MERA for Spin Chains with Continuously Varying Criticality

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by  Jacob C. Bridgeman

Multiscale Entanglement Renormalisation Ansatz for Spin Chains with Continuously
Varying Criticality
Jacob C. Bridgeman, Aroon O’Brien, Stephen D. Bartlett, and Andrew C. Doherty
Centre for Engineered Quantum Systems, School of Physics,
The University of Sydney, Sydney NSW 2006, Australia
We use the multiscale entanglement renormalisation ansatz (MERA) to numerically investigate
three critical quantum spin chains with Z Z on-site symmetry: a staggered XXZ model, a
2 2
×
transverse field cluster model, and the quantum Ashkin-Teller model. All three models possess a
continuousone-parameterfamilyofcriticalpoints. Alongthiscriticalline,thethermodynamiclimit
of these models is expected to be described by classes of c = 1 conformal field theories (CFTs) of
5 twopossibletypes: theS1freebosonanditsZ2-orbifold. OurnumericsusingMERAwithexplicitly
1 enforcedZ2 Z2 symmetryallowustoextractconformaldataforeachmodel,withstrongevidence
×
0 supportingtheidentificationofthestaggeredXXZmodelandcriticaltransversefieldclustermodel
2 withtheS1bosonCFT,andtheAshkin-TellermodelwiththeZ -orbifoldbosonCFT.Ourfirsttwo
2
modelsdescribethephasetransitionsbetweensymmetryprotectedtopologicallyorderedphasesand
r
p trivialphases,whichlieoutsidetheusualLandau-Ginsburg-Wilsonparadigmofsymmetrybreaking.
A Our results show that a range of critical theories can arise at the boundary of a single symmetry
protected phase.

6
2
I. INTRODUCTION three related spin models along a line of criticality: a

] staggeredXXZmodel,atransversefieldclustermodel15,
l
e and the quantum Ashkin-Teller model16,17. These mod-
- Quantumspinmodelsareanareaofextensivetheoret-
r els are of interest because they have critical exponents
ical and numerical study, due to their relative simplicity
t that vary continuously as a function of model parame-
s and wide descriptive power. Simple spin models can ex-
. ters. All three models possess an on-site Z Z sym-
at hibitavarietyofexoticgroundstatesproperties,suchas metry that will play an important role in ou2r×ana2lysis.
m topological order1 and symmetry-protected topological
For the first two models of interest, the critical line
order2. Studying the critical behaviour that describes
- corresponds to a phase transition between a phase pos-
d a system at the transition point between two quantum
sessing nontrivial symmetry protected topological or-
n phases can also be investigated with quantum spin mod-
der (SPTO) for this symmetry group18 and a trivial
o els but brings new challenges. At these critical points,
c phase. Such systems are also of particular interest
themodelsbecomegaplessandmanyoftheexactresults
[ due to their connection to measurement-based quantum
that have been proven for gapped systems break down.
computation15,19,20. There has been extensive investi-
2 In particular, for critical systems the correlation length
gation into symmetry protected topological phases, but
v in the ground state diverges and the area law for en-
7 tanglement entropy is violated3. In the thermodynamic only a few studies have investigated transitions out of
1 them21,22. Such phase transitions are not part of the
limit, the behaviour at the critical point is described by
8 usual Landau-Ginsburg-Wilson paradigm of phase tran-
a conformal field theory (CFT)4,5.
2 sitions because there is no broken symmetry on either
0 A wide range of numerical methods have been devel- side of the transition. Despite the fact that the mod-
1. oped to study the behaviour of quantum spin chains, in- els we consider here involve a transition between a fixed
0 cludingtheircriticalbehaviour,despitethedifficultiesin SPTO phase and the trivial phase, the boundary is de-
5 working with a Hilbert space that grows exponentially scribed by a range of critical theories.
1 in the chain length. Numerical methods based on tensor In contrast, the Ashkin-Teller model possesses a con-
v: networks6,7 havebeenhighlysuccessfulinrecentyearsas ventionalsymmetrybrokenphaseandforthatmodelthe
i anefficientmethodforstudyingawiderangeofspinlat- phasetransitionofinterestisbetweenthesymmetrybro-
X
tice models. In particular, for one dimensional gapped kenandtrivialphases. Again,arangeofcriticaltheories
ar systems, methods based on matrix product states are describe the phase boundary.
known to efficiently describe ground state properties8.
The thermodynamic limit of these three models at
However, this behaviour does not generalise to gapless
their critical points are described by a special class of
systems9 (although c.f. Ref. 10). For a tensor network
CFT, namely those with central charge c = 1. These
description to naturally describe a critical spin chain,
CFTs do not correspond to any member of the series of
is should capture the area law violation, and one such
so-called minimal models that have been the main fo-
descriptionistheMultiscaleEntanglementRenormalisa-
cus of numerical research that has used MERA to study
tion Ansatz11 (MERA). The MERA has previously been CFTs12,14. ThisclassofCFTischaracterisedbyarenor-
shown to accurately and efficiently reproduce conformal
malisation group with an exactly marginal field, leading
data of critical spin chains12–14.
to a single parameter family of theories that reproduces
In this paper, we use the MERA to numerically study thecontinuously varyingcriticalexponents oftheunder-
2
lying spin models. By varying the coupling in the spin
models, we can tune the system along this line of criti-
cality. This continuous variation of scaling dimensions is
a distinctive feature of c=1 CFTs23.
Here, we show that the MERA allows us to extract
conformal data from all three models described above
possessing a continuous line of criticality, and show that
these data agree with the expected c = 1 CFTs includ-
ing the variation of the conformal data along the line of
Relevant Irrelevant Marginal
criticality. Specifically, the staggered XXZ model and
the transverse field cluster model are both shown to be
consistent with the c = 1 S1 free boson CFT, whereas
the quantum Ashkin-Teller model gives data consistent Figure 1. Illustrative phase space diagrams and renormali-
sation group (RG) flows for models with a unique RG fixed
with the c=1 orbifold CFT. Our results provide further
point(left)andalineofcriticality(right). RGrelevantopera-
evidenceoftheusefulnessofMERAindescribingcritical
torshave∆<2andgrowunderrenormalization,whilstthose
systems in one dimension.
with ∆ > 2 are called irrelevant, since a CFT deformed by
This paper is structured as follows. In Section II, we
such an operator flow back to the fixed point. Deformations
review CFTs as they pertain to the three spin models byanexactlymarginaloperatorleadtoanewconformalfixed
studied. InSectionIIIwereviewthedetailsoftheMERA point. Asthesizeofthedeformationisvaried,theconformal
algorithm. In Section IV we present results from the dimensions vary continuously.
numeric simulations for each model, and compare with
the expected CFT data.
termgivesanotherCFT,leadingtocontinuouslyvarying
families of theories (see Figure 1). This phenomenon is
II. c=1 BOSON CFTS AND ASSOCIATED SPIN notseeninunitaryCFTswithc<1. Thesetwocontinu-
MODELS ous families of CFTs will correspond to the critical lines
in our various quantum spin chains.
Wewillconsiderthreedifferentonedimensionalquan- The first relevant family of CFTs, the S1 boson CFT,
tum spin models at their respective critical points. In is the theory of a massless free boson ϕ(x). We will be
eachofthesemodelsthereisalineofcriticalpointsalong interested in periodic boundary conditions and we will
which the critical exponents vary continuously. These choose a parameterisation such that x [0,2π). We
criticalpointsarebelievedtobedescribedbycertaincon- will focus on a compactified version of ∈the free boson
formal field theories with marginal operators that gener- where φ itself takes values on a circle of radius R , so
C
ate flow along these lines of criticality. In this section, that ϕ(x) ϕ(x)+2πR . As a result when we move
C
we briefly review these conformal field theories and in- around a c≡ircle in the x co-ordinate, ϕ does not need to
troduce our three spin chain models. bestrictlyperiodicbutmaytwistaroundmtimessothat
ϕ(x+2π) = ϕ(x)+2πmR . The primary field scaling
C
dimensions are known for this theory and generally have
A. c=1 CFTs a non-trivial dependence on R . The free boson has a
C
natural internal SO(2) symmetry given by the transla-
Conformal field theories (CFTs) are field theories tion ϕ(x) ϕ(x)+θRC, where θ [0,2π), as well as a
≡ ∈
that are invariant under all conformal (angle preserv- Z2 symmetry given by ϕ(x) ϕ(x).
→−
ing)space-timetransformations. Theydescribethether- Theorbifold bosonCFTonceagaininvolvesabosonic
modynamic limit of critical lattice models4,5. Due to field ϕ(x) compactified on a circle of radius R , so that
O
the abundant symmetry, a (1+1)D CFT is completely ϕ(x) ϕ(x)+2πR 4,5. Thetheoryhastwosectors. The
O
≡
specified by knowledge of (i) its central charge c; (ii) first corresponds to the subspace of symmetric states of
the primary fields φ and their scaling dimensions (h,h¯), the free boson that map to themselves under ϕ(x)
→
which are eigenoperators of the scaling transformation ϕ(x). The second sector corresponds to quantising the
with eigenvalues ∆=h+h¯; and (iii) the operator prod- −freebosonwith“twisted”boundaryconditionsforwhich
uct expansion coefficients for these fields. It is the pri- ϕ(x+2π)= ϕ(x)andprojectingoutthestatesthatare
−
mary field scaling dimensions ∆ that predict the critical symmetric under ϕ(x) ϕ(x)4,5. A characteristic of
→ −
exponents of the associated critical lattice model. thetwistedsectoroftheorbifoldbosonisthatitcontains
We focus our attention on CFTs with c = 1, which a number of primary fields whose scaling dimension is
mainlyfallintotwocategories: thefreebosontheoryand independent of R . Due to this construction the O(2)
O
its Z -orbifold4,5. These c = 1 boson CFTs are charac- symmetry of the free boson model is broken down to a
2
terised by the presence of an exactly marginal primary Z symmetry given by ϕ(x) ϕ(x) + πR , and the
2 O
→
field, with scaling dimension 2, meaning it is fixed un- modelretainsitssymmetryunderϕ(x) ϕ(x)sothat
→−
der rescaling. Perturbing a CFT by an exactly marginal the natural symmetry group is Z Z .
2 2
×
3
InthisworkwewillbemainlyconcernedwithaZ Z
2 2
×
subgroup of these symmetries that will remain on-site in
all of our models, as we will incorporate this symmetry
explicitly into our numerical MERA simulations. This
symmetryisalsoimportantforconsiderationsofsymme-
try protected topological order2. We can choose genera-
tors for this subgroup as follows:
(j 1) j (j+1)
− N N
(cid:89) (cid:89)
S = σXτX S = σYτY. (2)
1 j j 2 j j
Figure 2. Each of the spin models is defined on a pair of
j=1 j=1
parallel chains. The σ operators act only on the red (top)
chain and the τ act only on the blue (bottom). The green The generator S1 corresponds to the spin-flip symme-
region indicates a single site. Each of the models is then try of the staggered-XXZ, and therefore to the ϕ ϕ
→ −
defined by a 1D nearest-neighbour Hamiltonian. symmetry of the free boson model. The generator S
2
corresponds to rotating each spin by π about the Z-axis
and then flipping the spins. Thus it corresponds to the
transformation ϕ ϕ+π on the free boson.
For a more detailed examination of these CFTs, in- For β = 0,λ >→−1 the ground state corresponds to
cluding expressions for their spectra of primary fields, pairing the red and−blue spins at each site in a maxi-
we refer to Ref. 5. mally entangled state, and thus has the structure of a
product state. For β ,λ > 1 the ground state of
→ ∞ −
this model pairs each blue spin with the red spin on the
B. Spin models at criticality site to the right in a maximally entangled state. If the
spin chain is defined on open boundary conditions there
In the following, we present three critical spin models is a four-fold ground state degeneracy associated with
whose thermodynamic limit is described by c=1 CFTs. unpaired spins at each end of the chain. This state is in
All three models will be defined on a line with periodic thenon-trivialsymmetryprotectedtopologicallyordered
sbyomunmdeatrryy.conditions, and will possess an on-site Z2×Z2 sthtaetceaonfoZn2ic×aZlr2espyrmesmenettartyi2v.esTohfetsheettwroivwiaalvaenfdunncotnio-tnrsivairael
Each of these models have a line of critical points sep- (respectively) symmetry protected topologically ordered
arating an ordered phase from a disordered phase. For phases with this symmetry25.
two of the models, the staggered XXZ model and the At the phase transition β = 1 separating these two
transversefieldclustermodel,theorderedphasedisplays phases, this model is the well-studied XXZ model and is
symmetry protected topological order19,24. In contrast, solvable via a Bethe Ansatz26. This information can be
for the quantum Ashkin-Teller model the ordered phase used to identify how different values of λ correspond to
displays conventional symmetry breaking of the Z2 Z2 the parameter RC that specifies the free boson model.
symmetry. × The critical line defined by
To clarify the on-site nature of this symmetry, we will β =1, λ [ √2/2,1), (3)
assigntwospin-1/2particlestoeachsite,andcolourthem ∈ −
red and blue (Fig. 2). At site j, the Pauli spin operators is known to be described by the S1 free boson27 com-
for each spin are denoted σα and τβ for α,β =X,Y,Z. pactified on a circle of radius
j j
2
R2 = cos 1( λ). (4)
C π − −
1. Staggered XXZ Model Duetotheexactsolution,thisantiferromagneticmodelis
used extensively in the study of quantum critical points
The staggered XXZ model is described by the Hamil- and critical ground states26–30. We also note that the
tonian XXZmodelwithnext-nearestneighbourinteractionshas
recently been studied in the context of SPT phases31.
N MoreoverthestudyofphasetransitionsoutoftheSPTO
H = (cid:88)(cid:0)σXτX +σYτY λσZτZ(cid:1) phase of SO(3) symmetric systems in one dimension in
XXZ − j j j j − j j
j=1 Ref. 21 corresponds to the Heisenberg model in one di-
N 1 mension and the value λ=1 in the free boson theory.
β (cid:88)− (cid:0)τXσX +τYσY λτZσZ (cid:1). (1)
− j j+1 j j+1− j j+1
j=1
2. Transverse field cluster model
Note that this model has an O(2) symmetry group, cor-
responding to rotating each spin about the z-axis and a Cluster states are highly entangled many body states
Z symmetry corresponding to flipping each spin. of spin-half particles that arose in the theory of quan-
2
4
tum computing. It is possible to simulate any quan- Finally the mapping (7) maps the symmetry operations
tum computation by making only local measurements S ,S of the staggered XXZ model to the correspond-
1 2
on cluster states on an appropriate lattice32. The clus- ing symmetry operators for the transverse field cluster
ter state corresponding to a linear arrangement of spins model. Thus we can identify these two symmetry opera-
is the ground state of the following local Hamiltonian tionsintermsofthemappingsϕ ϕandϕ ϕ+π,
H = (cid:80)σZ σXσZ . Recently, cluster state mod- respectively, in the free boson mo→de−l. →−
elscon−sistingµo−f1thµisHµ+a1miltonianwithvariousadditional
terms have been studied, from the perspective of quan-
tum computing15,19,20, their phase structure as models 3. Quantum Ashkin-Teller model
with symmetry protected topological (SPT) phases19,33,
natural models for the gapless edge physics of 2D SPT ThequantumAshkin-Teller (AT)model16,17 isdefined
models33,34, and nonequilibrium dynamics arising in by the Hamiltonian
these models35.
We define the transverse field cluster model (TFCM) H = (cid:88)N (cid:0)σZ +τZ +λσZτZ(cid:1)
by the Hamiltonian AT − j j j j
j=1 (8)
N N 1
H = (cid:88)(cid:0)σX +τX +λσXτX(cid:1) β (cid:88)− (cid:0)σXσX +τXτX +λσXτXσX τX (cid:1).
TFCM − j j j j − j j+1 j j+1 j j j+1 j+1
j=1 j=1
βN(cid:88)−1(cid:0)σZτXσZ +τZσX τZ This model consists of a pair of transverse-field Ising
− j j j+1 j j+1 j+1 chainsσ,τ coupledbytwo-andfour-spinterms. Inpar-
j=1 ticular, when λ = 0, we have a decoupled pair of Ising
+λσZτYσY τZ (cid:1). (5) chains.
j j j+1 j+1
Thismodelalsopossessesanon-siteZ Z symmetry,
2 2
This model possesses an on-site Z2 Z2 symmetry gen- generated by ×
×
erated by
N N
(cid:89) (cid:89)
N N S = σZ S = τZ. (9)
S = (cid:89)σX S = (cid:89)τX, (6) 1 j 2 j
1 j 2 j j=1 j=1
j=1 j=1
We note that with open boundary conditions this
andpossessestwodistinctSPTphasesforthissymmetry. modelisalsounitarilyrelatedtothetransversefieldclus-
For β = 0, the ground state is a product state, i.e., in a ter model, but now through a nonlocal unitary transfor-
trivial phase, whereas if β the ground state is the mation
→ ∞
well-studied cluster state, known to be within a Z Z
2 2
symmetry protected topological phase19. × σX σZ τX τZ (10)
j → j j → j
The transverse field cluster model can be obtained (cid:32)j 1 (cid:33)  N 
from the staggered-XXZ model through the transforma- σZ (cid:89)− τZ σX τZ τX (cid:89) σZ.
tion j → k j j → j k
k=1 k=j+1
σjX →σjXτjZ τjX →τjZ (7) Under this mapping the four-fold degenerate ground
σZ σYτZ τZ σZτY. space of the transverse field cluster model maps to the
j →− j j j → j j four-fold degenerate groundspace of the Ashkin-Teller
As this mapping is unitary, the ground state energy and model. Although this mapping is nonlocal, in the
phase transitions of the TFCM model are identical to bulk it maps local Z Z -symmetric terms to local
2 2
×
those of the XXZ model. The mapping (7) preserves lo- Z Z -symmetric terms, and thus can be viewed as a
2 2
×
cality, meaning that local operators confined to k sites generalisation37 of the map by Kennedy and Tasaki38,39,
of the staggered XXZ model map to local operators of mapping the SPTO phase of the transverse field cluster
the transverse field cluster model confined to the same modeltothesymmetrybrokenphaseoftheAshkin-Teller
k sites, and that this transformation can be performed model. Note, however, that boundary terms are trans-
using local unitaries applied to each site. Finally, this formed nontrivially by this map; local boundary terms
mapping respects the Z Z symmetry of the models. canmaptononlocalones. Inparticular,periodicbound-
2 2
×
As a result the ground states of the two models must be aries in the XXZ and TFCM models pick up extensive
in the same phase36. This is one way of seeing that the stringlike terms in the AT model.
ordered phase of the transverse field cluster model pos- (NotethattheAshkin-TellermodelalsopossessesaZ
2
sesses symmetry protected topological order. Moreover symmetry associated to swapping σ τ, however this
↔
we expect that the thermodynamic limit of TFCM on does not map to a local symmetry in the other models.)
the line defined by Eq. (3) likewise corresponds to the Becausethemodelsareunitarilyrelated,theATmodel
S1 boson with radius R determined by λ as in Eq. (4). will possess the same phase structure and ground state
C
5
and unitary disentanglers u
u(cid:96) :H(cid:96)⊗2 →H(cid:96)⊗2, u†(cid:96)u(cid:96) =I⊗(cid:96)2, (13)
where istheHilbertspaceofdimensionχ ofonespin
(cid:96) (cid:96)
H
(one index) on layer (cid:96).
Togetherthesetensorsperformarealspacerenormali-
sationgroup(RG)transformation,witheachlayerofthe
w structure corresponding to a description of the model on
a different length scale. The free indices correspond to
u the degrees of freedom associated with the microscopic
model of interest, and the isometries map 3 spins on
layer (cid:96) onto an effective spin on layer (cid:96) + 1. The ap-
Figure 3. Tensor network diagram for the scale invariant proximation involved in this transformation is controlled
ternaryMERA.Isometriesareindicatedbytriangles(green), by the bond dimension χ; the dimension of the effective
while disentanglers are rectangles (blue). The layered struc- spin. If χ < χ3, the full microscopic physics cannot
(cid:96)+1 (cid:96)
ture accurately represents a wide range of length scales and be captured and a variational algorithm is used to en-
captures the scale invariant physics of critical spin chains. sure the low energy subspace is retained. The maximum
bond dimension used in the MERA is labelled χ. The
unitary tensors rearrange the local Hilbert space, locally
energy as the transverse field cluster and staggered XXZ
removing entanglement and allowing χ to be relatively
models (and, in particular, will possess the same line of
small11.
criticality described by Eq. (3)). However, the phases
that arise have quite different properties and the CFT Generically, all tensors in the MERA may be differ-
correspondingtothecriticallinewhereβ =1willhavea ent, however for translationally invariant states a sin-
different spectrum of primary fields. Based on finite size gle pair of tensors u(cid:96),w(cid:96) characterise layer (cid:96). For
{ }
simulations40,41andCFTarguments42,43,thecriticalline scale invariant models, such as critical systems, the net-
oftheATmodelhasbeenidentifiedastheorbifoldboson workbecomessimplerstill. Thedescriptionofthemodel
with radius shouldbescale-invariant,andthereforealllayersbecome
identical12 and the entire MERA is described by a single
RO2 =RC−2, (11) pair of tensors {u,w}, and all bond dimensions are χ.
The isometries and disentanglers are chosen to min-
where R is defined in Eq. (4).
C imise the energy of the spin model Hamiltonian using
techniques that are described in Refs. 44 and 45. This
resultsinaniterativealgorithmwhereeachstepupdates
III. SCALE-INVARIANT MERA WITH
either the isometries or the disentanglers and each step
SYMMETRIES
can be performed at a cost (χ8), although this can
O
be reduced by modifying the network to include further
Following the work of Refs. 44 and 45, we have inde- approximations, such as spatial symmetries45 or on-site
pendently developed code to optimise multi-scale entan- symmetries46.
glement renormalisation ansatz (MERA)11. This class
of tensor networks can be used to efficiently describe
ground states of critical quantum lattice models. Specif-
ically, we give a brief introduction to the scale invariant
ternary MERA, closely following Ref. 45, and describe
B. -symmetric tensors
the adiabatic crawling method which we used to stream- G
line the calculation of critical exponents at various lo-
cations along the critical line of our c = 1 spin chain The incorporation of symmetries can decrease the re-
models. sources required to optimise the MERA. If a model pos-
sessesanon-sitesymmetry,suchastheZ Z symmetry
2 2
×
occurring in the models described in section II, these in-
A. Elements of the MERA ternalsymmetriesmaybeexploitedtofurtherreducethe
resource requirement by using -symmetric tensors46.
G
TheternaryMERAshowninFig.3isconstructedfrom A tensor Tβ1,β2,...,βm is said to be -symmetric if it is
two kinds of tensors: isometries w invariant undαe1r,αt2h,e...,aαcntion of the grouGp on each index.
G
For example, the requirement for the isometry described
w(cid:96) :H(cid:96)+1 →H(cid:96)⊗3, w(cid:96)†w(cid:96) =I(cid:96)+1, (12) above to be -symmetric is as follows
G
6
1. Reduction in variational parameters
Xg† Xg† Alargenumberoftheblocksarefixedtobezerobythe
symmetry, so do not have to be stored or manipulated
duringoptimisation. Thisdecreaseinvariationalparam-
= g , (14) eters leads to a decrease in the storage space needed for
∀ ∈G
the network. It also leads to a multiplicative decrease
in the computational time (aχ8 aχ8 in the case of
Ug Vg Wg Ug Vg Wg Z2 Z2) allowing either larger χ→to4 be accessed or a
×
decrease in the overall runtime.
where U,V,W,X are unitary representations of .
G
By Schur’s lemma, the tensor decomposes into blocks,
2. Selection of symmetry sector
whereeachblocktransformsasoneoftheirreduciblerep-
resentations (irreps) of . In general, due to the block
structure of the tensorsG, we can decompose the indices By decomposing the tensors into blocks, any operator
α (c,d), where c labels the irrep and we will call it applied to the spin chain can be classified according to
the→charge index, and d is the degeneracy index46. The howittransformsunderthesymmetry. AoperatorOq1,q2
charge structure is completely specified by the group . has charge (q1,q2) under the Z2 Z2 symmetry if
G ×
The degrees of freedom for the particular model of inter-
est are completely described by the index d. S1OS1† =q1O, S2OS2† =q2O, (16)
Inourcasewehavethegroup =Z Z = x,y x2 =
G ∼ 2× 2 ∼{ | where S , S are the symmetry generators described in
y2 =e,xy =yx whose irreps are given by the character 1 2
} section II. Operators carry a unique charge, so operators
table
of the form O =a O +a O are forbidden.
1 1,1 2 1, 1
−
e x y xy
D(1,1) 1 1 1 1 3. Nonlocal operators
D 1 1 1 1 ,
(1, 1)
− − −
D 1 1 1 1 Asweknowtheactionofthesymmetryonthetensors
( 1,1)
D − 1 −1 1 −1 u,w , we can trivially compute expectation values for
(−1,−1) − − a{ clas}s of highly nonlocal operators, with computational
where the D are the irreps and the possible charges cost which barely exceeds that of local operators13. This
c
c = (q ,q ) are (1,1),(1, 1),( 1,1),( 1, 1). Since is discussed below with respect to scale invariant opera-
1 2
the group is Abelian, tens−or pro−ducts o−f ir−reps results tors and conformal data.
in a new irrep that can be obtained by elementwise
multiplication in the character table. For example,
D D = D . This product of taking C. Optimising a MERA for a spin model
( 1,1) (1, 1) ( 1, 1)
ten−sor p⊗roduct−s results−in−multiplicative operation on
charges c c = c that is often called the fusion rule. Given a local, critical spin Hamiltonian H =(cid:80) h , a
1⊗ 2 3 j j
Usingthisfusionrule,wecanconstructhigherorderten- MERA description of the ground state ψ can be gen-
sors from vectors. Condition (14) becomes a charge con- erated by optimising the tensors. Here w|e(cid:105)briefly review
servation rule47. In numerics it is possible to identify the optimisation algorithm, following Ref. 45.
the nonzero blocks of a given tensor by checking that The MERA for a critical model is built from two sec-
the block conserves charge. The condition for a nonzero tions. First,anumber(usually2-3)oftransitional layers
block Bi is areused,whicharetranslationallyinvariantbutnotscale
invariant, with tensors u ,w . These allow the bond
(cid:96) (cid:96)
{ }
dimension of the remaining (scale invariant) tensors to
c c
n+1 n+m bechosenindependentlyofthephysicaldimensionofthe
spins. Generically, the Hamiltonian H will contain RG
··· irrelevant terms, breaking scale invariance. Since these
(cid:89) (cid:89)
Bi (cid:54)=0 ⇐⇒ c(cid:96) = c(cid:96). (15) termsbecomesuppressedatlargerscales,thetransitional
(cid:96) In (cid:96) Out layers reduce their effect.
∈ ∈
··· Above the transitional layers, the MERA is built from
c c c
1 2 n a unique pair of tensors u,w , characterising the scale
{ }
invariant nature of the critical model. We allow the di-
Using these block tensors provides us with three main mension of the upper and lower indices of u to differ; it
advantages in optimising the network and extracting is not unitary, but Eq. (13) holds. This preserves the es-
physical information. sential structure of the MERA, but allows for increased
7
values(scalingdimensionsforrelevantoperatorsandcen-
tral charge) is defined as v = C C , where C is
j j 1 j
the conformal data on iteration j.−Wh−en this falls be-
low some threshold, the point is declared converged, and
ρ
‘
the algorithm is repeated on the next point, using the
previouslyconvergedtensorsasastartpoint. Ifthevari-
ation is sufficiently small, the ground state of the new
Hamiltonianisverysimilartothepreviousgroundstate,
and few convergence iterations are required. Typically,
Figure 4. The descending superoperator. This computes the convergence of the remaining 60-100 points can be com-
reduceddensitymatrixofthemodelatlevel(cid:96)oftheRG.Just pleted in a time similar to that required to converge the
oneofthecontractionswhichmustbeperformedtooptimise
initial point.
the MERA tensors. The cost is (χ8), however this can be
O
reduced as discussed in the text.
numericalefficiency13. Apairofbonddimensions,χ and
l E. Extracting conformal data from the MERA
χ on the lower and upper indices of u characterise the
u
MERA.
To initialise the MERA, a pair of bond dimensions The properly converged MERA provides a compact
χ ,χ ischosen,andrandomtensorsgenerated. Asin- approximation to the ground state, and we can extract
l u
{ }
gle transitional layer is initialised to allow χ to differ physical properties of the CFT that describes the spin
l
from the physical dimension d. These tensors are then latticemodelatcriticalitybyoptimizingaMERA.Here,
iteratively optimised to minimise the energy ψ H ψ = wedescribehowtoobtaintheconformalparametersfrom
(cid:104) | | (cid:105)
Tr(H ψ ψ ) as discussed in Refs. 44 and 45. the tensors in the MERA, following Refs. 12 and 13.
| (cid:105)(cid:104) |
Optimising the MERA requires many contractions of
One part of the conformal data for a CFT is the spec-
networks analogous to those shown in Fig. 4, where ρ is
(cid:96) trumofscalingdimensions ∆ associatedwithaprimary
thereduceddensitymatrixdescribingthesystemonlayer φ
field φ. These are the eigenvalues of the rescaling opera-
(cid:96). Thiscontractioncanbecomputedintime (χ4χ4). In
O l u torinthefieldtheory. IntheMERA,theisometrictensor
numerical implementation, χ < χ allows for decreased
u l w performs a rescaling operation on the spin model, so
runtimewithoutapparentdegradationoftheresults. By
allowsusto extractthescalingdimensions12. Byfinding
assumingthatmostoftheeigenvaluesofthereducedden-
the operators which are fixed under the one site scaling
sitymatrixρaresmall,wecanmodifythenetworktoim-
superoperator
prove the scaling as follows45. We assume we can choose
a projector P = vv such that ρ = Pρ+(cid:15), where (cid:15) is
†
a small error. Then χ¯ is the rank of the projector. In
thenetworks, ρisreplacedwithPρ, givingtheimproved
efficiency at the expense of optimising the tensor v. Nu-
mericalevidenceshowsthatχ¯maybechosen (χ)rather (φ)= φ =λφ φ , (17)
S
O
than (χ2), thus providing a scaling of (χ6χ¯).
O O
As the optimisation proceeds, the change in the ten-
sors between iterations decreases. Once the network is
changing sufficiently slowly, a new transitional layer is
added by promoting the lowest level of the scale invari- we find the scaling operators and their dimensions. The
ant portion. This process is repeated until it does not eigenvaluesλ of arethenrelatedtothescalingdimen-
φ
S
produce significant improvement of the ansatz (usually sions via ∆ =log (λ ).
φ 3 φ
once there are 2-3 transitional layers).
Wecanalsocomputethescalingdimensionsforaclass
of nonlocal scaling operators by making use of the sym-
metry. Nonlocal scaling operators take the form
D. Adiabatic crawling
N 1
We are simulating families of models with properties (cid:79)−
O = (U ) o , (18)
that vary continuously along a critical line, and so we g j ⊗ N
can make use of the solution from a converged MERA j=−∞
at one point as a starting point to speed up convergence
at a neighbouring point. We call this approach the adia- whereU isaunitaryrepresentationof . Sincethesym-
g
G
batic crawling technique. First, the MERA is converged metry operators commute with the MERA tensors, ap-
from random tensors at a start point somewhere along plying one layer to an operator of this form and utilising
the line. The velocity rate of variation of the conformal Eq. (12-13) gives a new operator of the form Eq. (18).
8
Eigenoperators of the g-twisted scaling superoperator 2
−
AT
TFCM
2.5 XXZ
−
Exact
E
S
g(φ)= Ug φ =λφ φ , (19) G
S 3
−
3.5
−
allow extraction of the conformal spectrum of the CFT 0.5 1 1.5 2
arising from taking the thermodynamic limit of the spin S1 CFT Radius R2
C
model with boundary conditions which are twisted by
the group element g; see Ref. 13. Setting g = e, is 12
S
recovered, giving the spectrum on periodic boundaries. )
5
The spectrum of can be extracted from the con- − 8
g 0
verged MERA for nSegligible additional cost simply by 1×
diagonalizing g12,13. E( 4
ThecentralSchargeofthetheorycanbeextractedfrom ∆
the fixed-point density matrix using the scaling of the
0
entanglement entropy12 0.5 1 1.5 2
c S1 CFT Radius RC2
S = log (L/a)+k, (20)
critical 3 2
Figure 5. Ground state energies (GSE) extracted from the
where c is the central charge of the CFT, L is the block MERAandtheirrelativeerrors∆E. Forthebonddimension
length, a is the lattice scaling (here, a=1 by definition) usedhere,theseremain (10−4)astheHamiltonianparame-
O
and k is some constant. The one-site reduced density ter λ is varied. The numerical GSE remains an upper bound
matrixρfpisobtainedbysymmetrisingoverthetwoways on the true GSE. Note that the CFT radius RC is related to
1
of tracing out one site of the fixed-point density matrix the spin model coupling parameter via Eq. (4)
ρfp (Fig. 4). The central charge is then obtained by
2
(cid:16) (cid:16) (cid:17) (cid:16) (cid:17)(cid:17)
c=3 S ρfp S ρfp , (21)
2 − 1 three models considered here, and this provides a useful
where S is the von Neumann entropy of the density ma- benchmark; see Fig. 5. The relative error in the ground
trix. state energy is given by ∆E =(E E )/E .
exact MERA exact
−
The exact solution is obtained by numerically integrat-
ing the Bethe Ansatz solution26. The MERA is an ex-
IV. NUMERICAL RESULTS plicitwavefunction,andassuch,itsenergycannotbeless
than the true ground state energy; thus ∆E 0. The
≥
In this section, we present results obtained from a ground state energy obtained using the MERA for all
Z Z symmetric MERA with χ ,χ = 20,12 and three models is consistent with the exact solution, with
2 2 l u
χ¯ =×60 (see Sec. IIIC) applied t{o the t}hree{mode}ls de- relativeerrorpositiveandoforder10−4 forthefullrange
scribed in Sec. II. Three transitional layers are used to of λ considered; see Fig. 5.
ensure scale invariance of the Hamiltonian. We use the
adiabatic crawling technique, starting at R2 = 1.25 and
using a velocity threshold of v =9 10 5. Both ground
t −
×
stateenergiesandconformaldataarepresentedalongthe
critical lines of the three models given by Eq. (3). The
conformal spectrum is shown to be consistent with the
B. Conformal data
identifications made in Sec. II. We also compare these
MERA simulations with the result of exact diagonalisa-
tion studies for the models of interest. We now present the conformal data obtained from the
MERA for our three models. We show that this data
is consistent with the CFTs identified as the thermody-
A. Ground state energy namic description, namely the Z -orbifold boson CFT
2
for the Ashkin-Teller model; and the S1 boson CFT as-
As a result of the Bethe Ansatz solution for the XXZ sociated with both the XXZ and transverse field cluster
model, the ground state energy is known exactly for all models.
9
1.02 1/2
c
e 1 AT
g
r TFCM
ha 0.98 XXZ
c
al 0.96 ∆φ Orb.
ntr 0.94 AT S1
e TFCM
C
0.92 XXZ 1/8
0.9 Exact
0
0.5 1 1.5 2
0.5 1 1.5 2
S1 CFTRadiusR2
S1 CFT Radius R2 C
C
Figure 7. The lowest scaling dimensions (∆ ) of the (1, 1)
φ
Figure6. CentralchargesextractedfromMERAsimulations. −
charge sector of the three models. The XXZ and transverse
The CFTs associated with these models are expected to be field cluster models agree well with the S1 boson, whilst the
c=1(shownassolidline)forthefullrangeofR2 considered
C AT model is consistent with the fixed scaling operator from
here. theorbifoldtheory. NotethattheS1 andorbifoldCFTradii
are related via Eq. (11).
1. Central Charge
2. Scaling dimensions
One of the pieces of data required to specify a CFT
is the central charge. It labels classes of CFT and is
We now turn to the scaling dimensions ∆ extracted
identically c=1 for all values of R in both the S1 boson φ
fromtheMERA.Thesescalingdimensionscanbeclassi-
and orbifold boson CFTs. The values obtained from the
fied according to their Z Z symmetry sector. As the
2 2
MERAsimulationsareshowninFig.6. Theseareusually ×
parametersofthemodelsarevariedalongthelinesofcrit-
within 2% of the expected value c = 1 for the full range
icality, the scaling dimensions of the two CFTs can vary
ofR2. Wenoteanincreaseddeviationfromc=1atthe
C continuously. As noted in section II, the orbifold CFT
left (R2 0.5) end of the critical line. Here, fields in
C ∼ containsafixedsectorofprimaryfieldswhosescalingdi-
the Z Z symmetric sector are crossing ∆ = 2, the
2 2 φ mensionsdonotchangewithR ,whereasforthefreebo-
× O
RG relevant/irrelevant threshold. Some of these fields
son CFT all scaling dimensions (other than the identity
arenotsymmetricundertheO(2)symmetrydiscussedin
and related operators) will vary. The behaviour of these
sectionII29. NearR2 =1.6,wealsoseealargedeviation
C scalingdimensionsprovidesagoodpointerastothetype
from c=1 for the TFCM data. This region occurs close
ofc=1CFTthatcorrespondstoeachofthespinmodels.
toacrossingoftwofieldsintheZ Z symmetricsector
2 2 InFig.7, wefocusonfieldswithsmallscalingdimension
×
of the S1 theory. Again, this crossing does not occur in in the (1, 1) symmetry sector. Previous evidence13,45
the O(2) symmetric theory29. −
indicates that the accuracy of the MERA decreases with
Oursimulationsapproach, butdonotinclude,theend increasing scaling dimension, so it is expected that the
point λ = 1, (R2 = 2). At this point, the XXZ model agreement will be better in this region. The distinction
C
becomes the spin-1/2 Heisenberg model and the line of between the CFTs is already evident, and the scaling di-
criticality terminates. As we approach this point, the mensionsobtainedfromtheXXZandTFCMmodelsare
CFTs have an irrelevant primary operator with a scal- consistentwiththeS1 theory, andtheATmodelhasthe
ingdimension thatisdecreasingas R increases and be- expected fixed scaling dimension present in the orbifold
C
comesmarginalpreciselyatR2 =2. Thisoperatorleads theory.
C
to corrections to the scale invariant behaviour that, as
Wenextexaminethefullrelevantspectrumofthescal-
has previously been noted45, complicate studies of this
ingoperatorsforallthreemodels,uptoamaximumvalue
model using MERA simulations.
of∆ =2. AllsectorsoftheXXZandTFCMmodelsare
φ
This limiting case could be further investigated in two expected to have the same conformal dimensions; those
ways. One approach would be adding more transitional of the S1 theory. Close agreement is seen in the MERA
layers to the MERA, which is expected to move the simulations of the two models (Fig. 8 a,b), although the
HamiltoniantowardstheRGfixedpoint. However,many accuracy deteriorates for larger scaling dimensions. The
layers may be required because this term only decays conformal field theory has an infinite number of scal-
logarithmically. Alternatively, modifying the model by ing operators, however the one-site scaling superopera-
adding finely tuned terms (for example, a next nearest torusedtoobtain∆ hasafinitenumberofeigenvalues.
φ
neighbour coupling) could remove the marginally irrele- This results in only a finite subset of the dimensions be-
vantoperatorwithoutchangingthecontinuumlimit48,49. ing recoverable, with the smallest being most accurately
Bothoftheseapproachesarebeyondthescopeofthecur- reproduced. In particular, ∆ = 0 in the (1,1) sector
φ
rent investigation. (Fig. 9) corresponds to the identity operator. This scal-
10
2 2
3/2 3/2
φ φ
∆ 1 ∆ 1
1/2 a) ( 1, 1) sector 1/2 c) ( 1, 1) sector
− − − −
TFCM XXZ S1 CFT
0 0
0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2
S1 CFT Radius RC2 Orbifold CFT Radius RO−2
2 2
AT Orb. CFT
3/2 3/2
φ φ
∆ 1 ∆ 1
b) ( 1,1) sector d) ( 1,1) sector
1/2 − 1/2 −
0 0
0.6 0.8 1 1.2 1.4 1.6 1.8 2 0.6 0.8 1 1.2 1.4 1.6 1.8 2
S1 CFT Radius RC2 Orbifold CFT Radius RO−2
Figure 8. The behaviour expected from the S1 CFT is recovered by the XXZ and TFCM (a,b), whilst AT simulations recover
the orbifold behaviour (c,d). The accuracy of the recovered scaling dimensions ∆ degrades for larger ∆ . Note that the S1
φ φ
and orbifold CFT radii are related via Eq. (11).
ing dimension is recovered perfectly due to the isometric length. At this length, finite-size effects are still present
constraint on the tensor w. but we are nevertheless close to full convergence. As can
TheATmodelisexpectedtoreproducetheorbifolded be seen in Fig. 9a, the gap is closing like 1/L, as is ex-
theory,withthesignaturefixedsectorbeingequallysplit pected of a critical model5.
over the (1, 1) and ( 1,1) charge sectors. We see this Thespectrumofthequantumspinchaincanberelated
− −
indeed occurs in Fig. 8c, with good agreement for the tothatoftheCFToncethegroundstateenergy(E )and
0
lower scaling dimensions. the overall normalisation of the Hamiltonian are chosen
In the (1,1) sectors (Fig. 9), we see the deviation in- appropriately51. In our problem we need to choose a
crease as we move towards R2 = 1/R2 = 2. This also normalisation of H for each and we achieve this simply
C O
occurs in the XXZ and TFCM models. Recall that, at by rescaling the gap (E1 E0) of the spin model. Then
−
thisendpoint,allthreemodelsareunitarilyequivalentto the rest of the conformal spectrum can be inferred from
the Heisenberg model. the spectrum of the spin chain as follows
∆ =L α(λ)(E E ), (22)
k k 0
× −
C. Exact diagonalisation of the quantum where, anticipating the orbifold CFT, the α(λ) is cho-
Ashkin-Teller model
sen to fix the gap ∆ = 1/8. This value corresponds to
1
the scaling dimension of the operators with fixed scal-
It is well known that the low energy spectrum of finite ing dimension in the ( , ) sectors (Fig. 8d). Thus the
± ∓
size spin chains should correspond to the spectrum of first two scaling dimensions of the CFT are correct as
primary and descendant fields of the associated CFT5. a result of our choice of normalisation and the zero of
As such the results of exact diagonalisation (ED) with energy. Obtaining the remaining levels represents a con-
periodic boundary conditions can be directly compared firmation of the CFT identification. The same approach
withthoseofbothCFTandtheMERAsimulations. For was taken in previous studies of the quantum Ashkin-
a review of the ED technique, see for example Ref. 50. Teller model40,41, although there an analytic form for α
As a further test of the MERA simulations, we per- was proposed.
formedEDcalculationsfortheAshkin-Tellermodelwith We see good qualitative agreement between the two
periodic boundary conditions for chains up to L=12 in methods. A selected Z Z symmetry sector is pre-
2 2
×