Contractor renormalization (CORE) is a real-space renormalization-group
method to derive effective Hamiltionians for microscopic models. The original
CORE method is based on a real-space decomposition of the lattice into small
blocks and the effective degrees of freedom on the lattice are tensor products
of those on the small blocks. We present an extension of the CORE method that
overcomes this restriction. Our generalization allows the application of CORE
to derive arbitrary effective models whose Hilbert space is not just a tensor
product of local degrees of freedom. The method is especially well suited to
search for microscopic models to emulate low-energy exotic models and can guide
the design of quantum devices.Comment: 5 pages, 4 figure

A. Fabricio Albuquerque

Helmut G. Katzgraber

K. Lanczos

Matthias Troyer

English

arXiv

Albuquerque, A. Fabricio

Katzgraber, Helmut G.

Troyer, Matthias

Texas A&M University

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ENCORE: An Extended Contractor Renormalization algorithm
A. Fabricio Albuquerque,1,2 Helmut G. Katzgraber,1, 3 and Matthias Troyer1
1Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland
2School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia
3Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA
(Dated: September 18, 2018)
Contractor renormalization (CORE) is a real-space renormalization-group method to derive effective Hamil-
tionians for microscopic models. The original CORE method is based on a real-space decomposition of the
lattice into small blocks and the effective degrees of freedom on the lattice are tensor products of those on the
small blocks. We present an extension of the CORE method that overcomes this restriction. Our generalization
allows the application of CORE to derive arbitrary effective models whose Hilbert space is not just a tensor
product of local degrees of freedom. The method is especially well suited to search for microscopic models to
emulate low-energy exotic models and can guide the design of quantum devices.
PACS numbers: 87.55.kd,03.67.Ac,02.70.-c,03.67.Pp
I. INTRODUCTION
Identifying the emergent low-energy degrees of freedom in
a strongly-correlated system is a highly nontrivial problem re-
quiring considerable physical intuition and a careful analysis
of available experimental data [1]. The contractor renormal-
ization (CORE) method introduced by Morningstar and We-
instein [2, 3] is a tool to systematically perform this task: by
suitably selecting low-energy local degrees of freedom and
applying a real-space renormalization procedure, one can in
principle obtain an effective Hamiltonian which is simpler
than the original one and therefore (ideally) more amenable
to subsequent analytical or numerical treatment. For recent
applications of CORE see, for example, Refs. 4, 5, 6, 7, 8, 9.
The idea behind CORE is to divide the lattice on which
the model is defined into blocks and to retain only a small
number of suitably chosen low-lying block eigenstates. The
low-energy eigenstates of the full Hamiltonian defined on a
cluster formed by two or more blocks are then projected onto
the restricted basis formed by tensor products of the retained
block states. By requiring that the low-energy spectrum of
the full problem is exactly reproduced, an effective Hamilto-
nian is obtained. The mapping onto a coarser-grained lattice
with redefined degrees of freedom is done at the expense of
having longer-range effective interactions. A successful ap-
plication of the method relies on a fast decay of the effective
interactions, which in turn depends on the correct choice of
the effective degrees of freedom and on the particular way the
lattice is divided into blocks, as well as on how the retained
block eigenstates are chosen. Because physical intuition and a
good idea of the relevant local degrees of freedom are needed
to obtain physically sound results, we believe that this is the
main reason why CORE has not found a more widespread use.
The “inverse” problem of using CORE to find microscopic
models that map well onto a desired effective low-energy
Hamiltonian does not suffer from the aforementioned prob-
lems: because the emergent degrees of freedom are known a
priori and their adequacy in describing the low-energy physics
of the device is enforced by design, the aforementioned limi-
tations of CORE can be used to our advantage. Whenever the
effective Hamiltonian includes sizable long-range interactions
and/or states with a vanishing projection on the restricted basis
are present in the low-energy spectrum, one can conclude that
the considered microscopic model is not well approximated
by the proposed low-energy effective model. Given current
interest in the emulation of exotic phases via physical models
(e.g., by using Josephson junctions or cold atomic/molecular
gases), we expect this approach to be useful when designing
manipulable quantum tool boxes. Finally, we note that the step
of dividing the lattice into blocks is no longer required or even
desirable within this context and we thus extend the method to
models built from geometrically-constrained degrees of free-
dom, such as quantum dimer models [10, 11], where the emer-
gent degrees of freedom cannot be described in terms of tensor
products of local states. Below we introduce an extension of
the CORE method applicable to arbitrary basis states of the
effective model and illustrate the application of the method
with an array of quantum Josephson junctions [12] used to
implement a topologically protected qubit [13].
II. EXTENDED CORE METHOD
We first review the standard CORE algorithm [2, 3, 14, 15]
and then contrast it to the extended CORE (dubbed ENCORE)
method proposed here.
A. Standard CORE algorithm
Given a Hamiltonian H defined on a lattice L, the standard
CORE algorithm can be described as follows:
1. Divide the lattice L into disconnected small blocks
B and diagonalize the Hamiltonian H within a single
block, keepingM low-lying eigenstates {|φm〉}M1 . The
subspace spanned by tensor products of these block
eigenstates on a cluster C—formed by joining a num-
ber of elementary blocks—defines the reduced Hilbert
space within which the effective model is derived.
22. Diagonalize H on a cluster C consisting of N con-
nected blocks retaining the M = MN lowest eigen-
states {|n〉}M
1
with energies ǫn and project them
onto the basis formed by the tensor products of the
block states, {|φm1 , ..., φmN 〉}M1 , forming the pro-
jected states {|ψn〉}M1 .
3. Orthonormalize the obtained projected states {|ψn〉}M1
using a Gramm-Schmidt procedure
|ψ˜n〉 =
1
Zn
(
|ψn〉 −
∑
m<n
|ψ˜m〉〈ψ˜m|ψn〉
)
, (1)
where Zn stands for the normalization of the orthogo-
nalized state.
4. The range-N renormalized Hamiltonian is then
HrenN =
M∑
n
ǫn|ψ˜n〉〈ψ˜n|. (2)
By construction, this Hamiltonian has the same low-
energy spectrum as the original one.
5. Writing Eq. (2) in terms of the tensor product states
{|φm1 , ..., φmN 〉}
M
1
, we obtain the range-N effective
interactions between the blocks forming the cluster af-
ter subtracting the previously calculated shorter-range
interactions
hi1...iN = H
ren
N −
N−1∑
N ′=1
∑
〈i1,...,iN′〉
hi1...iN′ , (3)
where 〈i1, ..., iN ′〉 denotes the set of all connected
range-N ′ subclusters. The effective range-N Hamilto-
nian can then be written as
HeffN =
∑
i
hi +
∑
〈i,j〉
hij +
∑
〈i,j,k〉
hijk + . . . , (4)
where hi is the block self-energy, hij the interaction be-
tween nearest-neighbor blocks, hijk a three-block cou-
pling, etc. up to range-N interactions.
The successful application of the above procedure relies on a
fast decay of the long-ranged effective interactions appearing
in Eq. (4) and therefore one chooses the restricted set of de-
grees of freedom by specifying the elementary blocks B and
the retained block states {|φm〉}M1 .
B. ENCORE algorithm
It is possible to extend the ideas presented in Sec. II to
constrained effective models—e.g., quantum dimer models,
loop models, and string nets—for which the relevant degrees
of freedom are no longer formed by tensor products of block
states but, instead, by the set of configurations on a given clus-
ter satisfying the constraints of the Hamiltonian to be emu-
lated. We thus present an extended algorithm using alternative
ways of selecting the restricted degrees of freedom.
1. Choose a finite-size cluster C and build a basis
{|φm〉}
M
1
for the Hilbert space of the effective model.
In the standard CORE method this effective basis is
a tensor product of the relevant states on the blocks,
whereas here it is comprised by all constrained config-
urations on C. For example, for a quantum dimer model
we generate all M possible dimer coverings on the clus-
ter C.
2. Diagonalize H on the cluster C, calculating the M low-
est eigenstates {|n〉}M1 with energies ǫn and project
them onto the restricted basis {|φm〉}M1 , forming the
projected states {|ψm〉}M1 [16].
3. Orthonormalize by means of a Gramm-Schmidt proce-
dure as in Eq. (1).
4. The Hamiltonian within the restricted space is then
given by Eq. (2).
5. Writing this Hamiltonian in the restricted basis
{|φm〉}
M
1 we obtain the effective model
Heff =
M∑
m,m′,n
ǫn|φm〉〈φm|ψ˜n〉〈ψ˜n|φm′〉〈φm′ |. (5)
It is again possible to perform a cluster expansion
within ENCORE by using Eqs. (3) and (4).
Note that the above discussion is for an orthonormal restricted
basis {|φm〉}M1 , such as in the example discussed in Sec. III.
Small changes in the procedure are required if this is not the
case [10].
III. APPLICATION: EMULATION OF THE QUANTUM
DIMER MODEL
A. Array of quantum Josephson junctions
We apply the algorithm described in Sec. II B to extract the
dimer flip amplitude t for a Josephson-junction array intro-
duced by Ioffe et al. [12] to emulate a quantum dimer model
(QDM) [10] on a triangular lattice. This model—first investi-
gated by Moessner and Sondhi [11]—has the desired proper-
ties needed to implement a topologically protected qubit and
is given by H = H +H +H with
H = −t
∑[
| 〉〈 |+ | 〉〈 |
]
+v
∑[
| 〉〈 |+ | 〉〈 |
]
,
(6)
with similar definitions for H and H . Parallel dimers on
the same rhombus (henceforth we refer to such configurations
as flippable rhombi) flip with an amplitude t and interact via a
potential strength v; the sum runs over all rhombi with a given
orientation. Moessner and Sondhi showed that a topologically
ordered liquid phase exists over a finite region of the model’s
3CX
Ci
C
C , Jh h
FIG. 1: (Color online) Josephson junction array used to emulate
the quantum dimer model on a triangular lattice (shaded lines, el-
lipses represent the dimers; see Refs. [12] and [13] for details). X-
shaped superconducting islands (thick black lines) form a kagome
lattice with normal-state star-shaped islands (thin black lines) placed
at the center of every hexagon of the kagome lattice. Cooper pairs
hop between nearest-neighbor X-shaped islands with an amplitude
given by the Josephson current Jh. A large ratio between the capac-
itances Ci and Ch ensures an on-hexagon repulsion Ehex to emulate
the hard-core dimer constraint. Figure adapted from Ref. [13].
(a)
(c)
(b)
(d)
(a)
(c)
(b)
(d)
FIG. 2: Open-boundary clusters studied: (a) N × 2 (here N = 3)
hexagon ladders; (b) ten-hexagon cluster.
phase diagram (0.82 . v/t ≤ 1), something confirmed in a
number of subsequent studies [12, 17, 18, 19].
The Josephson-junction array (JJK) can be described by the
generalized Bose-Hubbard model
H =
1
2
∑
j,k
njCˆ
−1
j,knk − Jh
∑
〈j,k〉
(b†jbk + b
†
kbj), (7)
where the positions of the X-shaped islands in the array are
denoted by the indices j and k and 〈j, k〉 represents nearest-
neighbor pairs on the kagome lattice (see Fig. 1). nj = b†jbj
is the bosonic occupation number at site ~rj , Jh is the Joseph-
son current between two X-shaped islands, and Cˆ is the ar-
ray’s capacitance matrix. We restrict the analysis to the case
of hard-core bosons [13].
B. Two-dimer flips
In this example we focus on the off-diagonal dimer flip term
t in Eq. (6) from the microscopic model [Eq. (7)] with the fol-
lowing set of capacitances: C∗ = 1, CX = 0.25, Ci = 2.5,
a)
b)
c)
d)
FIG. 3: (Color online) Nonequivalent dimer flips in the ten-hexagon
cluster [Fig. 2(b)], comprising (a) two, (b) three, (c) four, and (d) five
dimers. Dimer flips are represented by their associated transition
graph: dimers (thick black lines) flip to new positions (thick light
lines) while observing the hard-core constraint. Only the underlying
triangular lattice of the JJK array is shown (shaded lines in Fig. 1).
The quantity Σ used for gauging the validity of the mapping onto a
QDM is defined via the multi dimer flips enclosed by dashed lines.
and Ch = 0.5 (see Fig. 1). Using ENCORE these are ob-
tained from Eq. (5): the transition amplitude between dimer
configurations |φm〉 and |φm′〉 (m 6= m′) is the matrix el-
ement Heff(m,m′). Our results have been obtained on the
clusters shown in Fig. 2.
The dominant off-diagonal term in the effective Hamilto-
nian is the two-dimer flip with amplitude t. This process in-
volves the creation of a virtual state with a doubly-occupied
hexagon, with energy Ehex, in the kagome lattice and occurs
with amplitude t ≈ J2
h
/Ehex [12, 13]. Two-dimer flips in
the cluster with ten hexagons are shown in Fig. 3(a). Al-
though the amplitudes for all these are ideally equal, there
are small deviations, e.g., by the configuration of the neigh-
boring dimers (effects of Coulomb interactions) or the open
boundaries. All two-dimer flips depicted in Fig. 3(a) can be
seen as being correlated and are considered individually at the
algorithmic level.
Figure 4(a) shows results for t as a function of the Joseph-
son coupling Jh for the various clusters and in comparison to
second-order perturbative results. Results for the ten-hexagon
cluster are obtained by averaging the amplitudes for the pro-
cesses depicted in Fig. 3(a); amplitudes for the individual pro-
cesses are shown in Fig. 4(b). The results agree up to a point
[vertical dashed lines in Figs. 4(a) – 4(c)] where the mapping
onto a QDM fails. This agreement is an indication that, for ca-
pacitances and Josephson currents leading to a valid mapping,
the low-energy physics of the JJK array is indeed described
by a QDM with local dimer resonances. Furthermore, it also
points to the absence of sizable finite-size effects in our re-
sults.
40.5 1.0 1.5 2.0 2.5 3.0
Jh (mK)
0
0.5
1.0
1.5
2.0
2.5
t (
mK
)
Perturbative results
2 x 2
3 x 2
4 x 2
5 x 2
10 Hex.
C
*
 = 1 - CX = 0.25 - Ci = 2.5 - Ch = 0.5
(a)
0.5 1.0 1.5 2.0 2.5 3.0
Jh (mK)
0
0.5
1.0
1.5
t (
mK
)
10 Hex.
0.5 1.0 1.5 2.0 2.5 3.0
Jh (mK)
0
0.05
0.1
0.15
0.2
0.25
Σ 
/ t
(b) (c)
FIG. 4: (Color online) (a) Amplitude for the two-dimer flip t in the
JJK array obtained from the ENCORE analysis of the finite clus-
ters shown in Fig. 2. The (red) solid curve represents second-order
perturbation results. (b) Results for the ten hexagon cluster are ob-
tained as the average (triangles) of the amplitudes of the two-dimer
processes depicted in Fig. 3(a). (c) Added absolute values for the
amplitudes associated to multi dimer flips (Σ). When Σ is large the
mapping onto the QDM breaks down (vertical dashed lines). Data
for C∗ = 1, CX = 0.25, Ci = 2.5, and Ch = 0.5 (adapted from
Ref. [13]).
C. Multi dimer flips: Breakdown of the mapping
Whereas a standard CORE expansion proceeds by consid-
ering clusters comprising an increasing number of sites, an
ENCORE expansion for the JJK array, due to the dimers’
hard-core constraint, is performed in terms of the number of
dimers in a cluster. The analysis of multi dimer terms can
be used to gauge the validity of the mapping onto a QDM:
large amplitudes for multi dimer flips indicate that the device
is not properly described by the effective model. We denote
the summed absolute value of the amplitudes associated to
these multi dimer flips by Σ, which are directly obtained as
the off-diagonal matrix elements of the effective Hamiltonian
associated to the multi dimer flips enclosed by dashed lines
in Figs. 3(b) – 3(d). Figure 4(c) shows Σ as a function of the
Josephson current Jh. A sudden increase in Σ at the same
value of Jh for which different results for t start to deviate
from each other in Figs. 4(a) and 4(b) indicates the breakdown
of the mapping. The appearance of “intruder states” in the
low-energy spectrum with negligible overlap with the hard-
core dimer configurations also indicate the breakdown of the
mapping. The vertical dashed line in Fig. 4(c) indicates the
point where the first intruder state appears. As Jh increases
and charge fluctuations start to dominate, intruder states dis-
playing multiply-occupied hexagons in the JJK array violat-
ing the hard-core dimer constraint have their energy lowered,
eventually causing some of the projected states {|ψm〉}M1 to
vanish.
IV. SUMMARY
We have presented an ENCORE algorithm suitable for con-
strained effective models whose basis states are not simply
tensor products of local block states. We find that CORE is
very effective in the design of quantum devices for emulating
exotic phases. The inadequacy of the restricted set of degrees
of freedom in accounting for a system’s low-energy behavior
is reflected by the presence of long-range terms in the effective
Hamiltonian obtained from CORE and is used as a criterion in
deciding on whether successful emulation is achieved.
Acknowledgments
We thank A. Abendschein for fruitful discussions.
A.F.A. acknowledges financial support from CNPq (Brazil),
NIDECO (Switzerland), and ARC (Australia). H.G.K. ac-
knowledges support from the Swiss National Science Foun-
dation under Grant No. PP002-114713. We would like to
thank the ETH Zurich Integrated Systems Laboratory and
especially Aniello Esposito for computer time on the large-
memory workstation “schreck.”
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Physical Review E

ENCORE: An extended contractor renormalization algorithm

OAKTrust Digital Repository (Texas A&M Univ)

arXiv:0805.2290v2  [cond-mat.str-el]  29 Apr 2009ENCORE: An Extended Contractor Renormalization algorithmA. Fabricio Albuquerque,1,2 Helmut G. Katzgraber,1, 3 and Matthias Troyer11Theoretische Physik, ETH Zurich, 8093 Zurich, Switzerland2School of Physics, The University of New South Wales, Sydney, NSW 2052, Australia3Department of Physics, Texas A&M University, College Station, Texas 77843-4242, USA(Dated: September 18, 2018)Contractor renormalization (CORE) is a real-space renormalization-group method to derive effective Hamil-tionians for microscopic models. The original CORE method is based on a real-space decomposition of thelattice into small blocks and the effective degrees of freedom on the lattice are tensor products of those on thesmall blocks. We present an extension of the CORE method that overcomes this restriction. Our generalizationallows the application of CORE to derive arbitrary effective models whose Hilbert space is not just a tensorproduct of local degrees of freedom. The method is especially well suited to search for microscopic models toemulate low-energy exotic models and can guide the design of quantum devices.PACS numbers: 87.55.kd,03.67.Ac,02.70.-c,03.67.PpI. INTRODUCTIONIdentifying the emergent low-energy degrees of freedom ina strongly-correlated system is a highly nontrivial problem re-quiring considerable physical intuition and a careful analysisof available experimental data [1]. The contractor renormal-ization (CORE) method introduced by Morningstar and We-instein [2, 3] is a tool to systematically perform this task: bysuitably selecting low-energy local degrees of freedom andapplying a real-space renormalization procedure, one can inprinciple obtain an effective Hamiltonian which is simplerthan the original one and therefore (ideally) more amenableto subsequent analytical or numerical treatment. For recentapplications of CORE see, for example, Refs. 4, 5, 6, 7, 8, 9.The idea behind CORE is to divide the lattice on whichthe model is defined into blocks and to retain only a smallnumber of suitably chosen low-lying block eigenstates. Thelow-energy eigenstates of the full Hamiltonian defined on acluster formed by two or more blocks are then projected ontothe restricted basis formed by tensor products of the retainedblock states. By requiring that the low-energy spectrum ofthe full problem is exactly reproduced, an effective Hamilto-nian is obtained. The mapping onto a coarser-grained latticewith redefined degrees of freedom is done at the expense ofhaving longer-range effective interactions. A successful ap-plication of the method relies on a fast decay of the effectiveinteractions, which in turn depends on the correct choice ofthe effective degrees of freedom and on the particular way thelattice is divided into blocks, as well as on how the retainedblock eigenstates are chosen. Because physical intuition and agood idea of the relevant local degrees of freedom are neededto obtain physically sound results, we believe that this is themain reason why CORE has not found a more widespread use.The “inverse” problem of using CORE to find microscopicmodels that map well onto a desired effective low-energyHamiltonian does not suffer from the aforementioned prob-lems: because the emergent degrees of freedom are known apriori and their adequacy in describing the low-energy physicsof the device is enforced by design, the aforementioned limi-tations of CORE can be used to our advantage. Whenever theeffective Hamiltonian includes sizable long-range interactionsand/or states with a vanishing projection on the restricted basisare present in the low-energy spectrum, one can conclude thatthe considered microscopic model is not well approximatedby the proposed low-energy effective model. Given currentinterest in the emulation of exotic phases via physical models(e.g., by using Josephson junctions or cold atomic/moleculargases), we expect this approach to be useful when designingmanipulable quantum tool boxes. Finally, we note that the stepof dividing the lattice into blocks is no longer required or evendesirable within this context and we thus extend the method tomodels built from geometrically-constrained degrees of free-dom, such as quantum dimer models [10, 11], where the emer-gent degrees of freedom cannot be described in terms of tensorproducts of local states. Below we introduce an extension ofthe CORE method applicable to arbitrary basis states of theeffective model and illustrate the application of the methodwith an array of quantum Josephson junctions [12] used toimplement a topologically protected qubit [13].II. EXTENDED CORE METHODWe first review the standard CORE algorithm [2, 3, 14, 15]and then contrast it to the extended CORE (dubbed ENCORE)method proposed here.A. Standard CORE algorithmGiven a Hamiltonian H defined on a lattice L, the standardCORE algorithm can be described as follows:1. Divide the lattice L into disconnected small blocksB and diagonalize the Hamiltonian H within a singleblock, keepingM low-lying eigenstates {|φm〉}M1 . Thesubspace spanned by tensor products of these blockeigenstates on a cluster C—formed by joining a num-ber of elementary blocks—defines the reduced Hilbertspace within which the effective model is derived.22. Diagonalize H on a cluster C consisting of N con-nected blocks retaining the M = MN lowest eigen-states {|n〉}M1with energies ǫn and project themonto the basis formed by the tensor products of theblock states, {|φm1 , ..., φmN 〉}M1 , forming the pro-jected states {|ψn〉}M1 .3. Orthonormalize the obtained projected states {|ψn〉}M1using a Gramm-Schmidt procedure|ψ˜n〉 =1Zn(|ψn〉 −∑m<n|ψ˜m〉〈ψ˜m|ψn〉), (1)where Zn stands for the normalization of the orthogo-nalized state.4. The range-N renormalized Hamiltonian is thenHrenN =M∑nǫn|ψ˜n〉〈ψ˜n|. (2)By construction, this Hamiltonian has the same low-energy spectrum as the original one.5. Writing Eq. (2) in terms of the tensor product states{|φm1 , ..., φmN 〉}M1, we obtain the range-N effectiveinteractions between the blocks forming the cluster af-ter subtracting the previously calculated shorter-rangeinteractionshi1...iN = HrenN −N−1∑N ′=1∑〈i1,...,iN′〉hi1...iN′ , (3)where 〈i1, ..., iN ′〉 denotes the set of all connectedrange-N ′ subclusters. The effective range-N Hamilto-nian can then be written asHeffN =∑ihi +∑〈i,j〉hij +∑〈i,j,k〉hijk + . . . , (4)where hi is the block self-energy, hij the interaction be-tween nearest-neighbor blocks, hijk a three-block cou-pling, etc. up to range-N interactions.The successful application of the above procedure relies on afast decay of the long-ranged effective interactions appearingin Eq. (4) and therefore one chooses the restricted set of de-grees of freedom by specifying the elementary blocks B andthe retained block states {|φm〉}M1 .B. ENCORE algorithmIt is possible to extend the ideas presented in Sec. II toconstrained effective models—e.g., quantum dimer models,loop models, and string nets—for which the relevant degreesof freedom are no longer formed by tensor products of blockstates but, instead, by the set of configurations on a given clus-ter satisfying the constraints of the Hamiltonian to be emu-lated. We thus present an extended algorithm using alternativeways of selecting the restricted degrees of freedom.1. Choose a finite-size cluster C and build a basis{|φm〉}M1for the Hilbert space of the effective model.In the standard CORE method this effective basis isa tensor product of the relevant states on the blocks,whereas here it is comprised by all constrained config-urations on C. For example, for a quantum dimer modelwe generate all M possible dimer coverings on the clus-ter C.2. Diagonalize H on the cluster C, calculating the M low-est eigenstates {|n〉}M1 with energies ǫn and projectthem onto the restricted basis {|φm〉}M1 , forming theprojected states {|ψm〉}M1 [16].3. Orthonormalize by means of a Gramm-Schmidt proce-dure as in Eq. (1).4. The Hamiltonian within the restricted space is thengiven by Eq. (2).5. Writing this Hamiltonian in the restricted basis{|φm〉}M1 we obtain the effective modelHeff =M∑m,m′,nǫn|φm〉〈φm|ψ˜n〉〈ψ˜n|φm′〉〈φm′ |. (5)It is again possible to perform a cluster expansionwithin ENCORE by using Eqs. (3) and (4).Note that the above discussion is for an orthonormal restrictedbasis {|φm〉}M1 , such as in the example discussed in Sec. III.Small changes in the procedure are required if this is not thecase [10].III. APPLICATION: EMULATION OF THE QUANTUMDIMER MODELA. Array of quantum Josephson junctionsWe apply the algorithm described in Sec. II B to extract thedimer flip amplitude t for a Josephson-junction array intro-duced by Ioffe et al. [12] to emulate a quantum dimer model(QDM) [10] on a triangular lattice. This model—first investi-gated by Moessner and Sondhi [11]—has the desired proper-ties needed to implement a topologically protected qubit andis given by H = H +H +H withH = −t∑[| 〉〈 |+ | 〉〈 |]+v∑[| 〉〈 |+ | 〉〈 |],(6)with similar definitions for H and H . Parallel dimers onthe same rhombus (henceforth we refer to such configurationsas flippable rhombi) flip with an amplitude t and interact via apotential strength v; the sum runs over all rhombi with a givenorientation. Moessner and Sondhi showed that a topologicallyordered liquid phase exists over a finite region of the model’s3CXCiCC , Jh hFIG. 1: (Color online) Josephson junction array used to emulatethe quantum dimer model on a triangular lattice (shaded lines, el-lipses represent the dimers; see Refs. [12] and [13] for details). X-shaped superconducting islands (thick black lines) form a kagomelattice with normal-state star-shaped islands (thin black lines) placedat the center of every hexagon of the kagome lattice. Cooper pairshop between nearest-neighbor X-shaped islands with an amplitudegiven by the Josephson current Jh. A large ratio between the capac-itances Ci and Ch ensures an on-hexagon repulsion Ehex to emulatethe hard-core dimer constraint. Figure adapted from Ref. [13].(a)(c)(b)(d)(a)(c)(b)(d)FIG. 2: Open-boundary clusters studied: (a) N × 2 (here N = 3)hexagon ladders; (b) ten-hexagon cluster.phase diagram (0.82 . v/t ≤ 1), something confirmed in anumber of subsequent studies [12, 17, 18, 19].The Josephson-junction array (JJK) can be described by thegeneralized Bose-Hubbard modelH =12∑j,knjCˆ−1j,knk − Jh∑〈j,k〉(b†jbk + b†kbj), (7)where the positions of the X-shaped islands in the array aredenoted by the indices j and k and 〈j, k〉 represents nearest-neighbor pairs on the kagome lattice (see Fig. 1). nj = b†jbjis the bosonic occupation number at site ~rj , Jh is the Joseph-son current between two X-shaped islands, and Cˆ is the ar-ray’s capacitance matrix. We restrict the analysis to the caseof hard-core bosons [13].B. Two-dimer flipsIn this example we focus on the off-diagonal dimer flip termt in Eq. (6) from the microscopic model [Eq. (7)] with the fol-lowing set of capacitances: C∗ = 1, CX = 0.25, Ci = 2.5,a)b)c)d)FIG. 3: (Color online) Nonequivalent dimer flips in the ten-hexagoncluster [Fig. 2(b)], comprising (a) two, (b) three, (c) four, and (d) fivedimers. Dimer flips are represented by their associated transitiongraph: dimers (thick black lines) flip to new positions (thick lightlines) while observing the hard-core constraint. Only the underlyingtriangular lattice of the JJK array is shown (shaded lines in Fig. 1).The quantity Σ used for gauging the validity of the mapping onto aQDM is defined via the multi dimer flips enclosed by dashed lines.and Ch = 0.5 (see Fig. 1). Using ENCORE these are ob-tained from Eq. (5): the transition amplitude between dimerconfigurations |φm〉 and |φm′〉 (m 6= m′) is the matrix el-ement Heff(m,m′). Our results have been obtained on theclusters shown in Fig. 2.The dominant off-diagonal term in the effective Hamilto-nian is the two-dimer flip with amplitude t. This process in-volves the creation of a virtual state with a doubly-occupiedhexagon, with energy Ehex, in the kagome lattice and occurswith amplitude t ≈ J2h/Ehex [12, 13]. Two-dimer flips inthe cluster with ten hexagons are shown in Fig. 3(a). Al-though the amplitudes for all these are ideally equal, thereare small deviations, e.g., by the configuration of the neigh-boring dimers (effects of Coulomb interactions) or the openboundaries. All two-dimer flips depicted in Fig. 3(a) can beseen as being correlated and are considered individually at thealgorithmic level.Figure 4(a) shows results for t as a function of the Joseph-son coupling Jh for the various clusters and in comparison tosecond-order perturbative results. Results for the ten-hexagoncluster are obtained by averaging the amplitudes for the pro-cesses depicted in Fig. 3(a); amplitudes for the individual pro-cesses are shown in Fig. 4(b). The results agree up to a point[vertical dashed lines in Figs. 4(a) – 4(c)] where the mappingonto a QDM fails. This agreement is an indication that, for ca-pacitances and Josephson currents leading to a valid mapping,the low-energy physics of the JJK array is indeed describedby a QDM with local dimer resonances. Furthermore, it alsopoints to the absence of sizable finite-size effects in our re-sults.40.5 1.0 1.5 2.0 2.5 3.0Jh (mK)00.51.01.52.02.5t (mK)Perturbative results2 x 23 x 24 x 25 x 210 Hex.C* = 1 - CX = 0.25 - Ci = 2.5 - Ch = 0.5(a)0.5 1.0 1.5 2.0 2.5 3.0Jh (mK)00.51.01.5t (mK)10 Hex.0.5 1.0 1.5 2.0 2.5 3.0Jh (mK)00.050.10.150.20.25Σ / t(b) (c)FIG. 4: (Color online) (a) Amplitude for the two-dimer flip t in theJJK array obtained from the ENCORE analysis of the finite clus-ters shown in Fig. 2. The (red) solid curve represents second-orderperturbation results. (b) Results for the ten hexagon cluster are ob-tained as the average (triangles) of the amplitudes of the two-dimerprocesses depicted in Fig. 3(a). (c) Added absolute values for theamplitudes associated to multi dimer flips (Σ). When Σ is large themapping onto the QDM breaks down (vertical dashed lines). Datafor C∗ = 1, CX = 0.25, Ci = 2.5, and Ch = 0.5 (adapted fromRef. [13]).C. Multi dimer flips: Breakdown of the mappingWhereas a standard CORE expansion proceeds by consid-ering clusters comprising an increasing number of sites, anENCORE expansion for the JJK array, due to the dimers’hard-core constraint, is performed in terms of the number ofdimers in a cluster. The analysis of multi dimer terms canbe used to gauge the validity of the mapping onto a QDM:large amplitudes for multi dimer flips indicate that the deviceis not properly described by the effective model. We denotethe summed absolute value of the amplitudes associated tothese multi dimer flips by Σ, which are directly obtained asthe off-diagonal matrix elements of the effective Hamiltonianassociated to the multi dimer flips enclosed by dashed linesin Figs. 3(b) – 3(d). Figure 4(c) shows Σ as a function of theJosephson current Jh. A sudden increase in Σ at the samevalue of Jh for which different results for t start to deviatefrom each other in Figs. 4(a) and 4(b) indicates the breakdownof the mapping. The appearance of “intruder states” in thelow-energy spectrum with negligible overlap with the hard-core dimer configurations also indicate the breakdown of themapping. The vertical dashed line in Fig. 4(c) indicates thepoint where the first intruder state appears. As Jh increasesand charge fluctuations start to dominate, intruder states dis-playing multiply-occupied hexagons in the JJK array violat-ing the hard-core dimer constraint have their energy lowered,eventually causing some of the projected states {|ψm〉}M1 tovanish.IV. SUMMARYWe have presented an ENCORE algorithm suitable for con-strained effective models whose basis states are not simplytensor products of local block states. We find that CORE isvery effective in the design of quantum devices for emulatingexotic phases. The inadequacy of the restricted set of degreesof freedom in accounting for a system’s low-energy behavioris reflected by the presence of long-range terms in the effectiveHamiltonian obtained from CORE and is used as a criterion indeciding on whether successful emulation is achieved.AcknowledgmentsWe thank A. Abendschein for fruitful discussions.A.F.A. acknowledges financial support from CNPq (Brazil),NIDECO (Switzerland), and ARC (Australia). H.G.K. ac-knowledges support from the Swiss National Science Foun-dation under Grant No. PP002-114713. We would like tothank the ETH Zurich Integrated Systems Laboratory andespecially Aniello Esposito for computer time on the large-memory workstation “schreck.”[1] R. B. Laughlin and D. Pines, PNAS 97, 28 (2000).[2] C. J. Morningstar and M. Weinstein, Phys. Rev. Lett. 73, 1873(1994).[3] C. J. Morningstar and M. Weinstein, Phys. Rev. D 54, 4131(1996).[4] M. S. Siu and M. Weinstein, Phys. Rev. B 75, 184403 (2007).[5] A. Abendschein and S. Capponi, Phys. Rev. B 76, 064413(2007).[6] M. S. Siu and M. Weinstein, Phys. Rev. B 77, 155116 (2008).[7] A. F. Albuquerque, M. Troyer, and J. Oitmaa, Phys. Rev. B 78,132402 (2008).[8] J. D. Picon, A. F. Albuquerque, K. P. Schmidt, N. Laflorencie,M. Troyer, and F. Mila, Phys. Rev. B 78, 184418 (2008).[9] A. Abendschein and S. Capponi, Phys. Rev. Lett. 101, 227201(2008).[10] D. S. Rokhsar and S. A. Kivelson, Phys. Rev. Lett. 61, 23765(1988).[11] R. Moessner and S. L. Sondhi, Phys. Rev. Lett. 86, 1881 (2001).[12] L. B. Ioffe, M. V. Feigel’man, A. Ioselevich, D. Ivanov,M. Troyer, and G. Blatter, Nature 415, 503 (2002).[13] A. F. Albuquerque, H. G. Katzgraber, M. Troyer, and G. Blatter,Phys. Rev. B 78, 014503 (2008).[14] E. Altman and A. Auerbach, Phys. Rev. B 65, 104508 (2002).[15] S. Capponi, Theor. Chem. Acc. 116, 524 (2006).[16] Although the Lanczos [20] algorithm could in principle beused, it has problems with degenerate eigenvalues. Thus, for thesmallest clusters a dense matrix eigensolver is recommended,whereas for larger clusters we use the Davidson algorithm [21].[17] A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys.Rev. B 71, 224109 (2005).[18] F. Vernay, A. Ralko, F. Becca, and F. Mila, Phys. Rev. B 74,054402 (2006).[19] A. Ralko, M. Ferrero, F. Becca, D. Ivanov, and F. Mila, Phys.Rev. B 74, 134301 (2006).[20] C. Lanczos, J. Res. Natl. Bur. Stand. 45, 225 (1950).[21] E. R. Davidson, J. Comput. 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ENCORE: An Extended Contractor Renormalization algorithm

ENCORE: An Extended Contractor Renormalization algorithm

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