= 0.2}each direction. We suggest{ Ttwo
possibleorigins of thewith
below
algorithm, which {is Toneof theﬃcientemethods forHshowsðSz1;i decrease
Sz2;i þ</UNITCELL>
Sy1;i decreasing
Sy2;i þ Sx1;itemperature
Sx2;i Þ¼ J 1 a slight= 0.5} most.
Oneistheexistenceoﬁ
nter-dimer
gradual increaseinM?
<LATTICEGRAPH
name="squarelattice">about 20i K, k shows an upturn below about 5 K.
simulating quantumsystems.
The linear size along the{ Tspin= 1.0}
X
X
<FINITELATTICE>
interactions, namely,along
thelegThe Figure 3 shows theprocesses
at 1.3 K inx
chain L isinteractions
64 in the presentcase,andwedirection.
adopt a periodicz magnetization<LATTICEþ J 2 ﬁelds upðStoSz ;iþþT HforSkya;iref="squareSy);iþandH
þ ?Saxlattice"/>;i11;i S ;iþ 1Þother is a mixing
betweena innon-magnetic
ground¼S,
wheremagneticabout
53(Mboundary
condition
this direction, Si.e.,;iþsinglet
L;ik(0,1)<EXTENT dimension="1"size="L"/>iis the spin operatorsite
i on the( ¼1; 2). In (M? ). For
increases
graduallyup
toS ;i magnetic
X HXk a, magnetization
state and exitedstates at
causedbythachainlarge
anisotropy.
<EXTENT dimension="2" size="L"/>~addition,
itis
ascertained
that
thereisnosize
dependence.
about
5T,
thenshowsasteep
increase type="periodic"/>
up to about 10 T. On ð1ÞgH;þS<BOUNDARYB;iThe ESR spectra of a single crystal of SIO at 1.3 K in
Details of the algorithm are given in refs.
13 and14.
the other
H ? a,</FINITELATTICE>
it shows a gradual increase up
i hand, formagnetic ﬁelds up to about 53 T for
H k a are shown in<UNITCELL
ref="simple2d"/>
(0,0)
(1,0)
The density matrix renormalization group (DMRG)to the saturation magnetizationof approximately
30 T. Thewheretheinterandintrachain
exchangeconstantsJareFig. 4. We observestrongandweakresonance
signals</LATTICEGRAPH>
1
method used here is an inﬁnite system size algorithmlinear
at magnetization above the saturation ﬁeld with almost</LATTICES>
andJ 2 (J 1slopes, J 2 >in0,both
antiferromagnetic),
respectively,indicated by closed
and openarrows, respectively,
and plot
T ¼ 0. However,a conventional
inﬁnite-system method
withidenticaldirections could be attributed
to the< 1are usedanisotropy
constant,whoseassuming
identical
them in the frequency‒ﬁeld
as showninandFig.often
5.thenumber cannot
be usedfails
the total-Sz quantumdiagram,
Van
Vleck paramagnetism,susceptibility
isvalues
estimated2þT) (¼ 0:011emu/mol).
Thisin the convergence oﬁ terations, since the externalfor
ﬁeld
tonot
be only
aboutthe0.01
x- andy-components
but also
the intra- andB /(Fe17)
4. Analyses andDiscussion.
Foranvalueisalmost
thesameasthatofRbFeCldirections
in our experiments are
ﬀerentdi from the principal
3g~ the ginterchain interactions, B the Bohr magneton,
isolated
antiferromagneticHeisenberg
dimer
formedbyR asaxes.
Therefore,
weemploytheimproved
algorithm
based
tensor, andH the external magnetic ﬁeld. We deﬁne
4.1 Two-leg ladder modelneighboring ions on a rung, a steplike change in magneton the matrix product representation of the2þ DMRG waveion
It is well known that the lowest three states of theFe R ¼ J 2=J 1. The diagonal components for the principal axes
ization due to level crossing is expected to be observed infunction to obtain highly accurate magnetization processFrequency (GHz)|3z 2 −r 220

H = 100 Oe

Lattice XML Filesquare latticeParameter Fileparameter2xml
toolo -diagonalof the g-tensor aregx, gy, and gz, and the ﬀ
in an octahedral crystal ﬁeld are well separated from the124701-2
components are zero. We consider thata-axis
the of the SIO,
other excited states owing to the crystal ﬁeld and spin‒orbit
2þ
can often be similarly to that of SCO, is tilted from the magnetic principal
coupling. Thus, the magnetic properties of Fea-axis and thedescribed by a ﬁctitious spin
S ¼ 1 with anisotropic
g values axes.8) We deﬁne the angles between the18,19)
In thisand exchange interactions atlow temperatures.
Parameter
XML
File principal axes as shown in Fig. 6(a).
case, the spin Hamiltonian of a two-leg ladder model isFigures 6(a) and 6(b) show the calculated results ofmagnetic susceptibilities and magnetizations obtained bywritten as

Model XML File
H = JΣ [ S iz S jz + ( S i+ S j- + S i- S j+ )/ 2 ] - h Σi S iz

< i,j >

<MODELS><BASIS name="spin"><SITEBASIS name="spin"><PARAMETER name="local_S" default="1/2"/><QUANTUMNUMBER name="S" min="local_S" max="local_S"/><QUANTUMNUMBER name="Sz" min="-S" max="S"/><OPERATOR name="Sz" matrixelement="Sz"/><OPERATOR name="Splus" matrixelement="sqrt(S*(S+1)-Sz*(Sz+1))"><CHANGE quantumnumber="Sz" change="1"/></OPERATOR><OPERATOR name="Sminus" matrixelement="sqrt(S*(S+1)-Sz*(Sz-1))"><CHANGE quantumnumber="Sz" change="-1"/></OPERATOR></SITEBASIS></BASIS><HAMILTONIAN name="spin"><PARAMETER name="J" default="1"/><PARAMETER name="h" default="0"/><BASIS ref="spin"/><SITETERM>-h * Sz(i)</SITETERM><BONDTERM source="i" target="j">J * (Sz(i)*Sz(j) + (Splus(i)*Sminus(j)+Sminus(i)*Splus(j)))/2</BONDTERM></HAMILTONIAN></MODELS>

Quantum Lattice Model

124701-3

application programs
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Source Code Management System• SVN (Subversion)
Bug Tracking System• Trac1
J z T iz T iz+ γ + 3J x T ix T ix+ γ4i, γ= x,y√
± 3J mix (T iz T ix+ γ + T ix T iz+ γ) −J z T iz T iz+ z

H 120 = −i0.05|x 2 −y2and orbital degrees of freedom• Effective Hamiltonian for orbital degrees of freedom (120 degree model)H⊥a0.100.00

A Simulation with ALPSXML manipulation Python binding GUItools

0.05

H// ay0.15 0.15
0.10

H//a

Fig. 1. (Color online) (a) Crystal structure of Na
Fig. 2. Temperature5 dependences
suceptibilities
of2Na2 Fe2 (C 2 O4 )3 (H 2O)2 .
2 Fe10 of magnetic
15202530Magnetic Field (T)2þions areHydrogen and sodium atoms are omitted for clarity.
FeO)
(HO)
atH
¼100Oefor
HkaandH?
a.
(C24322Temperature (K)Yamaguchi
et al (2009)
connected by oxalate molecules and form two-leg ladders along theFig. 4. (Color online)
ESR
Na
Koctahedrafor
2 Fe
2 (C 2 O4 ) 3 (H(b)
2 O)
2 at 1.36FeO
a-axis,as spectra
shown byofred
brokenlines.
Distorted(b)H k a. The closedbridgedand openarrows
indicatethestrongandweaksignals,
by oxalate ion. (c) Arrangement of the ladders inbc-plane,thez
H// aand red broken lines corresponding to rungs of the ladders.
respectively.
4 5H// a1600

The ALPS Framework

a0.20

977.2 GHzOzd-shells• Non-tivial interplay of charge, spin,

Jmix / Jz

1288.1 GHz(b)

(a)

0.20

H. Y AMAGUCHI et al.

• Mott insulators with partially filled

orbiton gap

1623.6 GHz

Susceptibility (emu/mol)

J. Phys. Soc. Jpn., Vol. 78, No. 12(a)ALPS• community codes• generic implementations• common libraries simplify codedevelopment• common file formats

Motivation• established algorithms• increased demand for reliable
simulations from theorists andexperimentalists• Fe2+ ions in octahedral crystal field→ effective S=1 spins at low T• Fitting experimental data by QMC results for several theoretical models
J. Phys. Soc. Jpn., Vol. 78, No. 12
H. Y AMAGUCHI et al.
(chain, ladder, dimer, etc)

Magnetization (μ B/ion)

The status quo• individual codes• model-specific implementations• growing complexity of methods• outputs in non-portable formats

Transmission (arb.units)

Open source libraries and simulation
code for strongly correlated quantumsystems• quamtum Monte Carlo• exact diagonalization• DMRG• DMFT, etc

Quantum Monte Carlo

Exact Diagonalization

DMRG

DMFT

Python based evaluation tools
Plots

Outputs in HDF5 & XML

Windows & Mac OS X Support

ALPS + VisTrails

Lisence Issue

Ready-made Binary Installers

Integration with Workflow and Provenance Management System

The "cite-me" Lisence
Applications code
• published under ALPS
Application Lisence• free for non-commercial use• based on GNU public license• citation requirements

for Mac OS X

Library code
• less restrictive• partially available under a freelicense
Modification/improvements of codes• should be integrated into ALPS
• not obligatory to publish

ALPS Paper• “The ALPS project release 2.0: open source software for strongly
correlated systems”, B. Bauer et. al., JSTAT P05001 (2011).

for Windows

http://www.vistrails.org/powered by ALPS
http://alps.comp-phys.org

contact:
Synge Todo <wistaria@phys.s.u-tokyo.ac.jp>
Department of Physics, University of Tokyo