Entanglement entropy in quantum spin chains with broken reflection symmetry

Entanglement entropy in quantum spin chains with broken reflection symmetry

Abstract

We investigate the entanglement entropy of a block of sites in quasifree translation-invariant spin chains concentrating on the effect of reflection symmetry breaking. The majorana two-point functions corresponding to the Jordan-Wigner transformed fermionic modes are determined in the most general case; from these it follows that reflection symmetry in the ground state can only be broken if the model is quantum critical. The large asymptotics of the entropy is calculated analytically for general gauge-invariant models, which has, until now, been done only for the reflection symmetric sector. Analytical results are also derived for certain non-gauge-invariant models, e.g., for the Ising model with Dzyaloshinskii-Moriya interaction. We also study numerically finite chains of length with a non-reflection-symmetric Hamiltonian and report that the reflection symmetry of the entropy of the first spins is violated but the reflection-symmetric Calabrese-Cardy formula is recovered asymptotically. Furthermore, for non-critical reflection-symmetry-breaking Hamiltonians we find an anomaly in the behavior of the ”saturation entropy” as we approach the critical line. The paper also provides a concise but extensive review of the block entropy asymptotics in translation invariant quasifree spin chains with an analysis of the nearest neighbor case and the enumeration of the yet unsolved parts of the quasifree landscape.

I Introduction

Understanding the entanglement properties of systems with many degrees of freedom, such as quantum spin chains, has been one of the main recent research topics connecting quantum information theory and condensed matter physics (1); (2); (3); (4); (5). Huge amount of results have been accumulated about translation-invariant systems. However, the results almost exclusively correspond to reflection symmetric systems, despite the fact that models violating reflection invariance play a prominent role in many-body theory, e.g., in describing interactions of Dzyaloshinskii-Moriya type or non-equilibrium steady states.

Considering a subsystem of a system, which is in a pure state, the entanglement between the subsystem and its environment is characterised by the von Neumann entropy

where denotes the density matrix of the subsystem. In the case of infinite one-dimensional critical chains, this entanglement entropy belonging to a block of contiguous spins was shown to grow asymptotically as (1); (2)

(1)

where is the conformal charge of its universality class and is a non-universal constant. For non-critical chains the asymptotics of the entanglement entropy is bounded. This saturation value of the entropy diverges as one approaches the critical point: it increases as (2)

(2)

where is the correlation length. In the case of finite chains (with open boundary conditions) consisting of spins, the conformal field theoretic prediction for the entanglement entropy of the first spins (at criticality) is (2); (6); (7)

(3)

where is the boundary entropy introduced by Affleck and Ludwig (8).

In this paper, we will study the asymptotics of the entanglement entropy in chains with broken reflection symmetry. We consider quasifree models (with finite range coupling): their Hamiltonian can be mapped to quadratic fermionic chains by the Jordan-Wigner transformation 1

(4)

Throughout the paper we will assume either open boundary conditions or ”fermionic” periodic boundary conditions () 2. The requirement of translation-invariance implies that and are Toeplitz matrices ( and for any ), hermiticity of implies that is a (possibly complex) hermitian matrix, and is (a possibly complex) anti-symmetric matrix. Finite-ranged interaction means that there exists a positive integer such that if . Such a spin-chain Hamiltonian is not invariant with respect to the reflection transformation , iff is not a real matrix. (One might think that the term , with also breaks the translation invariance of the spin chain, but a short calculation shows that its image under the Jordan-Wigner transformation is the following reflection-invariant term ). One of the most studied quantum spin chain with broken reflection symmetry is the Ising model with transverse magnetic field and Dzyaloshinskii-Moriya (DM) interaction (in the -direction) (10); (11); (12); (13):

(5)

Another type of model that has been studied extensively in the literature is the model

(6)

whose ground states are used to describe the energy current carrying eigenstates of the model (14); (15); (16). Certain non-reflection invariant quasifree states also appear as invariant states of reflection-invariant quantum cellular automata (17).

The entanglement entropy asymptotics of the models given by Eq. (4) has been studied by many authors (18); (19); (20); (22); (21). The main analytic tool for tackling this problem was expressing the entropy in terms of the determinant of a Toeplitz matrix, applied first by Jin and Korepin (18). Until now the most general results have been achieved by Keating and Mezzadri (19), who gave a general analytic expression for the entropy asymptotics when is real and , and by Its, Mo, and Mezzadri (22), who gave an analytic (although less explicit) expression even for the case of general (finite-ranged) real and matrices, while certain results about the dimensional case can be found in (23). However, none of these studies concerned reflection symmetry breaking cases, i.e., when is complex.

We will generalize the above mention results by deriving an analytic expression for the general gauge-invariant case (i.e., when is a ”general” complex Hermitian finite-ranged Toeplitz matrix, while ). This includes, as a particular case, the model described in Eq. (6). Moreover, we will also introduce a multitude of transformations between models of Eq. (4), which allows for deriving analytic expressions for cases with non-vanishing . A remarkable result that we obtained is that for these ”quasifree” models reflection invariance can only be broken in the ground state if the model is critical. If the model is non-critical the ground state of the model does not change if we replace with Re() in the Hamiltonian. From this, as we will show, it follows that scaling in Eq. (2) may be violated. However, we will discuss how we can reinterpret this equation to keep its validity. Furthermore, we will present numerical results in non-reflection-symmetric spin chains providing an example of broken reflection symmetry in the finite size scaling of the entropy breaking the symmetry of Eq. (3), but we will see that that this deviation goes to zero as we increase the system size.

The paper is structured as follows. In Section II we calculate the majorana two-point functions of these general (finite-ranged) quasifree models and recapitulate how one can obtain the entanglement entropy from the two-point functions. The results already known about the entanglement asymptotics of certain types of quasifree models are collected in Section III. We derive an analytic formula for the entanglement entropy for general gauge-invariant models in Section IV, whereas in Section V we show how we can extend our results for certain types of non-gauge-invariant models too. Section VI is an application of the above to models with nearest neighbor interactions, while in Section VII we discuss how some of our analytic and numerical results conflict with the formulas (2) and (3) and how we can ”resolve” this discrepancy. Finally, Section VIII is devoted to the summary and the remaining open questions.

Ii Two-point function of the Majorana operators and entanglement entropy

The entanglement entropy asymptotics of the models described by the quadratic Hamiltonians in Eq. (4) can be calculated from the ground-state expectation values , where ’s denote the so-called majorana operators defined as

(7)

In this section we will first derive these majorana two-point functions in terms of the matrices and that define the Hamiltonian Eq. (4). Then we describe how to calculate (in this quasifree setting) the entanglement entropy alone from two-point functions, and finally we recapitulate the ”determinant trick” of Jin and Korepin, which will allow us later to obtain analytical results.

ii.1 The majorana two-point functions

Let us fix our conventions used in the calculation. We will consider the ”fermionic” periodic boundary condition: . The Fourier and inverse transforms of the one-particle annihilation operators read

(8)
(9)

the summation runs in the set of integers () for N odd (even) and the transform of the one-particle creation operators are to be computed by means of taking the adjoint of the above formulae. For Toeplitz matrices we define the Fourier transform as

(10)
(11)

here stands for either or , and the summation again runs in the set of integers () for N odd (even). Using these definitions, the Hamiltonian (4) can be written as

(12)

To bring this Hamiltonian into a diagonal form , one performs a Bogoliubov transformation

(13)

where the coefficients have to satisfy

(14)
(15)

so that the canonical anticommutation relations are satisfied. The consistency conditions for the commutator give

(16)

One readily extracts the one-particle spectrum

(17)

having taken the relations (which are direct consequence of ) into account. The ground state correlations for the two-point functions of the new Fermi operators read

and all other correlations vanish. Now, using the inverse of (13), , one can compute the correlations among the Fourier components , and substituting the solution of (16) for we arrive at

(18)

where , and the two remaining two-point functions and can be calculated directly from the above equations. Ultimately, we would like to have a linear combination of the above, the two point functions of the self-adjoint majorana operators defined in Eq. (7). Before writing down the final result, let us introduce some notations. We will use the combinations

(19)
(20)

and the step functions

(21)

Note, that and . We now take the thermodynamic limit () and write the final result in a manner usually adopted in the literature

(22)

where the matrix has the following structure

(23)

The ’s are block entries that read

(24)

where , so all Fourier series become functions on the circle in the limit. This type of matrix is called block-Toeplitz and the matrix argument in (24) of the integral is called its symbol.

The -point majorana function can be obtained from the two-point functions by the Wick rule (9):

where the sum runs over all pairings of , i.e., over all permutations of the elements which satisfy for and for .

Before coming to the calculation of the entropy, let us analyse the obtained result. The one particle spectrum (17) has the form of a sum of a reflection invariant and a non-invariant term (Note that the real space reflection corresponds to the Fourier space one as follows from the Fourier transform (8)). The symbol (24) characterizing the correlation matrix has a dependence on the non-reflection invariant part of the spectrum only via . This term, however, vanishes identically unless at some . In other words, non-critical quasifree systems never break reflection invariance 3. We will discuss some implications of this important fact in Sections VI and VII.

ii.2 Calculation of the entanglement entropy from the two-point functions

Restricting the ground state to a subsystem consisting of consecutive sites one obtains a mixed state. Let us restrict the matrix defined in Eq. (22) (which describes the two-point majorana correlations) to consecutive modes, that is to a submatrix

(25)

where the ’s are matrices given by Eq. (24). Let us denote by the orthogonal matrix, the adjoint action of which brings the antisymmetric real matrix into its canonical form, i.e., for , we have

where () are the singular values of . (Due to the fact that is antisymmetric, the degeneracy of its singular values is always an even number, that is why we label them only from to .) The density matrix corresponding to the restricted state can be written as 4

where for all . (Actually, translation invariance is not used here, the density matrix of any quasifree state, i.e., of any state for which the Wick expansion applies, can be written in this form.) The entropy can now be easily calculated. It can be written in terms of the function

(26)

as

(27)

The trick (18) to obtain the asymptotics of the entanglement as the size of the block grows is computing the determinant

(28)

and exploiting the residue theorem by writing down the following integral

(29)

where the contour is shown in Fig. 1. That contour encircles all eigenvalues of , but bounds a region, in which is analytic.

Figure 1: The integration contour encloses a domain, where defined in (26) is analytic and contains all singular values of the matrix .

Hence, the main task in all cases is to compute the determinant of the block-Toeplitz matrix matrix

(30)

Finally, we should mention that in the gauge-invariant case (i.e., when ) there is an easier method for the calculation of the entanglement entropy. In this case, as can be seen from Eq. (18), the non-gauge-invariant two-point functions vanish (). If we restrict the state to consecutive sites, and denote by the corresponding restriction of the matrix and by the (not necessarily real) unitary, the adjoint action of which diagonalizes (), then the density matrix of the restricted state reads

where and . Hence the entropy of the restricted state is given by

(31)

Iii Summary of previously known cases

In this section we shortly recapitulate what has been previously known about the entanglement entropy for quasifree models.

iii.1 Gauge and reflection invariance

In the case of gauge- and reflection-invariant quasifree models, the matrix is zero while is real, which implies and the symbol of the majorana two-point functions (24) reduces to

Hence, can be factorized as

(32)

where is the restriction of the Toeplitz matrix with scalar symbol to an block on the diagonal. From this it follows that . To extract the entropy asymptotics, one only needs to calculate the ( asymptotics of the) determinant of using the Fisher-Hartwig theorem, and then use the residue theorem as described in Section II.2. This was done by Keating and Mezzadri (19); (20): they obtained the following result: Let there be number zeros of denoted by () in the semi-circle (implying another zeros in the other semi-circle : , ). Then the entanglement entropy asymptotics is given by

(33)

where

where is Euler’s constant, and , independent of (consult (18) for its derivation).

iii.2 Reflection invariance and real

The other case that has already been discussed in the literature is the case when both matrix and are real, i.e., when . In this case, the symbol reads

(34)

Here the idea, invented for the XY model in (21) and generalized for the present case in (22), is to extend the domain of to the complex plane and use a theorem of Widom (26), which yields a formula of the block Toeplitz determinant at hand expressed in terms of Wiener-Hopf factors of the symbol: , where the matrices are analytic inside (outside) the unit circle. The factorization resides on the fact, that due to the assumption of finite range interaction, the functions , and are Laurent polynomials. One writes

with being the roots of the polynomial , where is the range of the coupling (defined after (4)). Note, that the equality in the middle is a choice of analytic continuation as holds on the unit circle (as is obvious from the general form . The non-analytic behaviour of the above rational function is then the only thing that has to be taken care of and the factorization is done with the help of theta functions living on the hyperelliptic surface of genus given by

(35)

The -model has thus the underlying Riemann surface is a torus, while for for general finite ranged couplings can be any degree Laurent polynomial, which satisfies on the unit circle. The result (Theorem 3. of (22)) for the logarithmic derivative reads

(36)

and the difference (rhs.lhs.) where the constant satisfies (the complex numbers are the roots of and their reciprocals). The saturation entropy is given by

This formula depends on the surface (35) via the theta functions (which are uniquely defined by some quasi-periodicity properties along non-contractible curves on the surface); their definition and that of their arguments will be omitted here (see (22)). We only remark that it is exactly at criticality, when the above surface becomes degenerate and the formula diverges.

Iv Gauge invariant models in general

The reason why one could give an explicit formula for the entropy asymptotics in the reflection and gauge invariant case (when ) and a less explicit one in the case when was that the structure of the symbol was considerably simplified in both cases.

In the general quasifree case it is hard to find the Wiener-Hopf factorization of the symbol, since there is no identically zero entry of in the matrix function . This is true even in the restricted case of gauge invariant (but not reflection invariant) models. However, as we will show in this section, one can circumvent this problem in this restricted case. We have seen in section II.2 that we can extract the entropy also from the correlation matrix : it is given by , where are the eigenvalues of the matrix . Now, we can use the contour integral trick again with a small alteration and write the entropy as

(37)

where , the function and the contour were defined in Section II.2. Hence the situation is analogous to section III.1 except that is replaced by , which is also a Toeplitz matrix, but its symbol

(38)

is not necessarily symmetric ( implies ). Now we can use the Fisher-Hartwig conjecture (25): Suppose that the symbol of a Toeplitz matrix has the following form

(39)

with

where the function is smooth, non-vanishing and has zero winding number. Then the asymptotic formula for the determinant reads

where

and the so-called Barnes function is defined by

In our case the symbol, defined by Eq. (38) above, is a step function jumping between and , and the jumps occur at the zeros of . We can assume that , as the local transformation (which keeps the entanglement entropy invariant) yields Using the notation for the zeros of by , in an increasing order in the period we can write the factors in (39) for the symbol (38)

where

Indeed, one can easily check that the function given by (39) with the above defined ingredients has the value and alternates between with jumps at the zeros of . Now, substituting our data in the statement of the conjecture we get the expression for the determinant

From this point, the calculation of the contour integral (37) is entirely identical to that of (18); (19), and the result for entropy asymptotics reads

(40)

where the constants and were given at the end of section III.1.

V Exact results for the entropy asymptotics for certain non-gauge-invariant models

We now turn to discuss the cases of some non-gauge-invariant models. In the first two subsections we will determine the entropy asymptotics for chains that are Kramers-Wannier selfdual and for those that decouple to two independent majorana chains, by relating these cases to certain gauge-invariant models. In the last two subsections we will relate the entropy asymptotics of different non-gauge-invariant models, by generalizing the XY-Ising transformation and doing local rotations.

We will make use of the fact that one can write the general (translation-invariant) quasifree Hamiltonian (4) in terms of the majorana operators defined by (7) in the following way:

(41)

with the properties and for all . The transformation between the two descriptions reads