Particle partition entanglement of bosonic Luttinger liquids
We consider the Rényi entanglement entropy of bosonic Luttinger liquids under a particle bipartition and demonstrate that the leading order finite-size scaling is logarithmic in the system size with a prefactor equal to the inverse Luttinger parameter. While higher order corrections involve a microscopic length scale, the leading order scaling depends only on this sole dimensionless parameter which characterizes the low energy quantum hydrodynamics. This result contrasts the leading entanglement entropy scaling under a spatial bipartition, for which the coefficient is universal and independent of the Luttinger parameter. Using quantum Monte Carlo calculations, we explicitly confirm the scaling predictions of Luttinger liquid theory for the Lieb-Liniger model of -function interacting bosons in the one dimensional spatial continuum.
Present Address: ]Institute for Quantum Computing, University of Waterloo, Ontario, N2L 3G1, Canada
Entanglement is a fundamental non-classical property of quantum many-body systems, and provides a physical resource for quantum information processing Horodecki et al. (2009). The quantification of entanglement via entanglement entropy (EE) has been demonstrated to aid in the characterization of quantum matter Amico et al. (2008); Eisert et al. (2010), especially when the conventional use of local correlation functions may not be feasible due to a lack of symmetry breaking. In particular, the study of the EE between two spatially distinct regions (partitions) of a many-body system allows the canonical “area law” scaling to be probed, including its sub-leading corrections which have been demonstrated to reveal fundamental features of the underlying phase of matter Srednicki (1993); Hamma et al. (2005a, b); Levin and Wen (2006); Kitaev and Preskill (2006); Fradkin and Moore (2006); Casini and Huerta (2007); Metlitski et al. (2009); Stéphan et al. (2009); Hsu and Fradkin (2010); Zaletel et al. (2011); Metlitski and Grover (2011); Kallin et al. (2011); Ju et al. (2012); Stéphan et al. (2012). Systems of itinerant particles can alternatively be bipartitioned into subsets of particles rather than spatial regions Zanardi (2002); Shi (2003); Fang and Chang (2003); Zozulya et al. (2007); Haque et al. (2007); Zozulya et al. (2008); Haque et al. (2009), with the associated entanglement entropies providing complementary information on the role that interactions and exchange statistics play in generating the non-classical correlations in a quantum state.
The study of EE has been particularly fruitful in low dimensional quantum many-body systems, where strong fluctuations inhibit transitions to conventional ordered phases. For example, the EE of a spatial bipartition of a critical system in one-dimension (1D) scales logarithmically with subsystem size, where the prefactor is proportional to the central charge of the underlying conformal field theory Vidal et al. (2003); Calabrese and Cardy (2004). The low energy behavior of such systems is known to be described by Luttinger liquid (LL) theory, which characterizes the universal long distance physics that is unique to quantum mechanics in one spatial dimension Haldane (1981). A LL is characterized by a single non-universal dimensionless parameter which determines the nature of correlations functions, in addition to a velocity which sets the energy scale for quasiparticle excitations. While the central charge is a universal feature of LLs, depends on the specific details of the system, and in general, is not easily determined for a given microscopic model. Thus the leading spatial entanglement scaling gives access to a universal quantity of the LL, but is not sensitive to which characterizes the long-distance behavior of correlation functions. The Luttinger parameter does appear in the bipartite fluctuations of a Luttinger liquid, which scales logarithmically with subsystem size with a prefactor that is proportional to Song et al. (2010, 2012), as well as higher order corrections to the spatial partition EE Laflorencie et al. (2006) and the EE of a partition into disjoint spatial regions Furukawa et al. (2009); Calabrese et al. (2009).
In this paper we demonstrate that the finite-size scaling form of the -particle partition 2nd Rényi entanglement entropy of a -particle bosonic Luttinger liquid is:
thus the leadinger-order scaling coefficient of the particle partition EE is proportional to the the inverse Luttinger parameter. We evaluate the explicit form of the constant and correction terms for a bosonic LL, and confirm their accuracy for the Lieb-Liniger model of -function interacting bosons in 1D Lieb and Liniger (1963); Lieb (1963) via large scale quantum Monte Carlo simulations.
Ii Quantum hydrodynamics
The ubiquity of Luttinger liquid behavior manifests in its experimental observation in a diverse set of quasi-1D experimental systems including electronic conductance and tunneling in carbon nanotubes Bockrath et al. (1999) and quantum Hall edges Chang (2003) as well as in the energy and correlation functions of low density ultra cold bosonic gases Paredes et al. (2004); Kinoshita et al. (2004, 2005). Recent work has proposed that it may also be seen in confined superfluids of He Del Maestro et al. (2011); Eggel et al. (2011) with associated experiments already underway Taniguchi et al. (2010).
Here, we focus on spinless bosons, and the low energy effective Hamiltonian capturing the hydrodynamics of a LL with interacting particles on a line of length with density may be written in terms of bosonic fields and representing the phase and density fluctuations of a microscopic field operator as:
For Galilean invariant systems, the Luttinger parameter may be computed from the compressibility, as
where is the ground state energy and is the mass of the particles. Even at zero temperature, LLs display no true long-range order; the ground state of Eq. (2) has algebraically decaying correlations parametrized by . In the non-interacting limit, , and phase correlations dominate over density correlations. The reverse is true for impenetrable bosons with . In particular, the one-body reduced density matrix is a power law at long distances, with exponent proportional to :
With the exception of a few models where exact solutions are available Girardeau (1960); Lieb and Liniger (1963); Sutherland (1971), determining the precise value of for a given microscopic model is a difficult task, and may require a detailed analysis of the decay of correlation functions Astrakharchik and Giorgini (2003). We find that this information is encoded in the leading order scaling of the particle partitioned entanglement entropy, and can even be bounded by measurements of the condensate fraction Herdman et al. (2014a).
Iii Particle partition entanglement
Given a bipartition of a system into subsystems and , one can quantify the entanglement in describing the full pure state via Rényi entanglement entropies:
where is the reduced density matrix (RDM) of subsystem ,
and is the Rényi index. For the Rényi entropy is equivalent to the von Neumann entropy: .
Entanglement entropy of many-body systems is most often studied under spatial bipartitions. For example, if a translationally invariant 1D system is bipartitioned into contiguous subsystems, then subsystem is completely characterized by the length of the subsystem . For the ground states of critical 1D systems, the leading order scaling of the EE is logarithmic in Vidal et al. (2003); Calabrese and Cardy (2004),
where is the central charge of the underlying conformal field theory. In systems of itinerant particles, one may alternatively bipartition the system into subsets of particles (Fig. 1), rather than spatial subregions Zanardi (2002); Shi (2003); Fang and Chang (2003); Zozulya et al. (2007); Haque et al. (2007); Zozulya et al. (2008); Haque et al. (2009). An indistinguishable particle partition is uniquely characterized by the number of particles in the subsystem, with the RDM for the subsystem given by the -body reduced density matrix:
where are the positions of the particles and is the full -body density matrix. One can then quantify the entanglement entropies between subsets of particles by the particle partitioned EE Zozulya et al. (2008); Haque et al. (2009):
Note that since is non-local in space and contains no length scale, can capture physics that is distinct from, and complimentary to, the EE under a spatial bipartition.
The importance of particle partition EE, and in particular, its connection to physically utilizable entanglement for quantum information processing applications has only recently been uncovered Wiseman and Vaccaro (2003); Killoran et al. (2014). Zozulya et al. have proposed a general scaling form for particle EE which is linear in the subsystem size and logarithmic in the system size Zozulya et al. (2008):
The coefficients have only been computed in a limited number of models and one may ask what physical content is contained withinSantachiara et al. (2007); Zozulya et al. (2007); Haque et al. (2007); Zozulya et al. (2008); Haque et al. (2009). Additionally, previous work has demonstrated that in systems of itinerant bosons, the second Rényi EE is bounded by the experimentally accessible condensate fraction, thus offering a direct relationship between particle EE and experimentally measurable quantities Herdman et al. (2014a).
Iv One particle entanglement in bosonic Luttinger Liquids
iv.1 Asymptotic scaling for
We begin with an analysis of the one particle partition 2nd Rényi EE for a LL described by Eq. (2). Using the Galilean invariance of the one-body RDM, we take the general, appropriately normalized and regularized form of to be:
where we have used Eq. (6), is some short distance length scale and is a dimensionless function satisfying , that depends on the non-universal details of the microscopic model. Performing the trace in Eq. (7) (see Refs. Gritsev et al., 2006; Polkovnikov et al., 2006 for related calculations) for yields
Taking the log, the central finding of our work is that the finite size scaling of the one-particle 2nd Rényi EE for is described by
where and the constants depend on both as well as the short distance physics captured in . We note this result is consistent with previously reported results for the Tonks-Girardeau model of impenetrable bosons, which corresponds to where it was found that Santachiara et al. (2007); Haque et al. (2009). Thus the leading scaling coefficient of the particle EE is equal to the inverse of the Luttinger parameter which characterizes the universal quantum hydrodynamics in one dimension.
iv.2 Finite size corrections
Modifications can be determined for a finite system with periodic boundary conditions by replacing the separation in Eq. (6) with the distance between two points on a ring of circumference Cazalilla (2004); Cazalilla et al. (2011):
Proceeding as above, by assuming a short distance form and by employing normalization and translational invariance we may write:
Performing the trace over now yields:
where is the Hypergeometric function. Thus we find the Rényi entropy to be
Expanding for large and making use of the -function recursion relation: we find:
which exhibits identical scaling behavior as that in Eq. (1) and (13), albeit with different non-universal subleading coefficients . Up to this point, we have only demonstrated the accuracy of Eq. (1) for , but we now conjecture its leading order dependence for a Luttinger liquid under an arbitrary particle partition of size , and numerically test for in a specific microscopic model.
V Confirmation in the Lieb-Linger model
where and is an interaction strength with dimensions of energy length. We consider only repulsive interactions and as the Tonks-Girardeau Girardeau (1960) gas of impenetrable bosons is recovered. It is useful to parameterize finite interactions using a single dimensionless parameter , which has particular experimental relevance to ultracold Bose gases confined in quasi-1D optical traps.Kinoshita et al. (2005) The Lieb-Liniger model is analytically soluble via the Bethe ansatz, which provides access to the exact ground state energy in the thermodynamic limit . Lieb and Liniger (1963) At low energies, its emergent properties are well described by LL theory and moreover, the existence of an exact solution allows for access to the short distance properties of the one-body RDM parameterized here by the function Olshanii and Dunjko (2003).
Olshanii and Dunjko Olshanii and Dunjko (2003) have analyzed the short distance behavior of the one-body density matrix as a power series
where and the constants and were found to be:
with a dimensionless function of the interaction parameter appearing in the ground state energy that can be numerically determined from the Bethe ansatz solution Lieb and Liniger (1963). Both and are decaying functions of the interaction parameter as shown in Fig. 2.
v.1 Quantum Monte Carlo Method
We test our LL calculations for the particle partition EE of the Lieb-Liniger model using a path integral ground state quantum Monte Carlo (QMC) method Ceperley (1995); Sarsa et al. (2000) which provides unbiased access to ground state expectation values through imaginary time projection:
where an observable and is a trial wave function. In this section, for ease of presentation, we set . We Monte Carlo sample the discrete imaginary time worldlines of bosons in one spatial dimension with the -function interactions described by the Lieb-Linger Hamiltonian in Eq. (19). Using a discrete imaginary time representation, we approximate the propagator as the product of short time propagators where . We represent the imaginary time worldlines configurations in the position basis, such that each imaginary time slice is described by a state , where is a vector of length describing the position of all particles in continuous space. The short time propagator is approximately decomposed into the product of the free particle propagator, which can be sampled exactly, and an interaction propagator such that
where is the free particle propagator
To efficiently sample the imaginary time propagator, because of the short-ranged nature of the interactions we use a pair-product decomposition Ceperley (1995) which employs the exact two-body propagator for -function interacting bosons Gaveau and Schulman (1986); Casula et al. (2008); Manoukian (1989):
where is a weight that takes into account the pairwise interactions, and only depends on the relative separation of each pair. The form of for the Lieb-Linger model is given in Appendix A.1.
Rényi EE are accessible via QMC by sampling an extended configuration space consisting of two identical replicas of the physical system under consideration. The sampled ensemble has imaginary-time worldlines that are broken at the center of both paths corresponding to the “A” subsystems consisting of particles Hastings et al. (2010); Herdman et al. (2014b, a). The estimator for is related to the expectation value of the short-imaginary-time propagator which connects the broken worldlines across the replicas:
where and are the reduced propagators for the A subsystems which connect the broken beads to the same and other replica, respectively; see Ref. Herdman et al., 2014b for complete algorithmic details.
v.2 Numerical Results
Using QMC, we have computed for the ground state of the Lieb-Linger Model for and and ranging over two orders of magnitude.
Fig. 3 shows the finite-size scaling of at constant for and . The EE data has been analyzied via two approaches: (solid lines) a generic LL scaling form and (dashed lines) the microscopic finite-size form for the Lieb-Linger model. The generic LL scaling form is given in Eq. (13) where are taken to be free parameters. The coefficient of the term, allows us to determine numerically, directly from the QMC data, free of any assumptions regarding the microscopic model. Alternativerly, for each value of , we may use the short distance behavior of the Lieb-Linger one-body RDM Olshanii and Dunjko (2003) and computed from the Bethe ansatz Lieb (1963) in combination with Eq. (17), leaving only a single free parameter: the short distance cutoff . Fig. 4 shows the best fit values of as a function of ; for all considered here, we find that is of order , a natural short-distance length scale.
The agreement between these two independent analyses is seen in Fig. 3, with full details provided in Appendix A.3. The two-particle partition EE computed with QMC is also shown in Fig. 3, scaled by a factor of , demonstrating consistency with the conjectured generic scaling form:
The small offset between the and data suggests that the constant term in Eq. (1) may depend on .
Using the generic Luttinger liquid three parameter fit to the QMC finite size scaling data shown in Fig. 3, we have determined the Luttinger parameter from and plot it as a function of interaction strength in Fig. 5.
The agreement between the QMC derived data points and the analytical value of determined from the Bethe ansatz provides quantitative confirmation of the predictions from LL theory discussed above. We reiterate that this analysis uses no microscopic information and is applicable to any bosonic model displaying LL behavior.
We have demonstrated from bosonic Luttinger liquid theory that the Luttinger parameter may be extracted from the coefficient of the leading logarithmic term in the finite-size scaling of the -particle partition Rényi entanglement entropy. We note that this scaling is intuitively sensible: this leading term vanishes in the noninteracting limit where and converges to the particle partition EE of a Mott insulator as where the bosons are impenetrable. Additionally, this result is consistent with previously reported results for the Tonks-Girardeau model of impenetrable bosons, corresponding to Santachiara et al. (2007); Haque et al. (2009). It is interesting to contrast the particle partition EE scaling with that of a spatially bipartitioned Luttinger liquid, for which the coefficient of the EE has a universal value of Vidal et al. (2003); Calabrese and Cardy (2004). As another point of comparison, bipartite fluctuations in a LL scale logarithmically with subsystem size with a prefactor that is proportional to , rather that Song et al. (2010, 2012). Additionally, we have shown that may be numerically computed via direct quantum Monte Carlo simulations of a microscopic model without appealing to an analysis of the energy or the algebraic decay of correlation functions. Thus the particle partition entanglement entropy of a bosonic Luttinger liquid provides access to the sole non-universal dimensional parameter which characterizes its long distance behavior. This suggests that further studies of entanglement entropy under particle partitions offer the possibility of uncovering new insights into quantum phases of matter that are distinct from those employing spatial bipartitions.
Computations were performed on the Vermont Advanced Computing Core supported by NASA (NNX-08AO96G).
Appendix A Path-integral ground state Monte Carlo simulations of the Lieb-Linger model
a.1 Pair product approximation for the Lieb-Linger model
is determined by the exact two-body propagator for the relative motion:
where the relative pair propagator is defined as
and the center of mass Hamiltonian is
For a -function potential, the relative propagator is known exactly Gaveau and Schulman (1986); Lawande and Bhagwat (1988); Manoukian (1989); Cheng and da Luz (1993); Yearsley (2008) and can be written in dimensionless form as
where the two fundamental length scales are defined as and , and the associated dimensionless separations are . We then may write the interaction weight as
A similar pair-propagator was used in a quantum Monte Carlo method to study a system of harmonically trapped fermions with -function interactions in one dimension in Ref. Casula et al., 2008.
a.2 Convergence of particle partition Rényi entropy with QMC parameters
Here we demonstrate the convergence of to the exact ground state values with imaginary time length and finite time step which parametrize the systematic errors in the QMC method. We use a constant trial wave function in all cases; a variationally optimized trial wave function would be expected to improve convergence to the ground state at smaller .
To compute ground state properties with imaginary time projection, one must choose a larger than the finite-size gap to excited states. We assume an exponential decay to the ground state value of the form:
where is a dimensionless constant and has units of energy. For we benchmarked our code against the exact ground state value of computed from numerical integration of the Bethe ansatz ground state wave function. Fig. 6 shows the convergence of with for , ; we find , , and which is in agreement with the exact ground state value of down to order .
Fig. 7 shows the imaginary-time length scaling for , for which we find and exponential decay with , , and . When scaling at constant , we take due to the scaling of the finite-size Luttinger liquid gap.
We assume a power-law scaling of the systematic error due to finite time-step , for fixed and , of the form
where and are dimensionless constants (that may in general depend on and ). Fig. 7 shows the convergence of with for , for which we find , , and . The required for a particular systematic error scales primarily as a function of interaction and density and therefore and thus is independent of system size.
a.3 Analysis of QMC data
We now discuss in detail the analysis of the finite-size scaling of as computed by QMC. We consider three scaling functions: the generic Luttinger liquid form given in Eq. (13), the finite-size Lieb-Linger specific form using Eq. (17), and a pure logarithmic form, valid only at large .
Luttinger liquid: In fitting to the generic Luttinger liquid form given in Eq. (13), we take all three parameters to be free parameters determined numerically by the fit; this approach uses no assumptions about the microscopic model. Thus this fit allows us to extract numerically from the finite-size scaling data by taking .
Finite-size Lieb-Liniger: The finite-size Lieb-Linger specific form given in Eq. (17) is employed, where the short distance behavior of the one-body RDM is taken to be that of the Lieb-Linger model given in (20). In this case, for each interaction strength we fix , and to take their known Bethe ansatz values, leaving only a single free parameter, the short distance cutoff . This approach allows us to test the Lieb-Linger predicted form against the QMC data.
Pure logarithmic: We also consider a generic two parameter fit to a simple logarithmic scaling in , i.e. taking in (13). This again allows us to numerically extract the from with no assumptions about the microscopic model.
Fits to the finite-size scaling QMC data for using all three of these forms are shown in Fig. 8.
For the pure logarithmic case, we have only fit to the larger system sizes as there are clear deviations from the pure logarithmic scaling for small . In the other approaches we fit to all values of for . To extract numerically with the same precision shown in Fig. 5 (which uses the full generic Luttinger liquid scaling form where is not set to zero) with a pure logarithmic fit (i.e. ) would require studying larger system sizes due to the finite-size corrections that appear for small .
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