Entanglement spectra between coupled Tomonaga-Luttinger liquids:
Applications to ladder systems and topological phases
We study the entanglement spectrum (ES) and entropy between two coupled Tomonaga-Luttinger liquids (TLLs) on parallel periodic chains. This problem gives access to the entanglement properties of various interesting systems, such as spin ladders as well as two-dimensional topological phases. By expanding interchain interactions to quadratic order in bosonic fields, we are able to calculate the ES for both gapped and gapless systems using only methods for free theories. In certain gapless phases of coupled non-chiral TLLs, we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder. In gapped phases of coupled non-chiral TLLs, which are relevant to spin ladders and topological insulators, we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. Based on a calculation for coupled chiral TLLs, we are also able to provide a very simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies.
pacs:71.10.Pm, 03.67.Mn, 11.25.Hf
Quantum entanglement has been found to be useful in characterizing topological states of matter, which is not possible with conventional local order parameters. For example, the topological entanglement entropy (TEE)Kitaev and Preskill (2006); Levin and Wen (2006) encodes information of topolocically ordered phases. The TEE has been numerically and analytically calculated in many exotic systems including fractional quantum Hall systemsHaque et al. (2007); Zozulya et al. (2007); Läuchli et al. (2010a) and quantum spin liquids.Hamma et al. (2005); Furukawa and Misguich (2007); Flammia et al. (2009); Zhang et al. (2012); Jiang et al. (2012); Depenbrock et al. (2012) However, TEE does not give the complete picture of the entanglement. The entanglement spectrum (ES), as proposed by Li and Haldane,Li and Haldane (2008) contains more information than the TEE and is also a powerful tool in studying topological phases. A large amount of focus of the investigation of ES has been with a real space partition in various systems. Those systems include quantum Hall systems,Li and Haldane (2008); Thomale et al. (2010a); Läuchli et al. (2010b); Sterdyniak et al. (2012); Dubail et al. (2012a); Rodriguez et al. (2012, 2013) topological insulators,Turner et al. (2010); Fidkowski (2010); Kargarian and Fiete (2010); Alexandradinata et al. (2011); Fang et al. (2013) fractional Chern insulators,Regnault and Bernevig (2011) symmetry-protected topological phases,Pollmann et al. (2010); Turner et al. (2011); Fidkowski and Kitaev (2011) quantum spin chainsCalabrese and Lefevre (2008); Pollmann and Moore (2010); Franchini et al. (2011); Alba et al. (2012, 2012); Läuchli (2013); Giampaolo et al. (2013) and ladders,Poilblanc (2010); Peschel and Chung (2011); Läuchli and Schliemann (2012); Schliemann and Läuchli (2012); Tanaka et al. (2012); Lundgren et al. (2012); Chen and Fradkin (2013) and other spin and fermionic systems.Yao and Qi (2010); Lou et al. (2011); Cirac et al. (2011); Santos (2013); Deng and Santos (2011); Schliemann (2013); Dubail and Read (2011); Grover (2013).
For the system’s ground state , the ES is obtained by first partitioning the system into a subregion and the rest and then forming the reduced density matrix on . The reduced density matrix is conventionally written in the thermal form , where is referred to as the entanglement Hamiltonian. The ES is simply the full set of eigenvalues of the entanglement Hamiltonian.
A key result found in many topological phases and some other states (often with an energy gap) has been the remarkable correspondence between the ES and the edge-state spectrum. In two-dimensional (2D) topological phases, the ES has been found to exhibit gapless structures which are related to the conformal field theory (CFT) describing low-energy edge excitations.Li and Haldane (2008); Thomale et al. (2010a); Läuchli et al. (2010b); Sterdyniak et al. (2012); Dubail et al. (2012a); Turner et al. (2010); Fidkowski (2010); Kargarian and Fiete (2010); Regnault and Bernevig (2011) An analytic proof for this correspondence has been proposed by Qi, Katsura, and LudwigQi et al. (2012) using the “cut and glue” approach. For model wave functions of quantum Hall systems, different proofs have been proposed by using clustering propertiesChandran et al. (2011) and a connection to CFT.Dubail et al. (2012b) A geometric proof for two- and three-dimensional topological phases has also been put forward.Swingle and Senthil (2012) In the approach of Qi et al.,Qi et al. (2012) it was suggested to start from two pieces and of a topological phase which both support gapless chiral edge states, and then to glue them along the edges by switching on an interaction Hamiltonian that couples and as in Fig. 1. Restricting the analysis to a coupling between the gapless edge modes only, boundary CFT was applied to prove the correspondence between the entanglement Hamiltonian and the single chiral edge Hamiltonian.
The “cut and glue” approach suggests that the entanglement properties of topological phases are closely related to those of coupled one-dimensional (1D) gapless systems. We now give a quick review of results for the ES of coupled 1D systems (“ladders”), which have been shown to exhibit various interesting features depending crucially on the interchain couplings. PoilblancPoilblanc (2010) numerically calculated the ES in gapped phases of a spin- Heisenberg ladder, where the entanglement cut was introduced between the two periodic chains. He observed that the ES remarkably resembles the energy spectrum of a single Heisenberg chain. Peschel and ChungPeschel and Chung (2011) and Läuchli and SchliemannLäuchli and Schliemann (2012) explained this result using perturbation theory from the strong-rung-coupling limit. They also generalized it to the model with an XXZ anisotropy and showed that the entanglement Hamiltonian is given by an XXZ chain with its anisotropy modified from the physical single-chain Hamiltonian. Schliemann and LäuchliSchliemann and Läuchli (2012) also studied the cases of higher-spin ladders (such as spin-), and again found that in the isotropic case, the entanglement Hamiltonian is proportional to the single-chain Hamiltonian. We note that unlike the spin- case, the spectrum of the single-chain Hamiltonian is gapped for certain spin lengths. ES in a variety of coupled 1D systems, including fermionic and spin ladders, was studied by Chen and Fradkin.Chen and Fradkin (2013) Their result for gapped coupled systems supported the correspondence between the ES and the single-chain spectrum. They also investigated a system of two Tomonaga-Luttinger liquids (TLLs) coupled by a marginal interaction, and interestingly found an ES with a flat dispersion. These results open the question, what condition is necessary for finding the correspondence between the ES and the single-chain spectrum. Lundgren, Chua, and FieteLundgren et al. (2012) studied the ES of the Kugel-Khomskii model, which can be regarded as a spin ladder system, with spin degrees of freedom on one leg and orbital degrees of freedom on the other. In that work an entanglement gap was found at the gapless point. The origin of this gap and the structure of the levels below the entanglement gap, which could have universal features, has yet to be explained.
In this paper, we study the ES between two coupled (chiral or non-chiral) TLLs on parallel periodic chains for a variety of interchain interactions. In addition to having direct applications to ladder systems, this problem is also closely related to the entanglement properties of 2D topological phases via the “cut and glue” picture of Fig. 1. By expanding interchain interactions to quadratic order in bosonic fields, we are able to obtain the ES in both gapped and gapless phases using only methods for free theories. This is similar to the approach of Chen and FradkinChen and Fradkin (2013) for a gapless phase but extends it to a wider variety of phases of coupled TLLs. We also carefully treat zero modes, which turn out to be important to the full structure of the ES. Based on the calculation for coupled chiral TLLs, we provide a simple proof for the correspondence between the ES and the edge-state spectrum in quantum Hall systems consistent with previous numerical and analytical studies. Using this result, we are easily able to obtain the TEE, which is equal to , for the quantum Hall state at the filling fraction .Kitaev and Preskill (2006); Dong et al. (2008); Fendley et al. (2007) In gapped phases of coupled non-chiral TLLs, which are relevant to spin ladders and time-reversal-invariant topological insulators,Kane and Mele () we show that the ES consists of linearly dispersing modes, which resembles the spectrum of a single-chain TLL but is characterized by a modified TLL parameter. When the system becomes partially gapless (either in the symmetric or antisymmetric channels of bosonic fields), we interestingly find an ES with a dispersion relation proportional to the square root of the subsystem momentum, which we relate to a certain long-range interaction in the entanglement Hamiltonian. We numerically demonstrate the emergence of this unusual dispersion in a model of hard-core bosons on a ladder.
While the edge-state picture has been widely used for ES in gapped systems, few works have successfully addressed the universal features of ES in gapless systems. For 1D critical systems, CFT has been applied to reveal the universal distribution function of the ES.Calabrese and Lefevre (2008) In that case, the partition is a block of some finite length in a longer finite- or infinite-length chain. There have been some works for understanding the ES of gapless systems with broken continuous symmetry.Alba et al. (2013); Kolley et al. (2013) Another avenue to explore entanglement in gapless systems is to partition the system in momentum space instead of real spaceThomale et al. (2010b); Mondragon-Shem et al. (2013); however such work is beyond the scope of this paper. Gapless phases of coupled TLLs investigated in Ref. Chen and Fradkin, 2013 and here provide novel examples in which critical correlations manifest themselves in unusual dispersion relations in the ES.
The rest of the paper is organized as follows. In Sec. II, we consider a system of two coupled chiral TLLs and show that the ES is proportional to the spectrum of a single chiral TLL. This result gives a simple proof for the correspondence between the edge states and the ES in quantum Hall states. In Sec. III, we consider two coupled non-chiral TLLs, which are relevant to spin ladders, Hubbard chains, and topological insulators. We calculate the ES in both gapped and gapless phases and show that it has a variety of low-energy features depending on the phase. In Sec. IV, we numerically demonstrate some of the predictions of Sec. III in a model of hard-core bosons on a ladder. Finally, in Sec. V, we present our summary, conclusions and open questions.
Ii Two coupled chiral Tomonaga-Luttinger liquids
ii.1 Introduction of model
In this section, we study the ES between two coupled chiral TLLs. This provides the simplest illustration of our approach. Furthermore, this problem is closely related to the ES in quantum Hall states, as we explain below.
To be specific, we consider the quantum Hall states at filling faction , where is odd for fermions and even for bosons. These systems are described by the -dimensional Chern-Simons gauge theory with the action
where is a -dimensional coordinate and is a Levi-Civita symbol. The gauge field is related to the particle current via .
We define the system on a cylinder of circumference and study the ES between the upper and lower halves of the system as in Fig. 1(a). To this end, following the “cut and glue” approach of Qi et al.,Qi et al. (2012) we first physically cut the system as in Fig. 1(b). Along the new edges of the two subsystems, there appear 1D right- and left-moving chiral modes described by the chiral TLL Hamiltonians
where is the velocity. These Hamiltonians are derived from the Chern-Simons action (1) by imposing certain gauge-fixing conditions at the edges.Wen (2004) Here, the bosonic field is related to the particle density fluctuation (relative to the ground state) via . The annihilation operators of an original fermionic/bosonic particle at the edges are given by 111For fermionic systems, in fact, some modifications are necessary in order to have the anticommutation relations of these operators. One simple way to do it is to multiply the factors and to the expressions of and , respectively. In this case, the argument of the cosine term in Eq. (9) is shifted by . This does not change the subsequent argument by focusing on the sector of fixed total particle number .
We now consider the Hamiltonian , which connects between the decoupled system () and the original system (). Here, and describe and regions, respectively, and describes the coupling between them. At , the system is decoupled into two independent quantum Hall systems on regions and . Then gapless chiral TLLs appear as the edge states of the two systems. When the coupling is small enough with respect to the bulk gap, it can be regarded as a perturbation to the gapless edge modes. In particular, when is a relevant perturbation in the renormalization group (RG) sense, the coupling effectively increases as the energy scale is lowered. If the RG flow simply connects the decoupled limit to the strong coupling limit, the low-energy limit of the system with an arbitrary small coupling may be considered essentially the same as that of the original system with . Thus the entanglement between the two regions and in the original system could be essentially understood in terms of the entanglement between the two coupled chiral TLLs, as pointed out by Qi et al.Qi et al. (2012)
The bosonic fields in Eq. (2) have the mode expansions
Here, with () are bosonic operators describing oscillator modes and are the changes in the numbers of particles relative to the ground state. The zero modes satisfy the commutation relations
We introduce the interedge coupling consisting of particle tunneling and a density-density interaction:
Using Eq. (3) and defining
the total Hamiltonian can be recast into a sine-Gordon Hamiltonian
Here is the renormalized velocity and is the TLL parameter. The cosine term in Eq. (9) is relevant in the RG sense when its scaling dimension becomes smaller than .
In the absence of the density-density interaction (), the inter-edge tunneling is RG relevant only for and RG irrelevant for fractional quantum Hall states with (marginal for ). However, for sufficiently large , the inter-edge tunneling can be made RG relevant even for . Even when the inter-edge tunneling is RG irrelevant, there may be a critical value such that the system flowsQi et al. (2012) to the strong tunneling limit when . The present 1D formulation would be still valid in such a case if but is still sufficiently small with respect to the bulk gap.
ii.2 Free-field description of the coupled system
In the rest of Sec. II, we aim to calculate the ES in two coupled chiral TLLs described by Eq. (9). Let us first consider the situation where the low-energy limit of the system is given by the strong inter-edge tunneling limit . This is the case when the inter-edge tunneling is relevant under RG, or even when the infinitesimal inter-edge tunneling is irrelevant.
Our approach is based on the simple observation that, in the limit , is locked into the minimum of the cosine potential. In order to describe the nontrivial entanglement, we can expand the cosine term around its minimum as
where is the locking position. The total Hamiltonian then becomes a Klein-Gordon Hamiltonian with a mass gap . Within this approach, one can also study the case of by setting . We note that for small but finite , the expansion in Eq. (11) is not well justified (When exactly, however, the expansion is not necessary and the approach is justified again). This is due to the presence of the higher order terms in the expansion and possible tunneling among the minima of the cosine. Therefore, when the bare value of is small, this expansion should be done after it grows to a sufficiently large value (comparable to ) under RG. This means that our approach is valid when the system size is sufficiently larger than the correlation length . Since is now quadratic in bosonic fields, one can use the methods for free theories to address the entanglement properties of the system (as done by Chen and FradkinChen and Fradkin (2013) for a gapless case). We note that after the locking, the winding mode for is suppressed, i.e., , since can no longer have a large spacial dependence.
Plugging in the mode expansions (4) into the total Hamiltonian , we see that can be decomposed into zero-mode and oscillator parts:
Focusing on , we observe that it can be diagonalized via a Bogoliubov transformation
We then obtain
as expected for the Klein-Gordon model with a mass gap . The ground state of is specified by the condition that for all .
We note that Qi et al.Qi et al. (2012) used boundary CFT to describe the ground state of Eq. (9). The advantage of our approach lies in its simplicity and versatility. Our approach relies only on the methods for free theories and does not require the knowledge of boundary CFT. Furthermore, as we will see, our approach can treat both gapped and gapless systems in a unified manner while only the gapped case was discussed in Ref. Qi et al., 2012.
ii.3 Reduced density matrix
Now we calculate the reduced density matrix for the right-movers by tracing out the left-movers (or one leg of the ladder). Since the Hamiltonian is decoupled into the zero-mode and oscillator parts, the reduced density matrix is of the form . Below we first calculate the oscillator part and then the zero-mode part .
ii.3.1 Oscillator part
The oscillator part can be calculated by using Peschel’s methodPeschel (2003) for free particles. We first calculate the two-point correlation functions for the chain with right-moving momentum . Using the ground state of , a non-zero correlation function is found as
We introduce the ansatz
This gives the Bose distribution
We now focus on the low-energy part of the entanglement Hamiltonian . Such part gives a high weight in the reduced density matrix . To this end, we expand for small . For and small , we find and therefore
Namely, the entanglement spectrum has a linear dispersion at low energies, which resembles the spectrum of the edge Hamiltonian .
ii.3.2 Zero-mode part
Next we calculate the zero-mode part . Defining and , we can cast our zero-mode Hamiltonian (13) as
Combining the commutation relations (5) for the zero modes, we find , which allows us to identify . Therefore, Eq. (28) can be identified with the Hamiltonian of a harmonic oscillator with discrete momentum . The discreteness becomes irrelevant when is sufficiently localized for large . In this case, the ground state in the basis can be simply approximated by a gaussian
Using (due to the suppression of the winding mode in ), this ground state can be written in a Schmidt decomposed form as
where is a normalization constant. Tracing over the left-moving degrees of freedom, we obtain the reduced density matrix
We then finally arrive at the main result for Sec. II.3, the low-energy expression of the total entanglement Hamiltonian :
This is precisely proportional to the physical edge Hamiltonian in Eq. (6a), in both zero and oscillator modes. This completes the proof for the correspondence between the entanglement spectrum and the edge-state spectrum.
ii.4 Topological Entanglement Entropy
The entanglement entropy is obtained as the thermal entropy of the entanglement Hamiltonian at the fictitious temperature . Kitaev and PreskillKitaev and Preskill (2006) have calculated by assuming a general CFT Hamiltonian for and using the modular transformation property of the partition function. We here obtain by explicitly calculating the partition function for the entanglement Hamiltonian (32). This provides a simple illustration of the result of Ref. Kitaev and Preskill, 2006 in free-boson .
By performing the -function regularization for the infinite constant term, the entanglement Hamiltonian (32) can be rewritten as
with . We consider the partition function , where is a fictitious inverse temperature. Introducing the modular parameter
the partition function is calculated as
where is the third Jacobi theta function and is the Dedekind function. We are interested in the asymptotic behavior for large , namely, small . Using the modular transformation properties of and (see for example Ref. di Francesco et al., 1997), we find
where in the last line, we have used the fact that in the small- limit, . The entanglement entropy is then obtained as
which agrees with the well-known result of Ref. Kitaev and Preskill, 2006. The present calculation without an ultraviolet cutoff has yielded only the universal constant and did not produce a boundary-law contribution present in generic states. The latter is expected to appear by introducing a high-energy cutoff in the entanglement spectrum.
In the present calculation, we have implicitly assumed that the system is in the ground state of a fixed topological sector. In Abelian quantum Hall states, the constant part of does not depend on the choice of a topological sector. However, by superposing the ground states in different topological sectors, we in general obtain a different constant in the entanglement entropy. The dependence of on the choice of a ground state in a variety of closed geometries has been analyzed by Dong et al.Dong et al. (2008) using the Chern-Simons theory. Zhang et al.Zhang et al. (2012) has analyzed this dependence in further detail for a torus geometry and proposed to use it to extract quasiparticle statistics and braiding properties. Although interesting, it is beyond the scope of this work to extend to such cases.
Iii Two coupled non-chiral Tomonaga-Luttinger liquids
iii.1 Introduction of model
In this section, we study the ES between two coupled non-chiral TLLs. This setting is relevant to spin ladders and Hubbard chains. Furthermore, it has a close relation with 2D time-reversal-invariant topological insulators via a “cut and glue” approach in Fig. 1. The ES in gapped phases of spin- ladders have been studied by PoilblancPoilblanc (2010) for the SU-symmetric Heisenberg model and in Refs. Peschel and Chung, 2011 and Läuchli and Schliemann, 2012 for the model with XXZ anisotropy. For a system of two non-chiral TLLs coupled by a marginal interaction, the entanglement entropy and spectrum have been studied by Furukawa and KimFurukawa and Kim () and Chen and Fradkin,Chen and Fradkin (2013) respectively. Here we consider more general states of coupled TLLs, which include all of the above cases as special examples.
We assume that the two non-chiral TLLs are equivalent and are described by the gaussian Hamiltonian
where and are the velocity and the TLL parameter, respectively, in each chain. The dual pair of bosonic fields, and , satisfy the commutation relation
The field is related to the particle density fluctuation (relative to the ground state) via while represents the phase of a fermionic/bosonic operator. Here and are the compactification radii of and , respectively. One can fix these radii to certain values (while keeping the fixed product ) to adjust to different normalizations of bosonization. For later applications, we set
In this convention, () corresponds to a repulsive (attractive) intrachain interaction in the case of fermions.
The bosonic fields have the mode expansions
Here, with () are bosonic operators describing oscillator modes. For bosonic systems, the winding numbers and are both integers. For fermionic systems, they obey the following condition,Haldane (1981) called the twisted structure: , . The winding number gives the change in the number of particles in the th chain relative to the ground state. The constant terms, and , and the winding numbers satisfy the commutation relations
For the interchain coupling, we consider the following interactions:
Here, the and terms arise from particle tunneling and a density-density interaction, respectively. In the Hubbard chain, the term appears at half-filling and induces a Mott gap.Giamarchi (2013); Gogolin et al. (1998) In spin- ladders in the absence of a magnetic field, combination of the and terms induces rung-singlet and Haldane phases.Giamarchi (2013); Gogolin et al. (1998)
To describe the coupled Hamiltonian , it is useful to introduce the symmetric and anti-symmetric combinations of bosonic fields as
The total Hamiltonian can then be decoupled into sine-Gordon Hamiltonians defined for the symmetric and antisymmetric channels:
Here the renormalized velocities and the TLL parameters are given by
The and terms are relevant in the RG sense when their scaling dimensions, and respectively, become smaller than .
iii.2 Free-field description of the coupled system
where and are the locking positions. The Hamiltonian then becomes a Klein-Gordon Hamiltonian with a mass gap . As we have discussed in Sec. II.1, the above description actually applies even if the (infinitesimal) cosine terms are irrelevant in the RG sense, provided that the strong coupling limit is realized.
On the other hand, when either or both of the cosine terms is RG irrelevant, or if it is fine-tuned to zero, the system remains gapless. When one of the modes and is gapless, the low-energy limit of the system is a single-component TLL. Likewise, when the two modes remain gapless, it is a two-component TLL. Such cases can be also treated in the present formulation, simply by setting and/or . We emphasize that, the mode that remains gapless still can contribute to nontrivial entanglement, owing to the density-density interaction which mixes the original variables .
Using Eq. (41), we obtain the mode expansions of and as
where we have defined
Due to the locking of and , the winding modes of these two fields must be suppressed, namely, . This yields the constraints and .
By inserting the expansions (50) (with the condition ) into the Klein-Gordon Hamiltonians, we find that these Hamiltonians can be decoupled into the zero-mode and oscillator parts:
Here the zero-mode part is given by
where we have defined and . Using Eq. (42), we obtain the commutation relations
The oscillator part is given by
By performing a Bogoliubov transformation
the oscillator part is diagonalized as
The ground state of is specified by the condition that for all .
iii.3 Reduced density matrix
Using the ground state of , we calculate the reduced density matrix for the first chain by tracing out the degrees of freedom in the second chain. The reduced density matrix can be factorized into the zero-mode and oscillator parts: .
We first consider the zero-mode part . Using the analogy with a harmonic oscillator as in Sec. II.3.2, the ground state of is calculated as
where is a normalization factor. From this, the zero-mode part of the entanglement Hamiltonian is calculated as