Quantum Phase Transitions in Long-Range Interacting Hyperuniform Spin Chains in a Transverse Field
QQuantum Phase Transitions in Long-Range Interacting Hyperuniform Spin Chains ina Transverse Field
Amartya Bose
Department of Chemistry, Princeton University, Princeton, New Jersey 08544
Salvatore Torquato
Department of Chemistry, Princeton University, Princeton, New Jersey 08544Department of Physics, Princeton University, Princeton, New Jersey 08544Princeton Institute for the Science and Technology of Materials,Princeton University, Princeton, New Jersey 08544 andProgram in Applied and Computational Mathematics,Princeton University, Princeton, New Jersey 08544
Hyperuniform states of matter are characterized by anomalous suppression of long-wavelengthdensity fluctuations. While most of the interesting cases of disordered hyperuniformity are providedby complex many-body systems like liquids or amorphous solids, classical spin chains with cer-tain long-range interactions have been shown to demonstrate the same phenomenon. Such systemsinvolving spin chains are ideal models for exploring the effects of quantum mechanics on hyperuni-formity. It is well-known that the transverse field Ising model shows a quantum phase transition(QPT) at zero temperature. Under the quantum effects of a transverse magnetic field, classicalhyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations ofthese cases are complicated because of the presence of highly nontrivial long-range interactions. Weperform extensive analysis of these systems using density matrix renormalization group (DMRG) tostudy the possibilities of phase transitions and the mechanism by which they lose hyperuniformity.Even for a spin chain of length 30, we see discontinuous changes in properties like the “ τ ordermetric” of the ground state, the measure of hyperuniformity and the second cumulant of the totalmagnetization along the x -direction, all suggestive of first-order QPTs. An interesting feature of thephase transitions in these disordered hyperuniform spin chains is that, depending on the parametervalues, the presence of transverse magnetic field may remarkably lead to increase in the order ofthe ground state as measured by the “ τ order metric,” even if hyperuniformity is lost. Therefore, itwould be possible to design materials to target specific novel quantum behaviors in the presence ofa transverse magnetic field. Our numerical investigations suggest that these spin chains can showno more than two QPTs. We further analyze the long-range interacting spin chains via the Jordan-Wigner mapping on to a system of spinless fermions, showing that under the pairwise interactingapproximation and a mean-field treatment, there can be at most two quantum phase transitions.Based on these numerical and theoretical explorations, we conjecture that for spin chains with long-range pair interactions that have convergent cosine transforms, there can be a maximum of twoquantum phase transitions at zero temperature. I. INTRODUCTION
The notion of hyperuniformity provides a unifyingframework to characterize the large scale structure of sys-tems as disparate as crystals, liquids, and exotic disor-dered states of matter. A hyperuniform state of matteris one where the density fluctuations at very large lengthscales are suppressed, and consequently, the structurefactor, lim | k |→ S ( k ) = 0. All perfect crystals and perfectquasicrystals are hyperuniform and can be rank orderedin terms of their capacity to suppress large-scale densityfluctuations [1, 2]. Disordered hyperuniform materialsare exotic amorphous states of matter that are like crys-tals in the manner in which their large-scale density fluc-tuations are anomalously suppressed and yet behave likeliquids or glasses in that they are statistically isotropicwithout any Bragg peaks [1, 2].Classical disordered hyperuniform systems are attract-ing great attention because they are endowed with novelphysical properties [3–9]. There are far fewer studies of quantum mechanical hyperuniform systems. Some ex-actly solvable quantum systems, such as free fermion sys-tems [10], superfluid Helium [11], the ground state of thefractional quantum Hall effect [12], and Weyl-Heisenbergensembles [13] are hyperuniform [14]. Recently, quantumphase transitions (QPTs) have been studied in nearest-neighbor Ising-chains with hyperuniform couplings [15].The exploration of the interplay between hyperunifor-mity, order and quantum fluctuations is a fertile area forresearch.Stealthy hyperuniform systems are ones in which thestructure factor is zero for wavenumbers in the vicinity ofthe origin, i.e., there exists a critical radius K , such that, S ( | k | ) = 0 for all 0 < | k | < K [16–19]. For sufficientlysmall K , stealthy hyperuniform many-particle systemsare the highly degenerate disordered ground states of cer-tain long-ranged interactions [18]. Disordered materialsbased on such stealthy hyperuniform distribution of par-ticles often possess desirable physical properties, such asphotonic band gaps that are comparable in size to pho- a r X i v : . [ qu a n t - ph ] D ec tonic crystals [8, 20], transparent dense materials [21],and optimal transport characteristics [4, 22]. The collec-tive coordinate optimization technique has been used togenerate disordered stealthy hyperuniform many-particlesystems [16, 17, 19]. Such inverse statistical-mechanicalalgorithms have also been extended to deal with dis-crete spin systems [23], and to determine long-rangeIsing interactions that have stealthy hyperuniform classi-cal ground states [24]. More recently, inverse statistical-mechanical techniques have been applied to quantum me-chanical problems to construct the Hamiltonian from theeigenstate [25].Historically spin chains have proven to be fertilegrounds for exploration of effects of quantum mechanics.In this paper, we characterize the effects of a transversefield on spin systems whose ground states, in absence ofsuch fields, are stealthily hyperuniform. Quantum fluc-tuations are typically associated with a loss of order ingeneral. This is true for the usual interactions that de-cay monotonically with distance. Examples of this in-clude the loss of the ferromagnetic order in Ising modelin presence of transverse field and the broadening of thepeaks of the pair correlation function g ( r ) of quantum liq-uids as compared to their classical counterparts at simi-lar thermodynamic conditions. We demonstrate the veryinteresting possibility of optimizing magnetic materialssuch that the quantum effects of an external transversemagnetic field can be used to increase the order of theground state of the spin system, while still remainingmore disordered than the standard antiferromagnetic orferromagnetic materials.In presence of a transverse field, classical mechanicsalone cannot account for the physics of a spin chain.Since hyperuniformity is extremely sensitive to the exactnature of the ground state and the interactions involved,it is natural to expect that the transverse field would leadto a loss of hyperuniformity. It is, therefore, interesting tostudy the quantum mechanism by which hyperuniformityis lost, and the phase transitions involved. In this study,we consider spin chains with long-range interactions thathave been optimized to have disordered stealthy hy-peruniform ground states in the absence of magneticfield [24]. We explore these systems using Density Ma-trix Renormalization Group (DMRG) [26–35] with a par-ticular emphasis on the difference in the physics due tothe disordered nature of the classical ground state in ab-sence of the transverse magnetic field and the long-rangenature of the interactions. DMRG is one of the best al-gorithms for dealing with one-dimensional lattice-basedproblems that are not at the critical points. However, itis typically applied to systems with short-range interac-tions with open boundary conditions. Recently, DMRGhas been applied to study the physics of long-range sys-tems with monotonically decaying interactions with openboundaries [36]. Stealthy hyperuniform systems, how-ever, can only be generated in the presence of periodicboundary conditions. Hence, this study involves com-plex non-monotonic, long-range interactions on periodic boundary conditions, which make the simulations verychallenging. All DMRG calculations in this paper wereperformed using the ITensor library [37].In Sec. II, we describe the system under study, themethods employed and the observables calculated. InSec. III, we illustrate the most important classes of re-sults obtained through the DMRG simulations. Our re-sults demonstrate that it is possible for the τ order met-ric [18, 24, 38] of the ground state to increase with atransverse magnetic field. Our numerical simulations alsosuggest that these long-range spin systems can sustain nomore than two QPTs. Of these, we choose the case withtwo discontinuous QPTs and illustrate a basis set cal-culation in appendix A. We carry out further analyticalexplorations by mapping the system on to a system ofspinless fermions via the Jordan-Wigner transformation,which results in a fermionic Hamiltonian with more thanpairwise interactions that make the direct solution non-trivial. We analyze the resultant Hamiltonian using asimple approximate model with only the pairwise interac-tion terms as well as under a mean-field treatment of theterms involving more than pairwise interactions. Con-sistent with our numerical results, we show in Sec. III Cthat for both the approximate model and the mean-fieldHamiltonian here can be a maximum of two phase tran-sitions. This leads us to a conjecture that for spin chainswith long-range pair interactions with convergent cosinetransforms, there can be no more than two zero tempera-ture quantum phase transitions. We end this paper witha concluding remarks and outlook for further interestingexplorations in Sec. IV. II. DESCRIPTION OF SYSTEM ANDMETHODS EMPLOYED
We study one-dimensional (1D) spin systems withlong-range interactions. The basic Hamiltonian is evalu-ated with periodic boundary conditions (PBC) and hasthe following form on the integer lattice Z : H = − (cid:88) i (cid:88) ≤ r ≤ R J r ˆ σ ( i ) z ˆ σ ( i + r ) z + (cid:88) i − Γˆ σ ( i ) x . (1)where ˆ σ ( i ) z and ˆ σ ( i ) x are the Pauli spin matrices along thez and x directions respectively on the i th site, J r is thecoupling between two spins separated by r lattice points,and Γ is the strength of the transverse field.We simulate the system for various sets of J r in orderto understand how the systems with stealthy hyperuni-form ground states in the absence of a transverse fieldbehave with increasing Γ. These hyperuniform parame-ters were obtained by Chertkov et al. [24] (see Supple-mentary Material (SM) for the parameters). They areoften atypical in the sense that the interaction strengthdoes not necessarily decrease with distance. Therefore,we also show results for simulations where the couplingsdecay according to inverse power law with the distance.Due to the increased computational complexity of theDMRG algorithm for periodic systems, all the parame-ters considered have N = 30 spins. It is important tonote here that in case of simulating long-range interac-tions in these systems with periodic boundary conditions,one encounters strong finite-size effects. A system with N = 30 spins might not enough to get rid of these ef-fects. Other investigations have used Ewald summationtechniques to take care of this finite-size effect [39]. How-ever, here we do not attempt to alleviate this problem inorder to maintain consistency with the work on hyper-uniform spin chains [24].Because of the long-range interactions, that do not de-cay with distance and the presence of periodic bound-ary conditions, the entanglement entropy grows fasterthan for the regular Ising model in a transverse field.This makes DMRG calculations significantly more diffi-cult to converge and likely to get stuck in other low-lyinglocal minima. Therefore, we perform ten independentsimulations of the system. In each simulation, we runtwenty DMRG calculations with random initial start-ing points and taking the state with the minimum en-ergy. We compare results across the various runs andtake the lowest energy state as the ground state. SinceDMRG is variational in nature, none of the higher en-ergy states can be the true ground state. Performingmultiple DMRG calculations often allows us to accessother low-lying states. Having an idea of other low-lyingstates expedites the analysis of the problem using basisset expansions. Because of the multiple DMRG calcu-lations we run, we can converge the wavefunction de-spite the long-range interactions. However, the conver-gence gets increasingly difficult in the immediate vicinityof the critical points. This is a well understood limi-tation of DMRG. At the critical points, the correlationlengths become very large, reducing the effectiveness ofthe DMRG algorithm. Therefore, these phase transitionsand the critical exponents involved cannot be character-ized by DMRG. Other methods, such as the Multi-scaleEntanglement Renormalization Ansatz (MERA) [40, 41]and the family of Quantum Monte Carlo (QMC) meth-ods, especially the so-called projective QMC methods likediffusion Monte Carlo (DMC), prove to be useful in suchstudies. Because we are limiting ourselves to chains oflength, N = 30, the locations of the critical points areunlikely to be correct in the thermodynamic limit. How-ever, as our results show, the transitions show a remark-able sharpness, which suggests that these transitions arenot artifacts of the finite-size of our systems. Hence ourresults regarding the possibility of increase in the τ or-der metric of the ground state and the variable numberof phase transitions would continue to hold qualitatively,even for larger systems.For each parameter, we first investigate the variation ofbasic observables like the average energy, and the secondcumulant of the transverse magnetization, h x = 1 N (cid:16)(cid:10) M x (cid:11) − (cid:104) M x (cid:105) (cid:17) . (2) Typically one would use either the average magnetizationalong the z-direction, m z for the “standard” long-rangeIsing model, or the average magnetization along the x-direction m x defined respectively by m z = 1 N (cid:42)(cid:88) j ˆ σ ( j ) z (cid:43) (3) m x = 1 N (cid:42)(cid:88) j ˆ σ ( i ) x (cid:43) (4)However, as we will illustrate, we have found that h x suf-fers significantly less from finite-size effects than m z forthe long-range Ising models, and performs just as wellas m x for the hyperuniform cases in identifying the crit-ical points. We also report the structure factor at theorigin, the deviation from zero of which is a measure ofhyperuniformity, S ≡ lim k → S ( k ) (5)= 1 N (cid:16)(cid:10) M z (cid:11) − (cid:104) M z (cid:105) (cid:17) (6)where M z is the total magnetization along the z-direction, as a function of Γ. The structure factor atthe origin, S = 0 for hyperuniform systems. The devi-ation of S from 0 measures how far the system is frombeing hyperuniform. As Γ → ∞ , we should recover adisordered state, irrespective of the exact nature of J r and so, S would asymptotically tend to 1. The degreeof order of the ground state is measured using the τ ordermetric [18, 24, 38], defined as follows: τ = (cid:88) k ( S ( k ) − S ref ( k )) N , (7)where S ref ( k ) is a reference structure factor. In this pa-per, we use the structure factor for a Poisson point pat-tern, S ref ( k ) = 1 as the reference. The normalizationfactor of N is chosen to make most of the results for τ to be of order unity. With this normalization factor, theantiferromagnetic spin configuration has τ ≈
1, and theferromagnetic spin configuration has τ ≈
2. The samemetric has been used without this normalization factorby Chertkov et al. [24] The structure factor of a givenground state is defined as: S ( k ) = 1 N N (cid:88) l =1 N (cid:88) j =1 ˆ σ ( l ) z ˆ σ ( k ) z exp ( ik ( l − j )) . (8)In addition to the hyperuniformity of the ground state,we also want to study the loss of the stealthiness of thehyperuniformity of the ground state. Since, for the ex-amples demonstrated here, the stealthiness extends onlyto the first non-zero wave-vector, we use the structurefactor at the first non-zero wave-vector as a measure ofstealthiness in the ground state: S = S (∆ k ) = S (cid:18) πN (cid:19) . (9)Finally, for the spin systems with hyperuniform groundstates, we also report the plots of S ( k ) as a function ofΓ. This allows for a greater clarity in the changes thathappen to the ground state before and after the phasetransitions. III. RESULTS
This section is organized in the following manner.First, we report results for “standard” long-range Isingmodels with interactions that decay with the distancebetween the spins. These results allow us to set a pointof comparison for the hyperuniform spin chains. We il-lustrate the finite-size effect on the various observables,and demonstrate how the quantum effects of a trans-verse field reduces the order of the ground state of thesystem. Thereafter, we examine the systems with inter-actions optimized to give stealthy hyperuniform groundstates. We demonstrate that for systems with these non-trivial long-range interactions, it is possible to generateorder from disorder using the quantum effects of a trans-verse magnetic field. Numerically, we observe no morethan two QPTs. To further theoretically explore of thenature of these phase-transitions, we map the spin sys-tems onto chains of spinless fermions. The long-rangespin-spin interaction manifests not only in long-distancepairwise interaction terms in the Hamiltonian, but re-sults also in terms involving higher-order non-pairwiseinteraction terms (interactions involving triplets of spins,quadruplets of spins, and so on). We analyze the resul-tant Hamiltonian under a very simple long-range pair-wise interaction approximation, and a subsequent mean-field treatment of non-pairwise interaction terms, show-ing that in both cases, a maximum of two phase transi-tions is possible. This is consistent with our numericalresults, and leads us to conjecture that for Ising mod-els with long-range interactions that have convergent co-sine transforms, there can be a maximum of two zero-temperature quantum phase transitions.
A. Interactions that Decay with Distance
For the purposes of comparison to parameters thatwere specifically optimized to have stealthy hyperuni-form ground states, we consider systems with interac-tions that decay with distance. These models have beenextensively studied using various analytic and numericaltechniques [42, 43]. Qualitatively, as decay of the inter-action becomes faster, we expect the system to asymp-totically approach the nearest neighbor interaction limit,that is the standard Ising model. So, we would expectthese systems to undergo a continuous phase transitionlike the standard Ising model. It is trivial to show usingEq. (6) that all of these long-range Ising models have or-dered hyperuniform ground state at Γ = 0, owing to theground states being direct products of eigenvectors of ˆ σ z . First, we consider the family of J r = r − a for a ∈{ , , } and J r = ( − r r − a up to a cutoff radius, R = 14with N = 30. As a point of comparison, we also includethe simulation result for the standard ferromagnetic andantiferromagnetic Ising model. These sets of parameterslead to a ferromagnetic and antiferromagnetic groundstates respectively for low values of Γ. In Fig. 1, weshow the effect of finite-size on some observables. Theaverage magnetization along the z-axis, m z happens tobe a very convenient observable to study the ferromag-netic systems. However, m z suffers from finite-size effect.We note that h x can also be used as a order parameter.This is a measure that is unity for all cases where theground state is a direct product of eigenstates of the ˆ σ z operator, that is, when in the ground state, all the spinspoint either “up” or “down.” These “direct-product”states can either be “ordered,” that is either ferromag-netic or antiferromagnetic, or “disordered” as we shall seein Sec. III B. However as we increase Γ → ∞ , h x decaysto zero. It is seen that the second cumulant of the trans-verse magnetization, h x , does not suffer as badly fromthe finite-size effect. The most notable change in h x asa function of the system size is that the maximum nearthe “critical point” for small systems seems to changeinto a point of non-differentiability as the system sizegets larger, resulting in a slight movement of the criticalpoint towards the infinite-size limit. The critical pointsas demonstrated by these different observables convergeto the correct thermodynamic value of the critical pointsin the limit of N → ∞ .In Fig. 2, we report our simulation results that de-scribe the phase transition in the various ferromagneticand antiferromagnetic generalized Ising models. The or-dered character of the ground state at Γ = 0 changesat the critical value of the transverse field. The criticalpoint for both the ferromagnetic and antiferromagneticmodels happen at exactly the same point. The τ or-der parameter decreases monotonically to zero and doesnot show a discontinuous change at the phase transition.The antiferromagnetic ground state is, of course, less or-dered than the ferromagnetic ground state, and this isreflected in the τ order metric. The degree of hyper-uniformity S is an order metric for the ferromagneticchains but not for the phase transition in the antiferro-magnetic chains. This is reflected in the sudden, sharprise in S for the ferromagnetic chains shown in Fig. 2. Inall of the inverse power-law interaction Hamiltonians, wesee a transition from an ordered hyperuniform state to adisordered non-hyperuniform state. The degree of hype-runiformity, S , like m z , shows a large finite-size effect.Therefore, the critical point should be obtained from the h x curve. Moreover, note that the behavior of all theferromagnetic and the corresponding antiferromagneticmodels in terms of h x is identical, showing that it is avalid order parameter in both cases. Γ m z N = 30N = 50N = 80 (a) m z Γ τ N = 30N = 50N = 80 (b) τ order metric Γ h x N = 30N = 50N = 80 (c) h x Γ S N = 30N = 50N = 80 (d) S FIG. 1. Size dependence of the critical point using various observables as the order parameters for the ferromagnetic Isingmodel: (a) Average magnetization along z -axis, Eq. (3); (b) τ order metric, Eq. (7); (c) Second cumulant of the transversemagnetization, h x , Eq. (2); (d) Measure of hyperuniformity, S , Eq. (6). The variation of the critical value of Γ with N is theleast for h x . The value of h x undergoes a maximum for small N , which changes to a point of non-differentiability as N becomeslarger. B. Interactions Optimized for Hyperuniformity
The interactions that are optimized for hyperuni-formity have disordered stealthy hyperuniform groundstates at Γ = 0. These ground states, unlike the ones forthe typical ferromagnetic long-range interactions that de-cay with distance, have an average magnetization m z = 0throughout the range of Γ and unlike the typical anti-ferromagnetic long-range interactions, are not ordered.They undergo phase transitions between various “disor-dered” phases with m z = 0. The stealthy hyperuniformground state at Γ = 0, which is a disordered direct-product state, is to be contrasted with the disorderedstate that is the ground state as Γ → ∞ . Now, theground state is a direct product of eigenstates of ˆ σ x op-erators. This state, also, has zero total magnetization inthe z-direction, but the individual spins are not point-ing along the z-direction. We call this the “disorderedquantum” state.The spin systems, that we simulated, can be groupedinto two broad classes: there are systems with one, ortwo first-order quantum phase transitions. Of course wewould need to increase the system size to truly charac-terize the phase transitions. However, the sharpness of the discontinuities observed even in the finite sized sys-tems seem to strongly suggest the existence of first-orderphase transitions. In the following subsections, we giverepresentative examples of each class. The behavior ofstealthiness as measured by S is universal across thethree classes. Stealthiness is a more sensitive propertythan hyperuniformity. We will show, in the followingsections, that S increases faster than S but the basicfeatures of S are identical to those of S in all the classes.
1. Parameters with one weak First-Order QPT: No Orderfrom Disorder
As a first example of the parameters that were opti-mized for hyperuniformity, consider a system with a weakfirst-order quantum phase transition (see J ( r ) in SM),where the qualitative features of the τ order metric aresimilar to that in the standard long-range Ising modeldiscussed in Sec. III A. Figure 3 shows the variation ofvarious basic observables as a function of the transversefield. There is a transition at Γ ≈ . τ order metric in Fig. 3(b) is smooth Γ -4-3.5-3-2.5-2-1.5-1 < H > / N (a) Energy per site Γ τ (b) τ order metric Γ h x (c) h x Γ S (d) S FIG. 2. Basic observables as functions of Γ: (a) Energy per site; (b) τ order metric; (c) Second cumulant of the transversemagnetization, h x ; (d) Degree of hyperuniformity, S . Black line: J r = r − . Red line: J r = r − . Green line: J r = r − . Blueline: ferromagnetic Ising model. Black markers: J r = ( − r r − . Red markers: J r = ( − r r − . Green markers: J r = ( − r r − .Blue markers: antiferromagnetic Ising model. The degree of hyperuniformity, S undergoes a sudden jump for the ferromagneticcases at the critical point. This jump in S is absent in the antiferromagnetic cases. and monotonically decreasing with the transverse fieldΓ, similar to the behavior depicted in Fig. 2(b). Thisis surprising because the interactions in case of Fig. 2decay rapidly with distance, whereas the ones in Fig. 3have been optimized to produce stealthy hyperuniformground states by long-range interactions. Of course, inthe limit of Γ → ∞ , all structure is lost and the groundstate of the Hamiltonian is a direct product of the groundstate of the local ˆ σ x operator. This indicates that thesystem starts at a disordered hyperuniform ground stateat Γ = 0 and continues to lose that order as well. Wesee that S shows a monotonic increase with Γ, implyingthat hyperuniformity is degraded. There is a very small“jump” in the measures of hyperuniformity, S and S ,around Γ ≈ .
5, which seems to suggest the presenceof a weak first-order QPT. S ( k ) as a function of Γ isshown in Fig. 4, which, however, seems to show a smoothtransition from the disordered classical ground state atlow values of Γ to the disordered quantum ground stateat high values. The smooth decay of the τ order metricand the small discontinuities in S and S as functions ofΓ at the critical point might be a result of the weaknessof the first-order QPT in this case.
2. Parameters with One First-Order QPT
Next, we consider the J r ’s which lead to systems witha single first-order phase transition (see J ( r ) in SM).The observables corresponding to one such parameter isshown in Fig. 5. There is a phase transition at Γ = 0 . S ( k, Γ) to demonstrate the changes in the structurefactor as a function of Γ.
3. Parameters with Two First-Order QPTs
Finally, there are cases with two first-order phase tran-sitions as demonstrated by the observables in Fig. 7 (see J ( r ) in SM). From the S plot, it is clear that there Γ -4-3.5-3-2.5-2-1.5-1 < H > / N (a) Energy per site Γ τ (b) τ order metric Γ h x (c) h x Γ S / S S S (d) S FIG. 3. Basic observables as functions of Γ: (a) Energy per site; (b) τ order metric; (c) Second cumulant of the transversemagnetization, h x ; (d) Degree of hyperuniformity, S and S , Eq. (9). Parameter demonstrates possibility of a weak first-orderQPT.FIG. 4. Ground state structure factor as a function of thewave-vector, k and the strength of the transverse field Γ. are two phase transitions: one between Γ = 0 . .
07, and another between Γ = 0 .
125 andΓ = 0 . | (cid:105) before the first phase transition (Γ < . | (cid:105) between the two phase transitions(0 . < Γ < . ≈ .
2, the structure becomes significantly more similarto | (cid:105) .These last two cases are extremely interesting from thefundamental perspective of understanding the impact ofa transverse field to these complicated long range inter-acting spin chains. It is curious that depending on theparameters of the system, phase transitions caused bytransverse magnetic fields can serve to increase the orderof the ground state as measured by the τ order parame-ter. Additionally, the order does not have to necessarilyincrease. Parameters can be defined where the phasetransitions involved have predefined effect on the ordermetric. As in this case, the system starts off from a dis-ordered hyperuniform state similar to the previous twocases. At the first QPT, there is a sudden increase in or-der, which decays smoothly till we encounter the secondphase transition. Then there is a sudden decrease in or-der from where the τ order parameter relaxes smoothlyto the disordered state as Γ → ∞ .We have numerically demonstrated that there are no Γ -1.2-1-0.8 < H > / N (a) Energy per site Γ τ (b) τ order metric Γ h x (c) h x Γ S / S S S (d) S FIG. 5. Basic observables as functions of Γ: (a) Energy per site; (b) τ order metric; (c) Second cumulant of the transversemagnetization, h x ; (d) Degree of hyperuniformity, S and S . Parameter demonstrates possibility of a first-order QPT indicatedby sharp discontinuities in (b), (c), and (d) around Γ ≈ . k and the strength of the transverse field, Γ. We have markedout the critical value of Γ where the phase transition occurs.Between Γ = 0 .
095 and Γ = 0 .
10, there is a phase change. cases with more than two QPTs. To provide some analyt-ical motivation for this conjecture, in the next section weanalyze the problem using the Jordan-Wigner transformand prove the conjecture for a very simple approximationof the model and the mean-field Hamiltonian.
C. Analysis using the Jordan-Wigner Mapping
Our goal, here, is to analytically diagonalize the Hamil-tonian and study the nature of the ground state undera couple of limiting cases. Using a unitary rotation, theHamiltonian defined in Eq. (1), can be written as: H = − (cid:88) i (cid:88) ≤ r ≤ R J r ˆ σ ( i ) x ˆ σ ( i + r ) x + (cid:88) i − Γˆ σ ( i ) z (10)The Pauli matrices on the same site anticommute and theones on different site commute. This makes it difficultto apply analytic tools to treat this system. A commonmethod of overcoming this difficulty is to apply a Jordan-Wigner string mapping[44] to convert the spin operatorsto fermionic creating and annihilation operators. Themapping can be summarized as:ˆ σ ( j ) z = 1 − c † j c j (11)ˆ σ ( j )+ = exp iπ (cid:88) l
065 and Γ ≈ . k S ( k ) |0>, Γ <0.060.07< Γ <0.120.13< Γ (a) k S ( k ) |1>|2>|3> (b) FIG. 8. Structure factors. Left: Ground states for select intervals of Γ. Ground state before the first critical point (Γ < . | (cid:105) . Right: Other low-lying states at Γ = 0, labeled | (cid:105) , | (cid:105) , and | (cid:105) respectively. Under this mapping the Hamiltonian, Eq. (10) would gettransformed as follows: H = − (cid:88) j (cid:88) ≤ r ≤ R J r (cid:16) ˆ σ ( j )+ + ˆ σ ( j ) − (cid:17) (cid:16) ˆ σ ( j + r )+ + ˆ σ ( j + r ) − (cid:17) − Γ (cid:88) j (cid:16) − c † j c j (cid:17) = − (cid:88) j (cid:88) ≤ r ≤ R J r l 07 and Γ ≈ . 1. Pairwise Interacting Approximation We begin the analysis by making the crudest simpli-fying assumption: we consider only the pairwise interac- tion terms in the Hamiltonian, and make an approximatelong-range fermionic model that can be directly solvedbecause it is quadratic: H = − (cid:88) j (cid:88) ≤ r ≤ R J r (cid:16) c † j c † j + r + c † j c j + r + c † j + r c j + c j + r c j (cid:17) − Γ (cid:88) j (cid:16) − c † j c j (cid:17) . (15)Now, transforming the creation and annihilation oper-ators into Fourier space using c j = √ N (cid:80) k s k e ikja and c † j = √ N (cid:80) k s † k e − ikja , where a is the lattice constant, weget H = (cid:88) 0) and antiferromagnetic( R = 1 , J r = J < 0) Ising models with the latticeconstant a = 1 as trivial examples of the mapping andcases where the model is exact. For 0 ≤ Γ < | J | , incase of the ferromagnetic Ising model, the k = π modeis populated. On the other hand, for the antiferromag-netic Ising model, the k = 0 mode is populated for0 ≤ Γ ≤ | J | . When Γ > | J | , in case of the ferromagneticIsing model, the k = 0 mode gets populated in additionto the k = π mode, and in case of the antiferromagneticcase, the k = π mode gets populated. This represents aphase transition, with the critical value of Γ = | J | , thathappens by different mechanisms in the two cases: theoccupation of the k = 0 mode in the ferromagnetic case, and the k = π mode in the antiferromagnetic case.Now, we come to the question of the number ofsuch phase transitions. If any one of (cid:80) ≤ r ≤ R J r and (cid:80) ≤ r ≤ R J r cos( πar ) is greater than zero, then therewould be one phase transition, and if both of them aregreater than zero, then there would be two phase transi-tions with the critical points being at these values of Γ.There is no other variable phase transition that is possi-ble because the k (cid:54) = 0 and k (cid:54) = π modes are always in thevacuum state. Thus, consistent with the numerical re-sults that we have encountered in the previous sections,this approximate model also allows a maximum of twophase transitions. We would like to emphasize that thiscondition on J r for the number of phase transitions isonly valid for the current approximate model. It is notvalid in general for the long-range spin system, for whichclosed-form analytic solutions do not exist.Observables can be calculated by mapping them ontothe spinless fermions by using the Jordan-Wigner map-ping. As an example, we choose N (cid:80) ≤ j ≤ N (cid:68) σ ( j ) z (cid:69) . Thiswould be useful in doing the mean-field analysis in thenext section. In the following γ = s and γ π = s π .1 N (cid:88) ≤ j ≤ N (cid:68) σ ( j ) z (cid:69) = 1 N (cid:88) k (cid:16) − (cid:68) s † k s k (cid:69)(cid:17) = 1 − N (cid:88) k (cid:68)(cid:16) u k γ † k − iv k γ − k (cid:17) (cid:16) u k γ k + iv k γ †− k (cid:17)(cid:69) = 1 − N (cid:88) k (cid:54) =0 ,π (cid:16) u k (cid:68) γ † k γ k (cid:69) − v k (cid:68) γ †− k γ − k (cid:69)(cid:17)(cid:124) (cid:123)(cid:122) (cid:125) term A − N (cid:88) k (cid:54) =0 ,π v k − N (cid:16)(cid:68) γ † γ (cid:69) + (cid:10) γ † π γ π (cid:11)(cid:17) (19)= 1 − N (cid:88) k (cid:54) =0 ,π v k − N (cid:16)(cid:68) γ † γ (cid:69) + (cid:10) γ † π γ π (cid:11)(cid:17) (20)= 1 − N (cid:88) k (cid:54) =0 ,π (cid:32) − α k (cid:112) α k + β k (cid:33) − N (cid:16)(cid:68) γ † γ (cid:69) + (cid:10) γ † π γ π (cid:11)(cid:17) (21)= 2 N + 1 N (cid:88) k (cid:54) =0 ,π α k (cid:112) α k + β k − N (cid:16)(cid:68) γ † γ (cid:69) + (cid:10) γ † π γ π (cid:11)(cid:17) (22)Since no k -modes apart from k = 0 and k = π arepopulated with Bogoliubov fermions, the term marked“A” is zero in Eq. (19). The equality between Eq. (20)and Eq. (21) is obtained by noting that tan( θ k ) = β k α k and v k = sin (cid:0) θ k (cid:1) , while the third equality is a result ofhaving N − k (cid:54) = 0 , π . Similar expressionscan be derived for other observables. 2. Mean-field Analysis For the mean-field analysis of Eq. (14), let usassume that at the i th iteration, the value of N (cid:80) ≤ l ≤ N (cid:68) − c † l c l (cid:69) = g i . We will use this assump-tion to solve the Hamiltonian as a function of g : H ( g ) = − (cid:88) j (cid:88) ≤ r ≤ R J r g r − (cid:16) c † j c † j + r + c † j c j + r + c † j + r c j + c j + r c j (cid:17) − Γ (cid:88) j (cid:16) − c † j c j (cid:17) (23)Of course, the analysis of Eq. (23) is simplified throughthe observation that it is isomorphic with Eq. (15) andconsequently, Eq. (16) under the transformation J r → ˜ J r = J r g r − . To start the process, we use the g ob-tained under the pairwise interaction approximation, us-ing Eq. (21). If we are only interested in the value of2 N (cid:80) ≤ l ≤ N (cid:68) − c † l c l (cid:69) , it might be possible to directlysearch for the root of the following equation: g = 2 N + 1 N (cid:88) k (cid:54) =0 ,π ˜ α (cid:113) ˜ α + ˜ β − N (cid:16)(cid:68) γ † γ (cid:69) + (cid:10) γ † π γ π (cid:11)(cid:17) (24)where ˜ α = Γ − (cid:88) ≤ r ≤ R J r g r − cos( kar ) (25)˜ β = (cid:88) ≤ r ≤ R J r g r − sin( kar ) . (26)However, here we are interested in characterizing thenumber of phase transitions in the ground state as thetransverse field is increased. This can be achieved muchmore simply by observing that because the Hamiltonianat every step is isomorphic to Eq. (15) at every step of theself-consistent field procedure, the constraint on the max-imum number of phase transitions in the model wouldhold. Thus, even after solving the long range problem,in a self-consistent manner, we expect that there cannotbe more than two phase transitions, which is consistentwith our numerical exploration. IV. CONCLUSIONS Long-range Ising models can exhibit very rich physics,especially when the long-range couplings do not decaywith distance. Such atypical long-range couplings are es-sential to ensuring that spin systems have stealthy hype-runiform ground states in the absence of any transversefield. Spin systems with hyperuniform ground states,thus, show very interesting physics that is qualitativelydifferent from standard long-range Ising models with in-teractions that decay with distance. We have presentedhere one of the first analyses of the physics of stealthy, hy-peruniform spin systems with nontrivial, non-monotonic,long-range interactions under a transverse magnetic field.We demonstrate numerically that for these systems, un-like standard long-range Ising models, the number ofphase transitions is not fixed. Their loss of hyperuni-formity, in presence of transverse fields, is not alwaysaccompanied by a loss of order, as measured by the τ order metric. This feature is very unusual and leads tothe possibility of designing hyperuniform materials whoseground-state in the presence of external transverse mag-netic fields can be more ordered than that in absence ofexternal the field. We also showed that the rate of lossof the property of stealthiness is much faster than thatof the loss of hyperuniformity, proving that stealthinessis a much more delicate property.To better theoretically understand the phase transi-tions we identified numerically, we have analyzed thelong-range Ising spin models using the Jordan-Wignermapping Hamiltonian. Under this mapping, the long-range spin-spin interactions manifest themselves as non- pairwise interaction terms. We showed that for a genericlong-range Ising model, under the pairwise interactionand mean-field “approximations,” there can be a maxi-mum of two phase transitions, which is consistent withour numerical results. Therefore, we conjecture that along-range pairwise interacting 1D Ising spin chain witharbitrary couplings for which a discrete cosine transformis convergent, can show at most two quantum phase tran-sitions at zero temperature.Future work on understanding the critical scaling ofthese phase transitions using MERA or QMC should leadto further insights into the nature of these long-rangehyperuniform spin chains. It would also be interesting tostudy these systems under a combination of longitudinaland transverse fields to map out the full phase diagram.Such studies would prove useful in the design of novelmaterials with simple models. ACKNOWLEDGMENTS We thank Roberto Car for many insightful discussions.A. B. acknowledges support of the Computational Chem-ical Center: Chemistry in Solution and at Interfacesfunded by the U. S. Department of Energy under AwardNo. DE-SC0019394. S. T. acknowledges the support ofthe National Science Foundation under Grant No. DMR-1714722. Appendix A: Explorations using ConfigurationInteraction (CI) expansions The Hamiltonian can be decomposed in the followingmanner: H = − (cid:88) i (cid:88) ≤ r ≤ R J r ˆ σ ( i ) z ˆ σ ( i + r ) z (cid:124) (cid:123)(cid:122) (cid:125) H ref + (cid:88) i − Γˆ σ ( i ) x (cid:124) (cid:123)(cid:122) (cid:125) V (A1) H ref | φ (cid:105) = E | φ (cid:105) (A2)Like in the DMRG simulations, we evaluate the Hamil-tonian in a PBC. We have already solved for the “refer-ence” | φ (cid:105) , which is the classical long-range Ising problemwith Γ = 0. Now, we expand the true ground state wavefunction | ψ (cid:105) in a configuration integral expansion: | ψ (cid:105) = c | φ (cid:105) + (cid:88) α c α | φ α (cid:105) + (cid:88) α,β c α,β | φ α,β (cid:105) + . . . (A3)where | φ α (cid:105) is obtained by flipping the α th spin with re-spect to | φ (cid:105) , | φ α,β (cid:105) is obtained by flipping the α th and β th spins, so on.We solve the eigenvalue equation representing theHamiltonian in the orthonormal basis described above.First, consider the matrix elements of H ref . Since H ref isa function of ˆ σ z and the basis vectors are eigenvectors ofˆ σ z , H ref is diagonal in the basis defined.3 Γ -1.125-1.1-1.075-1.05 < H > / N DMRG|0>|1>|2>|3> (a) CIS Energy Γ -1.125-1.1-1.075-1.05 < H > / N DMRG|0>|1>|2>|3> (b) CISD Energy Γ -1.125-1.1-1.075-1.05 < H > / N DMRG|0>|1>|2>|3> (c) CISDT Energy Γ -1.125-1.1-1.075-1.05 < H > / N DMRG|0>|1>|2>|3> (d) CISDTQ Energy FIG. 10. Energy per site with respect to different references. Though the basis consisting of all possible excitationsis complete, the smaller basis set obtained using a trun-cated number of excitations is not. As a method of ex-ploring the nature of the ground state qualitatively, weuse not just | (cid:105) as reference but also | (cid:105) , | (cid:105) , and | (cid:105) .Of course, in the full basis, the results should be inde-pendent of the reference used. To gain a better under-standing, we truncate the expansion such that, the basesdefined on all of the references are non-intersecting. So,the Hamiltonian matrix defined in terms of all the ref-erences would have a block diagonal structure, implyingthat we can solve the problem for each of the referencesindependently. Additionally this allows us to probe intothe nature of the ground state. We can now make quali-tative statements about which reference the ground statelooks like. Note in Fig. 10, for | (cid:105) , the inclusion of quarticexcitations includes vectors that are similar to | (cid:105) . Thatis why, though up to CISDT, | (cid:105) is higher in energy than | (cid:105) at Γ = 0, once the quartic excitations are included,using | (cid:105) as reference, we can get the correct behavior in the low Γ region as well.Obviously at Γ close to 0 . | (cid:105) over the rangeof Γ considered. However as soon as we introduce moreexcitations, phase transitions start appearing. We noticethat for the CISDT calculations, there is a cross-overfrom a | (cid:105) -like ground state to a | (cid:105) -like ground stateat around 0 . | (cid:105) -likeground state to a | (cid:105) like ground-state at 0 . 22. The | (cid:105) -like ground-state to | (cid:105) -like ground-state transition lieswithin the range of Γ where the calculations were betterconverged. Therefore, we get an estimate of the criticalvalue of Γ that is in good agreement with the estimatefrom DMRG calculations (0.0675 – 0.07). 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