Documentation for ALPS v2.0 TEBD Code

83 Pages·2015·1.24 MB·English

Appendix A
Documentation for ALPS v2.0 TEBD Code
This appendix contains the documentation for the TEBD code included
in the v2.0 release of the ALPS code, see Chap.10. At the time of the
writing of this thesis, this information was hosted on the ALPS wiki at
http://alps.comp-phys.org. Section A.1 is documentation about the code in
general, the background of the TEBD algorithm, and explanations of the
parameters used as input to the code. The original website for this section is
http://alps.comp-phys.org/mediawiki/index.php/Documentation:TEBD. Sections
A.2 and A.3 are tutorials on using the ALPS routines. The original websites
are http://alps.comp-phys.org/mediawiki/index.php/ALPS_2_Tutorials:TEBD-01
_bhquenchandhttp://alps.comp-phys.org/mediawiki/index.php/ALPS_2_Tutorials:
TEBD-02_kink,respectively.
A.1 Documentation:TEBD
A.1.1 Time-EvolvingBlockDecimation
Thetime-evolvingblockdecimation(TEBD)algorithmisamethodforsimulating
the time evolution of one-dimensional quantum lattice systems governed by a
Hamiltonian with at most nearest-neighbor interactions. It is closely related to the
densitymatrixrenormalizationgroup(DMRG)methodinthatbothmethodsoperate
on a class of states known as matrix product states (MPSs). In addition to real
time evolution, imaginary time evolution can also be used to find ground states.
Essentially,TEBDconsistsoftwoparts:acanonicalMPSrepresentationofamany-
bodystate,andaprotocolforfindingtheMPSclosesttoastatewhichisactedupon
byatwo-siteoperator.
©SpringerInternationalPublishingSwitzerland2015 291
M.L.Wall,QuantumMany-BodyPhysicsofUltracoldMoleculesinOptical
Lattices,SpringerTheses,DOI10.1007/978-3-319-14252-4
292 A DocumentationforALPSv2.0TEBDCode
The particular implementation of TEBD used in ALPS simulates a series of
global parameter quenches of the form g.t/ D g.t / C ..t (cid:2) t /=(cid:2)/p.g.t / (cid:2)
i i f
g.t //. The timescale (cid:2), power p, initial and final values g.t / and g.t /, and
i f i
Hamiltonian parameters g of each quench are all amenable to specification by the
user.Additionally,becausetheTEBDmethodproduceswavefunctions,awiderange
ofobservablesareavailable,includingentropies,correlationfunctions,andoverlaps
betweenthestateatdifferenttimes.
A.1.2 References
Vidal,G.:Efficientclassicalsimulationofslightlyentangledquantumcomputations.
Phys.Rev.Lett.91,147902(2003)
Vidal, G.: Efficient simulation of one-dimensional quantum many-body systems.
Phys.Rev.Lett.93,040502(2004)
Daley,A.J.,Kollath,C.,Schollwöck,U.,Vidal,G.:Time-dependentdensity-matrix
renormalization-groupusingadaptiveeffectiveHilbertspaces. J.Stat.Mech.2004,
P04005(2004)
A.1.3 TEBD-Specific Parameters
CHI_LIMIT
The maximum bond dimension of the MPS allowed during real time propagation.
Thedefaultvalueis50.
TRUNC_LIMIT
The maximum truncation error allowed for a specific two-site evolution. If the
bond dimension corresponding to this truncation is greater than CHI_LIMIT, then
CHI_LIMITischoseninstead.Thedefaultvalueis10(cid:2)12.
TAUS
Theelementsofthisvectorarethetimescales(cid:2) oftheglobalquenches.
GS
TheelementsofthisvectoraretheHamiltonianparametersgoftheglobalquenches,
given as character variables. Note that the elements of this vector may themselves
be vectors, which corresponds to quenching several parameters at the same time.
If this is so, the corresponding elements of POWS, GIS, and GFS must also be
vectorsofthesamelength.NotethatTAUS,NUMSTEPS,andSTEPSFORSTORE
A DocumentationforALPSv2.0TEBDCode 293
will not be vectors, as the timescale, number of time steps, and number of steps
betweenoutputsarethesameforeachparameterbeingquenched.
POWS
Theelementsofthisvectorarethepowerspoftheglobalquenches.
GIS
The elements of this vector are the initial values of the Hamiltonian parameters g
oftheglobalquenches.
GFS
TheelementsofthisvectorarethefinalvaluesoftheHamiltonianparametersg of
theglobalquenches.
CONSERVED_QUANTUMNUMBERS
Quantumnumbersconservedbythemodelofinterest.Forspinmodels“Sz”canbe
conserved,andforparticlemodels“N”canbeconserved.
NUMSTEPS
The elements of this vector are the number of time-steps of the global quenches.
Thisimplicitlydefinesthetimestepsdtofthequenches.
STEPSFORSTORE
The elements of this vector are the number of time-steps between the calculation
andoutputofobservables.
INITIAL_STATE
The state used at t D 0, before real time propagation begins. Currently, only two
valuesaresupported:“kink,”whichproducesaspecificinitialstatetobediscussed
furtherintutorial2a,and“ground,”whichcalculatesthegroundstateofaspecified
initialhamiltonian viaimaginary timepropagation. Thedefaultvalue is“ground”.
Seethetutorialsforexamples.
ITP_CHIS
The elements of this vector are the maximum bond dimensions used in iterations
of imaginary time propagation to find the group state. It is only referenced if
INITIAL_STATEis“ground.”
ITP_DTS
Theelements ofthisvector arethetimestepsusediniterationsofimaginarytime
propagation to find the group state. It is only referenced if INITIAL_STATE is
“ground.”
294 A DocumentationforALPSv2.0TEBDCode
ITP_CONVS
The elements of this vector are the convergence parameters used in iterations of
imaginary time propagation to find the group state. An iteration of imaginary
time propagation exits if the maximal difference between singular values at
some time interval is less than the convergence parameter. It is only referenced
ifINITIAL_STATEis“ground.”
SIMID
An optional integer input which differentiates a series of simulations and can
simplifyplottingcommands.
NUM_THREADS
ThenumberofOpenMPthreadsused.
VERBOSE
If set to “true,” then the code willoutput, time values, truncation errors,and other
runningmessages.Thedefaultvalueis“false.”
A.2 Tutorials:TEBD-01 bhquench
A.2.1 The HardcoreBosonModel
Inthisfirsttutorialweinvestigatethebehaviorofthehardcorebosonmodel
LX(cid:2)1 LX(cid:2)1
H D(cid:2)t .bi(cid:3)biC1Cbibi(cid:3)C1/CV niniC1 (A.1)
iD1 iD1
astheparameterV ischangedintime.ItiswellknownthatforlargeV=t theground
state of the hardcore boson model at half filling is a charge-density wave (CDW)
insulatorwhileforsmallV=t thegroundstateisasuperfluid(SF).Itisinterestingto
considerwhathappenstothesystemifwebegininonephaseandthendynamically
change, or “quench,” one of the Hamiltonian parameters t or V such that we are
in the other phase. As a simple first foray into the rich physics of quenches, we
willconsiderquenchingfromonephasetotheotherandthenbackintotheoriginal
phase. A particularly stringent criterion for adiabaticity of such a quench is how
closethefinalstateistotheinitialstate,i.e.
L.tI(cid:4)/(cid:3)jh .t/j .0/ij2 (A.2)
whichwecalltheLoschmidtEcho.Notethatthet inthisexpressionisthetimeand
notthehoppingparametert.Theparameter(cid:4) ismeanttoconveythatthisquantity
ingeneraldependsonthemannerinwhichthesystemisquenched.
A DocumentationforALPSv2.0TEBDCode 295
The general structure of a quench in the ALPS TEBD routines is given by the
parameterization
g.t/Dg.t /C..t (cid:2)t /=(cid:2)/p.g.t /(cid:2)g.t //; (A.3)
i i f i
where g is some Hamiltonian parameter. In the present case we will take g to be
the interaction parameter V. We will begin our system in the CDW regime with
V=t D 10,quenchtotheSFregimewhereV=t D 0,andthenquenchbacktothe
CDWregimewithV=t D10.Inthethreepartsofthistutorialwewillinvestigate(a)
theeffectsofthetimescale(cid:2) ontheLoschmidtechoduringalinearquench,(b)the
effectsof“holding”thesystemintheSFphaseforatime(cid:2) beforereturningto
hold
theCDWphase,and(c)theeffectsofchangingthepowerpofthequenchfunction.
A.2.2 Linear Quench
First, we will investigate the effects of the quench rate (cid:2) on the adiabaticity of a
linearquenchfromtheCDWtotheSFphaseandback.
A.2.2.1 PreparingandRunningtheSimulationUsingPython
TosetupandrunthesimulationinPythonweusethescripttutorial1a.py.Thefirst
partsofthisscriptimporttherequiredmodulesandthenpreparetheinputfilesasa
listofPythondictionaries:
import pyalps
import matplotlib . pyplot as plt
import pyalps . plot
parms=[]
count=0
for A in [5.0 , 10.0, 15.0, 25.0, 50.0]:
count+=1
parms.append({
’L’ : 10,
’MODEL’ : ’
hardcore boson’ ,
’CONSERVED_QUANTUMNUMBERS’ : ’N’ ,
’N’ : 5,
’t ’ : 1.0,
’V’ : 10.0,
’ITP_CHIS’ : [20, 30, 35],
’ITP_DTS’ : [0.05 , 0.05 ,0.025] ,
296 A DocumentationforALPSv2.0TEBDCode
’ITP_CONVS’ : [1E(cid:2)8, 1E(cid:2)8, 1E(cid:2)9],
’INITIAL_STATE’ : ’ground’ ,
’CHI_LIMIT’ : 40,
’TRUNC_LIMIT’ : 1E(cid:2)12,
’NUM_THREADS’ : 1,
’TAUS’ : [A, A] ,
’POWS’ : [1.0 , 1.0] ,
’GS’ : [ ’V’ , ’V’] ,
’GIS’ : [10.0 , 0.0] ,
’GFS’ : [0.0 , 10.0] ,
’NUMSTEPS’ : [500, 500],
’STEPSFORSTORE’ : [5, 3],
’SIMID’ : count
})
Let’s go through the TEBD-specific parameters in more detail (see the TEBD
documentation, Sect.A.1, for a list of all such parameters). The parameter
INITIAL_STATE is set to ground, which means that we begin from the ground
state of our Hamiltonian with user-specified parameters. The parameters t and V
specify that the initial Hamiltonian parameters t D 1 and V D 10 are used to
find the ground state. In order to find the ground state, TEBD performs evolution
in imaginary time. We refer to this step as ITP, and so all parameters containing
ITP deal with the ground state properties. The vectors ITP_CHIS, ITP_DTS, and
ITP_CONVSaretheentanglementcutoffparameters,timesteps,andconvergence
criteriaforsuccessiveapplicationsofimaginarytimepropagation.Theseconstitute
the main convergence parameters for TEBD, and convergence should always
be carefully checked in each parameter. For now, don’t worry too much about
theiractualvalues,we’llseehowerrorsarecontrolledinthenextsetoftutorials.
Now we turn to the real-time propagation parameters. We wish to perform
a series of two quenches. First we want to quench the parameter V linearly in
time from its initial value 10 to 0. Comparing with the general form of a quench
g.t/Dg.t /C..t(cid:2)t /=(cid:2)/p.g.t /(cid:2)g.t //weseethatthiscorrespondstog DV,
i i f i
g.t / D 10, g.t / D 0, p D 1, and (cid:2) is the free parameter whose effects are to
i f
beinvestigated. Lookingattheparameter list,weseethatthefirstelements ofthe
vectorsGS,GIS,GFS,andPOWScorrespondtog,g.t /,g.t /,andp,respectively.
i f
The first element of the vector TAUS is looped over using the variable A, which
meansthatwewillperformaseriesofsimulationswith(cid:2) D 5,10,15,25,and50.
The second quench is essentially the reverse of the first, with g D V, g.t / D 0,
i
g.t /D10,p D1,and(cid:2) thesameasthefirst.Comparingwiththeparameterslist,
f
weseethatthiscorrespondstothesecondelementsofthevectorsGS,GIS,etc.as
above.
Timeevolutionissimulatedbybreakingthefullpropagatorapproximatelyinto
aseriesofoperationswhichactonlyontwoneighboringsitesatatime.Theerror
in using this approximate propagator is second order in the “infinitesimal” time-
step dt. TEBD gives a protocol for updating the canonical form of our state after
A DocumentationforALPSv2.0TEBDCode 297
suchatwo-siteoperationhasbeenapplied.Theerrorinthisprocedureiscontrolled
by CHI_LIMIT, which is directly related to the amount of spatial entanglement,
and TRUNC_LIMIT, which is akin to the TRUNCATION_ERROR in the DMRG
routines. The parameter vector NUMSTEPS specifies how many time-steps are
takeninperformingeachquench,whichtogetherwith(cid:2) implicitlydefinesthetime-
stepdt.TheoverallerrorisanontrivialfunctionofCHI_LIMIT,TRUNC_LIMIT,
and NUMSTEPS which will be investigated in the next set of tutorials, so we
won’t worry about the choice of these much for now. Finally, STEPSFORSTORE
determines how many time steps are taken before observables are computed and
storedandSIMIDisanintegerdifferentiatingthesimulationswithdifferent(cid:2).
Wenowmoveontotheactualcomputation.Thelines:
baseName=’tutorial_1a ’
#write output files
nmlnameList=pyalps . writeTEBDfiles(parms , baseName)
#run the application
res=pyalps .runTEBD(nmlnameList)
preparetheinputfilesfortheTEBDroutinesandrunthesimulationsfortherange
of(cid:2) specifiedintheparameters.WenowloadtheLoschmidtEchoandinteraction
parameterU asfunctionsoftimevia:
#Load the loschmidt echo and V
LEdata=pyalps . load . loadTimeEvolution(pyalps . getResult
Files ( prefix=’tutorial_1a ’) , measurements=
[ ’Loschmidt Echo’ , ’V’])
Finally,weplotthecollecteddatausing:
LE=pyalps . collectXY(LEdata , x=’Time’ , y=’Loschmidt
Echo’ ,foreach=[’SIMID’])
for q in LE:
q. props[ ’label ’]=r ’$\ tau=$’+str (q. props[ ’TAUS’
][0])
plt . figure ()
pyalps . plot . plot (LE)
plt . xlabel( ’Time $t$ ’)
plt . ylabel( ’Loschmidt Echo $|< \ psi (0) |\ psi(t) > |^2$’
)
plt . title ( ’Loschmidt Echo vs. Time’)
plt . legend(loc=’lower right ’)
Ufig=pyalps . collectXY(LEdata , x=’Time’ , y=’V’ ,foreach
=[’SIMID’])
for q in Ufig:
q. props[ ’label ’]=r ’$\ tau=$’+str (q. props[ ’TAUS’
][0])
298 A DocumentationforALPSv2.0TEBDCode
plt . figure ()
pyalps . plot . plot (Ufig)
plt . xlabel( ’Time $t$ ’)
plt . ylabel( ’V’)
plt . title ( ’Interaction parameter $V$ vs. Time’)
plt . legend(loc=’lower right ’)
plt .show()
A.2.2.2 PreparingandRunningtheSimulationUsingVistrails
TorunthesimulationinVistrailsopenthefiletutorial1a.vtandlookattheworkflow
labeled“tutorial1a.”Clickon“Execute”topreparetheinputfile,runthesimulation
andcreatetheoutputfigure.
A.2.2.3 Questions
• Howdoesthebehavioroftheoverlapchangeasthequenchratedecreases?
• Roughly how slowly do you have to perform the quench in order for it to be
adiabatic?
• Isiteasierorharderforalargersystemtobeadiabatic?Why?
• Are these properties changed depending on whether the intermediate phase is
gapped or not? One can test this by changing from the hardcore boson model
to the (softcore) boson Hubbard model, and then quenching from the Mott-
Insulating (MI) phase at large U=t and unit filling to the CDW phase with
large V. As you quench from the Mott insulating to the CDW phase and back,
howdifficultisittobeadiabatic?
A.2.3 LinearQuenchwith Hold
Inthissectionwewillinvestigatetheeffectsof“holding”thesystemintheSFphase
foratime(cid:2) beforequenchingbacktotheCDWphase.
hold
A.2.3.1 PreparingandRunningtheSimulationUsingPython
TosetupandrunthesimulationinPythonweusethescripttutorial1b.py.Thefirst
partsofthisscriptimporttherequiredmodulesandthenpreparetheinputfilesasa
listofPythondictionaries:
A DocumentationforALPSv2.0TEBDCode 299
import pyalps
import matplotlib.pyplot as plt
import pyalps.plot
#prepare the input parameters
parms=[]
count=0
for A in [5.0, 10.0, 15.0, 25.0, 50.0]:
count+=1
parms.append({
’L’ : 10,
’MODEL’ : ’hardcore boson’,
’CONSERVED_QUANTUMNUMBERS’ : ’N’,
’N’ : 5,
’t’ : 1.0,
’V’ : 10.0,
’ITP_CHIS’ : [20, 30, 35],
’ITP_DTS’ : [0.05, 0.05,0.025],
’ITP_CONVS’ : [1E(cid:2)8, 1E(cid:2)8, 1E(cid:2)9],
’INITIAL_STATE’ : ’ground’,
’CHI_LIMIT’ : 80,
’TRUNC_LIMIT’ : 1E(cid:2)12,
’NUM_THREADS’ : 1,
’TAUS’ : [10.0, A, 10.0],
’POWS’ : [1.0, 0.0,1.0],
’GS’ : [’V’, ’V’, ’V’],
’GIS’ : [10.0,0.0, 0.0],
’GFS’ : [0.0, 0.0, 10.0],
’NUMSTEPS’ : [500, int(A/0.05) , 500],
’STEPSFORSTORE’ : [5,5, 3],
’SIMID’ : count
})
NotethatinthiscasewehavethreequenchesasGS,GIS,etc.areallvectorsof
lengththree.ThesecondquenchkeepstheHamiltonianparametersfixedatt D 1,
V D 0 for a time (cid:2) before quenching back. We write the input files, run the
hold
simulations,getoutputs,andplotasabove:
baseName=’tutorial_1b ’
#write output files
nmlnameList=pyalps . writeTEBDfiles(parms , baseName)
#run the application
res=pyalps .runTEBD(nmlnameList)
#Load the loschmidt echo and U
LEdata=pyalps . load . loadTimeEvolution(pyalps .
getResultFiles ( prefix=’tutorial_1b ’) , measurements
=[’Loschmidt Echo’ , ’V’])
300 A DocumentationforALPSv2.0TEBDCode
LE=pyalps . collectXY(LEdata , x=’Time’ , y=’Loschmidt
Echo’ ,foreach=[’SIMID’])
for q in LE:
q. props[ ’label ’]=r ’$\tau_{\mathrm{hold}}=$’+str
(q. props[ ’TAUS’][1])
plt . figure ()
pyalps . plot . plot (LE)
plt . xlabel( ’Time $t$ ’)
plt . ylabel( ’Loschmidt Echo $|< \ psi (0) |\ psi(t) > |^2$’
)
plt . title ( ’Loschmidt Echo vs. Time’)
plt . legend(loc=’lower right ’)
Ufig=pyalps . collectXY(LEdata , x=’Time’ , y=’V’ ,foreach
=[’SIMID’])
for q in Ufig:
q. props[ ’label ’]=r ’$\tau_{\mathrm{hold}}=$’+str
(q. props[ ’TAUS’][1])
plt . figure ()
pyalps . plot . plot (Ufig)
plt . xlabel( ’Time $t$ ’)
plt . ylabel( ’V’)
plt . title ( ’Interaction parameter $V$ vs. Time’)
plt . legend ()
plt .show()
A.2.3.2 PreparingandRunningtheSimulationUsingVistrails
TorunthesimulationinVistrailsopenthefiletutorial1b.vtandlookattheworkflow
labeled“tutorial1b.”Clickon“Execute”topreparetheinputfile,runthesimulation
andcreatetheoutputfigure.
A.2.3.3 Questions
• Howdoesthebehavioroftheoverlapchangeastheholdtimeincreases?
• Isthisbehaviormonotonicintheholdtime?Whyorwhynot?
A.2.4 NonlinearQuenches
In this section we will investigate the effects of varying the power of the quench
awayfrombeinglinear.