Effective Theory and Breakdown of Conformal Symmetry in a Long-Range Quantum Chain

Effective Theory and Breakdown of Conformal Symmetry in a Long-Range Quantum Chain

L. Lepori llepori81@gmail.com Dipartimento di Fisica e Astronomia, Università di Padova, Via Marzolo 8, I-35131 Padova, Italy icFRC, IPCMS (UMR 7504) and ISIS (UMR 7006), Université de Strasbourg and CNRS, Strasbourg, France    D. Vodola icFRC, IPCMS (UMR 7504) and ISIS (UMR 7006), Université de Strasbourg and CNRS, Strasbourg, France    G. Pupillo icFRC, IPCMS (UMR 7504) and ISIS (UMR 7006), Université de Strasbourg and CNRS, Strasbourg, France    G. Gori CNR-IOM DEMOCRITOS Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy    A. Trombettoni CNR-IOM DEMOCRITOS Simulation Center, Via Bonomea 265, I-34136 Trieste, Italy SISSA and INFN, Sezione di Trieste, Via Bonomea 265, I-34136 Trieste, Italy

We deal with the problem of studying the symmetries and the effective theories of long-range models around their critical points. A prominent issue is to determine whether they possess (or not) conformal symmetry (CS) at criticality and how the presence of CS depends on the range of the interactions. To have a model, both simple to treat and interesting, where to investigate these questions, we focus on the Kitaev chain with long-range pairings decaying with distance as power-law with exponent . This is a quadratic solvable model, yet displaying non-trivial quantum phase transitions. Two critical lines are found, occurring respectively at a positive and a negative chemical potential. Focusing first on the critical line at positive chemical potential, by means of a renormalization group approach we derive its effective theory close to criticality. Our main result is that the effective action is the sum of two terms: a Dirac action , found in the short-range Ising universality class, and an “anomalous” CS breaking term . While originates from low-energy excitations in the spectrum, originates from the higher energy modes where singularities develop, due to the long-range nature of the model. At criticality flows to zero for , while for it dominates and determines the breakdown of the CS. Out of criticality breaks, in the considered approximation, the effective Lorentz invariance (ELI) for every finite . As increases such ELI breakdown becomes less and less pronounced and in the short-range limit the ELI is restored. In order to test the validity of the determined effective theory, we compared the two-fermion static correlation functions and the von Neumann entropy obtained from them with the ones calculated on the lattice, finding agreement. These results explain two observed features characteristic of long-range models, the hybrid decay of static correlation functions within gapped phases and the area-law violation for the von Neumann entropy. The proposed scenario is expected to hold in other long-range models displaying quasiparticle excitations in ballistic regime. From the effective theory one can also see that new phases emerge for . Finally we show that at every finite the critical exponents, defined as for the short-range () model, are not altered. This also shows that the long-range paired Kitaev chain provides an example of a long-range model in which the value of where the CS is broken does not coincide with the value at which the critical exponents start to differ from the ones of the corresponding short-range model. At variance, for the second critical line, having negative chemical potential, only () is present for (for ). Close to this line, where the minimum of the spectrum coincides with the momentum where singularities develop, the critical exponents change where CS is broken.

thanks: luca.lepori@pd.infn.it

I Introduction

The study of classical and quantum long-range systems, both at and out of equilibrium, is a very active field of research libro (). One of the main reasons for this growing interest is that these systems have been predicted to exhibit new phases with peculiar properties, including the presence of correlation functions with both exponential and algebraic decay, even in the presence of a mass gap cirac05 (); tagliais (); nostro (); paperdouble (), the violation of the lattice locality hast (); hauke2013 (); eisert2014 (); metivier2014 (); noinf (); damanik2014 (); nbound (); storch2015 (); carleo (); kastner2015ent (); kuwahara2015 () and of the area-law for the von Neumann entropy tagliais (); nostro (), nonlinear growth of the latter quantity after a quench growth (), peculiar constraints on thermalization santos2015 (), .

Recently developed techniques in atomic, molecular and optical systems (such as Rydberg atoms, polar molecules, magnetic or electric dipoles, multimode cavities and trapped ions) provide an experimental playground to investigate the properties of phases and phase transitions for long-range models Childress2006 (); Balasubramanian2009 (); Weber2010 (); tech0 (); tech1 (); Schauss2012 (); Aikawa2012 (); Lu2012 (); Firstenberg2013 (); Yan2013 (); Dolde2013 (); Islam2013 (); exp1 (); exp2 (); tech5 () and motivated an intense theoretical activity Saffman2010 (); tech2 (); tech3 (); exp0 (); tech4 (); maghrebi2015tris (). In particular, Ising-type spin chains with tunable long-range interactions can be now realized using neutral atoms coupled to photonic modes of a cavity or with trapped ions coupled to motional degrees of freedom. In this latter case, the resulting interactions decay algebraically with the distance , with an adjustable exponent usually in the range .

A crucial issue in the investigation of many-body systems is the characterization and the study of the symmetries of their critical points. For classical short-range models in two and three dimensions the scale invariance hosted at the critical points where second order phase transitions arise is expected to be promoted to the larger conformal symmetry (CS), also including translations, Euclidean rotations, and special conformal transformations combining translations with spatial inversions dif (); muss (). This symmetry, conjectured long ago polyakov_1970 (), has in two dimensions far-reaching consequences dif (); muss (), as it fixes completely the universality class. Numerical studies both in two and three dimensions gave a clear evidence of the presence of CS for short-range models cardy_1994 (); langlands_1994 (); deng_2002 (); gori_2015 (); penedones_2015 ().

A related, natural question for long-range models is then whether their critical points possess (or not) CS and how its presence is eventually related to the range of the interactions. Referring to couplings decaying with the distance as , since for the short-range models and their CS at criticality are recovered, a central issue is up to what values of (and how) CS persists. This issue is related to another typical question arising in the study of long-range systems, where an important information is the determination of the upper limit value of (say ) such that above it the critical exponents of the short-range model are retrieved sak_1973 (); libro (). For example, classical spin systems with long-range couplings having power-law exponent display, for greater than a critical value , the same critical exponents of the corresponding short-range models (), while for , they exhibit peculiar long-range critical exponents libro (); sak_1973 ().
Finally, for classical long-range models with continuous symmetry one may have a finite critical temperature also in one spatial dimension spohn1999 (), this being not in contradiction with the Mermin-Wagner theorem MW (); lebellac (), which is valid for short-rang interactions (see e.g. the discussion in defenu2014 ()). As a further example, the classical long-range Ising model in dyson1969 (); thouless1969 (); anderson1970 () has and exactly at , a phase transition of the Berezinskii-Kosterlitz-Thouless universality class occurs cardy1981 (); frolich1982 (); lui2001 (); lui1997 ().

To deal with the interesting problems outlined above for interacting long-range quantum chains or long-range classical systems, namely the breakdown of the conformal symmetry at criticality and the behavior of the critical exponents, one has to eventually resort to computationally expensive numerical simulations. It is then clear that a qualitative understanding based on a simple, exactly solvable and possibly non-trivial model would be highly desirable.

The presence of CS at the critical point implies the Lorentz invariance of the theory. Typically, in short-range interacting models the resulting effective theory displays an effective Lorentz invariance (ELI) also near criticality, as it is seen for the quantum Ising chain in a transverse field muss (). Moreover, general perturbations of the critical points of short-range models allow for Lorentz invariance also in massive regimes, where CS is broken muss (). It is then a natural question if and to what extent the ELI and associated locality survive, both at and out of criticality, in long-range critical systems and, when they are present, how they affect the physical observables, as correlation functions.

To shed light on the issues described above, we decided to consider a long-range quadratic fermionic model in one quantum dimension exhibiting non-trivial quantum phase transitions, the Kitaev chain with long-range pairings power-law decaying with distance, recently introduced in nostro (). In particular we derive, via a renormalization group (RG) approach, an effective continuous theory for this model at and close to its critical points.

The Kitaev with long-range pairing (and hopping) is already experimentally realizable for for a particular type of helical Shiba chain Pientka2013 (); Pientka2014 () (while for example for the long-range Ising model one can engineer such that tech1 (); exp1 (); exp2 ()). Nevertheless, the long-range Kitaev chain provides an ideal playground to investigate the symmetries of the critical points of long-range systems, since:

  • It is a quadratic fermionic model, then exactly solvable.

  • For , where the short-range Kitaev chain kitaev () is recovered, it is mapped via Jordan-Wigner transformations to the short-range Ising model in a transverse field libro_cha () (a discussion of the comparison between the phases of the long-range Kitaev and long-range Ising chains can be found in paperdouble ()).

  • It displays non-trivial (non mean-field) quantum phase transitions at , where is the chemical potential appearing in the Hamiltonian (1).

  • As shown in nostro (); paperdouble (), it exhibits, in the presence of vanishing mass gap, a linear spectrum around the zero-energy points, which is the case considered in standard textbooks dif (); muss ().

  • It displays a hybrid decay behavior (exponential at short distances and power-law at longer ones, see Fig. 1) for the static correlation functions in gapped regimes nostro (). This hybrid behavior has also been observed e.g. in the Ising model with long-range interactions cirac05 (); paperdouble () and it seems to be characteristic of long-range systems.

In Ref. paperdouble () the origin of the hybrid decay behavior mentioned above was identified analytically as the result of the competition between two different sets of modes with well-separated energies. The short-distance exponential decay originates from the eigenmodes with energy near the minimum of the energy spectrum in the center of the Brillouin zone (as in short-range models), while, surprisingly, the long-distance algebraic decay originates from high-energy eigenmodes. A similar picture emerged for the long-range Ising model on the basis of a spin wave analysis cirac05 (). This fact suggests that the double contribution to the correlation functions from two different sets of modes can be not limited to free or weakly interacting long-range models. In the long-range paired Kitaev chain the modes at the edges of the Brillouin zone have been found responsible also for an anomalous scaling for the ground state energy at criticality nostro (); paperdouble (). For this reason, in this paper we put forward and develop an approximate renormalization group (RG) approach in which we explicitly take into account the contribution of the modes at the edges of the Brillouin zone during the decimation procedure.

Focusing mostly on the critical line , the main results that we obtained are:

Figure 1: Plot of the static two fermions correlation function defined in Eq. (17), showing the hybrid exponential and power-law decay. The first part on the left is described close to criticality by the Dirac action , while the algebraic tail is associated to CS breaking term [see Section IV].
  • Section III: The resulting effective action is found to be the sum of two terms: a Dirac action coming from contributions near the minimum of the energy spectrum, as in the short-range limit , plus an anomalous CS breaking term originating from the higher-energy contributions at the edges of the Brillouin zone, where singularities develop (in the -th derivative of the spectrum, labeling the integer part of ) due to the long-range nature of the model. This is one of the central results of the paper. Although the determination of effective theories for both classical and quantum long-range models has been the subject of a perduring interest (see e.g. Refs. laflorencie (); defenu (); Maghrebi2015 () and references therein), in our opinion the present example is particularly instructive, since the precise origin and consequence of the anomalous CS breaking term is directly identified and derived from the microscopics. Notably the appearance of the CS breaking action, due to high-energy modes, implies a different hierarchy for the quasiparticles weights along the RG, compared to the one generally assumed for short-range systems huang ().

  • Sections III and V: At criticality, for RG makes the CS breaking term flow to zero, while for it dominates over . This change of behavior is at the origin of the breakdown of CS, since is not conformal invariant.

  • Section III: The obtained effective theory allows to identify two new phases for . Close to criticality these phases display an additional emergent approximate symmetry under suitable anisotropic scale transformations.

  • Section IV: Out of criticality and do not decouple completely and they co-act to determine the physical quantities. For instance they are at the origin of the hybrid (exponential plus power-law) decay of the static two fermions correlation functions out of criticality. This coupling between the two terms and and the validity of the effective theory are probed against the lattice results. The asymptotic decay exponents of the lattice correlation functions are compared to the ones obtained from the effective theory, finding perfect agreement. Notably our study also unveils the central role played by RG subleading terms in , affecting even qualitatively the correlation functions in some regimes. In our knowledge this fact has no counterpart in short-range systems.

  • Section V.1: The hybrid decay of the two fermions static correlation functions out of criticality points to the breakdown of the Lorentz invariance for the effective theory. Indeed we infer that out of criticality the ELI, exact in the limit (where just acquires a mass term dif (); muss ()), is broken at every finite value of . Conversely, exactly at criticality the Lorentz group, belonging to the conformal group, is broken only below .

  • Section VI: By means of the effective theory we computed the von Neumann entropy ( being the size of a part of the bi-parted chain). Exactly at criticality follows the standard scaling law valid for short-range critical systems wilczek (); calabrese (). However, while above it is found , as for the short-range Kitaev (Ising) chain, below this threshold it holds . Correspondingly, out of criticality and above , we obtained no deviations from the so-called area-law (), valid for short-range gapped systems, while a logarithmic deviation (with ) is derived for . Our results are in agreement with the lattice results in Refs. nostro (); paperdouble (); ares () and confirm the presence of new phases at , as inferred by the effective theory (see Section III).

  • Section VIII: At every value of the decay exponent of the pairing in (1), the critical exponents (using the standard notation huang ()) (in classical short-range models related of the specific heat, not to be confused with decay exponent for the pairing of the Kitaev chain considered here) and (referring to the Ising order parameter) close to assume the same values as for the short-range Ising (Kitaev) model. This fact is allowed by the simultaneous presence of and and out of criticality.

In Section VII the critical line and (for the Hamiltonian (1) acquires a mass gap) is also analyzed for the sake of comparison. Since close to this line the minimum of the quasiparticle energy is now at , the effective action is composed by a single term only, unlike the situation around the line . The consequences of this fact on the two-points correlation functions, on the breakdown of conformal invariance, on the violation of the area-law for the von Neumann entropy and on the critical exponents for the Ising order parameter are discussed.

In the following we will obtain and describe in detail the results mentioned above. Conclusions and perspectives are discussed in Section IX, while more technical material is presented in the Appendices.

Ii The model

We start from the Kitaev Hamiltonian with long-range pairing nostro () defined on a lattice:


In Eq. (1), is the operator destroying a (spinless) fermion in the site , being the number of sites of the chain. For a closed chain, we define () if () and we choose antiperiodic boundary conditions nostro (). We measure energies in units of and lengths in units of the lattice spacing . We also set for simplicity , since as discussed in nostro () the critical values of the chemical potential does not depend on .

The spectrum of excitations is obtained via a Bogoliubov transformation and it is given by


In Eq. (2), with and . The functions can be also evaluated in the thermodynamic limit, where they become polylogarithmic functions grad (); abr (); nist ().

The spectrum Eq. (2) displays the critical point for every and the critical point for . The ground state of Eq. (1) is given by , with , while the ground state energy density is given by . We remind that no Kac rescaling libro () is needed for the Kitaev Hamiltonian of Eq. (1), since remains finite in the limit (see also Appendix D).

Iii Critical effective theory

In this Section we derive an effective theory for the long-range paired Kitaev chain Eq. (1) by using a RG approach, valid close to the critical lines .

First we focus on a region in proximity of the critical line , where the minimum of the spectrum in Eq. (2) is at (later in the text we will discuss the case ). For the Hamiltonian of Eq. (1) reads


where acts on the space of Nambu spinors and the are the Pauli matrices.

We now proceed to the RG procedure, which for the long-range Kitaev chain takes advantage of the quadratic nature of the model. We refer to Ref. continentino () for an exact RG treatment of the short-range Kitaev chain. Starting from a theory having an energy cut-off (here equal to ), the standard strategy divides in the following three steps huang (); shankar (): a) decimation: integrate out high-energy portions of the momentum space, that is in our model between and (with ). The resulting Hamiltonian is denoted by ; b) rescaling: restore the old integration domain for the momenta, by redefining ; c) renormalization: reabsorb the effect of in the normalizations of the fields and in the parameters appearing in .

The standard prescription for decimation described above would lead to write:


after integrating out the momenta far from the minimum of the spectrum at . However decimation must be performed with particular care for Eq. (1). Indeed one should not discard the contributions of the high-energy modes at the edges of the Brillouin zone, where singularities develop (in the -th derivative of the spectrum , labeling the integer part of ) due to the long-range nature of the model encoded in . Indeed previous works nostro (); paperdouble () have shown that these high-energy modes heavily affect various basic properties of Eq. (1). In particular they determine the peculiar hybrid decay of the correlation functions and the anomalous scaling of the ground-state energy density at criticality (see Section I and Appendix D). Thus we proceed keeping the contributions of the modes at the edges of the Brillouin zone during the decimation procedure. In Section IV we verify the validity of the proposed RG procedure by comparing the correlation functions obtained from the effective theory with the lattice correlation functions.

This way of performing RG implies a different hierarchy for the quasiparticles weights along the RG, compared to the one generally assumed for short-range systems huang (). Alternative hierarchies have been discussed in the context of RG theory (see Ref. janos () and references therein) and concerning localization in long-range systems with disorder germandis (). We also observe that if had one performed exactly the RG procedure, then the contribution to the effective theory by the modes at the edges of the Brillouin zone would have been automatically correctly reproduced: however, to perform analytically the procedure we resort to expansions of the energy spectrum and this requires to explicitly single out the contribution from the edges the Brillouin zone.

Using the procedure described above, and absorbing the factor in the normalization of the fermionic field, we write instead of Eq. (4):






Here, and have support close to the momentum and , respectively. We exploited the periodicity of the Brillouin zone and assumed in the cut-off for the momenta to be again , with .

In order to perform the rescaling and renormalization steps it is useful to expand and in Eqs. (6) and (7) in powers of (see Appendix A). We obtain from Eq. (6)


where , , and vanishes at criticality (the subscript on and on the constants in the following denotes bare quantities, and it is removed for the corresponding renormalized quantities). Moreover is the rescaled Fermi velocity in and, following the specific form of the tight-binding matrix from Eq. (1), we conventionally choose and (see Appendix B).

Equation (8) is the usual Dirac Hamiltonian, describing, as it is well known, the short-range Ising model at criticality muss (), and it provides the dynamics around the minimum at . A central point of this Section is that the contribution takes instead a different form for and . For one has


where and . The coefficient (not to be confused with the scaling exponent mentioned in the Introduction) is given by


In Eq. (9) the coefficients and () have the expressions given in Eq. (36). These expressions are -dependent, although in Eq. (9) and in the following this dependence is dropped for the sake of brevity. Again in Eq. (9) the sum over the odd ’s is up to the largest integer smaller than . The power in Eqs. (9) and (11) has to be interpreted here and in the following as . This definition comes directly from the two different expansions in -powers series of in Eq. (5), depending on , as discussed in Appendix A.

For we find


The Hamiltonian (8) commutes with those in Eqs. (9) and (11). This fact is at the origin of the good agreement between effective theory results and numerical findings not only at asymptotically large distances, but also at intermediate length scales.

Notice that while in Eq. (8) and Eq. (11) we retained only the leading term in the expansion of in -powers, in (9) also subdominant terms are kept for future convenience. Since these terms are suppressed along the RG flow they will be discarded in the following of the present Section, where RG is analyzed. As discussed in textbooks dif (); muss () and are Majorana fields majorana (); pal (); their canonical quantization is described in Appendix B.

Rescaling and renormalization [points b) and c)] in Eq. (5) proceed now as follow. Under rescaling , transforms as (omitting the primes)


while transforms as


for and as


for . Standard power-counting arguments peskin (); wei1 () show that the field has dimension in mass (with ). Therefore, under rescaling it transforms as . The same scaling law applies to , since the mass terms in Eqs. (8) and (11) have the same functional form. Equivalently, differs from only for the lattice momentum where is centered around ( and respectively).

With these scaling laws we obtain:


(similarly for ) and


In Eq. (15) the -dependence can be reabsorbed by the renormalization transformation (not related with the rescaling law) , also implying . Similarly for in we find: . Assuming again for the same renormalization as for , we obtain for Eq. (16): and . A different choice for the renormalization of does not change the evolution along RG of the ratio between the renormalized parameters and .

In order to better analyze the RG flow of Eq. (5), we discuss separately the cases and :

  • If , after neglecting the subdominant -powers in Eq. (9), both Eqs. (8) and (9) become Dirac Hamiltonians. If , then for Eq. (8), while for Eq. (9). Since these masses renormalize with the same factor , we conclude then that is dominant against along the RG flow, as an effect of the mass term . The role played by the modes at is suppressed for the dynamics close to criticality in this range of . One thus retrieves the Majorana field theory proper of the critical short-range Ising model.

  • If a direct comparison between the -powers of their kinetic terms shows that becomes dominant with respect to along the RG flow. Thus, due to the contributions at , the RG flow leads to a critical theory different from the Dirac one predicted around the minimum of the lattice spectrum. The analysis of the effective theory shows that one may further distinguish the case and :

    • For the ratio increases along the flow: fixing conventionally , one finds in this unit that , with . More precisely, since , we obtain that stays nonzero also on the critical line , which is massless for the lattice spectrum. The term produces important effects on the dynamics close to criticality: as discussed in Section IV.2, it allows for an asymptotic decay of correlations functions consistent with lattice calculations (Section IV.1). In particular, although , the decay of static correlation functions is still algebraic and with exponents in agreement with the ones derived in Section IV. We observe that if one sets a wrong decay would rather be obtained, providing a check for the soundness of our RG scheme.

    • If , then and tends to vanish along the RG. This behavior for , different from the one at , is a signature of a quantum phase transition arising along a line passing through at , as argued in nostro ().

The RG flow of in the three different ranges for is plotted in Fig. 2.

Figure 2: Typical evolution for under RG flow (encoded in the dependence on the decimation scale ) for , , and . The origin of the axes is in , where

We notice that Hamiltonians (or corresponding actions) with non integer -exponents, similar to Eqs. (9)-(11), have been presented in literature as effective theories of long-range models (see e.g. Refs. defenu (); Maghrebi2015 () and references therein). However to our knowledge the example in this paper is the first one in which is obtained by RG from microscopics for a quantum long-range model.

Finally, in one dimensional quantum systems with continuous symmetries a RG behavior as the one described in the present Section is expected related with the breakdown of the Mermin-Wagner theorem MW (); lebellac (), generally occurring at small enough values of dyson1969 (); thouless1969 (); anderson1970 (); cardy1981 (); frolich1982 (); lui2001 (); lui1997 (); Magh_bis ().

Iv Two-points correlation functions

In this Section we compare the two-fermions correlation functions computed from the effective theory close to the critical line with the ones directly computed on the lattice model.

iv.1 Lattice results

The lattice Hamiltonian in Eq. (1), being quadratic, allows for a straightforward computation of the correlation functions. We discuss in the following the behavior of the lattice correlation , referring to nostro (); paperdouble () and Appendix E for more details. Other correlation functions, as density-density ones, can be built from and (also calculated in Appendix E) by Wick’s theorem.

In the limit , reads


with . We focus on the range and at the beginning outside of the critical line . There the explicit calculation of the integral in Eq. (17) gives


with and

  • if ;

  • if ;

  • if .

The parameters and are zeros of and depend on and in an implicit way. and denote the Gamma and the Riemann zeta functions, respectively. The expression is just a multiplicative constant whose expression can be extracted from the results of Appendix E.

The first contribution in Eq. (18) decays purely exponentially outside of the massless line and it is due to momenta close to the minimum of the energy, at . The second contribution decays instead algebraically and it originates from the higher energy momenta around . When the two contributions become of the same order, the change of decay from exponential to algebraic takes place cirac05 (); paperdouble (). We notice that the algebraic tail occurs in the presence of a nonzero lattice mass gap, as found also in cirac05 (); tagliais (); nostro (). This hybrid behavior has also been observed in the Ising model with long-range interactions cirac05 (); paperdouble () and seems to be general for long-range systems.

Since it will be useful in the following, we write down the explicit expression for :


The asymptotical behavior of Eq. (19) can be calculated by integrating the main contribution of in the limit Ablowitz2003 (). This term can be evaluated exploiting a series expansion of entering in , as in Section III. The result is


where and are complex factors, while is real: their expression can be deduced from the expansion of given in Appendix A. In this way the term does not contribute to the imaginary part of in Eq. (19). In Eq. (20) the symbol labels terms with integer -powers larger than , not contributing to , as well as the higher non integer powers, whose contribution to is suppressed in the limit . Notice also that if the leading term for in the denominator of Eq. (20) is the one (with again ).

Exactly at criticality a similar calculation starting from Eq. (17) yields:


with (this decay, due to the modes close to the zero point of the energy spectrum, is expected, being the same as for the short-range Ising model at criticality) and trivially the same as in Eqs. (18), points , with . It is important to notice that for the dominating term for in the limit is , while for both and decay as in the same limit.

iv.2 Effective theory results

The analysis performed in Section III can be summarized as follows. Close to the critical point , the effective theory can be described as the sum of two actions:


with the Euclidean Dirac action corresponding to Eq. (8) and the actions corresponding to Eqs. (9) and (11) for and , respectively. For the Euclidean action corresponding to Eq. (11) reads (we set after global redefinition for the fermionic fields and without any loss of generality):


In order to extend the interval of integration in Eq. (16) to the whole real line, as in Eq. (23), we reintroduced the lattice spacing by replacing with , and then took the limit . The notation using the fractional derivative means that the inverse propagator of the effective action in Fourier space depends on , as customarily done in the treatment of long-range systems defenu ().

In this Subsection we calculate the correlation around using the effective theory in Eq. (22), and compare it to the lattice results previously given. From the effective theory we expect to reproduce the large separation behavior of the corresponding lattice correlations.

Depending on , and become dominant along the RG flow respectively for and . However the two terms in the action do not decouple completely, at least out of criticality, and they co-operate to determine all the dynamical quantities (as correlations or entanglement properties). In other words, the correlations have to be computed using the effective theory renormalized at the momentum scale . Only afterwards the quantity can be sent to infinite.

Since the actions in Eq. (22) are commuting, is composed again by the sum of two contributions. Let us start from the non critical correlation functions. The first contribution from is well known muss () to have in the massive regime the same decay (exponential) of the lattice term (see Subsection IV). The second one from can be evaluated by the propagator in the Minkowski space-time corresponding to the action (23):


(as in (23), we set ). Multiplying by both the numerator and the denominator of Eq. (24), exploiting the standard residue technique to perform the integration on and following the usual Feynman -prescription peskin (), we obtain the time-ordered correlation


with and . The symbol denotes the ground state of the Hamiltonian from .

We first focus on the two point static correlation function obtained setting in Eq. (25). In this case, as discussed in Appendix B, the matrix propagator Eq. (25) is the continuum equivalent of the lattice correlation matrix: .

Our goal is to evaluate the limit of Eq. (25) and compare it with the large distance behavior of the lattice correlations and found in the previous Subsection and in Appendix E. In the limit , if (), the dominant part of Eq. (25) is


since the term proportional to , involving -derivatives of Eq. (26), gives a next-to-leading-order contribution. This latter term corresponds to the anomalous correlation function , while Eq. (26) corresponds to . The term in Eq. (26) is the effective equivalent of Eq. (17) in the continuous space limit and , as clear from the matrix structure of the propagator in Eq. (24), and with from the renormalization of . By the substitution one gets


no additional term occurring due to the analytical continuation. The imaginary factor in Eq. (27) is matching the asymptotical limit for of in Eq. (19). Repeating the same calculations as for the lattice contribution in Eq. (18) (see Appendix E), it is easy to show that Eq. (27) reproduces the power-law behavior for denoted above by . This result confirms the correctness of Eq. (23) and the role of the high-energy modes at the edges of Brillouin zone for the critical dynamics of the Hamiltonian (1).

If the first two terms in Eq. (25) map into each others after the transformation . Repeating the same calculation done for , we have that also in this case reproduces the lattice decay behavior . Notice that, since in this range the energy spectrum diverges for , the theory (23) is well defined only since this divergence is integrable, a fact closely related to the lack of necessity for the Kac rescaling on the lattice (see Section II).

The same asymptotical decays and can be obtained as well directly from the first term in Eq. (25), taking respectively the limits and and integrating the leading terms of the resulting expansions (see Appendix F). Summing up, the obtained agreement between the lattice correlations in Eq. (18) and the ones from the effective theory in Eqs. (22) and (23) is a check of the correctness of Eqs. (22) and (23) themselves.

For , from Eq. (9) we find that the dominant part of is


and being real factors and where the symbol indicates terms with even integer -power exponents between 2 and (see Formula 37). The same calculation as before leads to an expression in agreement with the power-law decay found on the lattice.

We observe that the terms with integer -powers in the denominator of Eq. (28) do not contribute to , similarly to Eqs. (19), (20) and (27). Therefore not including in Eq. (28) the term would give rise to a second exponential tail, in disagreement with the lattice result . Thus we arrive to the remarkable conclusion that, although the terms with -power exponents larger than in Eq. (28) are strongly suppressed along the RG flow approaching the critical line , their effects are appreciable in nonlocal quantities, such as correlation functions at very large separations. Notably to discard these terms amounts to retain only the leading order for the -expansion in the Hamiltonian (9).

From the analysis in the present Section, we conclude that keeping RG-subdominant terms is necessary in general to correctly compute nonlocal quantities of long-range systems as the two point static correlation functions. However a general strategy to exactly take into account their weights in the effective theory is still missing. For instance, keeping in the Hamiltonian (9) only the terms and , or even only the one , would have led to the same asymptotical decay for the correlation above . A complete solution of the problem requires more constraints on the effective theory, beyond the mere reproduction of asymptotical static correlation functions, or an exact RG treatment, as done in continentino () for the short-range limit. Notice that RG subleading terms do not affect the phase diagram of the model (1), in particular the appearance of new phase(s) at .

Agreement for the exponents of the power-law decay of the correlation functions between lattice results and effective theory is also found exactly at criticality (), where the contribution from is well known muss () to decay , as in Subsection IV. The contribution coming from to the correlation function at criticality can be seen to have the same exponent of .

Following the same procedures described above, it is straightforward to show that for every the decay exponents of the non-diagonal parts of the propagator are in agreement with the ones of the anomalous correlations (calculated in Appendix E), both out and exactly at criticality. This confirms as well the correctness of our approach.

iv.3 Discussion

In the previous Subsection we examined the large distance behavior of the two fermions static correlation functions computed from the effective theory (22). The action gives both at and near criticality a power-law contribution to them, while Dirac action gives an exponential contribution near criticality and a power-law one at criticality. At criticality the power-law contributions from are negligible at large distances for every , in the sense that its decay exponents are larger than the exponents coming from the Dirac action.
Using the total action we were able to reproduce:

  • both the exponential and power-law contributions to the correlation functions out of criticality;

  • both the the exponential and power-law contributions to the correlations exactly at criticality.

A possible objection to our approach is that one could instead use only to reproduce the correlation functions. However, using only would have reproduced the leading decay behaviors at large separations only outside of the critical point, while one would have failed to reproduce the dominant terms at criticality, which are rather coming from . This implies that if one wanted to use only one of the two terms in , he would need to choose out of criticality and exactly at criticality, which is a rather ad hoc approach. The advantage of using both the terms in the action is then that both near and at criticality all the leading contributions of the correlations are reproduced in a natural way.

Another related important remark is that a careful inspection shows that in the range other exponentially decaying terms develop in general from Eq. (26), that means using . An explicit example is given in the Appendix G. To ask whether the action alone can reproduce the hybrid decay for the static correlation functions, at least in the range , amounts at the first level to ask whether such exponential terms may become algebraic at the critical point. One can see that that this not the case. Indeed for exponential terms from the Dirac action , the correlation length depends on the mass in the same action and it diverges at the critical point. Conversely, the correlation length in the exponentially decaying terms from Eq. (26) depend on (see Appendix G), and then they do not become algebraically decaying at criticality.

The latter observation enforces the picture drawn in the previous Sections, involving two commuting actions and , jointly with the fact that out of no exponential decaying term is found from (see Appendix G). Nevertheless in principle a residual possibility to overcome the problems described above and obtain hybrid static correlations from alone would rely on a possible dependence of on the Dirac mass , instead of on . A similar possibility has been considered in a very recent work Maghrebi2015 (), dealing with causality in long-range critical systems. There two dispersion contributions and (typically used in RG treatments of long-range systems cardybook (); sak_1973 (); dutta2001 (); defenu ()) are present for every in an effective action with a single mass term, allowing for a discussion of the correlation functions Maghrebi2015 (). To make a comparison with the present paper, we notice that in our approach, at variance, the two contributions and (the latter one containing the term ) derive respectively from the modes at the minimum of the energy spectrum and at the edges of the Brillouin zone, and that both the action terms have their own mass, scaling differently if . Assuming our point of view for the implementation of the RG implies that the choice in Maghrebi2015 () leads to mix the dispersions of the two sets of quasiparticles. This may be significant for (a case not treated by the authors in Maghrebi2015 ()), where the energy of the ground state is still extensive in the thermodynamic limit. Indeed a preliminary computation seems to indicate that extending the effective action introduced in Maghrebi2015 () to does not reproduce the correct area-law violation for the von Neumann entropy at and out of criticality (see Section VI). This fact points out to the need of a systematic comparison of the two effective actions and their predictions for the quantities of physical interest.

V Breakdown of the conformal symmetry

The action in Eq. (23), dominant term in Eq. (22) for , breaks explicitly the conformal group. This can be shown analyzing the behavior of Eq. (23) under the global part of the conformal group dif (); muss (). Summing up the results of this analysis, when , is increasing along the RG flow, thus one concludes that CS is broken by the mass term . Another source for the CS breakdown is the anomalous exponent for the spatial derivative in Eq. (23), , as detailed in Appendix C. For the same reason the CS breakdown arises also at , even if .

We observe that these results are consistent with our findings for the scaling of the ground state energy density, as discussed in detail in Appendix D: below the scaling law predicted by CS dif (); muss () starts to fail nostro (); paperdouble ().

Although the action is dominant with respect to for , a complete decoupling between the two actions cannot occur in this range, even exactly at criticality. This non complete decoupling is suggested by the fact that at criticality and for every the leading part of the lattice correlation functions in the limit is reproduced by the contribution from the Dirac action . Again we find that RG subleading contributions to the total effective action for the critical model affect its physical observable not only quantitatively. Conversely, for at criticality flows to zero along the RG and the leading contributions to correlation functions for come from . Nevertheless, although near criticality is dominated by , the leading contributions to the correlations come from .

We also notice that below , a new symmetry emerges for as , , provided the corresponding transformation holds. Notably this scaling law for still matches the one in the presence of CS (see Appendix C) and with the scaling analysis in Section III. The appearance of this symmetry, approximate because of the subleading presence of , is a further indication of a new phase (or two new phases on the two sides of the line ) below , as discussed in Section III.

v.1 Breakdown of the effective Lorentz invariance

It is known that for a Lorentz invariant theory a two point static correlation function must decay exponentially in the presence of an energy gap peskin (); fred (); nota2 (), since in this condition two space-like events can be correlated only exponentially peskin (). It is then clear that the hybrid exponential and algebraic decay of the static two points correlations in gapped regime and for every finite described in Section IV has to be related with the breakdown of the ELI, by Eq. (23), as well as by the action related to Eq. (9). This fact, likely general for long-range systems, is non-trivial, since, at variance, for short-range models ELI emerges for the effective theories around their critical points muss (). For one has that and the ELI is then restored in the short-range limit.

Intuitively, for our long-range model close to criticality Lorentz invariance breaks as follows. The Hamiltonian of the classical lattice model corresponding to Eq. (1), derived by means of an (inverted) transfer matrix approach muss (), has long-range terms in the direction and short-range terms in the direction. While the exact calculation to prove this fact appears to be difficult starting directly from Eq. (1), a similar conclusion can be inferred more easily analyzing the long-range Ising model

By a straightforward calculation, it is easy to check that this model translates in the classical system:


where are classical variables and labels the direction. Independently from the exact values of and , Eq. (29) displays the strong anisotropy mentioned above: this anisotropy cannot be reabsorbed entirely along the RG flux, at least for small enough, and thus the breakdown of rotational symmetry close to criticality may emerge. This fact amounts to have a mechanism to break Lorentz invariance, since rotations in a D Euclidean space correspond to Lorentz transformations in a Minkowski space-time obtained from it by the Wick rotation. It is clear that this argument is qualitative, and it does not fix the precise value of at which Lorentz invariance breakdown happens in the long-range Ising models around criticality.

Exactly at criticality CS and ELI are closely related, since the conformal group contains, as a global subset, the Euclidean rotations dif (); muss () (see also Appendix C). In spite of this fact, CS and (near criticality) ELI are not directly connected in general, since Lorentz invariance breaking terms possibly present in the non critical effective theory can vanish at criticality, due to the RG flow. Our model is an instance of this latter situation, since for out of criticality the system does not have ELI but it does (and it has as well CS) at the critical point. Notice however that the ELI for is broken in a ”soft” way, since in the RG sense. Our findings are summarized in Table 1.

Critical point Near criticality
CI and ELI Non ELI
Non CI and Non ELI Non ELI
Table 1: Summary of obtained results as a function of at and near criticality for the critical line : CI stands for Conformal Invariance, and ELI for Effective Lorentz Invariance.

Vi von Neumann entropy and area-law violation

In this Section, we focus on the behavior close to the critical line of the von Neumann entropy after a partition of the system into two subsystems containing and sites respectively. The von Neumann entropy is defined for a chain as , being the reduced density matrix of the subsystem with sites. A study of the von Neumann entropy in fermionic chains with long-range couplings is given in gori_15 ().

In particular, we analyze the deviations from the so-called area-law, that is generally valid for short-range gapped systems eisert () and states that saturates quite rapidly increasing . This fact is closely related to the short-rangedness of the entanglement between different points of the system. In Refs. tagliais () and nostro () it was found instead a deviation from this law for long-range systems outside of criticality, resulting in a increasing logarithmically with , as for critical short-range systems.

In this Section, will be derived out of criticality for , using the correlation functions computed in Section IV.2 by means of the effective theory. We refer to Ref. ares () for an analytic computation of in the long-range paired Kitaev chains based on the properties of Toeplitz matrices.

We focus at the beginning on the contribution to from , . This quantity can be calculated following Refs. callan (); casini (); peschel () via the formula




with , . Eq. (31) is obtained through a discretization of (25). The von Neumann entropy is then fitted by the formula , expected in critical short-range one-dimensional systems and introduced for long-range systems in Ref. tagliais () also near criticality.

We checked, as a first step, that the extracted values for converge quite rapidly with increasing, as discussed in Refs. callan (); casini (); nez (). The results of our study are the following:

  • if . Indeed no dependence arises in Eq. (31) in this case, so that we recover the value for a massless Majorana theory, as for the critical short-range Ising model wilczek (); calabrese () at ;

  • if : for the same reason as above, no dependence occurs in Eq. (31) and saturates, increasing , to a constant value, as for gapped short-range systems;

  • if is finite, passes abruptly (for ) from (as when ) to (as when ). The change of value for arises at a certain critical rapidly increasing with .

Recalling that for and for (as seen in Section III), we conclude that near criticality if and if .

The contribution to from the Dirac action is known calabrese () to scale as with in the massive regime. Thus, we can conclude that the parameter