Stripe melting, a transition between weak and strong symmetry protected topological phases

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by  Yizhi You

Stripe melting, a transition between weak and strong symmetry protected topological
phases
Yizhi You1,2 and Yi-Zhuang You2
1Department of Physics and Institute for Condensed Matter Theory,
University of Illinois at Urbana-Champaign, Illinois 61801
2Kavli Institute for Theoretical Physics, University of California Santa Barbara, Santa Barbara, California 93106
(Dated: March 23, 2016)
For a gapped disordered many-body system with both internal and translation symmetry, one
6
can define the corresponding weak and strong Symmetry Protected Topological (SPT) phases. A
1
strongSPTphaseisprotectedbytheinternalsymmetryGonlywhileaweakSPTphase,fabricated
0
by alignment of strong SPT state in a lower dimension, requires additional discrete translation
2
symmetry protection. In this paper, we construct a phase transition between weak and strong

r SPT phase in strongly interacting boson system. The starting point of our construction is the
a
superconducting Dirac fermions with pair density wave(PDW) order in 2d. We first demonstrate
M
that thenodal line of thePDW containsa 1d boson SPT phase. Wefurthershow that melting the
PDW stripe and condensing the nodal line provoke the transition from weak to strong SPT phase
2
in 2d. The phase transition theory contains an O(4) non-linear-σ-model(NLσM) with topological
2
Θ-termemergingfromtheproliferationofdomainwallsboundtoanSPTchain. Similarschemealso

applies to weak-strong SPT transition in other dimensions and predicts possible phase transition
]
l from 2d to 3d topological order.
e
-
r
t
s I. INTRODUCTION AND MOTIVATION strong interaction. One representative of weak boson
. SPTphasein2dcanbeconstructedbyalignmentofspin
t
a Recently years, lots of attention had been devoted 1Haldane chainalongy direction1,10,11. SuchweakSPT
m
to the classification and characterization of symmetry state is protected by both time-reversal(T) and transla-
- protected topological(SPT) phases1–5. An SPT state is tion symmetry in y direction.
d
n short-range entangled(SRE) in the bulk, but with non- Sofar,thephasetransitionandtheconnectionbetween
o trivial boundary spectrum. As long as certain symme- weak to strong SPT phase at interacting level is less ex-
c try is preserved,one can never adiabatically connect the plored. In particular, as weak SPT phase is built upon
[ SPT state to a trivial many body state(e.g. direct prod- the strong SPT state in a lower dimension, figuring out
2 uct state). the connection between strong and weak SPT state also
v In particular, for many-body system with an internal provides us a new way to relate SPT phases in different
7 symmetry, one can always define a corresponding weak dimensions.
5 andstrongSPTphase6–9. AstrongSPTphasein(n+1)d In this paper, we are going to answer the following
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isonlyprotectedbyaninternalsymmetryG. Meanwhile, question. Isthereastraightforwardwaytoconnectweak
0
a weak SPT phase in (n + 1)d can be constructed by andstrong bosonSPT states? In particular,how can we
0
. alignment of strong SPT state in (n)d protected by an characterize the phase transition theory between strong
1 internal symmetry G′ (G′ and G can be different inter- and weak SPT phases? The strategy we apply for these
0 nal symmetries). Such corresponding weak SPT phase question is to melt the superconducting stripe.12–16
6
is therefore protected by both the internal symmetry G′ We start from 2d pair density wave superconducting
1
: and the discrete translation symmetry along the align- state whose nodal line contains 8 copies of helical Ma-
v
ment direction. One of the well-known example is the jorana modes. These gapless modes can be gapped into
i
X weak and strong time-reversal(T) invariant topological a boson SPT phase, being equivalent to a spin 1 Hal-
r insulator(TI) in 3d6. A strong TI in 3d can not be adia- dane chain4,17,18. The pair density wave(PDW) state,
a batically connected to a trivialband insulator as long as as a stripe superconducting phase, contains alignment
T ispreserved. While aweakTIin3d,obtainedbylayer of nodal lines decorated with the Haldane chain which
stacking of 2d T invariant TI along z direction, requires therefore forms a weak SPT state. By stripe melting
both T anddiscrete translationsymmetry alongz direc- and nodal loop condensation, the system becomes a 2d
tiontoprotectthephase. Otherwise,withouttranslation strong SPT state protected by T ×Z2 symmetry. The
symmetry, one can always turn on strong tunneling be- similar construction also applies to 3d stripe supercon-
tweentwoTIlayersandthefermionbandbecomestrivial ductorwherea2dSPTphaselivesinsidethenodalplane
without gap closing in the bulk. Therefore, as long as T ofPDWorder. The condensationofnodalmembranein-
isimposed,weakandstrongTIbelongtodifferentphases duces a transitionfromweak to strongSPT phase in 3d.
and a connection between them requires gap closing. In addition, after gauging the symmetry of an SPT
Beyond the non-interacting fermion SPT phase de- state, the gauge flux can acquire nontrivial braiding
scribed by band theory, much effort had been made on statistics19,20. Inthispaper,wewouldshowthatthecon-
the exploration of strong/weak boson SPT phase with nection between weak and strong SPT phases by stripe
2
melting also provides a new way to generate the rela- Hamiltonian in the Majorana basis Ψ† =χ +iχ ,
↑ 1,↑ 2,↑
tion between intrinsic topological order in different di-
mensions. H =χTk(σxkx+σzky +O2σyτx+O1σyτz)χ−k
Ourstripemeltingapproachinheritsthedecorateddo-
∆=O +iO (2.2)
1 2
main wall idea raised by Chen et al.21. In the decorated
domain wall approach, they attach a lower dimension Here τ matrix acts on the two Majorana index while σ
SPT phase on the domain wall of an Ising variable and actingonspinindex. Imaginewe chooseaspecific gauge
the condensation of domain walls drives the system into where ∆ is real so O is zero.
2
a boson SPT state. In this work, we intend to find a Now we turn to the situation that the superconduct-
naturalandmicroscopicwaytolettheSPTstateemerge ing field is nonuniform in space. It contains PDW order
inside the domain wall. The PDW stripes contain nodal ∆ = |∆|cos(Qr) whose pairing amplitude modulates in
line which act as a domain wall and the gapless fermion space along the PDW wave vector and forms a stripy
modes inside domain wall can be driven into a boson superconductor. Such PDW can be realized if the Dirac
SPT state by interaction4,18,22. Indeed, the phase tran- fermiondispersionhassomenematicityandthefermive-
sition between the stripe phase to the disordered phase locity is anisotropic. Else, the proximity effect between
includes a topological Θ term which reflects the nature an s-wave PDW superconductor and the Dirac fermion
of SPT decorated nodal line. Such decorated domain can also induce the stripe superconducting Dirac cone.
wall approach can be extended to other decorated de- The fermions in the stripes are gapped but the nodal
fects, e.g. the decorated vortex lines23–27. In section IV, line as a domainwallofO contains1dhelicalMajorana
1
we start from the Abrikosov phase of 3d topological su- mode. AssumethePDWwavevectorisintheydirection
perconductor whose vortex line contains 1d SPT chain so the nodal line is extended along x, the helical mode
and the melting of vortex lattice give rise to a boson can be written as,
SPT phase in 3d.
H =χTk(i∂xσx)χ−k
χ=(χ ,χ ) (2.3)
1,↑ 2,↓
II. MELTING THE SUPERCONDUCTING
STRIPE IN PDW PHASE The only fermion bilinear mass χTσyχ which can gap
out this helical mode is associate with the imaginary
part(O ) of the superconductivity in the stripe. As the
Inthissection,wesystematicallyinvestigatethephase 2
nodal line has zero pairing strength, there is no way to
transition from weak to strong SPT phase in all dimen-
gap out the helical mode.
sions. The weakSPT statewestartwiththroughoutthe
Now assume we have 8 copies of the superconduct-
paper is embedded in a stripe superconducting phase.
ing Dirac fermionembellished as Eq.(2.2),the nodalline
The domain wall between different stripes contains a
therefore contains eight helical Majorana modes. First
strong SPT state in a lower dimension which therefore
wetakethe first4 copiesofthe helicalmodesandcouple
makes stripe phase identical to a weak SPT state. After
them to an O(3) vector~n=(n ,n ,n ),4,18,22
stripe melting and domain wallproliferation, the system 1 2 3
experiencesa transitionfromweaktostrongSPT phase. H =χTk(i∂xσ100+n1σ312+n2σ320+n3σ332)χ−k
T :χ→Kσ300χ, ~n→−~n (2.4)
A. stripe melting in 2d PDW σabc refers to the direct product of Pauli matrices σa ⊗
σb ⊗σc. Here T symmetry operator is redefined in an
The starting point of our construction is the super- unusualwayasithasT2 =1whenactingonthefermion
conductivity in massless Dirac fermion at 2d. Imagine sector. However,wewouldshowinourlatercontentthat
we have a semimetal whose low energy theory contains the T symmetry operator acts on the bosonic degree of
several Dirac cones, freedom ~n in a nontrivial way. As ~n becomes −~n under
T operation, any nonzero expectation value of ~n breaks
H =Ψ†k(σxkx+σzky)Ψ−k (2.1) T. When~nisdisordered,wecanintegraloutthefermion
and obtain the effective theory of~n,
Here σ acts onthe spin index of the Dirac fermion. Now 1 iπ
we turn on s-wave pairing ∆ = Ψ†↑Ψ†↓ − Ψ†↓Ψ†↑ of the L= g(∂ρ~n)2+ Ω2ǫijkǫµνni∂µnj∂νnk (2.5)
Dirac cone28. The s-wavepairingtermcanemergewhen
we slightly doped the semimetal and turn on fermionin- Ωd refers to the volume of Sd. The theory we obtain is
teraction at finite strength, or from proximity effect be- theO(3)NLσMwithatopologicalthetatermforΘ=π,
tween the Dirac fermion and an s-wave superconductor. which is equivalent to a critical spin 1/2 AF Heisenberg
Thes-wavesuperconductivityturnstheDiracsemi-metal chaindescribed by SU(2) CFT. The theory is invariant
1
into a gapped phase. Write the superconducting Dirac under the T symmetry we defined in Eq.(2.4).
3
Till now, we had coupled the first 4 copies of the he- The PDW phase is therefore a weak SPT phase pro-
lical Majorana modes with an~n vector and the effective tected by both T and discrete translation symmetry in
theoryfor~ninheritedfromthefermionmodeisacritical y direction. The nodal line of the PDW order contains a
spin 1/2 Heisenberg chain. The rest 4 copies of the heli- Haldane chain as a 1d SPT phase protected by T. Even
cal mode can be coupled to another ~n′ in the same way the PDW state we start with is a fermonic theory, when
as Eq.(2.4) which give rise to another O(3) NLσM with the SC amplitude ∆ and the O(3) vector magnitude is
a topological theta term for Θ = π. The eight helical large enough, there would be an absence of fermonic ex-
Majorana modes are therefore turned into two copies of citationatlowenergyspectrumsotheeffectivetheoryat
spin 1/2 Heisenberg chain described by SU(2) CFT. low energy is bosonic.
1
The two SU(2)1 CFT canbe fused into a gappedHal- The discrete translation symmetry prevents two Hal-
dane phase via marginal interaction as,29 dane chain dimerizing into a spin 2 chain which is triv-
ial under T. Each end point of the nodal line contains
1 iπ
L= [(∂ ~n)2+(∂ ~n′)2]+ ǫijkǫµνni∂ nj∂ nk a spin 1/2 degree of freedom from the Haldane chain
g ρ ρ Ω2 µ ν so the boundary along y direction consists of an inter-
iπ acting spin 1/2 chain. According to Lieb-Schultz-Mattis
+ ǫijkǫµνn′i∂ n′j∂ n′k+α ~n·~n′ (2.6)
Ω2 µ ν theorem32,33,theGSofthespin1/2chainshalleitherbe
gapless or symmetry breaking (breaks T or translation).
When α is positive, the ferromagnetic interaction be-
Hereandafter,wewoulddemonstratethat,aftermelt-
tweentwocriticalHeisenbergchaindrivesthetheoryinto
ingthesuperconductingstripe,thereoccursaphasetran-
the Haldane phase, which is an SPT phase protected by
sition from the PDW phase to a uniform superconduct-
T symmetry defined in Eq.(2.4). As the interaction is
ing state, which meanwhile transmutes the weak SPT to
marginal, we only need small interaction compared to
a strong SPT state in 2d.
the superconducting gap in the stripe so the stripe su-
Melting the stripe superconducting configuration re-
perconductingstateisnotaffected. Finally,theeighthe-
stores both translation and rotation symmetry. The
lical Majorana modes in the nodal line are mapped into
melting procedure can be fulfilled by dislocation and
a Haldane chain29–31 describe by NLσM with a Θ = 2π
disclination proliferation. The dislocation disorders the
theta term,
stripe configuration to restore the continuous transla-
1 i2π tion symmetry while the disclination bends the straight
L= g(∂ρ~n)2+ Ω2 ǫijkǫµνni∂µnj∂νnk (2.7) nodal lines into an arbitrary crooked configuration. The
stripe melting12,34–36 driven by dislocation and disclina-
The Haldane chain carries a free spin 1/2 zero mode on tion condensation would meanwhile condense the nodal
the boundary which is protected by T. Based on the loop. We assumethere issome thermalorquantumfluc-
NLσMdescriptionoftheHaldanechain,theboundaryof tuation which effectively generates positive interaction
the (1+1)d NLσM in Eq.(2.7) is a domainwallbetween between disclination/dislocations and therefore triggers
Θ = 2π to Θ = 0. The edge therefore contains an O(3) a tendency to condense them. During the condensation,
Wess-Zumino-Witten term at level one, with a two-fold the origin Goldstone mode(φ) in the stripe phase asso-
degenerate ground state. We can represent the free spin ciated with the translation symmetry broken is gapped
1/2 on the edge of the Haldane chain in terms of the by the vortex tunneling term cos(nφ). The coherence of
CP1 representation as ~n = z†~σz. z is a complex spinor the condensed dislocations and disclinations generates a
field z =(z ,z )T and T operatoract on the spinor field coherent state of all types of nodal loop configurations.
1 2
as T : z → Kiσ2z. Therefore, the T symmetry acts Meanwhile, since the nodal line separates the positive
projectively on the spinor z. and negative pairing amplitude, the nodal line must be
closed in the bulk. If the nodal line has an end point in
The above construction decorates the nodal line with
the bulk, there must be a half vortex of the pairing field
1d SPT state, as is illustrated in Fig. 1.
associate with it. We focus on the situation where the
system is vortex free so all the nodal line must form a
close-loop configuration.
After we proliferateand condensethe nodalloops, the
rotationsymmetryofthepairingfieldisrestoredandone
obtains a uniform phase. The ground state(GS) of the
uniform phase can be written in terms of the superposi-
tionofallclosenodalloopconfigurationsdecoratedwith
Haldane chains.
As each nodal line carries a Haldane chain, the GS
FIG. 1. The stripe configuration of PDW. The pairing field wavefunctionisasaturationofHaldaneloopwhichsepa-
in the white/blue stripes has opposite sign. The red line is ratesthepositiveandnegativepairingfield. Letusassign
thenodallinewhichcontainsaHaldanechain. Thegreendot the positive/negative pairing amplitude as a Z variable
2
is the free spin 1/2 edge state of theHaldane chain.
and label it with n so ∆ = O = n . In the PDW
4 1 4
4
symmetry transformation assignment) and each has a
Z classification1,37. Here since we uniquely assign the
2
symmetry action, in disordered phase the classification
of Eq.(2.8) is Z .
2
Hereweemphasize thateventhe systemwe startwith
is fermionic, the finally SPT phase is a bosonic theory
whose low energy effective theory can be absent from
fermion excitations. This conclusionis based onthe fact
thattheO(4)vectormagnitudewhichservedasthemass
of the fermion system is large enough so the fermion ex-
citationissuppressedatlowenergy. Inaddition,onecan
also confine the fermonic degree of freedom by coupling
FIG.2. TheGSwavefunctionafternodalloopcondensation. the theory with a Z gauge field4,18. The vison flux of
The system is saturated with all close nodal loops(red line) 2
the Z gauge field would trap 16 majorana zero modes
whichcarriesaHaldanechain. Thenodalloopseparatesposi- 2
whichcanbegappedoutwithoutbreakingtheZ andT
tive/negativeamplitudeofthepairingfield. Thebluedotson 2
symmetry. Consequently, visoncondensation would lead
theboundaryarethefreespin1/2edgestatesoftheHaldane
chain. to a fully gapped nondegenerate bosonic state.
The effective theory we wrote in Eq.(2.8) can also be
verified in a microscopic way from the fermion side. We
phase, n4 = cos(Qr) breaks Z2 symmetry and forms a firsttake4outofthe 8superconductingDiracconesand
stripe configuration. When the nodal loop condensed, couple them with an O(3) vector4,18,22,38,
theIsingscalarn isthendisorderedandspatialsymme-
4
try is restored. The phase transition ofsuch stripe order H =χTk(i∂xσ1000+i∂yσ3000+O1σ2300+O2σ2100
to disorder phase can be captured as, +n1σ2212+n2σ2220+n3σ2232)χ−k
3 T :χ→Kσ2200χ, Z2 :χ→σ0100χ (2.10)
1
L=κ(∂ n )2+α(n −n¯ )2+β(n )4+ (∂ n )2
µ 4 4 4 4 g µ a The first line illustrates the four copies of supercon-
X
a=1
ducting Dirac cone as Eq.(2.2), the second line is the
i2π
+ ǫijklǫµνρni∂ nj∂ nk∂ nl couplingbetweendifferentcopiesoffermionviaO(3)vec-
Ω3 µ ν ρ
tor ~n. The imaginary part of the pairing O is taken to
2
n¯ =cos(Qr) (2.8)
4 bezerobygaugechoicesothepairingfieldisarealscalar
field which canbe regardedas an Ising variable O . The
1
The totaldegreeof freedomwe haveis the Z Ising vari-
2 domainwall of O contains four helical Majoranamodes
1
able n between the nodal line and the O(3) vector ~n
4 coupling with ~n as Eq. 2.4.
which comes from the NLσM on the nodal line. The
The stripe melting and condensation of the nodal
first three terms in Eq.(2.8) are the common Landau-
loops restored the translation(also rotation) symmetry
Ginzburg type theory for the phase transition. The
and thus gapped the goldstone mode with respect to
fourth term is the fluctuation of the O(3) vector deco-
translation symmetry broken. The Ising variable O is
1
rated on the nodal line. The last term is a topological
therefore disordered. Label O as n , the four copies of
1 4
theta term which captures the exotic physics during the
Dirac cone now couples with an O(4) vector. When the
transition. Whenthe Z Isingvariablen is orderedinto
2 4 O(4)vectorisdisordered,T ×Z symmetryispreserved.
2
stripe configuration, each nodal line ∂ n as a domain
y 4 Integrating out the fermion give rise to a critical O(4)
wall of the Ising variable carries a Haldane chain which
NLσM with a theta term at Θ=π.
could be described as a O(3) NLσM with theta term
The rest 4 copies of the superconducting Dirac cones
2πǫijkǫµνni∂ nj∂ nk. The last term in Eq.(2.8) exactly
Ω2 µ ν can be manipulated in the same way which gives rise to
captures the bound state between a Z domain wall and
2 anothercriticalΘ=π O(4)NLσM.Wecanthenfusethe
the(1+1)dthetaterm. Whenα<0,then scalarisdis-
4 two Θ= π theory into Θ =2π theory via ferromagnetic
ordered, the boson variable becomes O(3)×Z ∼ O(4),
2 interaction between two O(4) vector,
and the theory becomes the (2+1)d O(4) NLσM with
Θ = 2π. This is known as an SPT phase protected by 1 iπ
L= [(∂ ~n)2+(∂ ~n′)2]+ ǫijklǫµνρni∂ nj∂ nk∂ nl
T ×Z2, g ρ ρ Ω3 µ ν ρ
iπ
T :n1,2,3 →−n1,2,3; Z2 :n1,2,3,4 →−n1,2,3,4 (2.9) + Ω3ǫijklǫµνρn′i∂µn′j∂νn′k∂ρn′l+α ~n·~n′ (2.11)
T actsonthe O(3)vectoronthe nodalline andZ sym- Which finally drives the phase into a gapped SPT phase
2
metry enforces the n Ising variable to be disordered protected by T × Z symmetry. (Here we point out
4 2
so the nodal loops saturate and condense. The boson that although we choose a specific gauge so the pairing
SPT phase with T × Z protection has (Z )2 classifi- field ∆ = O is real, the phase fluctuation of the su-
2 2 1
cation, which contains two root states (with different perconducting field would result in small fluctuation of
5
O . However, as long as the ferromagnetic interaction
2
between two O(4) vector is turned on, the theory is al-
ways in the gapped SPT phase even in the presence of
smallO . Thiswouldbedemonstratedintheappendix.)
2
1 i2π
L= (∂ ~n)2+ ǫijklǫµνρni∂ nj∂ nk∂ nl
g ρ Ω3 µ ν ρ
T :n →−n ; Z :n →−n (2.12)
1,2,3 1,2,3 2 1,2,3,4 1,2,3,4
The edge state of this Θ = 2π O(4) NLσM is the O(4)
Wess-Zumino-Wittentheoryatlevelonewhichisgapless.
The GS wave function of this NLσM can be written in
terms ofthe partitionfunctionof anO(4)Wess-Zumino-
Witten model3.
FIG.3. TheslabconfigurationofPDW.thegreen/whileslab
contains pairing strength with positive/negative sign. The
|GSi= D[~n]eiΩ2π3 Rdx2R01duǫijklǫµνρni∂µnj∂νnk∂ρnl|~ni nodal plane(blue) contains a 2d SPT state.
Z
(2.13)
Thewavefunctionisasuperpositionofallpossibleconfig- We first take 8 out of 16 Majorana cones and couple
urationof~n,eachconfigurationhasacoefficientinterms them to an O(4) vector in the same way as Eq.(2.10),
of the Wess-Zumino-Witten term. If we further break
H =χT(i∂ σ1000+i∂ σ3000+n σ2100
the O(4) variable to Z2, the wave function is equivalent k x y 4
to the Levin-Gu model19. +n1σ2212+n2σ2220+n3σ2232)χ−k
Z : χ→σ0300χ, ~n→−~n (2.16)
2
B. Membrane condensation in 3+1D
The O(4) vector ~n change sign under the Z symmetry
2
we defined. When ~n is disordered, the theory is Z in-
2
In the previous section, we studied the PDW melting
variant. Integrating out the fermions, the nodal plane
transition which connects a weak to strong SPT state
then becomes a critical O(4) NLσM with Θ=π.
in 2d. Such idea can be easily generalized into other
Therest8MajoranaconescoupletoanotherO(4)vec-
dimensions. In3d,westartwithWeylfermionsandturn
tor in the same way as Eq.(2.16), we eventually obtain
ons-wavepairing∆=Ψ†Ψ†−Ψ†Ψ† togapouttheWeyl
↑ ↓ ↓ ↑ two copies of the critical O(4) NLσM with Θ=π. Cou-
cone. TheHamiltonianofsuperconductingWeylfermion plethetwocriticaltheoryof~nthroughtheferromagnetic
can be written in Majorana basis as,
inter-copy interaction in the same way as we did in Eq.
H =χTk(i∂xσ30+i∂yσ10+i∂zσ22+O1σ21+O2σ23)χ−k 2O.1(41), NthLeσsMystweimthgtooepsoilnotgoicaalgtahpeptaedteprhmasaetdΘes=cr2ibπe.d by
T =Kiσ21, ∆=O +iO (2.14)
1 2
1 i2π
The time reversaloperatorT we definedhereis different L= g(∂µna)2+ Ω3 ǫijklǫµνρni∂µnj∂νnk∂ρnl
from the usual one. The imaginary part of the pair-
Z :~n→−~n (2.17)
ing field O is zero, by gauge choice. Imagine the su- 2
2
perconducting state has a PDW order where the ampli-
This effective theory describes a 2d SPT state protected
tude of the pairing field modulates along z direction as
by Z symmetry1,4. Till now we had figured out that
∆ = O = cos(Qz). The superconductivity then forms 2
1 interactioncandrivethe16Majoranaconesonthenodal
a slab configuration along z and Qz =(n+1/2)π is the
plane into a gapped 2d boson SPT phase protected by
nodal plane which separates the positive and negative
Z symmetry. In the PDW phase, a 2d boson SPT state
slab of the pairing field. In the nodal plane, the pair- 2
living on the nodal plane stacks along the z direction as
ing strengthis zerosothe nodalplanecontainsa gapless
Fig. 3, which accordinglyforms a weak SPT phase in 3d
Majorana cone,
protected by Z and discrete translation symmetry in z
2
H =χTk(i∂xσ3+i∂yσ1)χ−k (2.15) direction.
Restoring the spatial symmetry of the PDW order
The only fermion bilinear mass χTσ2χ for the Majorana would meanwhile drive the weak SPT to a strong SPT
cone is associate with the imaginary part of the pairing state. To demonstrate, first we melt the superconduct-
fieldinthebulk,whichisexpectedtobezeroatthenodal ing slabs. The melting procedure can be realized by dis-
plane. location and disclination proliferation, which bends the
Now we take 16 copies of such superconducting Weyl nodalplaneintoarbitraryclosemembraneconfiguration.
fermion with pair density wave order, each nodal plane The condensation of nodal membrane restores the rota-
then contains 16 Majorana cones. tion/translation symmetry. After slab melting, the GS
6
wave function of the isotropic phase can be written in termindicates the domainwallofn is bound to a topo-
5
termsofthesuperpositionofallpossibleclosemembrane logical Θ term in 2d, as a signature of the 2d SPT state
configurations. If we relabel O = n as an Z variable, decoratedin the nodal plane. When n has stripe order,
1 5 2 5
the nodalmembranecondensationprocedureiscaptured the nodal plane together with the 2d SPT situates along
by the Z order-disorder transition with a topological z direction and simultaneously forms a weak SPT in 3d.
2
theta term, At the disordered phase of n , (n ,n ,n ,n ,n ) to-
5 1 2 3 4 5
gether forms an O(5) vector. The effective theory is just
4 the O(5) NLσM with Θ = 2π topological theta term.
1
L=κ(∂ n )2+α(n −n¯ )2+β(n )4+ (∂ n )2 This model is known as a boson SPT phase in 3d pro-
µ 5 5 5 5 µ a
Xa=1g tected by Z2×T1,4. The symmetry operator acting on
i2π the O(5) vector is,
+ ǫijklmǫµνρλni∂ nj∂ nk∂ nl∂ nm
Ω4 µ ν ρ λ
T :n →−n ; Z :n →−n (2.19)
n¯ =cos(Qz) (2.18) 5 5 2 1,2,3,4 1,2,3,4
5
The effective theory in Eq.(2.18) can also be obtained
The first three terms are the Ginzburg-Landau theory inamicroscopicwayfromthefermionside. Take8outof
of an Ising transition, the fourth term characterizes the 16 superconducting Weyl cones and coupled them with
fluctuation of the O(4) vector (n ,n ,n ,n ). The last an O(4) vector4,
1 2 3 4
H =χTk(i∂xσ30000+i∂yσ10000+i∂zσ22000+O1σ21000+n1σ23212+n2σ23220+n3σ23232+n4σ23100)χ−k
T :χ→Kiσ21000χ,n →−n Z :χ→σ00300χ,n →−n (2.20)
5 5 2 1,2,3,4 1,2,3,4
O is the real part of the pairing field(the imaginary tions and we can therefore view the effective theory as a
1
part of the paring is zero by gauge choice). Af- bosonic one a low energy.
ter nodal membrane condensation, the PDW becomes Ingeneral,the3dZ ×T bosonSPTstatehasZ3 clas-
2 2
disordered(hχTkσ21000χ−ki = 0), the T ×Z2 symmetry sification,whichcontainsthreerootstatewithrespectto
we defined in Eq.(A1) is therefore restored. Further la- differentsymmetryassignmentandeachrootstatehasZ
2
bel O as n and integrating out the fermion gives the class. Herewehadspecifiedthesymmetryassignmentso
1 5
criticalO(5)NLσMwithΘ=π,whichisagaplessboson the classification is Z . The gapless surface state of this
2
theory. SPTphaseisdescribedbyanO(5)Wess-Zumino-Witten
Coupletherest8superconductingWeylconeswithan- model17,39. If we further break the O(5) vector on the
other O(4) vector in the same way and follow the same surface to U(1)×U(1) or Z2 ×Z2, the surface can be
manipulation subsequently, we obtain another critical gapped and exhibit Z2 topological order2,40.
O(5) NLσM with Θ = π. Take the two copies of gap- In summary, in this section we start from 3d PDW
less O(5) NLσM with Θ = π and turn on ferromagnetic phase whose nodal plane contains a 2d SPT state. Melt-
inter-copy interaction as we did in Eq. 2.11, the system ingthenodalplaneleadsatransitionfromweaktostrong
becomes a gapped phase described by O(5) NLσM with SPTstatein3d. ThestrongSPTphasecanbedescribed
Θ = 2π. (Here we point out that although we choose by O(5) NLσM with Θ = 2π. Such PDW melting con-
a specific gauge so the pairing field ∆ = O is real. As structionforweak-strongSPTtransitioncanbeextended
1
long as the ferromagnetic interaction between two O(5) in other dimensions following the similar strategy.
vector is turned on, the theory is always in the gapped The strong SPT state we obtained via PDW melting
SPT phase even in the presence of small O fluctuation. is very similar to the decorated domain wall construc-
2
This would be demonstrated in the appendix.) tion studied by severalpioneers21,23–27. In our work,the
domain wall is the nodal line(plane) of the PDW and
5 the Ising variable is the positive/negative amplitude of
1 i2π
L= g(∂µna)2+ Ω4 ǫijklmǫµνρλni∂µnj∂νnk∂ρnl∂λnm the pairing field. As the PDW nodal line(plane) carries
Xa=1 some gapless fermion mode, we can turn on interaction
T :n →−n ; Z :n →−n (2.21) to drive the gapless fermion state into a gapped boson
5 5 2 1,2,3,4 1,2,3,4
SPT state. In this way, the SPT state embellished on
This theory is a 3d SPT phase protected by Z × T the domain wall appears in a nature way. This deco-
2
symmetry. As we had emphasized in the last section, rated defect picture can be extended to a large class of
although the starting point of out model is a fermonic SPT states, some of which are beyond group cohomol-
system, once the O(5) vectormagnitude is largeenough, ogyclassification23,26,27. InourPDWmeltingtransition,
the low energy spectrum is absent from fermion excita- the change of ”topology” and the restoration of spatial
7
symmetry appears at the same time as the PDW nodal The Hamiltonian is the sum of all projection operators
line(plane)whichbreaksthecontinuoustranslationsym- and P projects three spins onto S =3. The three spins
3
metry is embeddedwith O(n)NLσM witha topological ijk live on the all elementary triangles where ij and jk
Θ term. The condensation of the nodal line restore the are the NN bond and ki are the NNN bond. The GS of
spatialsymmetry and meanwhile generatesa topological each triangle shall contain at least one singlet bond to
Θ term from the emergent O(n+1) vector. minimized the energy. The GS of such Hamiltonian is
Inparticular,itwaspointedoutthatthedecorateddo- an alignment of the AKLT chain42,44 along either x or y
main wall approach only works when the SPT state liv- direction (or x+y,x−y direction) as illustrated in Fig.
ing on the domain wallsatisfied the non-double-stacking 4. The GS has a bond nematic order which breaks C
4
condition (NDSC)24,41, otherwise one can always stack- rotation. The nematic order parameter can be written
ing a different counterpart to gap the boundary without as a director field v ,v ,
1 2
degeneracy. Therefore, our stripe melting approach for
weak-strong SPT phase transition is not general and is v2−v2 2v v
x y x y
v = ,v =
limitedtothebosonSPTstateswithinthetopologicalΘ 1 2
v2+v2 v2+v2
term scheme23. In addition, even the strong SPT state x y x y
q q
does not require discrete translation symmetry protec- v =S~ S~ −S~ S~
x x,y x+1,y x,y x−1,y
tion,itdemandsadditionalZ orT symmetrycompared
2
to the weak SPT phase in the stripe phase. The addi- vy =S~x,yS~x,y+1−S~x,yS~x,y−1 (3.3)
tionalsymmetry is essentialas it ensuresthe Z variable
2
between the nodal line(plane) is disordered so the nodal The line defect of the nematic order, as the domain wall
line(plane) is condensed at the strong SPT phase. Ac- ofv1,consistsaspin1/2chainwithAFinteraction. The
cordingly, our strong SPT phase after stripe melting is line defect then carries a critical Heisenberg chain de-
protected by an enlarged (internal)symmetry which en- scribed by SU(2)1 CFT illustrated in Fig. 4.
sures the domain wall proliferation.
III. LINE DEFECT CONDENSATION IN
NEMATIC PARAMAGNETIC
Besides the stripe melting of PDW in Dirac fermions,
hereweproposeanotherfeasiblerealizationonthephase
transition from weak to strong SPT state in frustrated
spinsystems. Ourstartingpointisaspinnematicparam-
agnetic state whose line defect contains a 1d SPT chain.
When the line defects align in parallel into stripe con-
FIG. 4. line defect of nematic paramagnetic state. Left and
figuration, the system as a 2d weak SPT state contains
right side refers to opposite bond nematic order v1. The red
alignment of 1d SPT chain. Once the line defects pro-
bond are the spin singlet bond in the AKLT chain. The line
liferate and condense, the effective theory is akin to a
defect in the centercarries a spin 1/2 chain.
strong 2d SPT phase.
Wefirststartfromaspin1nematicparamagneticstate
Imaginewehavetwocopiesofsuchbondnematicpara-
on square lattice studied by a series of theoretical and
magneticstate,thelinedefectcontainstwocopiesofcriti-
numerical papers,42,43.
calspin1/2Heisenbergchain. TurnonaweakFerromag-
H =J1 S~iS~j +J2 S~iS~j +... (3.1) netic interaction between the two copy, the two SU(2)1
X X CFTinthedefectlinecanthereforebegappedandtrans-
hiji hhijii
mute into a spin 1 Haldane chain. Consequently, for two
Thespinhasantiferromagneticinteractionbetweennear- copies of such nematic paramagnetic phase, the line de-
est and next nearest sites. When J1/J2 is in the inter- fect which separates opposite nematic order v1 carries a
mediateregime,somenumericalevidenceandtheoretical 1d SPT state protected by T. When the bond nematic
prediction42,43arguethatthereexistanematicparamag- variablehasstripeorderv =|v |cos(Qy),thedefectline
1 1
netic phasewhichbreaksspatialC4 symmetry toC2. To together with Haldane chain situates uniformly along y
verified this prediction, Wang et.al.42 raised an exactly directionandthereforeproduceaweakSPTphasein2d.
solvable Hamiltonian for the spin 1 nematic paramag- Hereweneedtomentionedthatwhenthe linedefecthas
netic state on square lattice, a snake configuration going along the corners, the spins
on defect line would also encounter with NNN antiferro-
H = P (S~ +S~ +S~ )
3 i j k magnetic interaction which have a tendency for dimer-
X
ijk ization. However, as long as the intercopy interaction is
P (S)= 1 S2(S2−2)(S2−6) (3.2) appropriateandstrongenough,onecanalwaysdrivethe
3 720 spins on the defect line into the Haldane phase29.
8
Takethepositiveandnegativevalueofv asaZ vari- densation. Beyond the domain wall defect, one can also
1 2
able, once we proliferate the line defect and condense start from other defects like vortex or skyrmion embel-
them, the defect line saturates in space and the Z sym- lished with an SPT state in the corresponded dimen-
2
metry is therefore restored. The GS wave function then sion. The condensation of vortex/skyrmion simultane-
becomes a superposition of close defect loop configura- ously drives the phase into a strong SPT state.
tions decorated with a 1d SPT. This defect line conden-
sationprocedureinanematicparamagneticstate isakin
In this section, we demonstrate another way to real-
to the nodal line condensation in PDW we discussed in
ize weak-strong SPT transition by vortex condensation.
previous sections. At this stage,we obtain a strong SPT
We startfrom 1dSPT states decoratedin the Abrikosov
phaseprotectedbyT ×Z symmetrywhoseeffectivethe-
2
lattice. By vortex condensation and Abrikosov lattice
ory is the same as the condensed nodal line phase we
melting,theweakSPTphase,originatefromthe1dSPT
explained in Eq.(2.10).
chainhostedinthevortexlattice,transmuteintoastrong
SPT state in 3d.
IV. ABRIKOSOV LATTICE MELTING
We first start with two Weyl cones with opposite chi-
In our previous discussion, we trigger the transition rality and turn on the s-wave intra-cone pairing, the ef-
fromweakto strongSPT by decorateddomainwallcon- fective theory is
H =χTk(i∂xσ103+i∂yσ303+i∂zσ223+O1σ210+O2σ230)χ−k
∆=O +iO ,Z :χ→σ020χ (4.1)
1 2 2
The vortex line of the s-wave superconductor traps a anO(3) topologicalΘ term, the vortex currentJv shall
µν
gapless Majorana helical mode. Now imagine the su- minimal couple with the O(3) theta term as,
perconductor is in the Abrikosov phase, where the vor-
tex line of ∆ = O1 + iO2 is pinned periodically along Jv B ∼ i2πǫabcǫµνρλO1∂ O2∂ na∂ nb∂ nc+....
x,y directionandaccordinglyformsanAbrikosovvortex µν µν Ω4 µ ν ρ λ
lattice45. At each vortex line along z direction, there is Jv =ǫijǫµνρλ∂ O ∂ O ,B = 2πǫabcn ∂ n ∂ n
a helical Majorana mode. µν ρ i λ j µν Ω2 a µ b ν c
(4.2)
Now we duplicate 8 copies of such system. The 8
copies of helical Majorana modes inside the vortex line
~n is the O(3) vector degree of freedom coming from the
can be turned into Haldane chain as section IIA. Thus,
Haldanechaininthevortexline. B isatwoformgauge
µν
with 8 copies of superconducting Weyl cone pair in the
field associated with the O(3) Θ term.
Abrikosov phase, each vortex line of the vortex lattice
When vortex condenses, the U(1) symmetry is re-
carries a Haldane chain. The Abrikosov phase therefore
stored. TheO(5)bosonfield(emergedfromO(3)×U(1))
forms a weak 3d SPT state protected by T and discrete
is described by the O(5) NLσM as Eq.(2.21). The effec-
translation symmetry in x−y plane. Once we melt the
tivetheoryaftervortexcondensationcanalsobe verified
Abrikosovlatticebyproliferationandcondensationofthe
by the microscopic fermion model we start with in Eq.
superconducting vortex, the system concurrently under-
4.1.
goes a transition from weak to strong SPT state.
We haveintotal8 copiesofthe superconductingWeyl
As the vortex line carries Haldane chain described by cone pair, take 4 of them to couple to an O(3) vector.
H =χTk(i∂xσ10300+i∂yσ30300+i∂zσ22300+O1σ21000+O2σ23000+n1σ02110+n2σ02130+n3σ02122)χ−k
T :χ→Kiσ02100χ,n →−n , Z :χ→σ02000χ,O →−O (4.3)
1,2,3 1,2,3 2 i i
Whenvortexcondenses,theZ ×T symmetryisrestored. withΘ=π. Therest4copiesaretreatedsimilarlywhich
2
Integrating out the fermion gives a critical O(5) NLσM givesanothercriticalO(5)NLσMwithΘ=π. Thesetwo
9
criticalbosontheorycanbe mergedintoa gappedboson Zb symmetry, each nodal plane exhibits 2d topological
2
SPT phaseinthe samewayassectionIIB.LabelO ,O order. However,thevisonloopofZa(orZb),boundwith
1 2 2 2
as n ,n , we finally have the effective theory as, a half vortex loop of (n ,n )(or (n ,n )), is contractible
4 5 1 2 3 4
so there is no nontrivial loop statistics in 3d19.
5
L=X1g(∂µna)2+ iΩ2π4 ǫijklmǫµνρλni∂µnj∂νnk∂ρnl∂λnm n5A,tfhteertwheeomryeilstath3edsSlaPbTasntdatreesptroorteecttheedZb2ysZy2ma×mZe2tbr×yZo2cf
a=1 symmetry,
T :n →−n ; Z :n →−n (4.4)
1,2,3 1,2,3 2 4,5 4,5
5
1 i2π
This is another root state of T ×Z2 SPT phase1 with L= g(∂µna)2+ Ω4ǫijklmǫµνρλni∂µnj∂νnk∂ρnl∂λnm
X
different symmetry assignment compared to Eq.(2.21). a=1
The Abrikosov lattice melting procedure drives the the- Za :n →−n ,Zb :n →−n ,Zc :n →−n
2 1,2 1,2 2 3,4 3,4 2 4,5 4,5
ory from weak to strong SPT state via the proliferation (5.2)
ofvortexlineembellishedbylowerdimensionSPTstate.
These decorated defect approach also shed light on the As we gauge the Z2a × Z2b × Z2c symmetry, each vison
relation between boson SPT phases in different dimen- of the Z2 gauge is bind with a half vortex and the
sions. Θ term in Eq.(5.2) indicates the three loop braiding
procedure20,46,47. Namely, when we braiding the vison
loop Za with the vison loop Zb, both penetrated by the
2 2
V. SLAB MELTING, A CONNECTION visonloop Zc, there accumulate a Berry phase of π/2 as
2
BETWEEN 2D TO 3D TOPOLOGICAL a signature of 3d topological order.
ORDERED AFTER GAUGING THE SYMMETRY This method provides us a new insight to connect the
topological order in different dimension. As one obtains
In the previous discussion, it turns out that a strong topologicalorder after gauging the symmetry of an SPT
SPT state can be obtained by condensation of domain state, we expect the 3d topological order46,47 with non-
wall(or vortex) decorated with lower dimension SPT trivialthreeloopstatistics(orloop-particlestatistics)can
phase. The decorated domain wall picture reveals the be obtained via condensation of domain wall decorated
connection between SPT state in different dimensions at with a 2d topological order.
interacting level. Then, it is natural to ask does such
decorateddomainwallpicturealsoappliesfortrue topo-
logical matter with intrinsic topological order? In this VI. WZW THEORY AS A MUTI-CRITICAL,
DECONFINED QUANTUM CRITICAITY
paragraph, we would try to make a connection between
topological order in different dimension by gauging the
symmetry and melting the slab. The phase transition from a boson SPT state to
InsectionIIB,westudiedthephasetransitionbetween a trivial state is always beyond Ginzburg-Landau-
3d weak and strong SPT phase. In the weak SPT side, Wilson(LGW)paradigm. Justasthe2dbosonSPTstate
the system is composed of a series of layers as domain inEq.(2.12),describedby the O(4)NLσM with topolog-
walls separating different Z variables in the slab. The icalthetatermatΘ=2π,hasthesamesymmetryasthe
2
domainwallitselfcontainsa2dSPTstate. Such2dSPT trivial O(4) disordered phase where Θ = 0. The critical
state,describedbyO(4)NLσM,canalsobeprotectedby theory connecting the SPT phase with Θ=2π to a triv-
the Za×Zb symmetry1. ial phase with Θ=0 involves a critical theory of NLσM
2 2
with O(5) Wess-Zumino-Witten term17,48. Meanwhile,
4 1 i2π the NLσM in 2d with O(5) Wess-Zumino-Witten term
L= g(∂µna)2+ Ω3ǫijklǫµνρni∂µnj∂νnk∂ρnl canalsoappearasthe deconfinedquantumcriticalpoint
X
a=1 which connects the two phases with different symmetry
Za :n →−n ,Zb :n →−n , (5.1) breaking.
2 1,2 1,2 2 3,4 3,4
Tomapthesephenomenonsinoneunifieddiagram,we
If we couple the O(4) vector to the Za × Zb gauge treattheNLσMwithO(5)Wess-Zumino-Wittentermas
2 2
field19,20, the vison excitation of the gauge field would a multicritical point in Fig. 5.
displaynontrivialbraidingstatistics. Imaginewedevelop When we go from the bottom to top point, there hap-
a pair of visons(π flux) of Za and Zb, eachvisionis bind pens a phase transition from the SPT phase(Θ = 2π)
2 2
with a half vortex of (n ,n )(or (n ,n )). The braiding to a trivial phase(Θ = 0) illustrated in the blue line
1 2 3 4
between two vison excitations can be calculated via the in Fig.5. The center purple dot is the transition criti-
braiding of the π vortex of (n ,n ) and the π vortex of cality represented by a NLσM with O(5) Wess-Zumino-
1 2
(n ,n ). The Θ term in Eq.(5.1) indicates the braiding Witten term. Meanwhile, if we go from the left to right
3 4
between these two vortices give rise to a phase of π/2. point(illustrated in the green line in Fig.5), there hap-
Now we go back to the 3d system. When we are in pensatransitionbetweentwophaseswithdifferentsym-
the weakSPTphase,the systemconsistslayersofthe 2d metry breaking49–52. Based on the 2d PDW system we
SPT state along z direction. When we gauge the Za × discussed in section IIA, the total symmetry we have is
2
10
wall(defect) approach21,23 as the SPT state embedded
inside the nodal line(vortex line) can emerge from the
gapless fermion mode via interactions. Our idea also
cast the relation between topological superconductivity
and boson SPT phases23. While the defects(e.g. nodal
lines, vortex lines) inside the topological superconduc-
tor can carry multiple helical fermion modes, one can
always gap out the fermion modes inside the defect into
a gapped boson SPT state, and the condensation of the
defects thereforegeneratesabosonSPTphase. We hope
this stripe melting idea could drive forward the explo-
ration of SPT state from a microscopic point of view.
Further, we also suggested possible transition from 2d
to 3d topological order via condensation of nodal planes
decorated with topological order. We hope this idea
wouldcastlightontheexplorationof3dtopologicalmat-
FIG. 5. phase diagram
ter in condensed matter system.
In particular, we also discussed the relation between
the three types of transition: an SPT to a trivial phase
O(3) × O(2). O(3) degree of freedom comes from the
transition, a weak SPT to strong SPT phase transition
Haldane chain we embellished in the nodal lines, while
andthe transitionbetweentwoorderedstate withdiffer-
O(2)is the realandimaginarypartofthe superconduct-
entsymmetrybroken. Wedemonstratedthatthesethree
ing pairing field. In the PDW phase(illustrated as the
type oftransitioncanbe concludedina unifieddiagram.
greendotinFig.5),the chargeU(1)symmetryisbroken.
The greenline in Fig.5 shows a transition from the O(3)
spin rotationsymmetry breaking state(e.g. AF order) to
ACKNOWLEDGMENTS
a PDW superconducting state, whose criticality is also
described the O(5) Wess-Zumino-Witten term.
The PDW phase itself is also a weak SPT phase as We are grateful to Thomas Scaffidi, Cenke Xu, Ed-
it contains alignment of nodal line decorated with Hal- uardo Fradkin, Andreas Ludwig, Steve Kivelson, Tarun
dane chain. After we melt the stripe and condense the Grover and Zhen Bi for insightful comments and dis-
nodal lines, the Z symmetry is restored and we go into cussions. This work was supported in part by the
2
the strong SPT phase(Θ = 2π) illustrated in the bot- National Science Foundation through grants DMR-
tomreddot. TheredlineinFig.5displaysthetransition 1408713 (YY) at the University of Illinois, grants NSF
fromweaktostrongSPTphase. This phasetransitionis PHY11-25915(YY,YZY)at The Kavli Institute for The-
withintheconventionalLGWtypewhichconnectsasym- oretical Physics, David and Lucile Packard Founda-
metry broken state to an isotropic state. The isotropic tion(YZY) and NSF Grant No. DMR-1151208 (YZY).
statecanbeobtainedviacondensationoflinedefects(e.g YY achokowledges the KITP Graduate fellowship pro-
nodalloop)hostingatopologicalΘterminlowerdimen- gram, where this work was initiated.
sion, which therefore induce an SPT phase.
As a result, we had consistently unified three types of
novel phase transition in a simple diagram. Appendix A: SPT phase in the presence of O2
fluctuation
VII. CONCLUSION AND OUTLOOK In section IIA and IIB, we choose a specific gauge so
the pairing field ∆ = O +iO is real and thus O = 0.
1 2 2
In this paper, we studied the transition from weak However, the phase fluctuation of the superconducting
to strong SPT phases. For all the examples consid- field would result in small fluctuation of O2. Here we
ered in this paper, we embellished the domain wall be- would demonstrated that as long as the ferromagnetic
tween different stripes with a topological theta term in interaction between two O(4) vector is turned on, the
lower dimension. The stripe melting procedure induced theory is always in the gapped SPT phase even in the
by domain wall saturation drives the system from a presence of small O2.
weak to strong SPT phase. This method can be view In section IIB, 8 copies of Weyl fermions with PDW
as a microscopic realization of the decorated domain order couples with O(4) vector as,