Variational Numerical Renormalization Group: Bridging the gap between NRG and Density Matrix Renormalization Group

0.89 MB·English

by  Iztok Pizorn

Variational Numerical Renormalization Group: Bridging the gap between NRG and
Density Matrix Renormalization Group
Iztok Piˇzorn and Frank Verstraete
University of Vienna, Faculty of Physics, Boltzmanngasse 5, A-1090 Wien (Austria)
(Dated: May 11, 2011)
The numerical renormalization group (NRG) is rephrased as a variational method with the cost
functiongivenbythesumofalltheenergiesoftheeffectivelow-energyHamiltonian. Thisallowsto
systematically improve the spectrum obtained by NRG through sweeping. The ensuing algorithm
hasalotofsimilaritiestothedensitymatrixrenormalizationgroup(DMRG)whentargetingmany
states, and this synergy of NRG and DMRG combines the best of both worlds and extends their
applicability. We illustrate this approach with simulations of a quantum spin chain and a single
impurityAndersonmodel(SIAM)wheretheaccuracyoftheeffectiveeigenstatesisgreatlyenhanced
as compared to the NRG, especially in the transition to the continuum limit.
2
1 PACSnumbers: 05.10.Cc,72.10.Fk,72.15.Qm,02.70.-c,03.67.-a,71.27.+a
0
2
The density matrix renormalization group [1] comesinaccurateforlongerchainswhichisseenasavio-
n
(DMRG), devised to improve on the Wilson’s numerical lationoftheFriedelsumrule[15]. Italsobecomeshighly
a
J renormalization group (NRG) [2], has become the expensive for many-band impurity problems whereas an
2 method of choice to simulate one-dimensional quantum additional self-consistency constraint in the context of
many-body systems at zero temperature and has found dynamical mean-field theory, see e.g. [16], calls for more

] a large number of applications in the fields of the accurate impurity solvers. What makes the NRG really
h
condensed matter physics, quantum chemistry and different from the DMRG is that it does not provide any
p
- quantum information theory where it turned out that feedback mechanism to optimize the matrices by sweep-
t
n DMRG is essentially equivalent to simulating quantum ing along the chain and a method along this line has
a systems in terms of matrix product states (MPS) [3]. already been used with quantum fields [17]. In this Let-
u
In all these methods, a quantum many-body state is ter, we identify the cost function in the NRG algorithm
q
represented by associating matrices to local configura- andintroduceafeedbackmechanismbywhichthestates
[
tions at sites and the coefficients in the expansion over can be optimized in a variational way under the orig-

2 the configuration states are given as products of the inal NRG cost function. We relate the NRG and the
v
corresponding matrices. The DMRG can be used to DMRG by identifying the common cost function in both
1
calculate not only the ground state but also the excited approaches show that the proposed scheme improves the
0
4 states through the concept of targeting where these results of both methods while retains the lower, NRG-
1 matrices are chosen such that they well represent many like, scaling of computational costs.
2. states at the same time. In the NRG, on the other Numerical renormalization group. In the context of
0 hand, the low energy states of a system are expressed the NRG, the lowest energy eigenstates of a quantum
1 in terms of the low energy states of a smaller system many-body system on n sites, Hˆ ψ = E ψ , are ap-
α α α
1 | (cid:105) | (cid:105)
which is a suitable description of impurity systems with proximatedbyorthonormalstates [n] ψ[n],...,ψ[n]
v: energy scale separation. In this context, the NRG gives defining an effective low energy HaSmilt≡on{ian1 Dn}
i remarkably good results and has retained the position
X
as a widely used impurity solver [4] whereas the DMRG
r Dn
a is rarely used to calculate the excited states [5] except Hˆ[n] = E[n] ψ[n] ψ[n] (1)
for the spectral gap. One of the reasons is that the eff α | α (cid:105)(cid:104) α |
α=1
(cid:88)
NRG is less costly as it provides D excited states at the
cost of O(D3) as compared to DMRG with targeting with effective energies E[n] = ψ[n] Hˆ ψ[n] . The set of
α α α
with the cost O(D4). Presently, the DMRG is mostly effective states [n] is obtained(cid:104)recu|rs|ively(cid:105)from [n−1],
used in its time dependent form (see e.g. [6]) which is the effective desScription of a system on n 1 sitSes, ex-
also true for impurity systems [7–11] where also other tendedbyanextrasiteofalocaldimension−d ,resulting
n
density-matrix related concepts are used [12–14]. Still, in an extended Hamiltonian H[n] C(Dn−1dn)×(Dn−1dn)
DMRG’s remarkable ability to calculate and optimize ext ∈
that, in order to prevent exponential growth, has to be
many excited states has not been used in this context.
projected to a subspace of a smaller dimension D . In
n
Even as an impurity solver, the NRG has certain limi- the NRG, this subspace is spanned by the lowest energy
tations: thehoppingtermsshouldfalloffsufficientlyfast eigenvectors as H[n] =UHH[n]U where H[n] CDn×Dn
which means lower resolution of the spectral densities at is a new effectiveeHffamiltoniaenxtand U C(Deffn−∈1dn)×Dn is
∈
higher energies e.g. Hubbard bands. Otherwise it be- anisometrymatrixUHU=1. Thesetofeffectivestates
2
L α A β R wα = e−βEα[n] (see Fig. 1). The cost (4) is minimized
usingtheconjugategradientmethodwithaunitarycon-
s
straint [18] in a variational way (i.e. the sum of energies
w
γ can only decrease) and the computational costs scale as
s’
A* O(D3),sameasintheoriginalNRG.Thenumberofopti-
α' β'
mization steps depends on the quality of the initial state
and the desired accuracy.
FIG.1. Tensornetworkrepresentationofthe(weighted)cost
function(4)intheoptimizationofatensorA[j](circles)under Density matrix renormalization group. Aspecialprop-
the Hamiltonian matrix product operator (squares). erty of the NRG-MPS states is a two-fold nature of
the external α-index which acts both as an enumerat-
ing index as well as a virtual bond in extending the
ψα[n] is given as a MPS with an external α-index [8] system for a site. For a fixed system size, the exter-
{ }
nal index can be associated with any site in the chain
ψ[n] = e A[1]s1A[2]s2 A[n]sne s ,...s (2)
| α (cid:105) 0· ··· α| 1 n(cid:105) and moved along the chain by means of the singular
{(cid:88)sj} valuedecomposition(SVD).Interestingly,theconceptof
with the right boundary vector e enumerating basis a moveable external index is intrinsic to the finite size
α
states. The set [n] is orthonormal due to the isometry DMRG algorithm with targeting [1]. The basic concept
constraint for mSatrices A[j] defined as [A[j]] A[j]s. of the finite size DMRG is as follows: split the system as
TheNRGprojectionisequivalenttorequir(ilns)grt≡hatltrhe 1,...,j 1 , j , j+1 , j+2,...,n ,findtheoptimal
{ − } { } { } { }
sum of the new effective energies E[n] is minimal which, tensoratsitej andmovetothenextsite j+1. Theway
α
according to (1), corresponds to the trace of H[n] and onefindstheoptimaltensoratsitej isbyconsideringthe
eff reduced density matrix (DM) for 1,...,j , obtained by
leads to a cost function { }
tracingtheDMofthecompletesystem(“universe”)over
f(A)=tr(AHH[n]A) where AHA=1 (3) the environment j+1,...,n whereas the universe can
ext
{ }
beeitherinapurestate(groundstate)orinamixedstate
whichisminimizedexactlybytheNRGisometryU. This
of lowest energy states as is the case with the targeting.
cost function is different from other DMRG-based ap-
Formally, the eigenstates of the universe can be written
proaches (e.g. [8, 12]) where the cost is determined from
as Ψ = G ϕ1,...,j−1 s s ϕj+2,...,n
the ground state alone. | α(cid:105) (l,sj,sj+1,r),α| l (cid:105)| j(cid:105)| j+1(cid:105)| r (cid:105)
where the index α enumerates the states and
Variational optimization. The identification of the the optim(cid:80)al tensor A[j] is obtained by the SVD
NRG cost function allows us to optimize not only
as G = A[j]sjσ V . Identifying
the tensor associated with the last site in the NRG- (lsjsj+1r)α c lc c csj+1αr
MPS (2) but an arbitrary tensor in the chain. The Al[j,r+1]sj+1α ≡ σlVlsj+1(cid:80)αr, we recover a NRG-MPS with
cost f(A[j]) = ψ[n] Hˆ ψ[n] can easily be under- the index α associated with the site j+1 (instead of site
stood as a tensorαn(cid:104)etwαo|rk|dαep(cid:105)icted on Fig. 1 where n). However, thispartisignoredintheDMRGsincethe
the α-index enum(cid:80)erating states Ψ[n] is contracted with tensor at site j +1 is obtained in the subsequent step.
α
the α-index in the conjugate |MPS(cid:105) Ψ[n] . The cost Therefore, the index α is intrinsic to but hidden in the
(cid:104) α | DMRG and is carried back and forth along the chain.
can thus be minimized by varying an arbitrary isome-
In the DMRG, only the central sites are optimized (the
try A[j] in the NRG-MPS (2) which we will illustrate
boundary parts are treated exactly). We can however
by expressing the Hamiltonian as a matrix product op-
sweep to the very end of the chain and end up exactly
erator (any 1D operator can be written as a MPO)
with the NRG-MPS (2) where the last tensor A[n] car-
Hˆ = {sj}e0·H[1]s1···H[n]sne0|s(cid:48)1,...,s(cid:48)n(cid:105)(cid:104)s1,...,sn| riestheαindexwhichnowallowsustoenlargethechain
where(cid:80)tensors H[j] are represented by squares on Fig. 1. for an extra site by a NRG step. Looking back, we re-
An arbitrary isometry A[j] is extracted out of the cost alize that the optimal tensor G is an isometry
(lsjsj+1r)α
function (3) which now takes a form matrix minimizing the same cost function (3) as in the
NRG context, just that it represents an arbitrary pair of
f(A)= tr(L[j]AR[j]TAH) for AHA=1 (4)
γ γ neighboringsitesinsteadofthelastsiteonly. Hence, the
(cid:88)γ DMRG minimizes the same cost function as the NRG.
where L[j] and R[j] correspond to the contractions of We also realize that the truncation in the SVD de-
the tensor networks on the left and right side of the creases the accuracy of states and it is not guaranteed
chosen site j, respectively (Fig. 1), explicitly defined as that the optimization at the next site will recover the
[Lγ[j]](α(cid:48)s(cid:48))(αs) = Lαsγs(cid:48)α(cid:48) and [Rγ[j]]β(cid:48)β = Rβγβ(cid:48). Fur- same accuracy (in fact the highest accuracy is reached
thermore,amodifiedweightedcostfunctionmaybecon- in the center of the chain). Therefore, the set of excited
sidered where the effective states are weighted accord- states obtained by the DMRG can be further optimized
ing the their importance, e.g. by a Boltzmann factor by means of the variational principles proposed earlier.
3
10−8
D=50
D=100 10−9 D=100
10−2 10−10 )
2Hi 2Hi10−11 nrg
−h −h10−12 dmrg D=150
2i 2i vnrg
Hh Hh10−13
10−3 10−14
vdnmrrgg 10−15
0 20 40 60 80 100 10−160 20 40 60 80 100 120 140
Spectrallevel Spectrallevel
FIG. 2. Accuracy of the low energy spectral levels for the FIG.3. Accuracyoftheeigenstatesfornon-interactingSIAM
tiltedquantumIsingchain(5)on200siteswithh =h =1, chain (6) on 70 sites (N = 68) for NRG, DMRG, and the
x z
obtainedbytheDMRG(red)andoptimizedbythevariational variational NRG, for D ∈ {100,150}. We have set Λ = 1.7,
method (blue), for various bond dimensions D. ξ =0.1, (cid:15) =−0.1.
0 f
coupled to a chain of fermions (c ) as
j,σ
Results. The variational optimization algorithm for a
NRG-MPS set of states is put to the test on two qual- N−1
itatively distinct models. First, we consider a quantum H =(cid:15) n+Un n + χf†c + t c† c +h.c.
N f ↑ ↓ σ 0σ j jσ j+1σ
Ising chain in a tilted magnetic field (cid:88)σ (cid:16) (cid:88)j=0 (cid:17)
(6)
with n f†f , n=n +n , χ ξ0, and t Λ−j/2
n−1 n σ ≡ σ σ ↑ ↓ ≡ π j ∝
H = σjxσjx+1+ (hxσjx+hzσjz) (5) (see [4] for details). The NRG Ham(cid:113)iltonian HN presents
j=1 j=1 an effective description of the Anderson problem in the
(cid:88) (cid:88)
energy interval [ΛN+1,ΛN] and only the complete se-
quence (H ,H ,...) gives the complete spectrum. As
0 1
which is a simple example of non-integrable quantum
mentioned in the introduction, the NRG works best for
spinchains. Theaccuracyoftheapproximateeigenstates
strong scale separation, i.e. Λ 1 which however take
of(5)isquantifiedbyameasure H2 H 2 whichputs (cid:29)
(cid:104) (cid:105)−(cid:104) (cid:105) one further away from the continuum limit Λ 1 and
a lower bound to the fidelity (see supplementary mate- →
yield lower resolution at higher frequencies. Contrar-
rial. We consider a chain of 200 sites, obtain the excited
ily, the DMRG does not rely on energy scale separa-
states by means of the DMRG and optimize them us-
tion and thus quickly falling hopping terms but is nu-
ing the variational approach. The DMRG provides pro- merically more costly, scaling as O(D4) (here M = D).
videshighlyaccurateresultsasseenfromtheFig.2,leav-
Thissuggestsagreatpotentialinoptimizingtheexisting
ing little space for improvements. However, the results
NRGsetofstatesusingthevariationalmethodproposed
are not optimal and can be improved by the variational in this Letter where the costs only scale as O(D3), as
NRG. Higher improvement is observed for smaller bond
in the NRG, but, like in the DMRG, no scale separa-
dimensions D which suggests a possibility of improving
tion is required. Let us first consider the exactly solv-
the excited states in the regime where sufficiently large
ablenon-interactingSIAMHamiltonianwithU =0with
D are not reachable due to higher entanglement, such as
a discretization parameter Λ = 1.7. We observe from
in two-dimensional or critical (see supp. material) sys-
Fig. 3 that for a fixed bond dimension of a NRG-MPS,
tems. It is also worth noting that the computational
the accuracy of eigenstates obtained by the DMRG and
costs of one DMRG sweep with targeting M states scale
the variational optimization (vNRG) is several orders of
as O(nd3MD3), compared to either O(MD3 +nd3D3)
magnitude better than the NRG results. Qualitatively
or O(nd3D3) for the variational optimization where the
the same results are obtained for the absolute errors of
α-index is associated with an inner or a boundary site,
energies (see supplementary material).
respectively. Thepre-factorstoO(MD3)aresimilar(re-
We now consider the interacting SIAM Hamilto-
lated to the number of Lanczos steps).
nian (6) with U = 2(cid:15) = 0.1 on 40 sites where, un-
f
−
As the second example we consider the Single Impu- like the non-interacting case, we employ the symmetries
rity Anderson model (SIAM) which, after a logarithmic (total particle number of either spin). We analyze the
discretization [4] of the conductance band, is described improvement of the first one thousand NRG states af-
by a semi-infinite linear chain where the impurity (f ) is ter variational optimization (Fig. 4) in the regime where
σ
4
10−6 sions with H. G. Evertz, A. Gendiar, K. Held, T. Pr-
uschke, G. Sangiovanni, A. Toschi, and R. Zˇitko, and
nrg/64/6000
financialsupportbytheEUprojectQUEVADISandthe
10−7 nrg/100/8000 FWF SFB project ViCoM. The computational results
2i nrg/128/10000 were in part achieved using Vienna Scientific Cluster.
H
h
−10−8
i
2
H
h vnrg/64/6000
10−9
vnrg/100/8000 [1] S. R. White, Phys. Rev. Lett. 69, 2863 (1992).
vnrg/128/10000
[2] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
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Spectrallevel (1995); G.Vidal,Phys.Rev.Lett.93,040502(2004); F.
Verstraete, D. Porras, and J. I. Cirac, Phys. Rev. Lett.
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SIAMchain(6)on40sites(N =38),forNRGandthevaria- [4] R.Bulla,T.A.Costi,andT.Pruschke,Rev.Mod.Phys.
tionalNRGwithsymmetries. Sub-sectorbonddimensionsare 80, 395 (2008).
taken D ∈ {64,100,128}, truncated to {6000,8000,10000} [5] I. Schneider, A. Struck, M. Bortz, and S. Eggert, Phys.
states of lowest energy. We have set Λ = 2, ξ = 0.01, Rev. Lett. 101, 206401 (2008).
0
(cid:15) =−0.05 and U =0.1. [6] U. Schollwoeck, Rev. Mod. Phys. 77, 259 (2005).
f
[7] S. Nishimoto, F. Gebhard, and F. Jeckelmann, J. Phys.:
Condens. Matter 16, 7063 (2004).
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NRGalreadyproducesaccurateresultsduetoscalesepa- Cirac,andJ.vonDelft,Phys.Rev.B80,165117(2009).
ration (Λ=2). Due to the symmetries, the bond dimen- [9] M. Karski, C. Raas, and G. S. Uhrig, Phys. Rev. B 77,
sion D 64,100,128 (which determines the computa- 075116 (2008).
∈{ } [10] E. Jeckelmann, Phys. Rev. B 66, 045114 (2002).
tional complexity) now refers to the maximal number of
[11] C.RaasandG.S.Uhrig,Eur.Phys.J.B45,293(2005).
states in individual subsectors whereas the total number
[12] H.Saberi,A.Weichselbaum,andJ.vonDelft,Phys.Rev.
ofkeptstatesisdenotedbyM 6000,8000,10000 . As
∈{ } B 78, 035124 (2008).
shown on Fig. 4, the accuracy of eigenstates is greatly [13] W. Hofstetter, Phys. Rev. Lett. 85, 1508 (2000).
enhanced by the optimization. While the cost of an op- [14] F.B.AndersandA.Schiller,Phys.Rev.Lett.95,196801
timization step is comparable to the cost of a NRG step, (2005).
a smaller bond dimension is needed to represent states [15] R. Zˇitko and T. Pruschke, Phys. Rev. B 79, 085106
(2009).
using the vNRG as compared to the NRG. This effect
[16] A.Georges,G.Kotliar,W.Krauth,andM.J.Rozenberg,
is more pronounced for small values of Λ while for larger
Rev. Mod. Phys. 68, 13 (1996).
valuesofΛtheNRGalreadygiveshighlyaccurateresults,
[17] R. M. Konik and Y. Adamov, Phys. Rev. Lett. 98,
only requiring little improvement. The results are con- 147205 (2007); G. P. Brandino, R. M. Konik, and G.
sistentalsoforthefidelityofexcitedstatesandtheabso- Mussardo, J. Stat. Mech. (2010) P07013.
lute errors of energies (see supplementary material). We [18] T. Abrudan, J. Eriksson, and V. Koivunen, Signal Pro-
stress that the logarithmic discretization is not needed cessing 89, 1704 (2009).
[19] H.G.Evertz,M.Ganahl,K.Held,I.Piˇzorn,T.Pruschke,
for the variational scheme proposed in this Letter.
and F. Verstraete, in preparation.
Conclusion. We have proposed a variational method
to simulate or optimize the effective low energy descrip-
tion of one dimensional quantum systems, introducing a
feedbackmechanismtotheNRGandidentifyingthecost
function. Furthermore, we have expressed the DMRG
method with targeting as a NRG method with a move-
able external index enumerating states and shown that
the DMRG results can be further improved by the vari-
ational optimization. We have tested the methods on
the quantum Ising chain in a tilted magnetic field and
the Single Impurity Anderson model where we observed
significant improvement of the approximate low energy
eigenstates. The proposed technique can be directly ap-
pliedtooptimizeexistingNRGresults[19]andcouldbea
beneficialimprovementofimpuritysolversinthecontext
of dynamical mean-field theory, see e.g. [16].
Acknowledgments. We acknowledge fruitful discus-
5
Supplementary material
In this section we sketch the algorithm proposed in the main text of the Letter, give a proof on the validity of the
measure used to quantify the accuracy of approximate eigenstates, and present some further results supporting the
accuracy of the proposed method.
Variational NRG algorithm block R[n] RM×1×M which now contains the sum over
∈
1. Perform n NRG steps to obtain {ψi,i=1,...,M} M energies and can be written as
(or just choose some random states).
2. Choose a site j :=n−1, initialize L[0] and R[n]. Rr[n,q],r(cid:48) ≡δq,0δr,r(cid:48)wr.
3. Calculate/retrieve tensors Ll,p,l(cid:48) and Rr[j,q],r(cid:48). where wr is some chosen weight function, e.g. wr = 1
4. Define G , R and X and minimize f(X) (no weighting) or w = e−r (position weighting) or
q q r
5. A[j]s :=[X] wr = e−(Er−E0) (energy weighting). In case of energy
l,r (l,s),r weighting, we use the initial approximation for the en-
6. If converged, then stop.
ergies and re-calculate the boundary block after each
7. if (j ==n), update wj and recalculate R[n]. sweep.
8. j :=j−1 or j :=j+1 and go to 3. For the unitary optimization we formally reshape ten-
sors L[j] R[j], O[j], and A[j] to matrices G , R and X
q q
TABLE I. Sketch of the VNRG algorithm for the variational as [G ] · = L[j] O[j]s(cid:48)s, [R ] R and
optimization of a NRG-MPS set of states. q (l,s),(l(cid:48),s(cid:48)) p l,p,l(cid:48) p,q q i,j ≡ i,q,j
[X] A[j]s. The idea behind this matrix reshap-
(l,s),r ≡ l,r (cid:80)
ingisthatwecannowusesomeblack-boxroutinewhich
Practical Algorithm minimizes the following cost function
f(X)= tr GTXR XT (A-2)
In Table I we provide a practical algorithm for the q q
variational optimization of the NRG states. Let us as- (cid:88)q (cid:0) (cid:1)
sume we are interested in the M lowest energy states of
under the constraint that XT X = 1. The algorithm
a system on sites 1,...,n which we write as [8] ·
which gives the best results according to our experience
n was proposed in Ref. [18].
ψ = IA[1]s1 A[n]sn µ s . (A-1) Relation to the NRG. The MPS description of our de-
α j
| (cid:105) (cid:104) | ··· | (cid:105) | (cid:105)
(cid:88)sj (cid:79)j=1 scription(A-1)istightlyconnectedtotheNRGalgorithm
where one diagonalizes an effective hamiltonian H
N+1
Here, the left boundary vector is a trivial vector I
(1,0,0,...) and µ (0,...,0,1,0,...) where the one≡is (Eq. 42 in [4]) and obtains eigenvalues EN+1 and eigen-
atthepositionµ ≡1,...,M . Wewillonlyconsiderthe vectors |w(cid:105)N+1 which are related to the eigenvectors of
∈{ } the smaller system (N sites) as (Eq. 43 in [4])
casewithoutemployingsymmetriesasthegeneralization
is straight forward. We will use a MPO representation
w = U(w,rs)r;s .
for the Hamiltonian operator, | (cid:105)N+1 | (cid:105)N+1
rs
(cid:88)
n
H = (cid:104)I|O[1]s(cid:48)1,s1···O[n]s(cid:48)n,sn|I(cid:105) |s(cid:48)j(cid:105)(cid:104)sj|. TtichaelctooetffihceietnentssoUr(cwo,erffisc)ieinnttshAe[aNb+o1v]seureseladtiionn(aAr-e1)id.en-
{s(cid:88)j,s(cid:48)j} (cid:79)i=1 Relation to DMRG with targer,twing. The main differ-
We assume real arithmetics, the generalization to com- ence between the DMRG with targeting (DMRGt) and
plex arithmetics is trivial. the VNRG is that the former optimizes the set of states
Inthealgorithm,weemploythefundamentalconcepts by moving the external bond along the chain whereas in
ofthematrixproductstateformalism,suchascalculating the latter the external bond is fixed to a chosen site. In
the blocks in a recursive way, theDMRGt,theenergies(andthusourcostfunction)do
notmonotonicallydecreasewhenmovingthebondalong
L[rj,q],r(cid:48) = L[lj,p−,l1(cid:48)]Al[j,r−1]sAl[j(cid:48),−r(cid:48)1]s(cid:48)Op[j,−q1]s(cid:48),s the chain but there exists a site in the chain where the
l,l(cid:48),s,s(cid:48),p energies are minimal. Typically, in models with trans-
(cid:88)
lation invariance such as the Ising model or the Heisen-
and similarly for the right block Rr[j,q],r(cid:48). The starting berg model, the highest accuracy is reached when at the
block is trivial, L[0] I . The only step which dif- center of the chain. In impurity models with falling hop-
1×1×1
≡
fers from the MPS formalism is initialization of the right pingterms, however, theoptimalsitetoputtheexternal
6
leg is (for long chains) the end of the chain - in which
case the cheaper variant (scaling as D3) of the VNRG is
preferable. We have described the cheaper variant of the 10−9 nrg/4000
VNRG in the manuscript but we have skipped the more nrg/6000
expensive one where the external leg is associated with 10−10
some inner site. In fact, to reduce computational com- F
−
plexity, we do not actually associate the external leg to 110−11
asitebutratherintroduceanadditional3-dimtensoron
thebondconnectingtwosites: thethirdleg(theexternal
one,similarlytothe“physical”leginthematrixproduct 10−12 vnrg/4000
state)playstheroleoftheexternalleg. Thisway,theop- vnrg/6000
timizationofthemoreexpensivevariantoftheVNRGis 10−130 200 400 600 800 1000
asfollows. Alltensorsassociatedtophysicalsitesareop- Spectrallevel
timized as in the cheaper VNRG, that is minimizing the
FIG. 5. Fidelity of the first 1000 approximate eigenstates
cost function (A-2) under a unitary constraint. This in
of the SIAM on 40 (N = 38) sites obtained by NRG and
total costs nO(D3) for one sweep along the chain. The
vNRG and total number of kept states M ∈ {4000,6000}.
additional tensor which selects the eigenstates is opti- We have set U = −2(cid:15) = 0.1, ξ = 0.01, and Λ = 2 and
f 0
mized in the same way as all tensors in the DMRG with used (sub-sector) bond dimension D = 5000. The basis for
targeting. TheDMRGwithtargetinginvolvesfindingM comparisonaretheNRGresultswithM =16000keptstates
lowest eigenstates of a sparse (Dd)2 (Dd)2 matrix and (and D=6000).
×
similarly in our case where we seek M lowest eigenstates
of a sparse D2 D2 matrix. The cost of the DMRGt is
thus nO(M˜(Dd×)3) for one sweep whereas the equivalent The second term on r.h.s is upper bounded by (cid:15)4,
step contributes O(M˜D3) to the VNRG, just once in a
G2
sweep. (cid:15)2(1 (cid:15)2) . (A-7)
− ≤ Φ (H E )2 Φ
j j j
(cid:104) | − | (cid:105)
On accuracy of the eigenstates The denominator is lower bounded by ∆2 which, assum-
j
ing (cid:15)2 < 1 and thus (1 (cid:15)2)−1 < 2, finally gives us
2 −
Statement. Let H be a real symmetric linear opera- the upper bound on (cid:15) and the lower bound on fidelity
tor with eigenvectors Ψ such that H Ψ = E Ψ , F =√1 (cid:15)2 1 (cid:15) (for (cid:15) 1) as
{ j} | j(cid:105) j| j(cid:105) − ≥ − ≤
and let Ψ˜ be an approximation to Ψ with fidelity
j j
F = Ψj Ψ˜j . If F > √1 , then (cid:15)2 <2G2 = F 1 √2 G . (cid:4) (A-8)
|(cid:104) | (cid:105)| 2 ∆2 ⇒ ≥ − ∆
j j
1−F ≤ √∆2j (cid:104)Ψ˜j|H2|Ψ˜j(cid:105)−(cid:104)Ψ˜j|H|Ψ˜j(cid:105)2 1/2 (A-3) The only assumption we made was (cid:15)2 < 21 and ∆j (cid:54)=0.
(cid:16) (cid:17) On Figure 5 we present the fidelity for the interacting
where ∆j =minj(cid:48)(cid:54)=j Ej Ej(cid:48) is the spectral gap to the single impurity Anderson model (SIAM) with respect to
| − |
nearest eigenlevel. We have assumed that all considered very precise results obtained with the total number of
states are normalized with respect to the Euclid norm. statesM =16000andabonddimensionD =6000inin-
We assume real algebra (generalization to complex alge- dividualsubsectors,suchthattheactualnumberofstates
bra is trivial). in individual subsectors after truncation to in total M
Proof. Let us write the approximation Ψ˜ as states is always smaller than D and thus no additional
j
truncationismadeinsubsectors. Wecomparetheresults
Ψ˜ = 1 (cid:15)2 Ψ +(cid:15)Φ (A-4) obtained with M = 4000 and M = 6000, in both cases
| (cid:105)j − | j(cid:105) | j(cid:105) again using a sufficiently large D. Let us stress that we
(cid:112)
where (cid:15) 0, ΦΨ = 0 and ΦΦ = Ψ Ψ = 1. It compare the NRG and vNRG states to the very precise
j j j
≥ (cid:104) | (cid:105) (cid:104) | (cid:105) (cid:104) | (cid:105)
is easy to show that the measure G2 Ψ˜ H2 Ψ˜ statesobtainedbytheNRGwhichwerenotfurtheropti-
j j
Ψ˜ H Ψ˜ 2 can be written as ≡ (cid:104) | | (cid:105)− mized in order to avoid possibly unfair comparison with
j j
(cid:104) | | (cid:105) preferenceforvNRG(nochangeinthefidelityisobserved
G2 =(cid:15)2 Φ (H E )2 Φ (cid:15)4( Φ H Φ E )2 (A-5) even if we do optimize the results for M = 16000). We
j j j j j j
(cid:104) | − | (cid:105)− (cid:104) | | (cid:105)− observethattheaccuracyoftheapproximateeigenstates
or as an expression for (cid:15) as isalwaysimprovedbythevariationalmethodproposedin
the manuscript. The improvement is most significant for
(cid:15)2 = G2 +(cid:15)4((cid:104)Φj|H|Φj(cid:105)−Ej)2. (A-6) those states with a low energy, since the higher excited
Φ (H E )2 Φ Φ (H E )2 Φ statesaremoreentangled(seee.g. V.Alba,M.Fagotti,P.
j j j j j j
(cid:104) | − | (cid:105) (cid:104) | − | (cid:105)
7
10−1
10−2 D=100 m.l.sp.] 100 D=50 dvnmrrgg
gy [
orofener1100−−43 ndrmgrg D=150 heenergy10−1 D=100
Absoluteerr1100−−65 vnrg uteerroroft10−2 D=150
10−7 bsol
10−80 20 40 60 80 100 120 140 A10−30 20 40 60 80 100 120 140 160
Spectrallevel Spectrallevel
100
vnrg
nrg/64/6000 10−2 dmrg
gy10−1 nrg/100/8000 D=50
er
errorofen10−2 vnrg/64/6000 nrg/128/10000 2Hi−hi10−3 D=100
Absolute1100−−43 vnrg/100/8000 2Hh10−4 D=150
vnrg/128/10000
10−50 200 400 600 800 1000 0 20 40 60 80 100 120 140
Spectrallevel Spectrallevel
FIG.6. Absolutedifferenceofenergiesofthenon-interacting FIG. 7. Absolute error of energies (top) and the measure
SIAM(top)andtheinteractingSIAM(bottom). Compareto (cid:104)H2(cid:105)−(cid:104)H(cid:105)2forthequantumIsingchainof200sitesincritical
Figs. 3 and 4 in the Letter. magnetic field.
Calabrese,J.Stat.Mech.0910,P10020,2009forananaly- field. Similar to the tilted Ising presented in the Letter,
sis on integrable spin systems) and require a larger bond wefirstcalculateDexcitedstateswithabonddimension
dimension in the description with matrix product states. D by means of the DMRG with targeting and then im-
A single approximate eigenvalue in general does not provetheresultsusingthevariationalscheme. Thequan-
tell anything about the quality of the corresponding ap- tum Ising chain in transverse field is a quadratic model
proximateeigenvectorexceptforthegroundstate(orthe and as such exactly solvable which allows us to compare
state on the opposite side of the spectrum). We consider the obtained energies (eigenvalues) to the exact values.
the first M lowest lying states, so the situation might We observe (Fig. 7) a very good qualitative agreement
be similar also in our case but we are not able to prove between the measure H2 H 2 which quantifies the
it. On the other hand, good eigenvectors automatically accuracyoftheeigenst(cid:104)ates(cid:105),a−nd(cid:104) th(cid:105)eabsoluteerrorsofthe
also mean good eigenvalues, seen from the fact that the energies. Note that in this case, the improvement on the
eigenvalue error is bounded from above by DMRGismoresignificant,sincethemodeliscriticaland
theexcitedstatescarrymoreentanglement-andthusde-
( Ψ˜ H Ψ˜ E)2 (cid:15)4∆2. (A-9)
(cid:104) | | (cid:105)− ≤ j mand a higher bond dimension in the MPS description.
Weobservethattheresultsforthedifferenceinenergies,
H E ,andthemeasure H2 H 2 arequalitatively
|(cid:104) (cid:105)− | (cid:104) (cid:105)−(cid:104) (cid:105)
very similar, as shown on Fig. 6 where we plot the error
Expectation values
of energies for the same data used as in Figs. 3 and 4 in
the Letter.
The accuracy of the expectation values is connected
to the fidelity. The zero temperature Green function is
Some additional results determined by matrix elements Ψ d 0 and the cor-
j σ
(cid:104) | | (cid:105)
responding energy differences E E where E is the
j 0 0
−
WealsotestthevariationalNRGmethodonanexam- energy of the ground state, and also Ψ d† 0 and the
(cid:104) j| σ| (cid:105)
pleofquantumIsingchainincriticaltransversemagnetic corresponding energy differences, see [4] for details. To
8
vnrg/10000 nrg/4000 vnrg/4000 nrg/6000 vnrg/6000 obtain the complete Green function, one would have
0.0122411109 0.0122409812 0.0122411285 0.0122410278 0.0122411168 to merge contributions from all NRG patches (i.e. for
0.0115338997 0.0115338108 0.0115339045 0.0115338277 0.0115339101 all system sizes) which is beyond the scope of this
0.0106204130 0.0106203467 0.0106204772 0.0106203414 0.0106204360 manuscript. We instead only calculate the ten largest
0.0088452899 0.0088452096 0.0088453015 0.0088452296 0.0088452934 matrix elements mentioned before and compare the ac-
0.0071215907 0.0071215287 0.0071215980 0.0071215428 0.0071215907 curacy before and after the variational optimization and
0.0066356189 0.0066202112 0.0066266177 0.0066353419 0.0066353341
showtheresultsinTableII.Weobservethatbythevaria-
0.0051098361 0.0051098100 0.0051098444 0.0051097957 0.0051098185
tionaloptimizationwegainroughlyonedigitofaccuracy
0.0049692462 0.0049637898 0.0049721638 0.0049693508 0.0049692198
in the matrix elements.
0.0039662029 0.0039661700 0.0039662269 0.0039661751 0.0039662175
0.0034311308 0.0033144374 0.0033174038 0.0034307370 0.0034306428
Asseenfromthematrixelements, theonlystatesthat
TABLEII.Thelargesttenvaluesof|(cid:104)ψ |d |0(cid:105)|fortheSIAM contribute to the Green functions are those which have
j σ
on 40 sites, U = 0.1, Λ = 2, ξ = 0.01, (cid:15) = −0.05. The one particles less or more than the ground state, that is
0 f
unreliable digits are colored red. the states which live in the subsectors d 0 and d† 0 ,
σ| (cid:105) σ| (cid:105)
respectively. We may therefore restrict the optimization
10−3 toonlythesetwosectorwhichenhancestheaccuracyand
reduces the computational costs. The ground state can
ergy10−4 nrg/4000 bMePoSpttiemchizneidquiensd.eIpnenFdige.nt8lywefroshmowthtahtabtyimthperoDveMmRenGtoorf
ofen10−5 nrg/6000 twhheereenetrhgeiebsaisnistohfecosumbpsaecritsoorncaornetatihneinNgRtGhersetsautletsdwσi|t0h(cid:105)
or
uteerr10−6 vnrg/4000 Msect=or1b6o0n0d0.dimInenthsiiosncaDse=w6e00u0sesuscuhffitchieanttnlyoaladrdgietisounba-l
ol
Abs10−7 vnrg/6000 tNrRunGcarteisounltsiswdhoenreeinintotthaelMsub-sec4t0o0r0s,6a0n0d0wsetattaeksewtehree
∈{ }
kept.
10−80 50 100 150 200 250 300 350 400 450
Spectrallevel For finite-temperature Green functions we also need
excitedstatesinothersymmetrysubsectorswithrespect
FIG. 8. Absolute error of energies in the subsector dσ|0(cid:105) for tothetotalparticlenumberandtotalSz. However,they
SIAM on 40 sites, U =0.1, Λ=2, ξ =0.01, (cid:15) =−0.05.
0 f can all be optimized independently and in parallel.