Exact asymptotic correlation functions of bilinear spin operators of the Heisenberg antiferromagnetic spin- chain
Exact asymptotic expressions of the uniform parts of the two-point correlation functions of bilinear spin operators in the Heisenberg antiferromagnetic spin- chain are obtained. Apart from the algebraic decay, the logarithmic contribution is identified, and the numerical prefactor is determined. We also confirm numerically the multiplicative logarithmic correction of the staggered part of the bilinear spin operators , and estimate the numerical prefactor as . The relevance of our results for ground state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless quantum phase transition points in one-dimensional systems is discussed at the end of our work.
pacs:64.70.Tg, 75.10.Jm, 75.10.Pq
The XXZ spin- chain is a paradigmatic one-dimensional quantum many-body system which can be studied using exact methods, while simultaneously describing the magnetic properties of real materials KM (). The Hamiltonian of the XXZ chain, written in terms of spin- matrices, reads,
where is an exchange coupling, that will be assumed to be positive, and is an anisotropy parameter. For , the XXZ chain reduces to the Heisenberg antiferromagnetic (AFM) chain.
Despite being exactly solvable, calculating correlation functions (objects that provide direct connection between theoretical calculations and experimental observations) from the microscopic XXZ model for is a formidable task, due to the complicated form of the wave functions Godin (). On the other hand, effective approaches, have allowed asymptotically exact calculation of the spin correlation functions in the gapless regime LZ (); Affleck98 (); Lukyanov99 (); LT ().
Effective theory, describing the low-energy properties of the XXZ spin-1/2 chain for , is given by the Gaussian model GNT (),
where is a real bosonic field with the compactification radius , , and is its conjugate momentum, . Spin-wave velocity and are known analytically as functions of from the exact solution of the model (1) JohnsonKrinskyMcCoy73 ().
In this work, using an effective approach, we determine exact asymptotic expressions of the uniform parts of two-point correlation functions of bilinear spin operators in the Heisenberg AFM spin- chain. Apart from the algebraic decay, we identify the logarithmic contribution and determine the exact numerical prefactor. Our calculations are similar to the ones that were performed by Affleck for obtaining exact asymptotic correlation functions of single-spin operators in the Heisenberg antiferromagnetic chain Affleck98 () by combining renormalization group (RG) improved perturbation theory with the exact asymptotic results of Lukyanov and Zamolodchikov LZ () conjectured for . However, in the case of the correlation functions of the bilinear spins, there are various operators of effective field theory that contribute equally at the antiferromagnetic point, though the exact asymptotic expression is known for for the correlation function involving only one of them LT (). We have to use additional symmetry arguments to obtain exact asymptotic expression of the complete uniform parts of the bilinear spin correlation functions at .
We also identify numerically the multiplicative logarithmic contribution of the staggered (leading) parts of the bilinear spin correlation functions, consistent with analytical prediction Affleck+89 (), and estimate the numerical prefactor.
Ii Single-spin correlation functions
Asymptotic expressions of the single-spin correlation functions of the XXZ spin- chain, , where and no summation with repeated indices is implied in this work, are known exactly in the gapless phase, including the numerical prefactors LZ (); Lukyanov99 (); LT ()
where and for .
These amplitudes appearing in Eqs. (3) have been checked numerically HF98 (). Amplitudes and diverge in the isotropic AFM limit, , since in this limit mapping of the spin- chain to the Gaussian model becomes singular due to the marginally irrelevant (cosine) term with the scaling dimension occurring in the low-energy effective theory from the “spin umklapp” processes Haldane (); BE (); dN (); Affleck85 ().
Effective theory description of the XXZ spin- chain, for , necessarily contains terms beyond the Gaussian model,
where and . The running coupling constants , with the bare values , are governed by the Kosterlitz-Thouless Kosterlitz74 () RG equations,
where the dot indicates a derivative with respect to the RG scale and is the running ultraviolet cutoff. At the AFM point , and the exact expression of amplitudes of the asymptotic correlation functions of spin operators were derived Affleck98 (); Lukyanov98 () by combining the expressions of LZ () and Lukyanov99 () for with RG improved perturbation theory Cardy86 (); Affleck+89 (),
Iii Bilinear spin correlation functions
We will generalize the approach leading to the exact assymptotic expressions of single-spin correlation functions at the Heisenberg AFM point Affleck98 (); Lukyanov98 (), Eq. (6), for the calculation of the uniform part of the correlation function of bilinear spin operators, .
Let us first address the correlation function of the bilinear in operator in the gapless region in the vicinity of (but not directly at) the AFM point. Up to the subleading corrections we have
The ’s, for , are amplitudes of the correlation functions of the following operators,
and is, similarly to other proportionality coefficients in (III), an -dependent factor such that
The scaling dimension of is , while those of and are and , respectively. In the limit of the AFM point and all of them become marginal.
The constant term in Eq. (7) can be easily fixed due to the translational symmetry, , where is the ground state energy density known exactly (together with its dependence on ) from the Bethe ansatz. However we will be interested in the following with the reduced correlation function,
Namely, the uniform part of the above reduced correlation function is the main quantity of our interest,
For the XXZ chain, for , the exact expression of amplitude has been obtained LT (),
and confirmed numerically away from the points HF04 (). In Appendix A we provide details of calculating , confirming expression (12). However, when , the expression for is only valid for evaluating correlations Eq.(11) at exponentially large distances, . In the limit of the AFM point we apply RG improved perturbation theory Affleck98 (). We note that obeys the following RG equation,
where are beta functions presented in Eq. (5) and is the anomalous dimension of the operator, calculated in Appendix B. This allows us to follow the approach Affleck98 (); Lukyanov98 () that led to the exact expression of the single-spin correlation function Eq. (6). Solving the RG Eq. (46) and integrating over in the solution, as shown explicitly in Appendix C, gives for the following behavior over an intermediate range : . Then taking the limit and using the limiting expression of the amplitude in Eq. (12), , we obtain the following exact asymptotic expression for ,
Let us consider now the mixed correlation function of bilinear spin operators at the Heisenberg AFM point,
for . Using bosonization GNT () one can show that the leading staggered part of the mixed correlation function behaves identically to comment0 (). To study the long-distance asymptotics of the uniform part of the mixed bilinear correlation function, for , it is useful to look at the correlation function of the Hamiltonian density,
where , and we will use the important property that the uniform part can not contain multiplicative logarithmic corrections due to energy conservation comment (). Since energy density does not pick up anomalous dimension due to marginally irrelevant perturbations, the correlation function of energy density behaves similarly to the correlation function of the energy-momentum tensor of the unperturbed conformally invariant Gaussian or Wess-Zumino model comment01 (), thus . This means that logarithmic contributions, such as in Eq. (14), all must be canceled by the mixed terms. Hence, at the AFM point, for the leading behavior of the uniform parts of the bilinear spin correlation functions, we obtain,
This exact asymptotic expression is our main result.
Iv Numerical results
In the remaining part we will present a numerical check of Eq. (18) based on our results obtained from the density matrix renormalization group (DMRG) method White (); Uli () implemented for systems with periodic boundary conditions.
Directly from the computation of the reduced bilinear spin correlation function Eq. (10) it is hardly possible to analyze the space dependence of its uniform part. The reason is that the reduced correlation function , is strongly dominated by the leading term, its staggered part, which is expected to behave as Affleck+89 (), and such decay is much slower than that of the uniform part (). We have performed numerical simulations of for different system sizes, ranging from (Lanczos) to and sites (DMRG), assuming periodic boundary conditions. In Fig. 1 we present the behavior of the reduced correlation function of the bilinear spin operators for the Heisenberg spin- AFM chain with and sites. We use conformal mapping of an infinite 2-dimensional plane on a cylinder comment01 () with finite circumference in the spatial direction to compare the analytic results for the thermodynamic limit with finite-size calculations for the systems with periodic boundary conditions. This implies that the distances are replaced by the chord distances on the circle, .
On the other hand, in the difference , , the leading oscillatory terms cancel comment0 () and from this quantity and Eq. (17) we can obtain the desired uniform part of the correlation function . In Fig. 2 we plot numerical data for , which includes both uniform and staggered components. For the uniform component our analytical result is , following from Eq. (18) and the relation , where dots indicate sub-leading contribution. We will calculate the leading oscillatory contribution in
In bosonization , where , up to sub-leading contributions. The anomalous dimension of is , giving
Hence, including the leading oscillatory contribution in , we obtain
where is a numerical constant estimated from fitting to DMRG data. We present in Fig. 2 comparison of our analytical curves, obtained separately for even and odd from Eq. (19), with our numerical data.
V Relevance for fidelity susceptibility
The asymptotically exact expression of the uniform part of the correlation function of the bilinear spin operators at , Eq. (18), confirms our previous work Vekua1 (), showing that computing the ground state fidelity susceptibility of the XXZ spin- chain by the effective Gaussian model gives a qualitatively wrong result at the Berezinskii-Kosterlitz-Thouless quantum phase transition point. Moreover, our approach allows us to explicitly follow the steps on how the divergence in fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless transition point arises in the thermodynamic limit due to the singular nature of the mapping of the Heisenberg spin- AFM chain on the Gaussian model and is not a property of either the microscopic or effective models. When taking into account marginally irrelevant corrections to the effective Gaussian model and resumming perturbation series with the help of the RG, the spurious divergence of fidelity susceptibility disappears, as explained in Appendix D.
Using an effective field-theory approach, exact asymptotic expressions of the uniform parts of the biliniar spin correlation functions of the Heisenberg antiferromagnetic spin- chain, and , have been computed. We have checked numerically analytical results and also estimated the numerical prefactor in front of the staggered part of the bilinear spin correlation function and identified the logarithmic contribution in accordance with the previous analytical investigations Affleck+89 ().
As a by-product, our studies confirm the finiteness of the ground state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless quantum phase transition points in one-dimensional systems.
This work has been supported by DFG Research Training Group (Graduiertenkolleg) 1729 and Center for Quantum Engeneering and Space-Time research (QUEST). We thank A. Furusaki, T. Hikihara, A. K. Kolezhuk, S. Lukyanov, and M. Oshikawa for helpful discussions.
viii.1 Calculating constant for
We provide details of calculating the exact expression of constant appearing in Eq. (12) in the main text. For this we introduce the Hamiltonian of the fully anisotropic XYZ spin- chain,
We will assume . Denoting , , and , we rewrite the Hamiltonian of the XYZ chain as follows,
We will put and measure energy in units of . Spin-wave velocity for the gapless, , case is
where is the lattice constant and . It is convenient to fix the spin-wave velocity equal to unity (hence also make dimensionless) for , independently of . For this, we will fix
In the following we will use the exact solution of the XYZ chain Baxter82 (); JohnsonKrinskyMcCoy73 (). In particular we will be interested in the limit , and take the so called scaling limit of the XYZ chain, where the spin gap behaves as Luther76 ()
where and to arrive from the second to the third line we used Eq. (22). The scaling limit is a continuous limit of the lattice model, , with additional requirements: the velocity Eq. (21) stays equal to unity and the gap Eq. (23) stays constant; hence so that . In this limit, the effective theory describing the XYZ chain is a massive relativistic sine-Gordon model Luther76 (),
To give explicit meaning to one has to specify normalization of fields. We will follow the approach developed by Zamolodchikov Zamolodchikov (), where the dimension of field and the fields are normalized as follows at short distances, where perturbation is irrelevant comment ():
Explicit connection between the coupling constant and the soliton mass of the sine-Gordon model was obtained Zamolodchikov () by using the Bethe ansatz integrability of the sine-Gordon model in external uniform gauge field, with amplitude , coupled to the conserved current JNW () and viewing the same model as a conformal field theory (Gaussian model for ) perturbed by a cosine term. In the Bethe ansatz approach, the ground-state energy of the quantum sine-Gordon model in strong external field, , can be expanded in a dimensionless parameter, the ratio of the soliton mass to the field amplitude . On the other hand, when viewing the sine-Gordon model in strong external field as a perturbation of conformal field theory with cosine field, the ground state energy, , can be expanded in the powers of dimensionless parameter . Note that without strong external field, ground-state energy can not be perturbatively expanded in , due to the infrared divergent integrals characteristic of the relevant cosine term. Matching the two ground-state energies in the first nontrivial power of gives
where the dimensionless parameter is called the ratio (proportionality constant between ultraviolet and asymptotic scales) and is given by Zamolodchikov ()
We wish to determine a proportionality constant ,
in order to obtain a precise value of the constant in the operator identification of bilinear spin operators in the scaling limit,
Equating gives us the following equation
Using the following property of functions,
Note that due to the symmetry at ,
Also note that at the effective theory enjoys conformal invariance and hence a unique normalization of correlation function is carried to all distances Eq. (25). Finally we obtain for ,
This expression agrees with the one obtained in [LT, ].
We note that one cannot use the effective representation of single-spin operators GNT () to obtain the short-distance correlation function for the XXZ spin- chain and in particular to obtain exact amplitudes of the correlation functions of bilinear spin operators with the fusion rules of underlying conformal theory. This is so, because conformal symmetry is only an effective property of the model and at short distances the XXZ chain is not conformally invariant, because of irrelevant, in infrared limit, corrections (the leading ones can be found in [Lukyanov98, ]). Due to this reason the constant is not related to coefficients and appearing in Eq. (3) of the main text in any simple way and also we cannot determine the exact numerical prefactor in front of the staggered part of the bilinear spin correlation function [the coefficient in Eq. (7) of the main text cannot be fixed with currently known methods].
viii.2 Calculating anomalous dimension
In this appendix we show how to calculate the anomalous dimension of the field picked up upon renormalization due to marginally irrelevant perturbations of the Gaussian model.
In the absence of perturbations, for , the effective theory given by Eq. (4) in the main text has conformal invariance, and hence
When marginally irrelevant perturbations are included on top of the Gaussian model, does not contribute to the anomalous dimension of the field to first order, since
Hence, at the lowest (first) order in , we can include into the quadratic part of the action and obtain
Perturbation can be included into the quadratic part of the action independently of the strength of . It is the strength of that must be small in order to use the anomalous dimension obtained from perturbative analyses at the lowest order.
From Eq. (39) we read off the anomalous dimension of the field at the lowest order in ,
Using the fixed-point value of in Eq. (39) reproduces the dependence of in Eq. (7) of the main text, .
Next we provide the details of calculating the exact long-distance asymptotics of at the antiferromagnetic point, given in Eq. (14) of the main text.
viii.3 RG improved perturbation theory approach for long-distance asymptotics of
Our aim is to compute the two-point correlation function for the effective action with the bare coupling constants (which carry information of the initial microscopic lattice model) . However, since the Hamiltonian is not Gaussian, one has to use some approximate methods for computing correlation functions. If one tries to perform a perturbation theory calculation in coupling constants, a standard method of interacting field theory, because of the logarithmic divergences that occur in the infrared limit, one cannot stop perturbative series at some finite order, even if initially . For example, in our case, the first order in the coupling constants contribution in the correlation function comes with ,
Combining this correction with Eq. (37) we obtain, up to the first order in coupling constants,
Hence, the effective expansion parameter of perturbation series increases logarithmically at large distances, .
RG is a way to resum the leading logarithmic divergences of the infinite perturbation series occurring in the limit. One can obtain from Eq. (42) directly at the antiferromagnetic point the double-logarithmic correction of the correlation function as follows. At we have and considering it as a small perturbation the following connection between the bare and renormalized couplings exists from the one-loop beta function,
Hence to the lowest order in coupling constant we can make a substitution,
and represent Eq. (42) in the following form,
From Eq. (43), at large distances, , and plugging this into Eq. (45) produces multiplicative double-logarithmic correction of the algebraic decay of the correlation function . Note that if the anomalous dimension of the operator does not depend on the coupling constants (which is the case for conserved quantities) there will be no multiplicative logarithmic corrections in the corresponding correlation function.
Moreover, apart from the logarithmic correction we can even determine the precise numerical prefactor, by comparing with the exact results for LT (). Since in the infrared limit the running coupling constant flows to zero, one can estimate the (renormalized) correlation function at large scale, from perturbative expansion in .
We note that obeys the following Callan-Symanzik (CS) RG equation,
where , are their beta functions presented in Eq. (5) of the main text and is the anomalous dimension of calculated in the previous section. The CS RG equation is equivalent to the one presented in the main text Eq. (13), up to the sign in front of the anomalous dimension (due to the fact that increase of the short-distance cutoff is equivalent to decreasing the distances measured in units of the new cutoff).
CS Eq. (46) defines the evolution of the two-point correlation function under variation of the length scale at which the theory is defined. Since the effective theory is derived from the original microscopic lattice model, the initial length scale is given by the lattice constant and is increased in the RG process of gradually eliminating high-energy degrees of freedom.
The following connection between the bare and renormalized correlation functions is provided by the CS Eq. (46),
We will choose large enough, so that can be expanded in powers of . This step is called the RG improvement of the perturbation theory. The zeroth-order term, evaluated by Gaussian fixed point action, with rescaled cutoff is
Note that increasing cutoff from to is equivalent to decreasing distance (measured in new units) by the same factor, .
Since at the lowest order , we need the solution of the Kosterlitz-Thouless RG equations (5), presented in the main text, only for ,
In obtaining Eq. (50) we used the following table integral and put . As an artifact of the finite order perturbation theory approximation (for correlation function, beta functions and anomalous dimension), in Eq. (50) contains some arbitrary number .
Choosing the ( dependent) normalization constant in such a way that the leading behavior of , for distances , becomes identical to that shown in Eq. (11) of the main text, we obtain for over an intermediate range of distances ,
Number in Eq. (51) can be estimated by going to higher order approximation in perturbation theory LT (); Lukyanov98 (); however it will contain some fitting constant depending on the order of truncation of the perturbative series. If the complete perturbation series could be summed then the correlation function naturally would not contain any fitting parameter.
viii.4 Ground state fidelity susceptibility at the Berezinskii-Kosterlitz-Thouless quantum phase transition
In quantum many-body systems, at zero temperature, phase transitions can be encountered when changing strength of the certain term in the Hamiltonian, . In the finite-size computational studies, a quantity that can be sensitive to the rapid change of the ground state is the overlap of two ground states at slightly different values of the parameter , ZanardiPaunkovic06 () and is called a ground-state fidelity. The ground-state fidelity susceptibility per site (FS) You+07 (); ZanardiGiordaCozzini07 (); Venut