Quantum phase diagram of two-dimensional transverse field Ising model: unconstrained tree tensor network and mapping analysis
aa r X i v : . [ c ond - m a t . s t r- e l ] J a n Quantum phase diagram of two-dimensional transverse field Ising model:unconstrained tree tensor network and mapping analysis
M. Sadrzadeh, R. Haghshenas, and A. Langari Department of Physics, Sharif University of Technology, P.O.Box 11155-9161, Tehran, Iran Department of Physics and Astronomy, California State University, Northridge, California 91330, USA
We investigate the ground-state phase diagram of the frustrated transverse field Ising (TFI)model on the checkerboard lattice (CL), which consists of N´eel, collinear, quantum paramagnet andplaquette-valence bond solid (VBS) phases. We implement a numerical simulation that is basedon the recently developed unconstrained tree tensor network (TTN) ansatz, which systematicallyimproves the accuracy over the conventional methods as it exploits the internal gauge selections. Atthe highly frustrated region ( J = J ), we observe a second order phase transition from plaquette-VBS state to paramagnet phase at the critical magnetic field, Γ c = 0 .
28, with the associatedcritical exponents ν = 1 and γ ≃ .
4, which are obtained within the finite size scaling analysis ondifferent lattice sizes N = 4 × , × , ×
8. The stability of plaquette-VBS phase at low magneticfields is examined by spin-spin correlation function, which verifies the presence of plaquette-VBSat J = J and rules out the existence of a N´eel phase. In addition, our numerical results suggestthat the transition from N´eel (for J < J ) to plaquette-VBS phase is a deconfined phase transition.Moreover, we introduce a mapping, which renders the low-energy effective theory of TFI on CLto be the same model on J − J square lattice (SL). We show that the plaquette-VBS phase ofthe highly frustrated point J = J on CL is mapped to the emergent string-VBS phase on SL at J = 0 . J . PACS numbers: 75.10.Jm, 75.30.Kz, 64.70.Tg
I. INTRODUCTION
Quantum phases of matter without magnetic long-range order have become an interesting field of researchin recent years. Frustrated magnetic systems are one ofthe best candidates to bring about such phases like spin-ice materials or spin liquids [1–3]. In fact, frustratedmagnetic models imply large degenerate classical groundstates (GS) that are very sensitive to perturbations suchas thermal or quantum fluctuations, spin-orbit interac-tions, spin-lattice couplings and impurities, all of whichmight be present in actual materials [4, 5]. Novel uncon-ventional phases such as valence bond solids and spin liq-uids can emerge from the effect of such purturbations onclassical frustrated systems. Moreover, the existence ofartificial square ice [6–8] and the realization of quantumspin ice with Rydberg atoms [9] demand a comprehensiveunderstanding of the associated models that are genericfor such materials.Generally, a spin system is frustrated whenever onecannot find a configuration of spins to fully satisfy theinteracting bonds between every pair of spins [5, 10]. Forinstance, a diagonal bond in addition to vertical and hor-izontal bonds construct a triangle, which makes frustra-tion on the spins sitting on triangle corners of a squareplaquette. In this respect, spin 1/2 antiferromagneticIsing models on the J − J square and half depletedsquare, i.e. checkerboard, lattices are generic 2D frus-trated magnets in which J and J , the strength of near-est and next nearest neighbor interactions, respectively,compete with each other (see Fig. 1). These are pro-totype models that low dimensionality makes them aneasier target for numerical/analytical approaches in con- FIG. 1. (color online) J − J model on (left) square lattice(SL) and (right) checkerboard lattice (CL). The solid anddashed lines are J and J bonds, respectively. trast to 3D counterparts [11–15]. Accordingly, CL canbe assumed as the 2D version of pyrochlore lattice oftrue spin ice materials [16]. Here, we focus particularlyon the role of quantum fluctuations on the ground statephase diagram of planar spin-ice, namely: CL and itslow-energy effective theory on the square lattice.In the case of Ising model on CL, quantum fluctuationsintroduced by both transverse magnetic field [16, 17] andin-plane XY interactions [18–21] lift the classical degener-acy of the highly frustrated point J = J toward a non-magnetic plaquette-VBS phase [16–18, 22] with brokentranslational symmetry, which shows two-fold degener-acy. The plaquette-VBS phase, which is mediated by an-harmonic quantum fluctuations as an order-by-disorderphenomenon [23–25], emerges from an exponentially de-generate classical background, which can not be observedwithin linear spin-wave theory [26, 27] due to strong frus-tration. In order to shed more light on the highly frus-trated region, in the first part of our paper, we obtainthe GS phase diagram of CL accurately by using a vari-ational tree tensor-network (TTN) ansatz and compareit with previous studies. We use a novel unconstrained(gauge-free) TTN, generalized to CL, to approximate theground state of the system with higher accuracy com-pared with previous isometric schemes [28]. By com-puting local correlations and plaquette operator expec-tations, we find that a plaquette-VBS state is establishedat the low magnetic field around J = J region of CL.Our results show that by increasing transverse magneticfield a second-order phase transition occurs at Γ c = 0 . ν = 1 and γ ≃ . ν reveals the divergence of correlation length and γ is an exponent, which governs the singularity in mag-netic susceptibility. We do not observe any other crit-ical point except the mentioned one, which rules out acanted N´eel phase predicted by the Monte-Carlo study[25] at J = J . Our results of unconstrained TTN are ingood agreement with the results of the cluster operatorapproach (COA) [22].On the other hand, the J − J TFI model on the squarelattice shows an emergent string-VBS phase at the fullyfrustrated point J = 0 . J [29, 30]. It can be expressedthat quantum fluctuations by means of transverse field,lift the classical degeneracy toward a doubly degener-ate VBS states along the horizontal or vertical directionsof the square lattice called string-VBS phase. However,there is a possibility that such a phase can be extendedto an intermediate region around the highly frustratedpoint J = 0 . J , which is sandwiched between a N´eeland striped antiferromagnetic states for small and large J /J , respectively [31]. Accordingly, in the second partof our paper we consider a different strategy to clarifythe quantum phase diagram of TFI model on the J − J SL. We introduce a mapping from CL to SL, which leadsto the GS phase diagram of the J − J SL in terms ofthe phase diagram of CL of the corresponding model. Inother words, we claim that the low-energy effective the-ory of frustrated TFI on CL is given by frustrated TFIon SL. This mapping suggests a string-VBS order at thehighly frustrated regime of SL, which is in agreementwith the results of COA [29]. It is worth mentioningthat the TFI model could represent the large easy-axisanisotropic limit of the antiferromagnetic J − J Heisen-berg model, where the true nature of a non-magnetic(VBS) phase is still under debate on SL [32–39] . Ourresults would be useful for further investigations in thelatter model.The paper is organized as follows. In the next section,we briefly introduce the model and different phases onCL. In Sec. III, we inaugurate a numerical TTN tech-nique to find accurately the quantum phase diagram ofCL. Then, in Sec. IV we establish the mapping from CLto SL and derive the corresponding quantum phase dia-gram of SL. Finally, the paper is summarized and con- p-VBS
Néel CollinearQuantum Paramagnet c o n t i n u o u s c o n t i n u o u s d e c o n fi n e d continuous c o n t i n u o u s / JJ / J Γ FIG. 2. (color online) A schematic phase diagram of theS=1/2 J − J TFI model on CL [22], including the infor-mation on the type of transitions between different phasesobatined within TTN numerical simulation, namely: contin-ious and deconfined phase transitions. cluded in Sec. V. The details of introduced mapping havebeen presented in Appendix A.
II. THE MODEL HAMILTONIAN
The Hamiltonian of J − J transverse field Ising modelon CL is, H = J X h i,j i S zi S zj + J X hh i,j ii S zi S zj − Γ X i S xi , (1)where h i, j i spans the nearest neighbor sites with J cou-pling, J > S x,z refer to x and z components of spin-1/2 oper-ators on the vertices of the lattice (see Fig. 1). It con-sists of four different phases, a N´eel and collinear orderedphases close to the non-frustrated points J /J = 0 and J /J = 2 respectively, a quantum paramagnet phase athigh fields and a plaquette-VBS phase for low magneticfields Γ . .
3, a narrow region around the highly frus-trated point J = J . The corresponding phase diagramis presented in Fig. 2, which has been obtained by COAapproach [22]. In fact, it has been concluded that the ex-ponential degeneracy of the classical ground state at thehighly frustrated point, J = J , (known as square ice[26]) is lifted toward a unique quantum plaquette-VBSstate that breaks translational symmetry of the latticewith two-fold degeneracy. It is a manifestation of or-der by disorder phenomena [23–25], which is induced byquantum fluctuations.In the next section, we use the unconstrained TTNapproach to further confirm the quantum GS phase dia-gram of J − J TFI model on CL. It has to be mentionedthat the plaquette-VBS exists in a narrow region on thehighly frustrated regime, which requires to be investi-gated within high accurate numerical simulations. In ad-dition, we apply TTN to find critical points and criticalexponents of the phase transitions from plaquette-VBSstate to the N´eel, collinear and paramagnet phases, whichcan classify the type of phase transitions.
III. UNCONSTRAINED TREE TENSORNETWORK ANSTAZ
The TTN states provide a variational ansatz [28, 40–43] to simulate large 2D lattice sizes, beyond the pos-sible sizes, which can be reached by exact diagonaliza-tion. We use an unconstrained TTN ansatz to varia-tionally approximate the ground-state wave function ofthe TFI model (Eq. 1) on the CL. The wave function ismade of the local tensors { w i } connected to each otherto form a tree-like graph as shown in Fig. 3-(a). Thetensors { w i } effectively map a number of spins to an ef-fective superspin by dimension Λ at each layer, making acoarse-graining transformation—each tensor w i defines aprojection from original (physical) Hilbert space onto therelevant subspace. That is the basic idea in the renor-malization group (RG) methodology invented by Wilsonand Kadanoff [44]. Here, the goal is to use an efficientvariational ansatz to minimize the ground-state energywith respect to tensors { w i } , finding the best variationalparameters (which grows like O (Λ )). In this paper, weuse a recently introduced novel ansatz [43] which, in con-trast to traditional schemes, releases the internal gaugesymmetry of the tensors (the isometry constraint) andprovides a computationally stable and efficient algorithmwith higher accuracy.We shortly explain the unconstrained TTN variationalansatz generalized to two-dimensional lattices. The opti-mization method is performed by minimizing the energywith respect to a specific tensor w i (while holding fixedother tensors), i.e.min w i {h Ψ w i | H | Ψ w i i − λ h Ψ w i | Ψ w i i = h w i | H eff | w i i − λ h w i |N | w i i} , where the so-called norm tensor N and effective Hamil-tonian H eff are obtained by removing tensor w i fromthe tensor-network representation of h Ψ w i | Ψ w i i and h Ψ w i | H | Ψ w i i . The solution is given by solving a gener-alized eigenvalue problem H eff | w i i = λ min N | w i i , whichis a standard equation in linear algebra. The optimiza-tion procedure is then completed by using an iterativestrategy: at each step, only one tensor is optimized whileothers hold fixed and then this task is repeated over alltensors till the variational energy does not change signif-icantly. In practice, the norm tensor N causes instabilityin the algorithm, while the condition number (i.e. small-est singular value) would be too small. In order to avoidthat, we need to use a ‘canonical normal form’ [45] forthe TTN state | Ψ w i i by making the norm tensor iden- w w w w w w w w w w w ′ Q Q Q Q Q Q Q Q Q R R R R R R R R R (a) (b) (c)(d) (e)(f) (g) = = = Λ ΛΛ
FIG. 3. (color online) Tensor-network representation of anunconstrained TTN state | Ψ i and its canonical form. (a) ATTN state for a 4 × { w i } connected by the so-called virtual bonds with dimen-sion Λ to form a tree-like geometrical graph. (b) A tensor-network representation of QR-decomposition applied to ten-sor w = Q R . One needs to fuse lower indices and thenrepresents it in a matrix form to do decomposition. (c-e) Theprocedure to transform a general TTN state to a canonicalnormal form by using a sequence of QR-decomposition. Thenorm tensor is defined by removing tensor w from tensor-network representation of h Ψ w | Ψ w i , denoted by N . A se-quence of QR decomposition is used to make norm tensoridentity N = I : tensors w , w , w are decomposed into QRforms, and (d) then tensors R , R , R are absorbed into ten-sor w , followed by a QR-decomposition by fusing the virtualbonds (last ones) w ′ = Q R . (e) The canonical procedure iscompleted by absorbing R into w , i.e. w ′ = w R . In thiscanonical form, one observes that h Ψ | Ψ i = h w ′ | I | w ′ i , i.e. thenorm tensor is identity N = I . The final optimum tensor w ′ is obtained by solving H eff | w ′ i = λ min | w ′ i , where H eff areobtained by removing tensor w ′ from h Ψ w ′ | H | Ψ w ′ i . tity N = I (which is the best conditioning). The basicidea to do that is to use an appropriate gauge transfor-mations similar to the case of matrix product states: itis obtained by using a sequence of QR-decomposition byfusing virtual bonds in a specific direction as shown inFig. 3-(b-d). In this figure, we have explained how touse QR-decomposition to end up with a canonical form.Once we obtain that, we replace the tensor w i by solv-ing standard eigenvalue problem H eff | w i i = λ min | w i i ,which could be efficiently solved without suffering frombad conditioning.The essential parameter Λ controls the accuracy ofTTN ansatz, as for Λ → ∞ the TTN state faithfullyrepresents the actual ground state of the system. Thecomputational cost of the algorithm scales like O (Λ )and O (Λ ) for running time and memory, respectively.In the present numerical TTN simulation, we considerclusters 4 ×
4, 6 × × → ∞ limit. The largestbond dimension that we could afford is Λ ∼ − (near the critical point, which is the lessaccurate case). A. J − J TFI model on the checkerboard lattice:TTN results
Before presenting the results, let us mention that theinteresting and controversial part of TFI model on the CLis in the low magnetic field limit around the highly frus-trated coupling J = J . This clarifies the reason thatwe concentrate on this region, while the other parts ofthe phase diagram are known by other methods withoutdoubt [22, 26]. To obtain an accurate phase diagram for J − J TFI model on the CL via TTN approach, we com-pute the first and second derivatives of the ground stateenergy by TTN simulation in two distinct directions onthe phase diagram. Firstly, we trace the phase diagramalong Γ /J at fixed J = J and then we consider anotherdirection along J /J at fixed magnetic field Γ /J = cte . J = J According to the following equations, the first and sec-ond derivatives of ground state energy with respect toΓ for the limit J = J are equivalent to the transversemagnetization and magnetic susceptibility, respectively, m x = − ∂ hHi /∂ Γ , (2) χ = ∂m x /∂ Γ = − ∂ H /∂ Γ . (3)Fig. 4-(a) and (b) show these quantities versus Γ /J (at J = J ) obtained from TTN data for different latticesizes. The transverse magnetization continuously reachesto its saturated value, which rules out any first ordertransition at this isotropic regime. However, we can seea peak on the magnetic susceptibility, which becomessharper and stronger by increasing the lattice size, corre-sponding to a continious second order phase transition.We use finite-size scaling theory to evaluate the criticalpoint and critical exponents for this transition [46]. Thescaling behavior of χ , which governs the singularity at Γ /J ∂ E / ∂ Γ =J (a) Γ /J ∂ E / ∂ Γ -1.5-1-0.500.5 J =J lnN l n ( Γ c − Γ m a x ) -4-3-2 (b) ν = 1 . Γ c = 0 . N / ν Γ − Γ c Γ c -10 -5 0 5 10 N − γ / ν χ -0.6-0.4-0.200.2 J =J lnN l n χ ( Γ m a x ) γ = 0 . ν = 1 . Γ c = 0 . (c) FIG. 4. (color online) (a) The first derivative of GS energywith respect to Γ corresponding to the transverse magnetiza-tion obtained from TTN data for different system sizes. (b)The second derivative of GS energy with respect to Γ, whichis the magnetic susceptibility obtained from TTN simulationfor different lattice sizes. It shows only a sharp peak indi-cating a phase transition from the plaquette-VBS phase inlow fields to the quantum paramagnet phase in high fields at(Γ /J ) c = 0 . ± .
01 with exponent ν = 1 . ± .
01. (c) Datacollapse of magnetic susceptibility obtained from TTN data,which shows the scale invariance of susceptibility governed byexponent γ = 0 . ± . the critical point is | Γ c − Γ max | ∼ N − / ν , (4) χ (Γ max ) ∼ N γ/ ν , (5)where Γ c is the critical field in the infinite size, Γ max isthe position of extermum of finite-lattice susceptibility, ν is the correlation length exponent i.e. ξ ∼ | Γ − Γ c | − ν and γ exhibits the trend of singularity in the magneticsusceptibility.We found a good scaling of TTN data, which gives thecritical field to be Γ c = 0 . ± .
01 in the thermodynamiclimit. Interestingly, Fig. 5 confirms that both open andperiodic boundary conditions lead to the same criticalfield Γ c ≃ .
28. This critical point is also in a good ac-cord with Γ c ≃ . ν = 1 . ± . γ = 0 . ± . J = J , which are separated at Γ c . This single peak canbe a signature for a quantum continuous phase transitionfrom the plaquette-VBS phase at low fields to the quan-tum paramagnetic phase of high fields. The continuousnature of such transition is also confirmed by the bro-ken lattice translational symmetry of the plaquette-VBSphase compared with symmetric quantum paramagneticphase, as we expect from a Landau-Ginzburg paradigm.The TTN results presented on the large two-dimensionallattices N = 4 × , × × J = J , which rulesout the existence of a N´eel order within 0 . . Γ . . C NN = h S zi S zj i , using TTN simulations on the8 × J = J . We obtained this correlationfunction for two different low and high values of trans-verse field Γ, shown in Fig. 6. Correlations for the lowfield regime depict a value close to the maximum valueof N´eel type ordering C maxNN = − .
25 on the bonds ofempty plaquettes with no corner sharing, while corre-lations have very small values on the other plaquettes.This is a clear signature of the plaquette formation as aVBS state, which breaks lattice translational symmetryleaving two-fold degeneracy. However, by increasing themagnetic field to the high field regime, we reach a quan-tum paramagnetic phase as it shows small correlationsalong vertical and horizontal directions of the lattice.Moreover, we plot in Fig. 7-(a), the translational orderparameter, defined by∆ T = h S zA S zB i − h S zB S zC i , (6)as a function of Γ /J for different system sizes, where thesites A, B, and C are shown in Fig. 6. It is observed that Γ m a x FIG. 5. (color online) The value of critical point versus inverseof lattice sizes. Both periodic and open boundary conditionsare presented, which are fitted by the scaling relation Γ( N ) =Γ c ( ∞ ) + aN . We obtain Γ c ( ∞ ) = 0 . ± .
01 and 0 . ± . -0.22 -0.22-0.025-0.22-0.22-0.22 -0.22-0.22-0.22 - . - . - . - . - . - . - . - . - . - . - . - . -0.025-0.025-0.025 -0.05-0.05-0.05-0.05 -0.05-0.05-0.05-0.05 - . - . - . - . - . - . - . - . - . - . - . - . -0.05-0.05-0.05-0.05 A B C
FIG. 6. (color online) Nearest-neighbor correlations, obtaindby TTN numerical simulation on the center of a 8 × J = J . Left: correlations at Γ /J = 0 .
2, whichshows the breaking of lattice translational symmetry corre-sponding to the plaquette-VBS phase. Right: correlations atΓ /J = 0 . by increasing system size the translational order parame-ter rapidly decreases (extrapolates to zero in the infinitesize limit) for Γ > . < . O ) [22, 25]. This operator is defined asˆ O = | ϕ ih ¯ ϕ | + | ¯ ϕ ih ϕ | , (7)where | ϕ i = | ↑↓↑↓i and | ¯ ϕ i = | ↓↑↓↑i are two pos-sible N´eel configurations of a single plaquette. In fact,ˆ O defines a measure of resonating magnitude between | ϕ i and | ¯ ϕ i on a plaquette. It is a suitable definitionas it avoids formation of magnetic long range orders likeN´eel and collinear states on the whole lattice. Hence, Γ /J ∆ T (a) Γ /J h ˆ O i (b) FIG. 7. (color online) (a) Expectation value of the transla-tional order parameter ∆ T and (b) the plaquette order pa-rameter operator h ˆ O i versus transverse magnetic field, ob-tained by unconstrained TTN ansatz on different lattice sizes. J = J Γ = 0 . . . . E/N -0.2525 -0.2607 -0.2770 -0.3050 h ˆ o i × J = J and open boundry condition. the expectation value of ˆ O is very close to one for a res-onating plaquette valence bond solid state, which has nomagnetic order in z-direction. Fig. 7-(b) shows the ex-pectation value of h ˆ O i obtained by TTN simulation ondifferent lattice sizes. It is evident that for J = J andlow fields, the value of h ˆ O i is very close to unity whichcorresponds to the presence of a plaquette-VBS state. J /J d E / d J -0.2-0.100.10.2 Γ /J =0.2 (a) J /J d E / d J -6-4-2024 Γ /J =0.2 lnN l n ( J c − J m a x ) -6.5-6-5.5 lnN l n ( J c − J m a x ) -7-6-5 ν = 0 . J c = 1 . ν = 1 . J c = 0 . (b) FIG. 8. (color online) (a) The first derivative of GS energywith respect to J at Γ /J = 0 .
2, obtained from TTN sim-ulation for different system sizes, (b) The second derivativeof GS energy with respect to J at Γ /J = 0 .
2, shows twosharp peaks indicating a phase transition from the N´eel andcollinear phases on both sides to the intermediate plaquette-VBS phase. The critical points occur at ( J /J ) c = 0 . ± . J /J ) c = 1 . ± . ν ≃ . Γ /J = 0 . To elucidate the structure of phase diagram close tostrong frustration, we fix the magnetic field in Γ /J = 0 . J /J . The first derivativeof GS energy, according to relation C (2) = h S zi S zj i hh i,j ii = ∂ hHi /∂J , is equivalent to the next-nearest neighborspin-spin correlation. Fig. 8-(a) presents C (2) versus J /J , which shows a change of sign at J = J . How-ever, the derivative of C (2) —that is the second deriva-tive of energy—represents two peaks as shown in Fig. 8-(b), which become sharper by increasing the lattice size.These peaks are interpreted as two critical points corre-sponding to two-phase transitions from the intermediateplaquette-VBS phase to the N´eel and collinear phases J = J Γ = 0 . J < J Γ = 0 . J > J Γ c ν γ J c ν J c ν on both sides of the phase diagram. The nature of quan-tum phase transition from the plaquette-VBS to N´eel andcollinear antiferromagnetic phases is an interesting fea-ture of our results. The N´eel and plaquette-VBS ordersbreak different kind of symmetries, i.e. N´eel order breaksa discrete Z J /J ) c = 0 . .
999 [22]. On the otherhand, as seen from Fig. 8, the plaquette-VBS to collinearphase transition is also continuous. However, it wouldbe a conventional second order phase transition, becauseboth the plaquette-VBS and collinear phases break trans-lational symmetry. The value of the latter critical pointis ( J /J ) c = 1 . .
001 obtained by COA. The insets of Fig. 8-(b)depicts finite size scaling data which reports correlationlength exponent to be ν ≃ . J = J for different valuesof transverse field Γ. In Table.II, we tabulate the cor-responding critical points and exponents obtained fromfinite-size scaling analysis on different parts of the phasediagram. IV. MAP FROM THE CHECKERBOARDLATTICE TO THE SQUARE LATTICE
Here, we establish our map from CL to SL. Let usconsider non-corner sharing set of crossed plaquettes onCL, as unit cells of our transformation (see Fig. 9-(a)).According to Fig. 9-(a), we assign a quasi spin-half toeach unit cell. These quasi spins form a new square lat-tice, whose lattice spacing is twice as the original lattice(see Fig. 9-(b)). Accordingly, the transverse field Ising (a) (b)
FIG. 9. (color online) Mapping from the CL to the squarelattice. (a) Hatched crossed plaquettes of the CL form theunit cells of transformation. Solid and dashed lines are J and J bonds, respectively. Green bullets represent quasi spins,which are associated to each unit cell. (b) A square latticeconstructed from quasi spins, by a lattice spacing twice as theoriginal checkerboard one. Solid and dashed lines represent J and J bonds for the square lattice, respectively. J /J E ne r g y / J -1.5-1-0.5 Γ /J =0.5 |u i |u i |u i |u i FIG. 10. (color online) The first four energy levels of a sin-gle crossed plaquette spectrum versus J /J in (an arbitrary)transverse field Γ /J = 0 . Hamiltonian Eq. 1 can be rewritten in the form H = H + H int ,H = X I H I , H int = X
2) devoted to the unit cell. Hence, we define | u i I = | τ zI = ↑i and | u i I = | τ zI = ↓i . On the otherhand, for J > J , the two eigenstates related to low-est eigenenergies are | u i and | u i , where | u i is twofolddegenerate, i.e. ǫ = ǫ . Therefore, for J > J , we con-sider two states | u i and | u ′ i = √ ( | u i + | u i ) as | ↑i and | ↓i quasi-spins, respectively.In the next step, we define projecting operators ontothe subspace spanned by the low-energy sector of unitcells. In fact, the terminology of effective theory, whichdescribes the low-energy behavior of a model is alwaysaccompanied by the reduction in the Hilbert space. Wedefine two projecting operators P I and P ′ I of unit celllabeled by I , for J < J and J > J , respectively.They read as, P I = | u i II h u | + | u i II h u | , (9) P ′ I = | u i II h u | + | u ′ i II h u ′ | . (10)These local operators act as Identity operator on otherunit cells. Therefore, the projecting operator for thewhole lattice is defined as P = N I P I and P ′ = N I P ′ I .Hence, the effective Hamiltonian in truncated subspacewill be obtained from the following relations, H eff = P ( H + H int ) P, ( J < J ) (11) H eff = P ′ ( H + H int ) P ′ , ( J > J ) . (12)The explicit form of H and H int in terms of original spinoperators are given in Appendix A.The original Hamiltonian is renormalized in truncatedsubspace according to Eqs. 11 and 12, which leads to theeffective Hamiltonian as follows, J < J : H eff = − α J X h I,J i τ xI τ xJ + α J X hh I,J ii τ xI τ xJ − ( ǫ − ǫ ) X I τ zI , (13) J > J : H eff = − α ′ J X h I,J i v τ xI τ xJ + 2 α ′ J X h I,J i h τ xI τ xJ − α ′ J X hh I,J ii τ xI τ xJ − ( ǫ − ǫ ) X I τ zI , (14)where, h I, J i h and h I, J i v run over horizontal and ver-tical nearest neighbor bonds on the effective square lat-tice. The coefficients α and α ′ are functions of J , J and Γ (see Appendix). Let us make a π -rotation aroundz-axis on the spins of one of the sublattices of the bi-partite square lattice defined in Eq. 13, which contractsthe the minus sign in the first term. Similarly, a π -rotation around z-axis on the spins sitting on even (orodd) labeled horizontal lines change the minus signs ofthe first and third terms of Eq. 14. Hence, all Ising terms( τ xI τ xJ ) in Eqs. 13, 14 have positive couplings. Now, it isclear from the sign of nearest and next-nearest neigh-bor interactions of the effective Hamiltonian, that thereis a N´eel and striped order for J ≪ J and J ≫ J limits, respectively. They correspond to well known clas-sical magnetic ordered phases of the Ising model on the J − J square lattice [48]. Hence, we can merge thetwo effective Hamiltonians 13 and 14 and write a unified J /J Γ / J quantumparamagnet stripedNéel Néel striped { string-VBS string-VBS FIG. 11. (color online) Quantum GS phase diagram of the J − J TFI model on square lattice obtained from the phasediagram of the CL [22] using the introduced effective theory.The inset indicates an opening of a narrow region of string-VBS phase, which fills the space between the N´eel and stripedphases around J /J = 0 . effective Hamiltonian in terms of the renormalized pa-rameters ˜ J − ˜ J that is a transverse field Ising model onthe effective square lattice, H eff = ˜ J X h I,J i τ xI τ xJ + ˜ J X hh I,J ii τ xI τ xJ − ˜Γ X I τ zI , (15)where, ˜ J ˜ J = 12 J J , ˜Γ˜ J = ǫ − ǫ α J , ( J < J ) , (16)˜Γ˜ J = ǫ − ǫ α ′ J , ( J > J ) . According to Eq. 15, the low-energy effective theory ofTFI model on CL is provided with the same model ona square lattice with renormalized parameters given inEq. 16. The effective Hamiltonian clearly shows thatat the zero field limit, the critical point J = J of CLis mapped to the critical point ˜ J = 0 . J of SL (seeEq. 16). Hence, the critical phase boundaries of ˜ J − ˜ J TFI model on SL can be achieved from the critical phaseboundaries of the J − J TFI model on CL.
A. GS phase diagram of J − J TFI model on thesquare lattice
We implement the mapping established in the previ-ous section and apply it to the GS phase diagram of TFI
FIG. 12. (color online) Plaquette-VBS phase of CL with bro-ken translational symmetry with two-fold degeneracy, whichis mapped to the string-VBS phase of square lattice with bro-ken rotational symmetry and two-fold degeneracy. model on CL— which has been obtained by COA, [22]—to get the GS phase diagram of J − J TFI model onSL. To this end, we insert the location of critical bound-aries of the CL phase diagram in Eqs. 16 to obtain thecorresponding critical boundaries of the SL phase dia-gram. The outcome of this map is shown in Fig. 11. Forinstance, the critical point Γ c /J = 0 . J = J onCL is mapped to Γ c /J = 0 .
32 at J = 0 . J on SL.This result is consistent with the result Γ c /J = 0 . J = 0 . J at low fields, exactlythe same as what appeared in the phase diagram of CLaround the highly frustrated point J = J at low fields,like Fig. 2. Hence, it can be deduced that quantum fluc-tuations of the weak transverse magnetic field induce anovel quantum state from the highly degenerate classicalGS of SL at J = 0 . J , before reaching to the quantumparamagnet phase at high fields.One of the smart features of the introduced mapping isto determine the structure of the novel state according tothe plaquette-VBS state on CL. Let us suppose that theCL is in the plaquette-VBS phase as shown by the colorplaquettes in Fig. 12. In fact, each color plaquette is sur-rounded by two close sites on the effective square lattice.Therefore, whenever color plaquettes of CL resonate be-tween two possible N´eel states, which comes from thenature of plaquette-VBS phase, then they bring abouta resonant situation on a set of sites on the effectivesquare lattice resembling the string formation. Moreover,as the plaquette-VBS state of CL breaks the translationalsymmetry of the lattice bearing two-fold degeneracy, theemergence of strings on the effective SL could be either invertical or horizontal directions, breaking the rotationalsymmetry of the lattice, which manifests the two-folddegeneracy of string formations. This is in agreementwith our earlier results in Ref. [29], which states that thehighly degenerate classical ground state of J − J TFImodel on SL at J = 0 . J goes to a unique string-VBSphase, when taking into account quantum fluctuations.This justifies the mapping procedure introduced here. V. SUMMARY AND CONCLUSIONS
Transverse field Ising model on two-dimensionalcheckerboard/square lattice would be a generic Hamilto-nian to represent uni-axial magnets driven by quantumfluctuations. It includes planar spin ice [16], artificialsquare ice [6–8] and even the realization of quantum spinice with Rydberg atoms [9] that offer the emegence ofnovel phases. We have investigated the phase diagramof the J − J TFI model on checkerboard lattice byan improved tree tensor-network algorithm. We devel-oped an unconstrained (gauge-free) tree tensor-networkansatz, adapted to two-dimensional systems up to thelattice size 8 ×
8, by relaxing isometry constraint. At thehighly frustrated point J = J , we confirm a plaquette-VBS phase at low fields, separated from a paramagnetphase at Γ c ∼ .
28. Utilizing finite-size scaling anal-ysis on N = 4 × , × × ν ≃ γ ≃ .
44. We did not observe a signature of a canted N´eelphase predicted by the Monte-Carlo study [25], which isin agreement with previous results based on cluster op-erator approach [22]. In addition, we found the natureand associated critical exponents of the quantum phasetransitions from the plaquette-VBS phase to the adjacentN´eel and collinear antiferromagnetic phases and also tothe quantum paramagnetic phase of high fields, summa-rized in table-II. It is shown that all transitions are ofthe second-order type except the transition from N´eel toplaquette-VBS, which is of deconfined type, where thefirst derivative of ground-state energy indicates no sin-gularity. The schematic structure of the phase diagramis given in Fig. 2.Our study justifies the importance of unconstrainedTTN ansatz as a promising numerical tool to addresssuch highly frustrated systems, where quantum MonteCarlo simulation fails due to the known sign problem forreaching ground state properties. Furthermore, we havedeveloped a mapping analysis to obtain quantum groundstate phase diagram of the J − J TFI model on squarelattice from the phase diagram of the J − J TFI modelon checkerboard lattice. An important outcome of ourmapping is to clarify the VBS nature of the intermedi-ate phase of square-lattice phase diagram at low fieldsaround the highly frustrated point J = 0 . J . In fact,we showed that the plaquette-VBS phase of the checker-board lattice is mapped to the string-VBS phase of sqaurelattice at the highly frustrated point J = 0 . J , com-pletely in agreement with the previous results of J − J TFI model on square lattice by cluster operator ap-proach, which describes such VBS ordering [29]. Briefly,we claim that the low-energy effective theory of J − J TFI model on checkerboard is given by the same modelon square lattice with renormalized parameters.0
VI. ACKNOWLEDGEMENTS
A.L. would like to thank the Sharif University of Tech-nology for financial support under grant No. G960208.R.H. was supported by the Department of Energy, Of-fice of Basic Energy Sciences, Division of Materials Sci-ences and Engineering, under Contract No. DE-AC02-76SF00515 through SLAC National Accelerator Labora-tory. We have used
Uni10 [49] library to build the TTNansatz.
Appendix A: Mapping from the checkerboard latticeto the square lattice
The details of mapping procedure is presented here. Aswe explained in the text, if we divide the CL into non-corner-sharing crossed plaquettes, the transverse fieldIsing Hamiltonian can be rewritten in the form H = P I H I + P
11 1 111 1 11233 333 3333 44 444 4 444 222 22222
I J(5)J(1)J(8)J(7)J(6)J(3) J(2)J(4)
FIG. 13. (color online) The CL: each isolated plaquette Iinteracts with eight neigboring plaquettes. The green andblue lines correspond respectively to J and J interactions ofplaquette I with its neighbors. H I = J ( s z ,I s z ,I + s z ,I s z ,I + s z ,I s z ,I + s z ,I s z ,I ) (A1)+ J ( s z ,I s z ,I + s z ,I s z ,I ) − Γ( s x ,I + s x ,I + s x ,I + s x ,I ) ,H IJ = J ( s z ,I s z ,J (1) + s z ,I s z ,J (1) ) + J ( s z ,I s z ,J (5) ) + J ( s z ,I s z ,J (2) + s z ,I s z ,J (2) ) + J ( s z ,I s z ,J (6) ) + J ( s z ,I s z ,J (3) + s z ,I s z ,J (3) ) + J ( s z ,I s z ,J (7) ) + J ( s z ,I s z ,J (4) + s z ,I s z ,J (4) ) + J ( s z ,I s z ,J (8) ) . (A2)Let us consider the case J > J , we consider the first twoeigenstates | u i and | u i of H I –corresponding to the first two energy levels of it– as two components of new quasi-spins assigned to each single plaquette. Then, we definethe projecting operator P as P = | u ih u | + | u ih u | to renormalize original spin operators in the truncatedsubspace according to the following equations, P s z P = P s z P = ατ xI ,P s z P = P s z P = − ατ xI ,P s x P = P s x P = P s x P = P s x P = ( β − γ ) τ zI (A3)where, α = 4 A B + 2 A B , β = 2 A ( A + 2 A + A )and γ = 2 B B in which the coefficients A , B are givenby the matrix elements of eigenvectors | u i and | u i , | u i = A A A A A A A A A A A A A A A A , | u i = B − B B B B − B − B − B B − B . (A4)These matrix elements are functions of J , J and Γ,which are lengthy and complicated expressions. The sim-plest one is B , which has the following form, B = − r (cid:16) √ +( J − J ) +2 J − J (cid:17) Γ + 32 . (A5)Now, we rewrite the Hamiltonians H I and H IJ ofEq.A1 and Eq.A2 in terms of new quasi-spins and finallyobtain the effective Hamiltonian, J < J : H eff = − α J X h I,J i τ xI τ xJ + α J X hh I,J ii τ xI τ xJ − ( ǫ − ǫ ) X I τ zI , (A6)where ǫ and ǫ are eigenenergies of a single plaquette,corresponding to eigenvectors | u i and | u i , respectively.We perform a π -rotation on spins on only even (or odd)1sites of bipartite square lattice. It finally leads to aneffective Hamiltonian for J < J as H eff = J ′ X h I,J i τ xI τ xJ + J ′ X hh I,J ii τ xI τ xJ − Γ ′ X I τ zI , (A7)where, J ′ J ′ = 12 J J , Γ ′ J ′ = ǫ − ǫ α , ( J < J ) . (A8) Similar procedure is also done for the case J > J . [1] M. J. Harris, S. T. Bramwell, D. F. McMorrow, T. Zeiske,and K. W. Godfrey, Phys. Rev. Lett. , 2554 (1997).[2] S. T. Bramwell and M. J. P. Gingras,Science , 1495 (2001).[3] C. Nisoli, R. Moessner, and P. Schiffer,Rev. Mod. Phys. , 1473 (2013).[4] C. Lacroix, P. Mendels, and F. Mila, Introduction toFrustrated Magnetism: Materials, Experiments, Theory (Springer, 2013).[5] H. T. Diep,
Frustrated Spin Systems , 2nd ed. (WORLDSCIENTIFIC, 2013).[6] R. F. Wang, C. Nisoli, R. S. Freitas, J. Li, W. McConville,B. J. Cooley, M. S. Lund, N. Samarth, C. Leighton, V. H.Crespi, and P. Schiffer, Nature , 303 (2006).[7] R. Wang, J. Li, W. McConville, C. Nisoli, X. Ke, J. Free-land, V. Rose, M. Grimsditch, P. Lammert, V. Crespi, et al. , Journal of applied physics , 09J104 (2007).[8] X. Ke, J. Li, C. Nisoli, P. E. Lammert, W. Mc-Conville, R. F. Wang, V. H. Crespi, and P. Schiffer,Phys. Rev. Lett. , 037205 (2008).[9] A. W. Glaetzle, M. Dalmonte, R. Nath,I. Rousochatzakis, R. Moessner, and P. Zoller,Phys. Rev. X , 041037 (2014).[10] R. Moessner, Canadian Journal of Physics , 1283 (2001).[11] R. Siddharthan, B. S. Shastry, A. P. Ramirez,A. Hayashi, R. J. Cava, and S. Rosenkranz,Phys. Rev. Lett. , 1854 (1999).[12] Y. K. Tsui, C. A. Burns, J. Snyder, and P. Schiffer,Phys. Rev. Lett. , 3532 (1999).[13] J. S. Gardner, S. R. Dunsiger, B. D. Gaulin, M. J. P.Gingras, J. E. Greedan, R. F. Kiefl, M. D. Lumsden,W. A. MacFarlane, N. P. Raju, J. E. Sonier, I. Swainson,and Z. Tun, Phys. Rev. Lett. , 1012 (1999).[14] R. G. Melko, B. C. den Hertog, and M. J. P. Gingras,Phys. Rev. Lett. , 067203 (2001).[15] J. P. C. Ruff, R. G. Melko, and M. J. P. Gingras,Phys. Rev. Lett. , 097202 (2005).[16] R. Moessner, O. Tchernyshyov, and S. L. Sondhi,Journal of Statistical Physics , 755 (2004).[17] R. Moessner and S. L. Sondhi,Phys. Rev. B , 224401 (2001).[18] N. Shannon, G. Misguich, and K. Penc,Phys. Rev. B , 220403 (2004).[19] O. A. Starykh, A. Furusaki, and L. Balents,Phys. Rev. B , 094416 (2005).[20] Y.-H. Chan, Y.-J. Han, and L.-M. Duan, Phys. Rev. B , 224407 (2011).[21] R. F. Bishop, P. H. Y. Li, D. J. J. Farnell, J. Richter,and C. E. Campbell, Phys. Rev. B , 205122 (2012).[22] M. Sadrzadeh and A. Langari,The European Physical Journal B , 259 (2015).[23] J. Villain, R. Bidaux, J.-P. Carton, and R. Conte,Journal de Physique , 1263 (1980).[24] L. Bellier-Castella, M. J. Gingras,P. C. Holdsworth, and R. Moessner,Canadian Journal of Physics , 1365 (2001).[25] L.-P. Henry and T. Roscilde,Phys. Rev. Lett. , 027204 (2014).[26] L.-P. Henry, P. C. W. Holdsworth, F. Mila, andT. Roscilde, Phys. Rev. B , 134427 (2012).[27] M. Sadrzadeh and A. Langari,Journal of Physics: Conference Series , 012114 (2018).[28] L. Tagliacozzo, G. Evenbly, and G. Vidal,Phys. Rev. B , 235127 (2009).[29] M. Sadrzadeh, R. Haghshenas, S. S. Jahromi, andA. Langari, Phys. Rev. B , 214419 (2016).[30] M. Sadrzadeh and A. Langari,Physica C: Superconductivity and its Applications , 1 (2018).[31] A. Kalz, A. Honecker, S. Fuchs, and T. Pruschke,Journal of Physics: Conference Series , 012051 (2009).[32] S. Sachdev and R. N. Bhatt,Phys. Rev. B , 9323 (1990).[33] M. E. Zhitomirsky and K. Ueda,Phys. Rev. B , 9007 (1996).[34] G.-M. Zhang, H. Hu, and L. Yu,Phys. Rev. Lett. , 067201 (2003).[35] L. Capriotti, F. Becca, A. Parola, and S. Sorella,Phys. Rev. B , 212402 (2003).[36] O. A. Starykh and L. Balents,Phys. Rev. Lett. , 127202 (2004).[37] M. Mambrini, A. L¨auchli, D. Poilblanc, and F. Mila,Phys. Rev. B , 144422 (2006).[38] R. Haghshenas and D. N. Sheng,Phys. Rev. B , 174408 (2018).[39] R. Haghshenas, W.-W. Lan, S.-S. Gong, and D. N.Sheng, Phys. Rev. B , 184436 (2018).[40] Y.-Y. Shi, L.-M. Duan, and G. Vidal,Phys. Rev. A , 022320 (2006).[41] P. Silvi, V. Giovannetti, S. Montangero, M. Rizzi, J. I.Cirac, and R. Fazio, Phys. Rev. A , 062335 (2010).[42] V. Murg, F. Verstraete, O. Legeza, and R. M. Noack,Phys. Rev. B , 205105 (2010). [43] M. Gerster, P. Silvi, M. Rizzi, R. Fazio, T. Calarco, andS. Montangero, Phys. Rev. B , 125154 (2014).[44] E. Efrati, Z. Wang, A. Kolan, and L. P. Kadanoff,Rev. Mod. Phys. , 647 (2014).[45] F. Verstraete, V. Murg, and J. Cirac,Adv. Phys. , 143 (2008).[46] H. Nishimori and G. Ortiz, Elements of Phase Transitions and Critical Phenomena (Oxford University Press, 2011).[47] T. Senthil, A. Vishwanath, L. Balents, S. Sachdev, andM. P. A. Fisher, Science , 1490 (2004).[48] J. L. Mor´an-L´opez, F. Aguilera-Granja, and J. M.Sanchez, Phys. Rev. B , 3519 (1993).[49] Y.-J. Kao, Y.-D. Hsieh, and P. Chen,Journal of Physics: Conference Series640