In the context of real-space renormalization group methods, we propose a
novel scheme for quantum systems defined on a D-dimensional lattice. It is
based on a coarse-graining transformation that attempts to reduce the amount of
entanglement of a block of lattice sites before truncating its Hilbert space.
Numerical simulations involving the ground state of a 1D system at criticality
show that the resulting coarse-grained site requires a Hilbert space dimension
that does not grow with successive rescaling transformations. As a result we
can address, in a quasi-exact way, tens of thousands of quantum spins with a
computational effort that scales logarithmically in the system's size. The
calculations unveil that ground state entanglement in extended quantum systems
is organized in layers corresponding to different length scales. At a quantum
critical point, each rellevant length scale makes an equivalent contribution to
the entanglement of a block with the rest of the system.Comment: 4 pages, 4 figures, updated versio

G. Vidal

L. P. Kadanov

P. Calabrese

English

arXiv

Crossref

Entanglement Renormalization

We propose a real-space renormalization group (RG) transformation for quantum systems on a D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with a computational effort that scales logarithmically in the system's size. The calculations unveil that ground state entanglement in extended quantum systems is organized in layers corresponding to different length scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the entanglement of a block

Vidal, G.

Caltech Authors - Main

Entanglement Renormalization
G. Vidal1,2
1School of Physical Sciences, the University of Queensland, QLD 4072, Australia
2Institute for Quantum Information, California Institute for Technology, Pasadena, California 91125, USA
(Received 1 December 2006; published 28 November 2007)
We propose a real-space renormalization group (RG) transformation for quantum systems on a
D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining
it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that
the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG
transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with
a computational effort that scales logarithmically in the system’s size. The calculations unveil that ground
state entanglement in extended quantum systems is organized in layers corresponding to different length
scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the
entanglement of a block.
DOI: 10.1103/PhysRevLett.99.220405 PACS numbers: 05.30.d, 02.70.c, 03.67.Mn, 05.50.+q
Renormalization group (RG), one of the conceptual
pillars of statistical mechanics and quantum field theory,
revolves around the idea of coarse-graining and rescaling
transformations of an extended system [1]. These so-called
RG transformations are not only a key theoretical element
in the modern formulation of critical phenomena and phase
transitions, but also the basis of important computational
methods for many-body problems.
In the case of quantum systems defined on a lattice,
Wilson’s real-space RG methods [2], based on the trunca-
tion of the Hilbert space, replaced Kadanoff’s spin-
blocking ideas [3] with a concise mathematical formula-
tion and an explicit prescription to implement RG trans-
formations. But it was not until the advent of White’s
density matrix renormalization group (DMRG) algorithm
[4] that RG methods became the undisputed numerical
approach for systems on a 1D lattice. More recently, such
techniques have gained renewed momentum under the
influence of quantum information science. By paying due
attention to entanglement, algorithms to simulate time-
evolution in 1D systems [5] and to address 2D systems
[6] have been put forward.
The practical value of DMRG and related methods is
unquestionable. And yet, they are based on a RG trans-
formation that notably fails to satisfy a most natural ex-
pectation, namely, to have scale invariant systems as fixed
points. Instead, for such systems, the size of an effective
site (as measured by the dimension of its Hilbert space)
increases with each iteration of the RG transformation.
This fact does not only conflict with the very spirit of RG
theory, but it also has important computational implica-
tions: after a sufficiently large number of iterations, the
effective sites are unaffordably large, and the numerical
method is no longer viable.
In this Letter, we propose a real-space RG transforma-
tion for quantum systems on a D-dimensional lattice that,
by renormalizing the amount of entanglement in the sys-
tem, aims to eliminate the growth of the site’s Hilbert space
dimension along successive rescaling transformations. In
particular, when applied to a scale invariant system, the
transformation is expected to produce a coarse-grained
system identical to the original one. Numerical tests for
critical systems in D  1 spatial dimensions confirm this
expectation. Our results also unveil the stratification of
entanglement in extended quantum systems and open the
path to a very compact description of quantum criticality.
Real-space RG and DMRG.—As originally introduced
by Wilson [2], real-space RG methods truncate the local
Hilbert space of a block of sites in order to reduce its
degrees of freedom. Let us consider a system on a lattice
L in D spatial dimensions and its Hilbert space
 VL 
O
s2L
Vs; (1)
where s 2 L denotes the lattice sites and Vs has finite
dimension. Let us also consider a block B  L of neigh-
boring sites, with corresponding Hilbert space
 VB 
O
s2B
Vs: (2)
In an elementary RG transformation, the lattice L is
mapped into a new, effective lattice L0, where each site
s0 2 L0 is obtained from a block B of sites in L by coarse
graining. More specifically, the space V0s0 for site s
0 2 L0
corresponds to a subspace SB of VB,
 V 0s0  SB  VB; (3)
as characterized by an isometric tensor w, Fig (1),
 w: V0s0  VB; w
yw  I: (4)
Isometry w can now be used to map a state ji 2 VL
[typically, a ground state] into a coarse-grained state
j0i 2 VL0 . A thoughtful selection of subspace SB is
essential. On the one hand, its dimension m should be as
small as possible because the cost of subsequent tasks, such
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S
week ending
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0031-9007=07=99(22)=220405(4) 220405-1 © 2007 The American Physical Society
as computing expectation values for local observables,
grows polynomially with m. On the other hand, SB needs
to be large enough that j0i retains all relevant properties
of ji. White identified the optimal choice as part of his
DMRG algorithm [4]. Let B denote the reduced density
matrix of ji on block B. Then the optimal subspace is
 SB  hj1i; 	 	 	 ; jmii; (5)
where jii are the m eigenvectors of B with largest
eigenvalues pi, and m is such that  
 1
Pm
i1 pi, with
 a preestablished truncation error,  1.
Notice that m refers to the amount of entanglement
between B and the rest of the lattice, LB, as charac-
terized by the rank of the truncated Schmidt decomposition
 ji 
Xm
i1

pi
p
jii  jii; jii 2 VLB: (6)
In other words, the performance of DMRG-based methods
depends on the amount of entanglement in ji.
Entanglement Renormalization.—We propose a tech-
nique to reduce the amount of entanglement between the
block B and the rest of the lattice L while still obtaining a
quasiexact description of the state of the system. This is
achieved by deforming, by means of a unitary transforma-
tion, the boundaries of the block B before truncating its
Hilbert space, see Fig. 1.
Let us specialize, for simplicity, to a 1D lattice and to a
block B made of just two contiguous sites s1 and s2. Let r1
and r2 be the two sites immediately to the left and to the
right of B. Then, we consider unitary transformations u1
and u2, the disentanglers, acting on the pairs of sites r1s1
and s2r2,
 u1: Vr1  Vs1 ! Vr1  Vs1 ; u
y
1u1  u1u
y
1  I;
u2: Vs2  Vr2 ! Vs2  Vr2 ; u
y
2u2  u2u
y
2  I:
(7)
Properly chosen disentanglers reduce the short-range en-
tanglement between the block B and its immediate neigh-
borhood [7]. The original state B of block B,
 B  trr1r2
r1s1s2r2; (8)
is now replaced with a partially disentangled state ~B,
 ~ B  trr1r2u1  u2
r1s1s2r2u1  u2y; (9)
which has a smaller effective rank ~m, ~m<m. Our RG
transformation consists of two steps: (i) First we decrease
or renormalize the amount of entanglement between block
B and the rest of L. This is achieved with disentanglers
that act locally around the boundary of B [8]. The block
does not become completely disentangled because only
entanglement localized near its boundaries can be re-
moved. (ii) Then, as in Wilson’s proposal, we truncate
the Hilbert space of block B, and we do so following
White’s idea to target the support of the block’s density
matrix. But instead of keeping the support of the original
B, we retain the (smaller) support of the partially disen-
tangled density matrix ~B.
The above steps, characterized by unitary and isometric
tensors u and w, produce an alternative coarse-grained
lattice ~L and a coarse-grained state j ~i 2 V ~L that con-
tains less entanglement than j0i 2 VL0 . Importantly, the
expectation value hjoji  troR, where o is an
observable defined on a small set of sites R  L, can be
efficiently computed from just j ~i, u, w, and o. There are
two ways: (i) as described in [9], R can be computed
from ~
~R  tr ~L ~Rj
~ih ~j, where ~R is the set of sites of
~L causally connected to R through u and w;
(ii) alternatively, we can use u and w to lift o from L to
~L, producing ~o, and exploit that hjoji  h ~j~oj ~i.
The lifting from L to ~L of operators generates, by
iteration, a well-defined RG flow in the space of
Hamiltonians [8,10] such that, in the 1D case, an interac-
tion term H3 acting on at most three consecutive sites of
L is mapped into an interaction term ~H3 acting also on at
most three consecutive sites in ~L [analogous rules apply in
D> 1 spatial dimensions]. Thus, in spite of the fact that
disentanglers deform the original tensor product structure
of VL, the above rescaling transformation preserves local-
ity, in that it maps local theories into local theories.
In the same way as DMRG is related to matrix product
states (MPS) [11], the above RG transformation is natu-
rally associated to a new ansatz for quantum many-body
states on a D dimensional lattice. The multiscale entangle-
ment renormalization ansatz (MERA) consists of a net-
work of isometric tensors (namely the isometries u and
disentanglers w corresponding to successive iterations of
the RG transformation) locally connected in D 1 dimen-
sions. The extra direction , related to the RG flow, grows
only as the logarithm of the lattice dimensions, see Fig. 2.
Several properties of the MERA together with its connec-
tion to quantum circuits are described in [9].
'V
V
w
1s 2s
's V
~
w
1u 2u
1s 2r1r 2s
s~
V V V V V
FIG. 1 (color online). Isometries and disentanglers. Left: a
standard numerical RG transformation builds a coarse-grained
site s0, with Hilbert space dimension m, from a block of two sites
s1 and s2 through the isometry w of Eq. (4). Right: by using the
disentanglers u1 and u2 of Eq. (7), short-range entanglement
residing near the boundary of the block is eliminated before the
coarse-graining step. As a result, the coarse-grained site ~s
requires a smaller Hilbert space dimension ~m, ~m<m.
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S
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Example.—Figures 3 and 4 study the ground state of the
1D quantum Ising model with transverse magnetic field,
 H 
X
hs1;s2i
xs1  
x
s2  h
X
s
zs; (10)
for an infinite lattice [12]. The entanglement entropy be-
tween a block B of L adjacent spins and the rest of the
lattice, defined in terms of the eigenvalues fpig of B as
SB  
P
ipilog2pi, scales at criticality (h  1) with the
block size L as [14]
 SL 
1
6
log2L: (11)
On the other hand, off criticality SB grows monotoni-
cally with L until it reaches a saturation value for a block
size comparable to the correlation length in the system.
Figure 3 shows the effect of disentanglers on the entangle-
ment entropy of a block. Most notably, the renormalized
entanglement of a block in the critical case is reduced to a
small value that is constant throughout the RG flow.
Correspondingly, as Fig. 4 shows, the Hilbert space dimen-
sion ~m of a block of size L  2 can be kept constant as we
increase , allowing in principle for an arbitrary number of
iterations of the RG transformation. Our calculations, in-
volving the reduced density matrix of up to L  214  16,
384 spins, have been conducted with Hilbert spaces of
dimension ~m  8, while keeping the truncation error  at
each step fixed below 5 107. We estimate that without
disentanglers, an equivalent error  requires a Hilbert space
of m  500–1000 dimensions for the largest spin blocks.
This exemplifies the computational advantages of using
disentanglers.
Discussion.—The above numerical exploration suggests
an appealing picture, also confirmed by subsequent calcu-
lations involving other 1D and 2D quantum models [10].
Entanglement in the ground state of the quantum spin chain
is organized in layers corresponding to different length
scales in the system. The entanglement of a given length
x
τ
0
1
2
3
x
yτ
FIG. 2 (color online). Left: MERA for a 1D lattice with
periodic boundary conditions. Notice the fractal nature of the
tensor network. Translational symmetry and scale invariance can
be naturally incorporated, substantially reducing the computa-
tional complexity of the numerical simulations. Right: building
block of a MERA for a 2D lattice. Disentanglers and isometries
address one of the x and y spatial directions at a time.
unrenormalized entropy
renormalized entropy  
4 8 16 64 256 1,024 4,096 16,384
1
1.5
2
2.5
4 8 16 64 256 1,024 4,096 16,384
0
0.5
1
1.5
2
2.5
unrenormalized entropy 
  renormalized entropy 
Block size 
B
lo
ck
 e
nt
ro
py
E = 1/6 log L + k
(b) 
(a) 
(i) 
(ii) 
FIG. 3 (color online). Scaling of the entropy of entanglement
in 1D quantum Ising model with transverse magnetic field. Up:
in a critical lattice [h  1 in Eq. (10)], the unrenormalized
entanglement of the block scales with the block size L according
to Eq. (11). Instead, the renormalized entanglement remains
constant under successive RG transformations, as a clear mani-
festation of scale invariance. Line (i) corresponds to using
disentanglers only in the first RG transformation. Line
(ii) corresponds to using disentanglers only in the first and
second RG transformation. Down: in a noncritical lattice [h 
1:001 in Eq. (10)], the unrenormalized entanglement scales
roughly as in the critical case until it saturates (a) for block
sizes comparable to the correlation length. Beyond that length
scale, the renormalized entanglement vanishes (b) and the sys-
tem becomes effectively unentangled.
5 10 20 30 40 50
10
−6
10
−3
100
B
lo
ck
 s
pe
ct
ra
 
5 10 20 30 40 50
10
−6
10
−3
100
Block 
size L: 
4 
8 
16
32
64
128
256
unrenormalized
 spectrum 
renormalized
 spectrum 
renormalized
 spectrum 
FIG. 4 (color online). Spectrum of the reduced density matrix
of a spin block. Up: as the size L of the spin increases, the
number m of eigenvalues fpig required to achieve a given
accuracy , see Eq. (5), also increases. In particular, m grows
roughly exponentially in the number   log2L of RG trans-
formations. The spectrum resulting from applying disentanglers
leads to a significantly smaller ~m invariant along successive RG
transformations. Down: spectrum of the reduced density matrix
of 2 spins immediately before and after using the disentanglers
at the th coarse-graining step. These spectra are essentially
independent of the value of   1; 	 	 	 ; 14.
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S
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scale can be modified by means of a disentangler that acts
on a region as large as that length scale. We detect two
situations: (i) Off criticality, the entanglement between a
block of sites B and the rest of the lattice L consists
roughly of contributions from layers of length scale not
greater than the correlation length in the system. Therefore,
after a sufficiently large number of RG transformations
with disentanglers, the effective ground state becomes a
product state, that is, completely disentangled. (ii) At criti-
cality, instead, the entanglement between B and the rest of
L receives equivalent contributions from all length scales
(smaller than the size of B)—see [9] for a justification of
the logarithmic scaling of Eq. (11). In this case, by apply-
ing disentanglers, we obtain a coarse-grained lattice iden-
tical to the original one, including an effective Hamiltonian
identical (up to a proportionality constant) to the original
one [10]. In either case, a fixed point of the RG trans-
formation is attained after a sufficient number of iterations.
Noncritical theories collapse into the trivial fixed point (a
product state) while critical theories correspond to a non-
trivial fixed point (an entangled state expected to depend
on the universality class of critical theory). We conjecture
that this picture, fully consistent with known facts of RG
[1], holds also for quantum lattices in D> 1 spatial di-
mensions. Interestingly, a third type of fixed point is pos-
sible, in the case of a ground state with either long-range
order or topological order [15].
In summary, we have presented a quasiexact real-space
RG transformation that, when tested in 1D systems, pro-
duces effective sites of bounded dimension and has (scale
invariant) critical systems as its nontrivial fixed points. The
key feature of our approach is a local deformation of the
tensor product structure of the Hilbert space of the lattice,
that identifies and factorizes out those local degrees of
freedom that are uncorrelated from the rest. This deforma-
tion is such that local operators are mapped into local
operators, so that relevant expectation values for the origi-
nal system can be efficiently computed from the coarse-
grained system.
We conclude with two remarks. First, a scale invariant
critical ground state leads to disentanglers u and isometries
w that are the same at each iteration of the RG trans-
formation. A single pair (u, w), depending on O ~m4
parameters, is thus seen to specify the ground state, leading
to an extremely compact characterization that deserves
further study. Second, renormalization group techniques
are extensively used also in classical problems, where the
MERA can efficiently represent partition functions of lat-
tice systems, extending the scope of this work beyond the
study quantum systems.
The author appreciates conversations with I. Cirac, J. I.
Latorre, T. Osborne, D. Poulin, J. Preskill, and, very spe-
cially, with F. Verstraete, whose advise was crucial to find a
fast disentangling algorithm. USA NSF Grant No. EIA-
0086038 and an Australian Research Council Grant
No. FF0668731 are acknowledged.
[1] M. E. Fisher, Rev. Mod. Phys. 70, 653 (1998).
[2] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
[3] L. P. Kadanov, Physics (Long Island City, N.Y.) 2, 263
(1966).
[4] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys.
Rev. B 48, 10345 (1993); U. Schollwoeck, Rev. Mod.
Phys. 77, 259 (2005).
[5] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); 93, 040502
(2004); S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93,
076401 (2004); A. J. Daley et al., Journal of Statistical
Mechanics: Theory and Experiment P04005 (2004).
[6] F. Verstraete and J. I. Cirac, arXiv:0407066.
[7] As an example, consider a lattice of spin 1=2 sites such
that the pairs of sites r1s1 and s2r2 are in singlet states
j i  j"#i  j#"i=

2
p
, namely j ir1s1  j 
is2r2 .
Then we can use disentanglers u1 and u2 to transform
the singlets into a product state, e.g., j""ir1s1  j""is2r2 .
[8] Explicit algorithms to find disentanglers and construct an
RG flow are presented in G. Vidal, arXiv:0707.1454.
[9] G. Vidal, arXiv:0610099.
[10] G. Evenbly and G. Vidal, arXiv:0710.0692.
[11] M. Fannes, B. Nachtergaele, and R. F. Werner, Commun.
Math. Phys. 144, 443 (1992); S. Östlund and S. Rommer,
Phys. Rev. Lett. 75, 3537 (1995).
[12] Here, we have obtained an MPS for the ground state of
Eq. (10) with the infinite TEBD algorithm [13] and then
transformed it using several rows of disentanglers and
isometries. An algorithm to compute the ground state at
each iteration of the RG transformation, as well to obtain
disentanglers, will be described elsewhere.
[13] G. Vidal, Phys. Rev. Lett. 98, 070201 (2007).
[14] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev.
Lett. 90, 227902 (2003); B.-Q. Jin and V. E. Korepin,
J. Stat. Phys. 116, 79 (2004); P. Calabrese and J. Cardy,
J. Stat. Mech. 0406, P002 (2004).
[15] M. Aguado and G. Vidal, Phys. Rev. Lett. 98, 070201
(2007).
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S
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220405-4


University of Queensland eSpace

Entanglement Renormalization
G. Vidal1,2
1School of Physical Sciences, the University of Queensland, QLD 4072, Australia
2Institute for Quantum Information, California Institute for Technology, Pasadena, California 91125, USA
(Received 1 December 2006; published 28 November 2007)
We propose a real-space renormalization group (RG) transformation for quantum systems on a
D-dimensional lattice. The transformation partially disentangles a block of sites before coarse-graining
it into an effective site. Numerical simulations with the ground state of a 1D lattice at criticality show that
the resulting coarse-grained sites require a Hilbert space dimension that does not grow with successive RG
transformations. As a result we can address, in a quasi-exact way, tens of thousands of quantum spins with
a computational effort that scales logarithmically in the system’s size. The calculations unveil that ground
state entanglement in extended quantum systems is organized in layers corresponding to different length
scales. At a quantum critical point, each relevant length scale makes an equivalent contribution to the
entanglement of a block.
DOI: 10.1103/PhysRevLett.99.220405 PACS numbers: 05.30.d, 02.70.c, 03.67.Mn, 05.50.+q
Renormalization group (RG), one of the conceptual
pillars of statistical mechanics and quantum field theory,
revolves around the idea of coarse-graining and rescaling
transformations of an extended system [1]. These so-called
RG transformations are not only a key theoretical element
in the modern formulation of critical phenomena and phase
transitions, but also the basis of important computational
methods for many-body problems.
In the case of quantum systems defined on a lattice,
Wilson’s real-space RG methods [2], based on the trunca-
tion of the Hilbert space, replaced Kadanoff’s spin-
blocking ideas [3] with a concise mathematical formula-
tion and an explicit prescription to implement RG trans-
formations. But it was not until the advent of White’s
density matrix renormalization group (DMRG) algorithm
[4] that RG methods became the undisputed numerical
approach for systems on a 1D lattice. More recently, such
techniques have gained renewed momentum under the
influence of quantum information science. By paying due
attention to entanglement, algorithms to simulate time-
evolution in 1D systems [5] and to address 2D systems
[6] have been put forward.
The practical value of DMRG and related methods is
unquestionable. And yet, they are based on a RG trans-
formation that notably fails to satisfy a most natural ex-
pectation, namely, to have scale invariant systems as fixed
points. Instead, for such systems, the size of an effective
site (as measured by the dimension of its Hilbert space)
increases with each iteration of the RG transformation.
This fact does not only conflict with the very spirit of RG
theory, but it also has important computational implica-
tions: after a sufficiently large number of iterations, the
effective sites are unaffordably large, and the numerical
method is no longer viable.
In this Letter, we propose a real-space RG transforma-
tion for quantum systems on a D-dimensional lattice that,
by renormalizing the amount of entanglement in the sys-
tem, aims to eliminate the growth of the site’s Hilbert space
dimension along successive rescaling transformations. In
particular, when applied to a scale invariant system, the
transformation is expected to produce a coarse-grained
system identical to the original one. Numerical tests for
critical systems in D  1 spatial dimensions confirm this
expectation. Our results also unveil the stratification of
entanglement in extended quantum systems and open the
path to a very compact description of quantum criticality.
Real-space RG and DMRG.—As originally introduced
by Wilson [2], real-space RG methods truncate the local
Hilbert space of a block of sites in order to reduce its
degrees of freedom. Let us consider a system on a lattice
L in D spatial dimensions and its Hilbert space
 VL 
O
s2L
Vs; (1)
where s 2 L denotes the lattice sites and Vs has finite
dimension. Let us also consider a block B  L of neigh-
boring sites, with corresponding Hilbert space
 VB 
O
s2B
Vs: (2)
In an elementary RG transformation, the lattice L is
mapped into a new, effective lattice L0, where each site
s0 2 L0 is obtained from a block B of sites in L by coarse
graining. More specifically, the space V0s0 for site s
0 2 L0
corresponds to a subspace SB of VB,
 V 0s0  SB  VB; (3)
as characterized by an isometric tensor w, Fig (1),
 w: V0s0  VB; w
yw  I: (4)
Isometry w can now be used to map a state ji 2 VL
[typically, a ground state] into a coarse-grained state
j0i 2 VL0 . A thoughtful selection of subspace SB is
essential. On the one hand, its dimension m should be as
small as possible because the cost of subsequent tasks, such
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending30 NOVEMBER 2007
0031-9007=07=99(22)=220405(4) 220405-1 © 2007 The American Physical Society
as computing expectation values for local observables,
grows polynomially with m. On the other hand, SB needs
to be large enough that j0i retains all relevant properties
of ji. White identified the optimal choice as part of his
DMRG algorithm [4]. Let B denote the reduced density
matrix of ji on block B. Then the optimal subspace is
 SB  hj1i; 	 	 	 ; jmii; (5)
where jii are the m eigenvectors of B with largest
eigenvalues pi, and m is such that  
 1
Pm
i1 pi, with
 a preestablished truncation error,  1.
Notice that m refers to the amount of entanglement
between B and the rest of the lattice, LB, as charac-
terized by the rank of the truncated Schmidt decomposition
 ji  X
m
i1

pi
p jii  jii; jii 2 VLB: (6)
In other words, the performance of DMRG-based methods
depends on the amount of entanglement in ji.
Entanglement Renormalization.—We propose a tech-
nique to reduce the amount of entanglement between the
block B and the rest of the lattice L while still obtaining a
quasiexact description of the state of the system. This is
achieved by deforming, by means of a unitary transforma-
tion, the boundaries of the block B before truncating its
Hilbert space, see Fig. 1.
Let us specialize, for simplicity, to a 1D lattice and to a
block B made of just two contiguous sites s1 and s2. Let r1
and r2 be the two sites immediately to the left and to the
right of B. Then, we consider unitary transformations u1
and u2, the disentanglers, acting on the pairs of sites r1s1
and s2r2,
 u1: Vr1  Vs1 ! Vr1  Vs1 ; uy1u1  u1uy1  I;
u2: Vs2  Vr2 ! Vs2  Vr2 ; uy2u2  u2uy2  I:
(7)
Properly chosen disentanglers reduce the short-range en-
tanglement between the block B and its immediate neigh-
borhood [7]. The original state B of block B,
 B  trr1r2r1s1s2r2; (8)
is now replaced with a partially disentangled state ~B,
 ~ B  trr1r2u1  u2r1s1s2r2u1  u2y; (9)
which has a smaller effective rank ~m, ~m<m. Our RG
transformation consists of two steps: (i) First we decrease
or renormalize the amount of entanglement between block
B and the rest of L. This is achieved with disentanglers
that act locally around the boundary of B [8]. The block
does not become completely disentangled because only
entanglement localized near its boundaries can be re-
moved. (ii) Then, as in Wilson’s proposal, we truncate
the Hilbert space of block B, and we do so following
White’s idea to target the support of the block’s density
matrix. But instead of keeping the support of the original
B, we retain the (smaller) support of the partially disen-
tangled density matrix ~B.
The above steps, characterized by unitary and isometric
tensors u and w, produce an alternative coarse-grained
lattice ~L and a coarse-grained state j ~i 2 V ~L that con-
tains less entanglement than j0i 2 VL0 . Importantly, the
expectation value hjoji  troR, where o is an
observable defined on a small set of sites R  L, can be
efficiently computed from just j ~i, u, w, and o. There are
two ways: (i) as described in [9], R can be computed
from ~ ~R  tr ~L ~Rj ~ih ~j, where ~R is the set of sites of
~L causally connected to R through u and w;
(ii) alternatively, we can use u and w to lift o from L to
~L, producing ~o, and exploit that hjoji  h ~j~oj ~i.
The lifting from L to ~L of operators generates, by
iteration, a well-defined RG flow in the space of
Hamiltonians [8,10] such that, in the 1D case, an interac-
tion term H3 acting on at most three consecutive sites of
L is mapped into an interaction term ~H3 acting also on at
most three consecutive sites in ~L [analogous rules apply in
D> 1 spatial dimensions]. Thus, in spite of the fact that
disentanglers deform the original tensor product structure
of VL, the above rescaling transformation preserves local-
ity, in that it maps local theories into local theories.
In the same way as DMRG is related to matrix product
states (MPS) [11], the above RG transformation is natu-
rally associated to a new ansatz for quantum many-body
states on a D dimensional lattice. The multiscale entangle-
ment renormalization ansatz (MERA) consists of a net-
work of isometric tensors (namely the isometries u and
disentanglers w corresponding to successive iterations of
the RG transformation) locally connected in D 1 dimen-
sions. The extra direction , related to the RG flow, grows
only as the logarithm of the lattice dimensions, see Fig. 2.
Several properties of the MERA together with its connec-
tion to quantum circuits are described in [9].
'V
V
w
1s 2s
's
V~
w
1u 2u
1s 2r1r 2s
s~
V V V V V
FIG. 1 (color online). Isometries and disentanglers. Left: a
standard numerical RG transformation builds a coarse-grained
site s0, with Hilbert space dimension m, from a block of two sites
s1 and s2 through the isometry w of Eq. (4). Right: by using the
disentanglers u1 and u2 of Eq. (7), short-range entanglement
residing near the boundary of the block is eliminated before the
coarse-graining step. As a result, the coarse-grained site ~s
requires a smaller Hilbert space dimension ~m, ~m<m.
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending30 NOVEMBER 2007
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Example.—Figures 3 and 4 study the ground state of the
1D quantum Ising model with transverse magnetic field,
 H  X
hs1;s2i
xs1  xs2  h
X
s
zs; (10)
for an infinite lattice [12]. The entanglement entropy be-
tween a block B of L adjacent spins and the rest of the
lattice, defined in terms of the eigenvalues fpig of B as
SB  Pipilog2pi, scales at criticality (h  1) with the
block size L as [14]
 SL  16 log2L: (11)
On the other hand, off criticality SB grows monotoni-
cally with L until it reaches a saturation value for a block
size comparable to the correlation length in the system.
Figure 3 shows the effect of disentanglers on the entangle-
ment entropy of a block. Most notably, the renormalized
entanglement of a block in the critical case is reduced to a
small value that is constant throughout the RG flow.
Correspondingly, as Fig. 4 shows, the Hilbert space dimen-
sion ~m of a block of size L  2 can be kept constant as we
increase , allowing in principle for an arbitrary number of
iterations of the RG transformation. Our calculations, in-
volving the reduced density matrix of up to L  214  16,
384 spins, have been conducted with Hilbert spaces of
dimension ~m  8, while keeping the truncation error  at
each step fixed below 5 107. We estimate that without
disentanglers, an equivalent error  requires a Hilbert space
of m  500–1000 dimensions for the largest spin blocks.
This exemplifies the computational advantages of using
disentanglers.
Discussion.—The above numerical exploration suggests
an appealing picture, also confirmed by subsequent calcu-
lations involving other 1D and 2D quantum models [10].
Entanglement in the ground state of the quantum spin chain
is organized in layers corresponding to different length
scales in the system. The entanglement of a given length
x
τ
0
1
2
3
x
yτ
FIG. 2 (color online). Left: MERA for a 1D lattice with
periodic boundary conditions. Notice the fractal nature of the
tensor network. Translational symmetry and scale invariance can
be naturally incorporated, substantially reducing the computa-
tional complexity of the numerical simulations. Right: building
block of a MERA for a 2D lattice. Disentanglers and isometries
address one of the x and y spatial directions at a time.
unrenormalized entropy
renormalized entropy  
4 8 16 64 256 1,024 4,096 16,384
1
1.5
2
2.5
4 8 16 64 256 1,024 4,096 16,384
0
0.5
1
1.5
2
2.5
unrenormalized entropy 
  renormalized entropy 
Block size 
Bl
oc
k 
en
tro
py
E = 1/6 log L + k
(b) 
(a) 
(i) 
(ii) 
FIG. 3 (color online). Scaling of the entropy of entanglement
in 1D quantum Ising model with transverse magnetic field. Up:
in a critical lattice [h  1 in Eq. (10)], the unrenormalized
entanglement of the block scales with the block size L according
to Eq. (11). Instead, the renormalized entanglement remains
constant under successive RG transformations, as a clear mani-
festation of scale invariance. Line (i) corresponds to using
disentanglers only in the first RG transformation. Line
(ii) corresponds to using disentanglers only in the first and
second RG transformation. Down: in a noncritical lattice [h 
1:001 in Eq. (10)], the unrenormalized entanglement scales
roughly as in the critical case until it saturates (a) for block
sizes comparable to the correlation length. Beyond that length
scale, the renormalized entanglement vanishes (b) and the sys-
tem becomes effectively unentangled.
5 10 20 30 40 50
10−6
10−3
100
Bl
oc
k 
sp
ec
tra
 
5 10 20 30 40 50
10−6
10−3
100
Block 
size L: 
4 
8 
16
32
64
128
256
unrenormalized
 spectrum 
renormalized
 spectrum 
renormalized
 spectrum 
FIG. 4 (color online). Spectrum of the reduced density matrix
of a spin block. Up: as the size L of the spin increases, the
number m of eigenvalues fpig required to achieve a given
accuracy , see Eq. (5), also increases. In particular, m grows
roughly exponentially in the number   log2L of RG trans-
formations. The spectrum resulting from applying disentanglers
leads to a significantly smaller ~m invariant along successive RG
transformations. Down: spectrum of the reduced density matrix
of 2 spins immediately before and after using the disentanglers
at the th coarse-graining step. These spectra are essentially
independent of the value of   1; 	 	 	 ; 14.
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending30 NOVEMBER 2007
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scale can be modified by means of a disentangler that acts
on a region as large as that length scale. We detect two
situations: (i) Off criticality, the entanglement between a
block of sites B and the rest of the lattice L consists
roughly of contributions from layers of length scale not
greater than the correlation length in the system. Therefore,
after a sufficiently large number of RG transformations
with disentanglers, the effective ground state becomes a
product state, that is, completely disentangled. (ii) At criti-
cality, instead, the entanglement between B and the rest of
L receives equivalent contributions from all length scales
(smaller than the size of B)—see [9] for a justification of
the logarithmic scaling of Eq. (11). In this case, by apply-
ing disentanglers, we obtain a coarse-grained lattice iden-
tical to the original one, including an effective Hamiltonian
identical (up to a proportionality constant) to the original
one [10]. In either case, a fixed point of the RG trans-
formation is attained after a sufficient number of iterations.
Noncritical theories collapse into the trivial fixed point (a
product state) while critical theories correspond to a non-
trivial fixed point (an entangled state expected to depend
on the universality class of critical theory). We conjecture
that this picture, fully consistent with known facts of RG
[1], holds also for quantum lattices in D> 1 spatial di-
mensions. Interestingly, a third type of fixed point is pos-
sible, in the case of a ground state with either long-range
order or topological order [15].
In summary, we have presented a quasiexact real-space
RG transformation that, when tested in 1D systems, pro-
duces effective sites of bounded dimension and has (scale
invariant) critical systems as its nontrivial fixed points. The
key feature of our approach is a local deformation of the
tensor product structure of the Hilbert space of the lattice,
that identifies and factorizes out those local degrees of
freedom that are uncorrelated from the rest. This deforma-
tion is such that local operators are mapped into local
operators, so that relevant expectation values for the origi-
nal system can be efficiently computed from the coarse-
grained system.
We conclude with two remarks. First, a scale invariant
critical ground state leads to disentanglers u and isometries
w that are the same at each iteration of the RG trans-
formation. A single pair (u, w), depending on O ~m4
parameters, is thus seen to specify the ground state, leading
to an extremely compact characterization that deserves
further study. Second, renormalization group techniques
are extensively used also in classical problems, where the
MERA can efficiently represent partition functions of lat-
tice systems, extending the scope of this work beyond the
study quantum systems.
The author appreciates conversations with I. Cirac, J. I.
Latorre, T. Osborne, D. Poulin, J. Preskill, and, very spe-
cially, with F. Verstraete, whose advise was crucial to find a
fast disentangling algorithm. USA NSF Grant No. EIA-
0086038 and an Australian Research Council Grant
No. FF0668731 are acknowledged.
[1] M. E. Fisher, Rev. Mod. Phys. 70, 653 (1998).
[2] K. G. Wilson, Rev. Mod. Phys. 47, 773 (1975).
[3] L. P. Kadanov, Physics (Long Island City, N.Y.) 2, 263
(1966).
[4] S. R. White, Phys. Rev. Lett. 69, 2863 (1992); Phys.
Rev. B 48, 10345 (1993); U. Schollwoeck, Rev. Mod.
Phys. 77, 259 (2005).
[5] G. Vidal, Phys. Rev. Lett. 91, 147902 (2003); 93, 040502
(2004); S. R. White and A. E. Feiguin, Phys. Rev. Lett. 93,
076401 (2004); A. J. Daley et al., Journal of Statistical
Mechanics: Theory and Experiment P04005 (2004).
[6] F. Verstraete and J. I. Cirac, arXiv:0407066.
[7] As an example, consider a lattice of spin 1=2 sites such
that the pairs of sites r1s1 and s2r2 are in singlet states
j i  j"#i  j#"i= 2p , namely j ir1s1  j is2r2 .
Then we can use disentanglers u1 and u2 to transform
the singlets into a product state, e.g., j""ir1s1  j""is2r2 .[8] Explicit algorithms to find disentanglers and construct an
RG flow are presented in G. Vidal, arXiv:0707.1454.
[9] G. Vidal, arXiv:0610099.
[10] G. Evenbly and G. Vidal, arXiv:0710.0692.
[11] M. Fannes, B. Nachtergaele, and R. F. Werner, Commun.
Math. Phys. 144, 443 (1992); S. O¨ stlund and S. Rommer,
Phys. Rev. Lett. 75, 3537 (1995).
[12] Here, we have obtained an MPS for the ground state of
Eq. (10) with the infinite TEBD algorithm [13] and then
transformed it using several rows of disentanglers and
isometries. An algorithm to compute the ground state at
each iteration of the RG transformation, as well to obtain
disentanglers, will be described elsewhere.
[13] G. Vidal, Phys. Rev. Lett. 98, 070201 (2007).
[14] G. Vidal, J. I. Latorre, E. Rico, and A. Kitaev, Phys. Rev.
Lett. 90, 227902 (2003); B.-Q. Jin and V. E. Korepin,
J. Stat. Phys. 116, 79 (2004); P. Calabrese and J. Cardy,
J. Stat. Mech. 0406, P002 (2004).
[15] M. Aguado and G. Vidal, Phys. Rev. Lett. 98, 070201
(2007).
PRL 99, 220405 (2007) P H Y S I C A L R E V I E W L E T T E R S week ending30 NOVEMBER 2007
220405-4


Entanglement renormalization

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Entanglement renormalization

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