Quantum Phase Transitions in Long-Range Interacting Hyperuniform Spin Chains in a Transverse Field
Amartya Bose Department of Chemistry, Princeton University, Princeton, New Jersey 08544
Salvatore Torquato Department of Chemistry, Princeton University, Princeton, New Jersey 08544 Department of Physics, Princeton University, Princeton, New Jersey 08544 Princeton Institute for the Science and Technology of Materials, Princeton University, Princeton, New Jersey 08544 and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544
Hyperuniform states of matter are characterized by anomalous suppression of long-wavelength density fluctuations. While most of the interesting cases of disordered hyperuniformity are provided by 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 systems involving 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, classical hyperuniform spin chains are expected to lose their hyperuniformity. High-precision simulations of these cases are complicated because of the presence of highly nontrivial long-range interactions. We perform extensive analysis of these systems using density matrix renormalization group (DMRG) to study 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 order metric” of the ground state, the measure of hyperuniformity and the second cumulant of the total magnetization along the x-direction, all suggestive of first-order QPTs. An interesting feature of the phase transitions in these disordered hyperuniform spin chains is that, depending on the parameter values, the presence of transverse magnetic field may remarkably lead to increase in the order of the ground state as measured by the “τ order metric,” even if hyperuniformity is lost. Therefore, it would be possible to design materials to target specific novel quantum behaviors in the presence of a transverse magnetic field. Our numerical investigations suggest that these spin chains can show no 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 interacting approximation 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 two quantum phase transitions at zero temperature.
I. INTRODUCTION quantum mechanical hyperuniform systems. Some ex- actly solvable quantum systems, such as free fermion sys- The notion of hyperuniformity provides a unifying tems [10], superfluid Helium [11], the ground state of the framework to characterize the large scale structure of sys- fractional quantum Hall effect [12], and Weyl-Heisenberg tems as disparate as crystals, liquids, and exotic disor- ensembles [13] are hyperuniform [14]. Recently, quantum dered states of matter. A hyperuniform state of matter phase transitions (QPTs) have been studied in nearest- is one where the density fluctuations at very large length neighbor Ising-chains with hyperuniform couplings [15]. arXiv:2012.06545v2 [quant-ph] 14 Dec 2020 scales are suppressed, and consequently, the structure The exploration of the interplay between hyperunifor- mity, order and quantum fluctuations is a fertile area for factor, lim|k|→0 S(k) = 0. All perfect crystals and perfect quasicrystals are hyperuniform and can be rank ordered research. in terms of their capacity to suppress large-scale density Stealthy hyperuniform systems are ones in which the fluctuations [1,2]. Disordered hyperuniform materials structure factor is zero for wavenumbers in the vicinity of are exotic amorphous states of matter that are like crys- the origin, i.e., there exists a critical radius K, such that, tals in the manner in which their large-scale density fluc- S(|k|) = 0 for all 0 < |k| < K [16–19]. For sufficiently tuations are anomalously suppressed and yet behave like small K, stealthy hyperuniform many-particle systems liquids or glasses in that they are statistically isotropic are the highly degenerate disordered ground states of cer- without any Bragg peaks [1,2]. tain long-ranged interactions [18]. Disordered materials Classical disordered hyperuniform systems are attract- based on such stealthy hyperuniform distribution of par- ing great attention because they are endowed with novel ticles often possess desirable physical properties, such as physical properties [3–9]. There are far fewer studies of photonic band gaps that are comparable in size to pho- 2 tonic crystals [8, 20], transparent dense materials [21], boundary conditions, which make the simulations very and optimal transport characteristics [4, 22]. The collec- challenging. All DMRG calculations in this paper were tive coordinate optimization technique has been used to performed using the ITensor library [37]. generate disordered stealthy hyperuniform many-particle In Sec.II, we describe the system under study, the systems [16, 17, 19]. Such inverse statistical-mechanical methods employed and the observables calculated. In algorithms have also been extended to deal with dis- Sec.III, we illustrate the most important classes of re- crete spin systems [23], and to determine long-range sults obtained through the DMRG simulations. Our re- Ising interactions that have stealthy hyperuniform classi- sults demonstrate that it is possible for the τ order met- cal ground states [24]. More recently, inverse statistical- ric [18, 24, 38] of the ground state to increase with a mechanical techniques have been applied to quantum me- transverse magnetic field. Our numerical simulations also chanical problems to construct the Hamiltonian from the suggest that these long-range spin systems can sustain no eigenstate [25]. more than two QPTs. Of these, we choose the case with Historically spin chains have proven to be fertile two discontinuous QPTs and illustrate a basis set cal- grounds for exploration of effects of quantum mechanics. culation in appendixA. We carry out further analytical In this paper, we characterize the effects of a transverse explorations by mapping the system on to a system of field on spin systems whose ground states, in absence of spinless fermions via the Jordan-Wigner transformation, such fields, are stealthily hyperuniform. Quantum fluc- which results in a fermionic Hamiltonian with more than tuations are typically associated with a loss of order in pairwise interactions that make the direct solution non- general. This is true for the usual interactions that de- trivial. We analyze the resultant Hamiltonian using a cay monotonically with distance. Examples of this in- simple approximate model with only the pairwise interac- clude the loss of the ferromagnetic order in Ising model tion terms as well as under a mean-field treatment of the in presence of transverse field and the broadening of the terms involving more than pairwise interactions. Con- peaks of the pair correlation function g(r) of quantum liq- sistent with our numerical results, we show in Sec.IIIC uids as compared to their classical counterparts at simi- that for both the approximate model and the mean-field lar thermodynamic conditions. We demonstrate the very Hamiltonian here can be a maximum of two phase tran- interesting possibility of optimizing magnetic materials sitions. This leads us to a conjecture that for spin chains such that the quantum effects of an external transverse with long-range pair interactions with convergent cosine magnetic field can be used to increase the order of the transforms, there can be no more than two zero tempera- ground state of the spin system, while still remaining ture quantum phase transitions. We end this paper with more disordered than the standard antiferromagnetic or a concluding remarks and outlook for further interesting ferromagnetic materials. explorations in Sec.IV. In presence of a transverse field, classical mechanics alone cannot account for the physics of a spin chain. II. DESCRIPTION OF SYSTEM AND Since hyperuniformity is extremely sensitive to the exact METHODS EMPLOYED nature of the ground state and the interactions involved, it is natural to expect that the transverse field would lead to a loss of hyperuniformity. It is, therefore, interesting to We study one-dimensional (1D) spin systems with study the quantum mechanism by which hyperuniformity long-range interactions. The basic Hamiltonian is evalu- is lost, and the phase transitions involved. In this study, ated with periodic boundary conditions (PBC) and has we consider spin chains with long-range interactions that the following form on the integer lattice Z: have been optimized to have disordered stealthy hy- X X (i) (i+r) X (i) peruniform ground states in the absence of magnetic H = − Jrσˆz σˆz + −Γˆσx . (1) field [24]. We explore these systems using Density Ma- i 1≤r≤R i trix Renormalization Group (DMRG) [26–35] with a par- (i) (i) ticular emphasis on the difference in the physics due to whereσ ˆz andσ ˆx are the Pauli spin matrices along the th the disordered nature of the classical ground state in ab- z and x directions respectively on the i site, Jr is the sence of the transverse magnetic field and the long-range coupling between two spins separated by r lattice points, nature of the interactions. DMRG is one of the best al- and Γ is the strength of the transverse field. gorithms for dealing with one-dimensional lattice-based We simulate the system for various sets of Jr in order problems that are not at the critical points. However, it to understand how the systems with stealthy hyperuni- is typically applied to systems with short-range interac- form ground states in the absence of a transverse field tions with open boundary conditions. Recently, DMRG behave with increasing Γ. These hyperuniform parame- has been applied to study the physics of long-range sys- ters were obtained by Chertkov et al. [24] (see Supple- tems with monotonically decaying interactions with open mentary Material (SM) for the parameters). They are boundaries [36]. Stealthy hyperuniform systems, how- often atypical in the sense that the interaction strength ever, can only be generated in the presence of periodic does not necessarily decrease with distance. Therefore, boundary conditions. Hence, this study involves com- we also show results for simulations where the couplings plex non-monotonic, long-range interactions on periodic decay according to inverse power law with the distance. 3
Due to the increased computational complexity of the Typically one would use either the average magnetization DMRG algorithm for periodic systems, all the parame- along the z-direction, mz for the “standard” long-range ters considered have N = 30 spins. It is important to Ising model, or the average magnetization along the x- note here that in case of simulating long-range interac- direction mx defined respectively by tions in these systems with periodic boundary conditions, * + 1 X one encounters strong finite-size effects. A system with m = σˆ(j) (3) z N z N = 30 spins might not enough to get rid of these ef- j fects. Other investigations have used Ewald summation * + 1 X techniques to take care of this finite-size effect [39]. How- m = σˆ(i) (4) ever, here we do not attempt to alleviate this problem in x N x j order to maintain consistency with the work on hyper- uniform spin chains [24]. However, as we will illustrate, we have found that hx suf- Because of the long-range interactions, that do not de- fers significantly less from finite-size effects than mz for cay with distance and the presence of periodic bound- the long-range Ising models, and performs just as well ary conditions, the entanglement entropy grows faster as mx for the hyperuniform cases in identifying the crit- than for the regular Ising model in a transverse field. ical points. We also report the structure factor at the This makes DMRG calculations significantly more diffi- origin, the deviation from zero of which is a measure of cult to converge and likely to get stuck in other low-lying hyperuniformity, local minima. Therefore, we perform ten independent S0 ≡ lim S(k) (5) simulations of the system. In each simulation, we run k→0+ twenty DMRG calculations with random initial start- 1 = M 2 − hM i2 (6) ing points and taking the state with the minimum en- N z z ergy. We compare results across the various runs and where M is the total magnetization along the z- take the lowest energy state as the ground state. Since z direction, as a function of Γ. The structure factor at DMRG is variational in nature, none of the higher en- the origin, S = 0 for hyperuniform systems. The devi- ergy states can be the true ground state. Performing 0 ation of S from 0 measures how far the system is from multiple DMRG calculations often allows us to access 0 being hyperuniform. As Γ → ∞, we should recover a other low-lying states. Having an idea of other low-lying disordered state, irrespective of the exact nature of J states expedites the analysis of the problem using basis r and so, S would asymptotically tend to 1. The degree set expansions. Because of the multiple DMRG calcu- 0 of order of the ground state is measured using the τ order lations we run, we can converge the wavefunction de- metric [18, 24, 38], defined as follows: spite the long-range interactions. However, the conver- 2 gence gets increasingly difficult in the immediate vicinity X (S(k) − Sref(k)) τ = , (7) of the critical points. This is a well understood limi- N 2 tation of DMRG. At the critical points, the correlation k lengths become very large, reducing the effectiveness of where Sref(k) is a reference structure factor. In this pa- the DMRG algorithm. Therefore, these phase transitions per, we use the structure factor for a Poisson point pat- and the critical exponents involved cannot be character- tern, Sref(k) = 1 as the reference. The normalization ized by DMRG. Other methods, such as the Multi-scale factor of N 2 is chosen to make most of the results for τ Entanglement Renormalization Ansatz (MERA) [40, 41] to be of order unity. With this normalization factor, the and the family of Quantum Monte Carlo (QMC) meth- antiferromagnetic spin configuration has τ ≈ 1, and the ods, especially the so-called projective QMC methods like ferromagnetic spin configuration has τ ≈ 2. The same diffusion Monte Carlo (DMC), prove to be useful in such metric has been used without this normalization factor studies. Because we are limiting ourselves to chains of by Chertkov et al. [24] The structure factor of a given length, N = 30, the locations of the critical points are ground state is defined as: unlikely to be correct in the thermodynamic limit. How- N N ever, as our results show, the transitions show a remark- 1 X X S(k) = σˆ(l)σˆ(k) exp (ik(l − j)) . (8) able sharpness, which suggests that these transitions are N z z not artifacts of the finite-size of our systems. Hence our l=1 j=1 results regarding the possibility of increase in the τ or- In addition to the hyperuniformity of the ground state, der metric of the ground state and the variable number we also want to study the loss of the stealthiness of the of phase transitions would continue to hold qualitatively, hyperuniformity of the ground state. Since, for the ex- even for larger systems. amples demonstrated here, the stealthiness extends only For each parameter, we first investigate the variation of to the first non-zero wave-vector, we use the structure basic observables like the average energy, and the second factor at the first non-zero wave-vector as a measure of cumulant of the transverse magnetization, stealthiness in the ground state: 2π 1 2 h = M 2 − hM i . (2) S1 = S(∆k) = S . (9) x N x x N 4
−a Finally, for the spin systems with hyperuniform ground First, we consider the family of Jr = r for a ∈ r −a states, we also report the plots of S(k) as a function of {2, 3, 4} and Jr = (−1) r up to a cutoff radius, R = 14 Γ. This allows for a greater clarity in the changes that with N = 30. As a point of comparison, we also include happen to the ground state before and after the phase the simulation result for the standard ferromagnetic and transitions. antiferromagnetic Ising model. These sets of parameters lead to a ferromagnetic and antiferromagnetic ground states respectively for low values of Γ. In Fig.1, we III. RESULTS show the effect of finite-size on some observables. The average magnetization along the z-axis, mz happens to This section is organized in the following manner. be a very convenient observable to study the ferromag- First, we report results for “standard” long-range Ising netic systems. However, mz suffers from finite-size effect. models with interactions that decay with the distance We note that hx can also be used as a order parameter. between the spins. These results allow us to set a point This is a measure that is unity for all cases where the of comparison for the hyperuniform spin chains. We il- ground state is a direct product of eigenstates of theσ ˆz lustrate the finite-size effect on the various observables, operator, that is, when in the ground state, all the spins and demonstrate how the quantum effects of a trans- point either “up” or “down.” These “direct-product” verse field reduces the order of the ground state of the states can either be “ordered,” that is either ferromag- system. Thereafter, we examine the systems with inter- netic or antiferromagnetic, or “disordered” as we shall see actions optimized to give stealthy hyperuniform ground in Sec.IIIB. However as we increase Γ → ∞, hx decays states. We demonstrate that for systems with these non- to zero. It is seen that the second cumulant of the trans- trivial long-range interactions, it is possible to generate verse magnetization, hx, does not suffer as badly from order from disorder using the quantum effects of a trans- the finite-size effect. The most notable change in hx as verse magnetic field. Numerically, we observe no more a function of the system size is that the maximum near than two QPTs. To further theoretically explore of the the “critical point” for small systems seems to change nature of these phase-transitions, we map the spin sys- into a point of non-differentiability as the system size tems onto chains of spinless fermions. The long-range gets larger, resulting in a slight movement of the critical spin-spin interaction manifests not only in long-distance point towards the infinite-size limit. The critical points pairwise interaction terms in the Hamiltonian, but re- as demonstrated by these different observables converge sults also in terms involving higher-order non-pairwise to the correct thermodynamic value of the critical points interaction terms (interactions involving triplets of spins, in the limit of N → ∞. 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- In Fig.2, we report our simulation results that de- ing that in both cases, a maximum of two phase transi- scribe the phase transition in the various ferromagnetic tions is possible. This is consistent with our numerical and antiferromagnetic generalized Ising models. The or- results, and leads us to conjecture that for Ising mod- dered character of the ground state at Γ = 0 changes els with long-range interactions that have convergent co- at the critical value of the transverse field. The critical sine transforms, there can be a maximum of two zero- point for both the ferromagnetic and antiferromagnetic temperature quantum phase transitions. models happen at exactly the same point. The τ or- der parameter decreases monotonically to zero and does not show a discontinuous change at the phase transition. A. Interactions that Decay with Distance The antiferromagnetic ground state is, of course, less or- dered than the ferromagnetic ground state, and this is For the purposes of comparison to parameters that reflected in the τ order metric. The degree of hyper- were specifically optimized to have stealthy hyperuni- uniformity S0 is an order metric for the ferromagnetic form ground states, we consider systems with interac- chains but not for the phase transition in the antiferro- tions that decay with distance. These models have been magnetic chains. This is reflected in the sudden, sharp extensively studied using various analytic and numerical rise in S0 for the ferromagnetic chains shown in Fig.2. In techniques [42, 43]. Qualitatively, as decay of the inter- all of the inverse power-law interaction Hamiltonians, we action becomes faster, we expect the system to asymp- see a transition from an ordered hyperuniform state to a totically approach the nearest neighbor interaction limit, disordered non-hyperuniform state. The degree of hype- that is the standard Ising model. So, we would expect runiformity, S0, like mz, shows a large finite-size effect. these systems to undergo a continuous phase transition Therefore, the critical point should be obtained from the like the standard Ising model. It is trivial to show using hx curve. Moreover, note that the behavior of all the Eq. (6) that all of these long-range Ising models have or- ferromagnetic and the corresponding antiferromagnetic dered hyperuniform ground state at Γ = 0, owing to the models in terms of hx is identical, showing that it is a ground states being direct products of eigenvectors ofσ ˆz. valid order parameter in both cases. 5
1 2 N = 30 0.8 N = 30 N = 50 1.5 N = 50 N = 80 N = 80 0.6 z
τ 1 m 0.4
0.5 0.2
0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Γ Γ
(a) mz (b) τ order metric
40
1 N = 30 N = 30 30 N = 50 N = 50 N = 80 N = 80 0 x 20 h S 0.5
10
0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Γ Γ
(c) hx (d) S0
FIG. 1. Size dependence of the critical point using various observables as the order parameters for the ferromagnetic Ising model: (a) Average magnetization along z-axis, Eq. (3); (b) τ order metric, Eq. (7); (c) Second cumulant of the transverse magnetization, hx, Eq. (2); (d) Measure of hyperuniformity, S0, Eq. (6). The variation of the critical value of Γ with N is the least for hx. The value of hx undergoes a maximum for small N, which changes to a point of non-differentiability as N becomes larger.
B. Interactions Optimized for Hyperuniformity the discontinuities observed even in the finite sized sys- tems seem to strongly suggest the existence of first-order phase transitions. In the following subsections, we give The interactions that are optimized for hyperuni- representative examples of each class. The behavior of formity have disordered stealthy hyperuniform ground stealthiness as measured by S1 is universal across the states at Γ = 0. These ground states, unlike the ones for three classes. Stealthiness is a more sensitive property the typical ferromagnetic long-range interactions that de- than hyperuniformity. We will show, in the following cay with distance, have an average magnetization m = 0 z sections, that S1 increases faster than S0 but the basic throughout the range of Γ and unlike the typical anti- features of S1 are identical to those of S0 in all the classes. ferromagnetic long-range interactions, are not ordered. They undergo phase transitions between various “disor- dered” phases with mz = 0. The stealthy hyperuniform 1. Parameters with one weak First-Order QPT: No Order ground state at Γ = 0, which is a disordered direct- from Disorder product state, is to be contrasted with the disordered state that is the ground state as Γ → ∞. Now, the As a first example of the parameters that were opti- ground state is a direct product of eigenstates ofσ ˆ op- x mized for hyperuniformity, consider a system with a weak erators. This state, also, has zero total magnetization in first-order quantum phase transition (see J (r) in SM), the z-direction, but the individual spins are not point- 1 where the qualitative features of the τ order metric are ing along the z-direction. We call this the “disordered similar to that in the standard long-range Ising model quantum” state. discussed in Sec.IIIA. Figure 3 shows the variation of The spin systems, that we simulated, can be grouped various basic observables as a function of the transverse into two broad classes: there are systems with one, or field. There is a transition at Γ ≈ 1.5 between the clas- two first-order quantum phase transitions. Of course we sically disordered ground state at “low” values of Γ and would need to increase the system size to truly charac- the disordered quantum state at “large” values of Γ. terize the phase transitions. However, the sharpness of Notice that the τ order metric in Fig.3(b) is smooth 6
-1 2
-1.5 1.5 -2
-2.5 τ 1
-4 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Γ Γ (a) Energy per site (b) τ order metric
20 1 15 0 x h S 10 0.5
5
0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Γ Γ
(c) hx (d) S0
FIG. 2. Basic observables as functions of Γ: (a) Energy per site; (b) τ order metric; (c) Second cumulant of the transverse −2 −3 −4 magnetization, hx; (d) Degree of hyperuniformity, S0. Black line: Jr = r . Red line: Jr = r . Green line: Jr = r . Blue r −2 r −3 r −4 line: ferromagnetic Ising model. Black markers: Jr = (−1) r . Red markers: Jr = (−1) r . Green markers: Jr = (−1) r . Blue markers: antiferromagnetic Ising model. The degree of hyperuniformity, S0 undergoes a sudden jump for the ferromagnetic cases at the critical point. This jump in S0 is absent in the antiferromagnetic cases. and monotonically decreasing with the transverse field 2. Parameters with One First-Order QPT Γ, similar to the behavior depicted in Fig.2(b). This is surprising because the interactions in case of Fig.2 Next, we consider the Jr’s which lead to systems with decay rapidly with distance, whereas the ones in Fig.3 a single first-order phase transition (see J2(r) in SM). have been optimized to produce stealthy hyperuniform The observables corresponding to one such parameter is ground states by long-range interactions. Of course, in shown in Fig.5. There is a phase transition at Γ = 0 .09. the limit of Γ → ∞, all structure is lost and the ground It is interesting to note the behavior of the order met- state of the Hamiltonian is a direct product of the ground ric in the vicinity of the phase transition. In this case, state of the localσ ˆx operator. This indicates that the the system starts from a relatively disordered, hyperuni- system starts at a disordered hyperuniform ground state form ground state. As the magnetic field is increased, at Γ = 0 and continues to lose that order as well. We initially the order metric does not undergo any substan- see that S0 shows a monotonic increase with Γ, implying tial change. Surprisingly, the new structure after the that hyperuniformity is degraded. There is a very small critical point is much more structured than the one with “jump” in the measures of hyperuniformity, S0 and S1, Γ = 0. This order is then smoothly lost as the magnetic around Γ ≈ 1.5, which seems to suggest the presence field is increased, leading asymptotically to a completely of a weak first-order QPT. S(k) as a function of Γ is disordered system. In Fig.6, we provide a surface plot shown in Fig.4, which, however, seems to show a smooth of S(k, Γ) to demonstrate the changes in the structure transition from the disordered classical ground state at factor as a function of Γ. low values of Γ to the disordered quantum ground state at high values. The smooth decay of the τ order metric and the small discontinuities in S0 and S1 as functions of 3. Parameters with Two First-Order QPTs Γ at the critical point might be a result of the weakness of the first-order QPT in this case. Finally, there are cases with two first-order phase tran- sitions as demonstrated by the observables in Fig.7 (see J3(r) in SM). From the S0 plot, it is clear that there 7
-1 0.2
-1.5 0.15 -2
-2.5 τ 0.1
-4 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Γ Γ (a) Energy per site (b) τ order metric
0.7 S 0.6 0 1 S 0.5 1
1 0.4 x / S h 0 0.3
0.5 S 0.2 0.1 0 0 0 0.5 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Γ Γ
(c) hx (d) S0
FIG. 3. Basic observables as functions of Γ: (a) Energy per site; (b) τ order metric; (c) Second cumulant of the transverse magnetization, hx; (d) Degree of hyperuniformity, S0 and S1, Eq. (9). Parameter demonstrates possibility of a weak first-order QPT.
tional references for the configuration-integral (CI) like basis set analysis of the phase transitions presented in appendixA. As shown in Fig.8, the ground state goes from |0i before the first phase transition (Γ < 0.06), to a structure like |2i between the two phase transitions (0.07 < Γ < 0.12). Finally, after the phase transition at Γ ≈ 1.2, the structure becomes significantly more similar to |1i. These last two cases are extremely interesting from the fundamental perspective of understanding the impact of FIG. 4. Ground state structure factor as a function of the a transverse field to these complicated long range inter- wave-vector, k and the strength of the transverse field Γ. acting spin chains. It is curious that depending on the parameters of the system, phase transitions caused by transverse magnetic fields can serve to increase the order of the ground state as measured by the τ order parame- are two phase transitions: one between Γ = 0.0675 ter. Additionally, the order does not have to necessarily and Γ = 0.07, and another between Γ = 0.125 and increase. Parameters can be defined where the phase Γ = 0.1275. We further analyze this particular parame- transitions involved have predefined effect on the order ter using configuration interaction (CI) like basis set ex- metric. As in this case, the system starts off from a dis- pansions to get a better intuition regarding the phase ordered hyperuniform state similar to the previous two transition. In the ten independent simulations that we cases. At the first QPT, there is a sudden increase in or- performed, there were occasions when the DMRG proce- der, which decays smoothly till we encounter the second dure produced energetically low-lying states which were phase transition. Then there is a sudden decrease in or- not the ground state. In Fig.8, we plot the structure der from where the τ order parameter relaxes smoothly factors of the ground state and the other low-lying states to the disordered state as Γ → ∞. that we encountered. These low-lying states are not rel- evant to the current discussion, but are useful as addi- We have numerically demonstrated that there are no 8
0.3
-0.8 0.25
0.2
-1 τ 0.15
-1.2 0.05
0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ Γ (a) Energy per site (b) τ order metric
1.2 0.8 S 0 S 1 0.6 1 1 x / S h
0 0.4
0.8 S
0.2 0.6 0 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Γ Γ
(c) hx (d) S0
FIG. 5. Basic observables as functions of Γ: (a) Energy per site; (b) τ order metric; (c) Second cumulant of the transverse magnetization, hx; (d) Degree of hyperuniformity, S0 and S1. Parameter demonstrates possibility of a first-order QPT indicated by sharp discontinuities in (b), (c), and (d) around Γ ≈ 0.1.
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 under a couple of limiting cases. Using a unitary rotation, the Hamiltonian defined in Eq. (1), can be written as:
X X (i) (i+r) X (i) H = − Jrσˆx σˆx + −Γˆσz (10) i 1≤r≤R i
FIG. 6. Structure factor as a function of the wave-vector, k The Pauli matrices on the same site anticommute and the and the strength of the transverse field, Γ. We have marked ones on different site commute. This makes it difficult out the critical value of Γ where the phase transition occurs. to apply analytic tools to treat this system. A common Between Γ = 0.095 and Γ = 0.10, there is a phase change. method of overcoming this difficulty is to apply a Jordan- Wigner string mapping[44] to convert the spin operators to fermionic creating and annihilation operators. The mapping can be summarized as:
(j) † σˆz = 1 − 2cjcj (11) (j) X Y † σˆ+ = exp iπ nl cj = 1 − 2cl cl cj (12) cases with more than two QPTs. To provide some analyt- l -1 0.3 0.25 -1.5 0.2 -2 τ 0.15 -3 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Γ Γ (a) Energy per site (b) τ order metric 0.6 S 0.5 0 1 S 1 0.4 1 x 0.3 / S h 0 0.5 S 0.2 0.1 0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 Γ Γ (c) hx (d) S0 FIG. 7. Basic observables as functions of Γ: (a) Energy per site; (b) τ order metric; (c) Second cumulant of the transverse magnetization, hx; (d) Degree of hyperuniformity, S0 and S1. Parameter demonstrates possibility of two first-order QPT indicated by sharp discontinuities in (b), (c), and (d) around Γ ≈ 0.065 and Γ ≈ 0.125. 15 15 |0>, Γ<0.06 |1> 0.07<Γ<0.12 |2> 0.13<Γ |3> 10 10 S(k) S(k) 5 5 0 0 0 1 2 3 4 5 6 0 1 2 3 4 5 6 k k (a) (b) FIG. 8. Structure factors. Left: Ground states for select intervals of Γ. Ground state before the first critical point (Γ < 0.06) is labeled |0i. Right: Other low-lying states at Γ = 0, labeled |1i, |2i, and |3i respectively. Under this mapping the Hamiltonian, Eq. (10) would get † X † +cj+rcj + cj+rcj − Γ 1 − 2cjcj (14) transformed as follows: j X X (j) (j) (j+r) (j+r) H = − J σˆ +σ ˆ σˆ +σ ˆ r + − + − † j 1≤r≤R Here c and c are the fermionic annihilation and cre- ation operators. The exponentials forσ ˆ± in the Hamilto- X † − Γ 1 − 2cjcj nian lead to more than pairwise interactions between the j fermions. To simplify analysis, we make two approxima- l tion terms in the Hamiltonian, and make an approximate long-range fermionic model that can be directly solved because it is quadratic: X X † † † † H = − Jr cjcj+r + cjcj+r + cj+rcj + cj+rcj j 1≤r≤R X † − Γ 1 − 2cjcj . (15) FIG. 9. Structure factor as a function of the wave-vector, k j and the strength of the transverse field, Γ. We have marked out critical values of Γ where the phase transitions occur. There are two phase transitions, one at Γ ≈ 0.07 and Γ ≈ 0.125, after which there is a steady continuous loss of order. 1. Pairwise Interacting Approximation Now, transforming the creation and annihilation oper- ators into Fourier space using c = √1 P s eikja and j N k k † † c = √1 P s e−ikja, where a is the lattice constant, we We begin the analysis by making the crudest simpli- j N k k fying assumption: we consider only the pairwise interac- get X † † X † † X H = 2 sksk + s−ks−k Γ − Jr cos (kar) + 2i s−ksk + s−ksk Jr sin (kar) 0 Here we have already gotten rid of terms with dou- in Eq. (16) is already diagonal. Putting everything to- ble creation (due to the Fermionic nature of the par- gether, the final diagonalized Hamiltonian is ticles) or double annihilation operators. This Hamil- X q 1 tonian can be diagonalized using a standard Bogoli- H = 2 α2 + β2 γ†γ − k k k k 2 ubov transform [45–47]. First, let solve the problem k6=0,π for the “inner” part of the Hamiltonian (k 6= 0 and P k 6= π). Let αk = Γ − Jr cos (kar) and X † 1≤r≤R − 2 Γ − Jr s0s0 P βk = 1≤r≤R Jr sin (kar). 1≤r≤R αk −iβk 0 0 sk X † − 2 Γ − Jr cos(πar) sπsπ (18) iβ −α 0 0 s† † † k k −k 1≤r≤R Hk = sk s−k s−k sk 0 0 αk iβk s−k 0 0 −iβ −α † Now, all the k-modes are independent. Therefore, we k k sk (17) can write down the ground state of the Hamiltonian in mixed Bogoliubov (space of sk) / non-Bogoliubov (space To diagonalize this Hamiltonian, consider the Bogoli- of γk) space. We only keep s0 and sπ in the standard † ubov transform defined by sk = ukγk + ivkγ−k, where space and talk about the rest of the modes in terms of γk. 2 2 We define the vacuum |i as a state which is annihilated uk = u−k, vk = −v−k and uk + vk = 1. We also de- θk by γk for all k 6= 0 and k 6= π, and also by s0 and sk. fine a single parameter, θk such that uk = cos and 2 Because the Hamiltonian is basically just the harmonic v = sin θk . The solution for all k 6= 0, π is given by k 2 oscillator Hamiltonian for all modes with k 6= 0 or k 6= π, βk tan (θk) = . The eigenvalues of the matrix are given αk the vacuum state is the ground state for the Hamiltonian. p 2 2 by ± αk + βk. For k = 0 or k = π, the Hamiltonian The phase transitions in the model are caused by changes 11 in the occupation of the k = 0 or the k = π modes. If and the k = π mode in the antiferromagnetic case. P Γ > Jr, then the k = 0 state being populated Now, we come to the question of the number of 1≤r≤R P lowers the energy, and the ground state must have this such phase transitions. If any one of 1≤r≤R Jr and state populated. Similarly if Γ > P J cos(πar), P 1≤r≤R r 1≤r≤R Jr cos(πar) is greater than zero, then there then k = π must be populated. This change in the nature would be one phase transition, and if both of them are of ground state is a phase transition. greater 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- Consider the nearest neighbor ferromag- ble because the k 6= 0 and k 6= π modes are always in the netic (R = 1,J = J > 0) and antiferromagnetic r vacuum state. Thus, consistent with the numerical re- (R = 1,J = J < 0) Ising models with the lattice r sults that we have encountered in the previous sections, constant a = 1 as trivial examples of the mapping and this approximate model also allows a maximum of two cases where the model is exact. For 0 ≤ Γ < |J|, in phase transitions. We would like to emphasize that this case of the ferromagnetic Ising model, the k = π mode condition on J for the number of phase transitions is is populated. On the other hand, for the antiferromag- r only valid for the current approximate model. It is not netic Ising model, the k = 0 mode is populated for valid in general for the long-range spin system, for which 0 ≤ Γ ≤ |J|. When Γ > |J|, in case of the ferromagnetic closed-form analytic solutions do not exist. Ising model, the k = 0 mode gets populated in addition Observables can be calculated by mapping them onto to the k = π mode, and in case of the antiferromagnetic the spinless fermions by using the Jordan-Wigner map- case, the k = π mode gets populated. This represents a 1 P D (j)E phase transition, with the critical value of Γ = |J|, that ping. As an example, we choose N 1≤j≤N σz . This happens by different mechanisms in the two cases: the would be useful in doing the mean-field analysis in the occupation of the k = 0 mode in the ferromagnetic case, next section. In the following γ0 = s0 and γπ = sπ. 1 X D E 1 X D E 2 X D E σ(j) = 1 − 2 s† s = 1 − u γ† − iv γ u γ + iv γ† N z N k k N k k k −k k k k −k 1≤j≤N k k 2 X D E D E 2 X 2 D E = 1 − u2 γ†γ − v2 γ† γ − v2 − γ†γ + γ† γ (19) N k k k k −k −k N k N 0 0 π π k6=0,π k6=0,π | {z } term A 2 X 2 D E = 1 − v2 − γ†γ + γ† γ (20) N k N 0 0 π π k6=0,π ! 1 X αk 2 D E = 1 − 1 − − γ†γ + γ† γ (21) N p 2 2 N 0 0 π π k6=0,π αk + βk 2 1 X αk 2 D E = + − γ†γ + γ† γ (22) N N p 2 2 N 0 0 π π k6=0,π αk + βk D E Since no k-modes apart from k = 0 and k = π are 1 P † N 1≤l≤N 1 − 2cl cl = gi. We will use this assump- populated with Bogoliubov fermions, the term marked tion to solve the Hamiltonian as a function of g: “A” is zero in Eq. (19). The equality between Eq. (20) βk X X † † † and Eq. (21) is obtained by noting that tan(θk) = r−1 αk H(g) = − Jrg cjcj+r + cjcj+r θ k j 1≤r≤R and vk = sin 2 , while the third equality is a result of having N − 2 modes with k 6= 0, π. Similar expressions X +c† c + c c − Γ 1 − 2c†c (23) can be derived for other observables. j+r j j+r j j j j Of course, the analysis of Eq. (23) is simplified through the observation that it is isomorphic with Eq. (15) and 2. Mean-field Analysis consequently, Eq. (16) under the transformation Jr → r−1 J˜r = Jrg . To start the process, we use the g0 ob- For the mean-field analysis of Eq. (14), let us tained under the pairwise interaction approximation, us- assume that at the ith iteration, the value of ing Eq. (21). If we are only interested in the value of 12 D E 1 P † pairwise interaction terms. We showed that for a generic N 1≤l≤N 1 − 2cl cl , it might be possible to directly search for the root of the following equation: long-range Ising model, under the pairwise interaction and mean-field “approximations,” there can be a maxi- 2 1 X α˜ mum of two phase transitions, which is consistent with g = + q our numerical results. Therefore, we conjecture that a N N 2 ˜2 k6=0,π α˜ + β long-range pairwise interacting 1D Ising spin chain with 2 D E arbitrary couplings for which a discrete cosine transform − γ†γ + γ† γ (24) N 0 0 π π is convergent, can show at most two quantum phase tran- X r−1 sitions at zero temperature. whereα ˜ = Γ − Jrg cos(kar) (25) Future work on understanding the critical scaling of 1≤r≤R these phase transitions using MERA or QMC should lead ˜ X r−1 β = Jrg sin(kar). (26) to further insights into the nature of these long-range 1≤r≤R hyperuniform spin chains. It would also be interesting to study these systems under a combination of longitudinal However, here we are interested in characterizing the and transverse fields to map out the full phase diagram. number of phase transitions in the ground state as the Such studies would prove useful in the design of novel transverse field is increased. This can be achieved much materials with simple models. more simply by observing that because the Hamiltonian at every step is isomorphic to Eq. (15) at every step of the self-consistent field procedure, the constraint on the max- ACKNOWLEDGMENTS imum number of phase transitions in the model would hold. Thus, even after solving the long range problem, in a self-consistent manner, we expect that there cannot We thank Roberto Car for many insightful discussions. be more than two phase transitions, which is consistent A. B. acknowledges support of the Computational Chem- with our numerical exploration. ical Center: Chemistry in Solution and at Interfaces funded by the U. S. Department of Energy under Award No. DE-SC0019394. S. T. acknowledges the support of IV. CONCLUSIONS the National Science Foundation under Grant No. DMR- 1714722. Long-range Ising models can exhibit very rich physics, especially when the long-range couplings do not decay Appendix A: Explorations using Configuration with distance. Such atypical long-range couplings are es- Interaction (CI) expansions sential to ensuring that spin systems have stealthy hype- runiform ground states in the absence of any transverse field. Spin systems with hyperuniform ground states, The Hamiltonian can be decomposed in the following thus, show very interesting physics that is qualitatively manner: different from standard long-range Ising models with in- X X X H = − J σˆ(i)σˆ(i+r) + −Γˆσ(i) (A1) teractions that decay with distance. We have presented r z z x i 1≤r≤R i here one of the first analyses of the physics of stealthy, hy- | {z } | {z } V peruniform spin systems with nontrivial, non-monotonic, Href long-range interactions under a transverse magnetic field. Href|φi = E0|φi (A2) We demonstrate numerically that for these systems, un- like standard long-range Ising models, the number of Like in the DMRG simulations, we evaluate the Hamil- phase transitions is not fixed. Their loss of hyperuni- tonian in a PBC. We have already solved for the “refer- formity, in presence of transverse fields, is not always ence” |φi, which is the classical long-range Ising problem accompanied by a loss of order, as measured by the τ with Γ = 0. Now, we expand the true ground state wave order metric. This feature is very unusual and leads to function |ψi in a configuration integral expansion: the possibility of designing hyperuniform materials whose X X ground-state in the presence of external transverse mag- |ψi = c0|φi + cα|φαi + cα,β|φα,βi + ... (A3) netic fields can be more ordered than that in absence of α α,β external the field. We also showed that the rate of loss th of the property of stealthiness is much faster than that where |φαi is obtained by flipping the α spin with re- th of the loss of hyperuniformity, proving that stealthiness spect to |φi, |φα,βi is obtained by flipping the α and is a much more delicate property. βth spins, so on. To better theoretically understand the phase transi- We solve the eigenvalue equation representing the tions we identified numerically, we have analyzed the Hamiltonian in the orthonormal basis described above. long-range Ising spin models using the Jordan-Wigner First, consider the matrix elements of Href. Since Href is mapping Hamiltonian. Under this mapping, the long- a function ofσ ˆz and the basis vectors are eigenvectors of range spin-spin interactions manifest themselves as non- σˆz, Href is diagonal in the basis defined. 13 -1.05 -1.05 -1.075 -1.075 DMRG DMRG -1.1 |0> -1.1 |0> 0 0.075 0.15 0.225 0.3 0 0.075 0.15 0.225 0.3 Γ Γ (a) CIS Energy (b) CISD Energy -1.05 -1.05 -1.075 -1.075 DMRG DMRG -1.1 |0> -1.1 |0> 0 0.075 0.15 0.225 0.3 0 0.075 0.15 0.225 0.3 Γ Γ (c) CISDT Energy (d) CISDTQ Energy FIG. 10. Energy per site with respect to different references. Though the basis consisting of all possible excitations the low Γ region as well. is complete, the smaller basis set obtained using a trun- Obviously at Γ close to 0.3 the results are not con- cated number of excitations is not. As a method of ex- verged. However we are interested in the nature of the ploring the nature of the ground state qualitatively, we “crossovers.” Figure 10 demonstrates that if we consider use not just |0i as reference but also |1i, |2i, and |3i. only single excitations, then there is no phase transition. Of course, in the full basis, the results should be inde- The ground state remains similar to |0i over the range pendent of the reference used. To gain a better under- of Γ considered. However as soon as we introduce more standing, we truncate the expansion such that, the bases excitations, phase transitions start appearing. We notice defined on all of the references are non-intersecting. So, that for the CISDT calculations, there is a cross-over the Hamiltonian matrix defined in terms of all the ref- from a |0i-like ground state to a |2i-like ground state erences would have a block diagonal structure, implying at around 0.074, followed by a cross-over from a |2i-like that we can solve the problem for each of the references ground state to a |1i like ground-state at 0.22. The |0i- independently. Additionally this allows us to probe into like ground-state to |2i-like ground-state transition lies the nature of the ground state. We can now make quali- within the range of Γ where the calculations were better tative statements about which reference the ground state converged. Therefore, we get an estimate of the critical looks like. Note in Fig. 10, for |2i, the inclusion of quartic value of Γ that is in good agreement with the estimate excitations includes vectors that are similar to |0i. That from DMRG calculations (0.0675 – 0.07). 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