Symmetry Breaking and Lattice Kirigami

Eduardo V. Castro,1, 2, 3 Antonino Flachi,4 Pedro Ribeiro,1 and Vincenzo Vitagliano4

1CeFEMA, Instituto Superior Tecnico, Universidade de Lisboa, Av. Rovisco Pais, 1049-001 Lisboa, Portugal2Centro de Fısica das Universidades do Minho e Porto,

Departamento de Fısica e Astronomia, Faculdade de Ciencias,Universidade do Porto, 4169-007 Porto, Portugal

3Beijing Computational Science Research Center, Beijing 100084, China4Department of Physics & Research and Education Center for Natural Sciences,

Keio University, 4-1-1 Hiyoshi, Kanagawa 223-8521, Japan(Dated: March 28, 2018)

In this work we consider an interacting quantum field theory on a curved two-dimensional manifoldthat we construct by geometrically deforming a flat hexagonal lattice by the insertion of a defect.Depending on how the deformation is done, the resulting geometry acquires a locally non-vanishingcurvature that can be either positive or negative. Fields propagating on this background are forcedto satisfy boundary conditions modulated by the geometry and that can be assimilated by a non-dynamical gauge field. We present an explicit example where curvature and boundary conditionscompete in altering the way symmetry breaking takes place, resulting in a surprising behaviour ofthe order parameter in the vicinity of the defect. The effect described here is expected to be genericand of relevance in a variety of situations.

Introduction. The theory of quantum fields in curvedspace has produced over its more than fifty years of exis-tence remarkable results [1, 2], the phenomena of particleproduction in gravitational fields [3] and that of blackhole evaporation [4] being, without doubt, amongst itsmost celebrated offsprings. From a more general per-spective, its semi-classical framework has established ahighly non-trivial connection between thermodynamics,gravity, and quantum field theory, and it is at this cross-road where non-trivial manifestations of the geometricaland topological attributes of curved space on the quan-tum domain occur.

A particularly interesting corner of this intersectionis that of quantum field theories featuring spontaneoussymmetry breaking, where effects of curved space are ex-pected to alter the way vacuum destabilization and phasetransitions take place. In absence of gravity, that is inflat space, the story has been known for a long time andthe Coleman-Weinberg mechanism has clarified the wayradiative corrections may destabilise the vacuum, withsymmetries being spontaneously broken as a result ofquantum effects [5].

On curved backgrounds there are differences that arenot difficult to anticipate. A first indication on howthings change comes from the same Coleman-Weinbergmechanism that, in flat space, predicts a first-order phasetransition from a broken to a restored symmetry phase inscalar electrodynamics, when the scalar mass is increased[5]. When lifted to a weakly curved space, renormaliza-tion theory implies the appearance of mass-like contri-bution proportional to the Ricci curvature, thus causingan effective increase of the mass of the scalar. It is thennatural to expect that the effect of a positive (negative)spacetime curvature would be to push the system towardsa phase of unbroken (broken) symmetry.

Additional insight comes from considering spacetimeswith horizons that are periodic in Euclidean (imaginary)

time. In such a situation, Green’s functions enjoy thisperiodicity with the period set by the horizon size, inanalogy with thermal Green’s functions that share thesame periodicity, but with the period set by the inversetemperature. This leads to the expectation that for asufficiently small horizon a transition from a broken to asymmetric phase may occur.

These arguments have been made quantitative in anumber of cases. Some of the initial discussions focusedon scalar fields and spatially homogeneous backgrounds(e.g., de Sitter space) and can be found in Refs. [6–8],where the direct evaluation of the effective potential hasshown that a positive curvature does indeed assists sym-metry restoration. Interestingly, it was also shown thatthe details of the scalar field theory (its conformal in-variance or lack of it) were responsible for a change inthe order of the curvature-induced phase transition fromfirst to second order [7]. Similar issues in relation tochiral symmetry breaking have also been discussed andan extensive review of earlier works is given in Ref. [9].The situation becomes more complicated when the back-ground is inhomogeneous or topologically non-trivial. Asample of early calculations in topologically non-trivialspacetimes can be found in Refs. [10–13]. In relationto spatially varying geometries a particularly interestingexample is that of black holes, for which the questionof symmetry restoration was discussed, for instance, inRefs. [14, 15]. There it was shown that a spontaneouslybroken symmetry is locally restored near a (sufficientlyhot) black hole (see also Ref. [16, 17]). Once again, theinterpretation is that the strong gravitational gradientnear the horizon is responsible for inducing symmetryrestoration.

The natural playground for contemplating how space-time topology and curvature might modify the stabilityof the vacuum in quantum field theory has always beendomain of early universe cosmology. Recently, however,

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other areas of physics are contributing to modernise theabove questions and to formulate new exciting problems.

One such area is related to recent advances in con-densed matter research at the nanoscale, particularly inconnection with layered materials. Graphene and, moregenerally, two-dimensional materials are the most spec-tacular example of the sort, owing to geometrical ver-satility coupled to an emergent relativistic behaviour offermions [18–20]. In these examples, the background ge-ometry is the two-dimensional lattice on top of whichfluctuations propagate and the relevance of curvature ef-fects has already been appreciated [21–25]. Other inter-esting examples can be found in Refs. [26, 27].

QCD physics is also fuelling novel research where theuse of language and methods of quantum field theory incurved space is becoming more common. Some inter-esting examples range from the more generic remarksof Refs. [28, 29] (and references therein), to the verypopular area of strongly interacting fermions and chi-ral symmetry breaking in rotating backgrounds (see, forexample,[30–36]), to applications of lattice QCD (see, forexample, [37–39]). In these contexts a range of pecu-liar geometry-induced phenomena are expected to occur(e.g, condensate suppression/enhancement, appearanceof new phases, changes in the critical points geography),whose physical relevance spans from relativistic heavy ioncollisions, to transport phenomena, to the astrophysics ofcompact stars.

The focus of this paper is to reconsider the role ofthe background geometry in affecting the stability of thevacuum. We will argue that, contrary to expectation,increasing the spatial curvature does not necessarily im-ply that the system moves closer to a phase of restoredsymmetry, and we shall present an explicit example ofthis. Although the example is non-trivial, it is amenableto simple explanation and anticipates the possibility ofappearance of exotic changes in the phase behaviour ofinteracting quantum field theories with a number of in-teresting implications that we will mention later.

Model and geometry. For the sake of concreteness, weshall consider here a specific class of (2 + 1)-dimensionalinteracting field theoretical models of the Hubbard-type,whose Hamiltonian H = H0 +HI is expressed as the sumof a free part,

H0 = −t∑

r, i, σ=±u†σ(r)vσ(r + bi) + H.C.,

plus an interacting sector,

HI =U

4

∑r,σ,σ′,i

(nσ(r)nσ′(r) + nσ(r + bi)nσ′(r + bi)) .

The above field theory is defined on an underlying latticethat we assume for the moment to be flat with hexago-nal cells and generated by linear combinations of a setof basis vectors as illustrated in Fig. 1 (r span a trian-gular sub-lattice and the vectors bi, i = 1, 2, 3 connectthe atom in r with the three nearest-neighbours). The

annihilation operators of the two sub-lattices are u andv and nσ is the number operator. The quantities t andU are positive numbers describing, respectively, the hop-ping and the interaction constant.

The above model is routinely used to describe manyof the properties of graphene and other layered materials[18, 40]. An important aspect is the possibility to locallyinduce curvature by deforming the lattice with the inser-tion of defects. Also, the continuum limit is not difficultto analyse and generalisations can be easily imagined.Finally, although here we will be concerned with the con-tinuum limit, carrying out lattice simulations should befeasible.

The specific type of symmetry breaking that we wishto discuss here is associated with the bipartite natureof the honeycomb lattice that the Hubbard model abovecaptures in the magnetisation that we shall properly de-fine below. Since our goal here is to scrutinise the ef-fect of curvature on the spontaneous breakdown of theabove sub-lattice symmetry, our first task is to covari-antize the model to curved space. For this it is convenientto work with the continuum Lagrangian counterpart thatcan be obtained using standard methods by expressingthe original Hamiltonian in terms of the SU(2) vectorS =

∑σ, σ′ u†σ(r)~τσ,σ′uσ′(r)/2, where ~τ is a vector with

the Pauli matrices as components, and then proceed bymeans of a Hubbard-Stratonovich transformation. In or-der to maintain our treatment as simple as possible wewill assume a scalar order parameter that can be mo-tivated by a rotational anisotropy favouring symmetrybreaking along the z axes (for graphene this could bedue to the presence of a substrate and related spin-orbitcoupling enhancement [41, 42]). This allows to gap outthe Goldstone modes that can be straightforwardly in-cluded in a more involved treatment. Choosing an auxil-iary field φ that breaks both the Z2 and the discrete sub-lattice symmetry, the Hamiltonian H can be mapped, atlow energies, onto the following (2 + 1)-dimensional fieldtheory

L = ψσı/∂ψσ +(σψσφψσ

)+φ2

2λ, (1)

where the first term is a free Dirac contribution and theremaining terms describe the interaction sector. Thesummation over repeated spin indices σ = ± is un-derstood and the four-component spinors ψσ are ar-ranged as ψTσ =

(ψA1σ , ψB1

σ , ψA2σ , ψB2

σ

), with ψIJσ (x) =

avF

∫d2p(2π)2 e−ıp·xzIJσ (p) and where zI,Jσ (p) = zIσ(KJ +

avF

p) represents the sub-lattice annihilation operators

(zA = u, zB = v) near the two Dirac cones KJ=1,2 of thedispersion relation. The spatial coordinates were rescaledby x = r/vF where vF = 3/2ta is the Fermi velocity anda is the lattice spacing. Finally, the coupling constantλ is proportional to the interaction strength λ ∝ U upto an unimportant factor, dependent on the particularregularization of the low energy theory.

The exchange of the sub-lattices can then be im-plemented by the simultaneous exchange x2 → −x2

3

(p2 → p2), leaving intact the Dirac points and thespin, and the Lagrangian (1) invariant as long as φ van-ishes. The order parameter for the above symmetry isφ = 2λ〈ψ−ψ− − ψ+ψ+〉 and it describes the staggeredmagnetization, i.e., φ 6= 0 indicates broken symmetry. Itis possible to arrive at the same expression (1) followingthe general decomposition of the Hubbard Hamiltonianas outlined in [43].

By means of a kirigami like procedure1 we introduce aspatial curvature in the model by inserting a disclinationthat warps the lattice locally. There are many ways todo this, but if we wish to isolate the interplay betweenquantum effects and geometry, we need to preserve thebipartite nature of the lattice at tree level, that is avoidfrustrating the lattice. This requirement restricts the al-lowed deformations to those induced by defects with aneven number of sides, as these are the only that preservethe above symmetry classically. Inserting a defect withns < 6 sides in an hexagonal lattice generates a deficitangle and a curvature that is locally positive (see Fig. 1).In contrast, adding a defect with ns > 6 generates an ex-cess angle and a locally negative curvature (see Fig. 1).

In the continuum model, the curvature can be intro-duced by specifying the background metric to be that ofa manifold with a conical singularity. The Riemanniangeometry of such manifolds is studied since, at least, [45].Refs. [46, 47] give details and additional bibliography onthe topic. Here, in order to model such a localised curva-ture, we use a Euclidean parametrisation for the metrictensor

ds2 = dτ2 + dr2 + α2r2dθ2 (2)

with r ≥ 0 and 0 ≤ θ < 2π being the polar coordinatescentred at the apex. We will not concern ourselves withfinite temperature effects here, but these can be includedin a straightforward manner by using the standard imagi-nary time formalism. Defining θ = α θ, it should be clearthat the metric is that of flat space with 0 ≤ θ < 2πα.If α < 1, then γ = 2π − 2πα describes a deficit angle.Removing the deficit angle and identifying the two sidesresults in a cone with opening angle 2 arcsinα. The closerto unity is α, the flatter is the cone. If α > 1, then thedeficit angle becomes an excess angle.

Since the curvature of conical manifolds diverges atthe apex, some regularisation is necessary to deal withthe singular behaviour. Here, we will regulate the geom-etry by replacing the singular space with a sequence ofregular manifolds as done in Ref. [46, 47]. Calculationsare done in the regularised geometry and results in theoriginal singular space are obtained as a limit, once theregularisation is removed at the end. In practise, thisprocedure can be implemented by replacing the original

1 Kirigami is a variation of origami that includes cutting of thepaper, rather than solely folding the paper [58].

b1

b2b3

r

FIG. 1: A honeycomb, planar lattice with bipartite nature,and some possible defective configurations. The top figuresillustrate the flat hexagonal lattice with a 2π/3 section high-lighted. The exchange symmetry between the two triangular(red and blue) sub-lattices is also illustrated. Defects areintroduced by a procedure of adding or cutting the latticeπ/3 sections and gluing the remaining sides along the cut.For example, subtracting a 2π/3 section, one obtains a pos-itive curved cone with a ns = 4-sides defect (bottom left).Adding a 2π/3 section, instead, generates a negative curvedsaddle geometry with ns = 8 (bottom right). These latticestructures are well known and form the basis of chiral curvedpoly-aromatic systems (n-circulene, see Ref. [44] for a reviewof the geometry of these lattice structures).

metric (2) with the following regular one:

ds2 = dτ2 + fε(r)dr2 + α2r2dθ2 (3)

where ε represents a regularisation parameter and fε(r) isa smooth function satisfying the following properties: 1)limε→0 fε(r) = 1; 2) fε(r) ≈ 1 for r � ε; 3) fε(r) = constfor r = 0. It should be noted that while the limit ofε → 0 corresponds to removing the regularisation, in aneventual comparison with a lattice simulation, ε shouldbe associated with the lattice spacing and it acquires thestatus of a physical cut-off.

The Lagrangian (1) is extended to curved space bya minimal covariantization procedure, i.e. letting theMinkowski metric, the derivatives, and the gamma matri-ces to the corresponding quantities in curved space. Weshall not include non-minimal couplings in the presenttreatment (see Ref. [48] for a discussion about this point).

The last element we need to take into account are theboundary conditions along the cut where the two sidesof the lattice have been glued after having removed or

4

added a portion of the lattice to accomodate the inser-tion of the defect. The same procedure that we shall usebelow has been discussed, for example, in Ref. [22]. Itis not difficult to realise that for a generic even-sided de-fect, the two sub-lattices are unchanged and the fermionwave function, after circulating around the defect, sat-isfies the following boundary condition: ψ(r, ϕ + 2π) =− exp (i(6− ns)πγ5/2)ψ(r, ϕ). (Here, we follow the sameconventions as Ref. [22], and choose to work in the stan-dard planar representation of the Clifford algebra of γ-matrices, where γ0 is diagonal). A transparent way to in-corporate these boundary conditions is by re-expressingthe fields as ψ′(r, ϕ) = exp (−iϕ(6− ns)γ5/4)ψ(r, ϕ),and by noticing that the primed fields obey the stan-dard periodicity condition ψ′(r, ϕ + 2π) = −ψ′(r, ϕ). Itis simple to prove that the effect of the above redefini-tion is to augment the Lagrangian by a non-dynamicalgauge connection Aµ = −δϕµ (6−ns)γ5/4. This term willbe crucial in altering the way symmetry breaking takesplace.

Methods and results. With all of the above in hands,we can examine whether the geometry and associatedboundary conditions, which we have implemented as de-scribed in the preceding section, favour a phase of brokenor restored symmetry in the region where curvature at-tains a positive (in the case of ns = 4 and deficit angle) ornegative (in the case of defects with even ns > 6 and ex-cess angle) value. As we have motivated at the beginning,since the curvature increases (decreases) as we approachthe defect for ns = 4 (ns = 8, 10, . . . ), the expectationis that curvature should favour symmetry restoration inthe vicinity of the defect for ns = 4, while for ns evenand larger than 6 broken symmetry should instead befavoured.

Below we shall address this question by computing theeffective action for the order parameter φ and by numer-ically solving the associated effective equations. Thereare a few technical steps that we should clarify in orderto allow anyone to reproduce our results. First of all,our analysis follows the large-N deformation of Ref. [43],where we pass from 2 to N flavours of the Dirac fields:

ψ+ψ+ →∑N/2σ=1 ψσψσ and ψ−ψ− →

∑Nσ=N/2+1 ψσψσ.

Then, the effective action at mean-field level can be ex-pressed as

Γ [φ] = −∫d3x√gφ2

2λ+

1

2

∑p=±

log det

(� +

R

4+ φ2p

),

where φ2± = φ2±√grrφ′ and the D’Alembertian is calcu-

lated from the spinor covariant derivative Dν = ∇ν+iAν .The tildes indicate that quantities are computed usingthe regularised metric (3).

Heat-kernel methods and zeta regularization tech-niques are used to perform the computation of the de-terminant above. The interested reader is referred toRefs. [2, 49–52]) for general background and to Refs. [16,48] where similar calculations have been done in thecontext of interacting fermion effective field theories in

λs = 10

λo = 500

λs = 500

λo = 10

φ

r

FIG. 2: The curves are obtained by numerical approximationand illustrate how the order parameter φ changes as the defectis approached. We have set the values of the coupling constantλs,o, in units of the renormalisation scale, to encompass boththe cases in which far away from the defect the system is in aphase of broken or unbroken symmetry. The indices s and ostand for square and octagon. In the numerical computationwe have varied the regularization parameter ε from 0.1 downto 0.01.

curved space. The algebra is handled by computersymbolic manipulation, here we summarise the essen-tial steps: 1) we first express the determinant in termsof a covariant derivative expansion of the propagatorin powers of curvature invariants and derivatives of φ(with scalar curvature contributions fully resummed asin Refs. [48, 53]); 2) we truncate this expansion to secondorder in the heat-kernel expansion and write the effectiveequations for φ arising from this truncated effective ac-tion; 3) finally, we use numerical approximation to solvethe resulting effective equations.

The results are illustrated in Fig. 2 for a few caseswith both locally positive (ns = 4) and negative cur-vature (ns = 8). The asymptotic value of the couplingconstant should be fixed by imposing specific renormal-ization conditions (see Ref. [9]), with its value adjustedto specific situations. In the present calculation, we havechanged its value in order to encompass both situationsin which symmetry is either broken or close to the criticalvalue far away from the defect (see Fig. 2). The numer-ical solutions show that φ develops a spatial variationand attains a value near the defect that is larger than itsasymptotic value for any ns 6= 6, signalling that in thevicinity of the defect curvature, irrespectively of its sign,seems to encourage an ordered phase.

To examine what is going on, let’s consider for illus-tration the case of ns = 4, corresponding to a locallypositive scalar curvature. In such a case, intuition sug-gests that a positive curvature (R > 0) should drive thesystem towards a symmetric phase with the order pa-rameter moving towards smaller values, thus making the

5

observed behaviour rather unexpected. However, as wehave seen earlier, the boundary conditions along the cutalso play an essential role by inducing the emergence ofa non-dynamical gauge field Aµ, whose origin is entirelygeometrical (the number of sides of the defect can be in-terpreted as an effective charge) and whose effect, in thefermion determinant, combines with that of the scalarcurvature. While the term proportional to R is responsi-ble for pushing the order parameter towards a symmetricphase in conformity with the arguments of Ref. [28], the

shift caused by Aµ competes with the effect of R. In ad-dition, such a term appear with a modulation factor thatoriginates from the metric tensor, ∼ gµνAµAν ∝ −n2s/r2,that amplifies its effect near the defect, explaining theobserved behaviour.

Conclusions. In this work we have engineered a curvedbackground starting from a flat 2D hexagonal lattice us-ing a kirigami-like procedure of removing (adding) a pieceof lattice and by gluing the parts along the cut, a way togeometrically inserting a defect in the lattice. In the con-tinuum this curved lattice corresponds to a Riemannianmanifold with a locally positive (negative) curvature anda conical-like singularity at the defect. We have consid-ered an interacting quantum field theory on this back-ground and analysed how symmetry breaking is alteredby the geometrical deformation caused by the defect.

The two important features of the story turn out to bethe increasing (or decreasing) curvature near the defectand the boundary conditions along the cut that can beassimilated by a non-dynamical gauge field modulatedby the conical structure. As a working example, we havelooked at the staggered magnetisation, the order param-eter associated with the discrete sub-lattice symmetry.The numerical results have shown an increase of the or-der parameter as the locally curved region is approached,a behaviour that signals a change towards order and thatgoes against the expectation that increasing the curva-ture should drive the system closer to a state of unbro-ken symmetry. This behaviour has been explained by thecompetition between curvature and the emergent gaugefield, i.e. between geometry and a feature of topologicalnature related to the boundary conditions.

The kirigami effect we have described should be genericand naturally expected to occur for different lattice struc-tures (i.e., different unit cells) as long as the same geo-metrical traits are maintained. We also expect the same

to occur for different field theory models. One intriguingpossibility is to consider a multi-defect configuration andsee whether there is any special arrangement where therelative weight of the geometry-induced gauge fields vscurvature can be adjusted, thus tailoring specific config-urations of the order parameter.

Amongst the various interesting implications, the pos-itive effect of curvature on the spontaneous breaking ofsub-lattice symmetry could be used to shed light on thelong lasting question regarding the semimetal-insulatorphase transition in graphene: even though graphene ispredicted to be very close to the transition point [40, 54],no experimental signature of the insulating behaviour hasbeen found in flat graphene so far [55]. Another promis-ing route in graphene would be the combination of cur-vature with adatom adsorption in order to enhance sym-metry breaking, in particular of magnetic order [56].

An interesting direction to extend the idea of this workis to look at higher dimensionality. A straightforwardapplication is to (straight) cosmic strings [57]. In thiscase, the effect discussed here should trigger fermion con-densation at the string. This would offer a mechanismof inducing a superconducting phase of different naturefrom usual arguments (see [57]). The idea should deservesome attention both in cosmology and condensed matterphysics (e.g., liquid crystals; see, for example, [59, 60]).

While beyond of the scope of this work, it is tempt-ing to relate the present ideas to gravity at the Planckscale, where spacetime may be discrete. In this case,the presence of defects in the background lattice wouldcause local changes in the geometry and topology, simi-lar to those described here, that could trigger a form ofgraviton condensation in the vicinity of these spacetimeglitches.Acknowledgments. The support of the Japanese Min-

istry of Education, Culture, Sports, Science Program forthe Strategic Research Foundation at Private Universities‘Topological Science’ (Grant No. S1511006) is gratefullyacknowledged. VV is supported by the Japanese Soci-ety for Promotion of Science (Grant No. P17763). EVCand PR acknowledge partial support from FCT-Portugalthrough Grant No. UID/CTM/04540/2013. AF is grate-ful to A. Beekman, K. Fukushima, T. Fujimori, M. Nitta,R. Yoshii for discussions on various aspects of symmetrybreaking and geometry and to A. Beekman for carefullyreading the manuscript and for various suggestions.

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