Open XXZ chain and boundary modes at zero temperature
We study the open XXZ spin chain in the regime and for generic longitudinal magnetic fields at the two boundaries. We discuss the ground state via the Bethe ansatz and we show that, in a regime where both boundary magnetic fields are equal and bounded by a critical field, the spectrum is gapped and the ground state is doubly degenerate up to exponentially small corrections in the length of the chain. We connect this degeneracy to the presence of a boundary root, namely an excitation localized at one of the two boundaries. We compute the local magnetization at the left edge of the chain and we show that, due to the existence of a boundary root, this depends also on the value of the field at the opposite edge, even in the half-infinite chain limit. Moreover we give an exact expression for the large time limit of the spin autocorrelation at the boundary, which we explicitly compute in terms of the form factor between the two quasi-degenerate ground states. This, as we show, turns out to be equal to the contribution of the boundary root to the local magnetization.
- 1 Introduction and main results
- 2 The integrable open XXZ spin chain
- 3 The form factor in the finite-size open chain
- 4 The ground state(s) in the thermodynamic limit
- 5 Form factors in the thermodynamic limit
- 6 Conclusion
- A The Bethe equations in logarithmic form and allowed quantum numbers for the real roots
- B Controlling the finite-size corrections in the large limit
1 Introduction and main results
The study of condensed matter theory involves understanding many-body systems starting from their elementary constituents. This protocol, which is in general notoriously hard, can be sometimes carried out in systems of one-dimensional spin chains. These constitute one the main theoretical playgrounds for the emergent physics of strongly correlated quantum systems, see for example the seminal work of Haldane [Haldane1983]. In particular, in the past years, a class of interacting spin chains which can be exactly solved by the so-called Bethe Ansatz [Bet31, FadST79] have been successfully applied to understand the dynamical response of real compounds [Mourigal2013, Wu2016] or to develop better numerical techniques [Nataf2018].
While the bulk physics of spin chains can be usually studied by considering the large-size limit of systems with periodic boundary conditions, a richer phenomenology can be observed in the presence of open boundaries. By tuning different parameters at the boundaries one can explore different phase transitions (also experimentally [Toskovic2016]), as well as probing the existence of boundary modes. Notoriously, in topological superconducting systems, the Majorana zero modes [Sarma2015] are boundary modes and they consist into two decoupled Majoranas fermions localized at the two edges of the system that can be combined to form a zero-energy regular fermion. As a consequence of their existence, all many-particle states are degenerate. While Majorana zero modes are present in the so-called Kitaev chain, which becomes the XY chain with a transverse field after a Jordan-Wigner transformation, it was recently shown by Fendley [Fen16] that also the gapped (massive) XYZ chain contains strong zero modes, namely operators defined at the two edges of the chain that commute with the Hamiltonian up to exponentially small corrections with the size of the chain. These operators, instead of being exactly localized at the two edges, are characterized by exponential tails that decay away from the edges and are related to the symmetry of the model. Their existence also implies an extensive number of degeneracies between the different many-body states in the spectrum.
From the physical point of view it is interesting to study the spin autocorrelation at the edge of the chain. Due to the presence of the aforementioned boundary modes, the latter should not decay to zero even at finite temperature , in the thermodynamic limit . Namely, given the Pauli spin operator at the left edge of the chain and its time evolution with the Hamiltonian containing a strong zero mode, one should find that, for any temperature ,
where denotes the connected correlator and the thermal expectation value. This prediction constituted a starting point of an active research field focused on the study of coherence time of edge spins in the open XXZ chain. When the Hamiltonian is perturbed by additional terms that do not preserve the symmetry of the zero mode, namely when the system is perturbed away from the integrable limit, it was shown [Pinto2009, KemYLF17, MacM18, ElsFKN17] that the dephasing time can still get very large and that the spin autocorrelation remains on a long-living plateau at large intermediate times.
We here consider the open XXZ Hamiltonian with anisotropy parameter and boundary longitudinal magnetic fields and ,
in the massive anti-ferromagnetic regime with (), which is indeed the regime of existence of the strong zero modes [Fen16]. We particularly focus on the case of a chain with an even number of sites . There are two critical values of the magnetic field at the boundary, and , where different crossings between eigenstates occur, see Fig. 3. In the regime where , as we shall see, the spectrum is gapped and the ground state is doubly degenerate in the large limit whenever , so that the zero-temperature spin-spin boundary autocorrelation function is expected to converge for large time to the form factor of the spin operator between these two degenerate ground states. Namely, by denoting with , , the two normalized ground states of the open chain with boundary magnetic fields , we expect that
In this paper, we explicitly compute the thermodynamic and large-time limit (1.3) of the boundary auto-correlation function at zero-temperature from the study of the open chain (1.2) in the algebraic Bethe ansatz (ABA) framework [Skl88]. By considering the large limit of the solutions of the Bethe equations and controlling the finite-size corrections up to exponentially small order in , we show that the difference of energy between the ground state and the first excited state becomes exponentially small in when ( ). Each of these two states is characterized by a Fermi sea of real Bethe roots and an isolated complex Bethe root which corresponds to a boundary mode and that we call boundary root. The latter represents a collective magnonic excitation pinned at one of the two edges of the chain, whose wave function has exponentially decreasing tails away from the boundary [KapS96]. We show that this boundary mode is responsible for the ground state degeneracy, which in particular has two main physical consequences:
The boundary magnetization in the ground state for even size depends on the value of both boundary fields, even in the infinite chain limit (thermodynamic limit). This is due to the fact that the presence of the boundary root in the Bethe solution for the ground state and its localization at one or the other edge of the chain depends on the values of both boundary fields. Moreover, when one of the fields is inside the interval , the boundary magnetization becomes a discontinuous function of the other field at , point at which the boundary root jumps from one edge of the chain to the other. We here provide an analytical derivation for the boundary magnetization at the left edge of the chain, and notably for the value of its discontinuity at (). The latter is given by , the (thermodynamic limit of the) contribution to the boundary magnetization carried by the boundary root at the left edge, which is non-zero only when :
see eq. (5.16) and (5.17) for an exact expression in terms of the parameters of the model. At exactly , the boundary root becomes delocalized between the two edges of the chain and contributes equally to the left or the right boundary magnetization, hence the factor in (1). In the particular case , we recover that [JimKKKM95, KitKMNST07]
where is the bulk magnetization [Bax73].
The degeneracy of the ground state implies that the matrix element of the spin operator in the first site of the chain between the two degenerate ground states and with fields (), is non-zero in the thermodynamic limit and its absolute value squared provides the infinite time limit of the boundary spin-spin autocorrelation function, see equation (1.3). We here exactly compute this matrix element in the ABA framework and explain how to derive its thermodynamic limit. We show that, in this limit, it is directly related to the contribution to the boundary magnetization carried by the boundary root at the left edge as
for any , so that it is given by half of the boundary magnetization discontinuity (1) at . For and the quantity , and so the matrix element (1.6), is non-zero, see (5.2). When both fields are zero () this reduces to the value (1.5):
In Fig. 1 and Fig. 2 the boundary spin autocorrelation function is computed numerically by tDMRG as a function of time for a chain with finite size : at large times the correlation attains the value given by (1.3)-(1.6).
Note that a similar relation to (1.7) exists also in the bulk of the chain. There, as shown first by Baxter [Bax73], the local magnetization of the ground state is staggered, namely we have
where is the global magnetic field. The value of is also equal to the form factor between the two degenerate ground states of the chain with periodic boundary conditions, see [IzeKMT99].
This article is organized as follows. In section 2, we recall the diagonalization of the Hamiltonian (1.2) in the framework of the boundary algebraic Bethe ansatz introduced by Sklyanin in [Skl88]. In section 3, we explain how to derive, in this framework, compact determinant representations for the finite-size matrix elements (form factors) of the operator between two Bethe eigenstates. In section 4 we study the solutions of the Bethe equations in the thermodynamic limit and explain how to control their finite-size corrections up to exponentially small order in . We identify the solution corresponding to the ground state for the different values of the boundary magnetic fields and . In the regime where both fields are between and , with , we show that the two states of lowest energy are given by a particular solution of the Bethe equations with real Bethe roots and one complex Bethe root which has to be chosen between the two possible boundary roots given in terms of the boundary parameter at the left or the right end of the chain. We moreover show that, when , the deviation between the two boundary roots becomes exponentially small in , and so does the difference of energy between the two corresponding states. In section 5, we compute the thermodynamic limit of the determinant representation that we obtained in section 3 in two particular cases: the mean value of in the ground state, which gives the boundary magnetization, and the form factor between the two degenerate ground states identified in section 4, which gives the infinite time limit of the boundary autocorrelation function.
2 The integrable open XXZ spin chain
The Hamiltonian (1.2) is integrable and can be diagonalized in the framework of the representation theory of the reflection algebra [Che84], by means of the boundary version of algebraic Bethe ansatz introduced by Sklyanin in [Skl88].
The key object in this approach is the boundary monodromy matrix where is the space of states of the system. It is such that satisfies the reflection equation111The monodromy matrix that we consider here corresponds to the matrix of [Skl88].,
where is the 6-vertex trigonometric -matrix,
The relation (2.1) has to be understood , and the subscripts parameterize the subspaces of on which the corresponding operators act non-trivially. The parameter is related to the anisotropy parameter of (1.2) as .
In the case of the spin chain (1.2) with longitudinal boundary fields, the boundary monodromy matrix solution of (2.1) can be constructed from the bulk monodromy matrix and a diagonal scalar solution of the reflection equation (2.1),
More precisely, we introduce two such boundary scalar matrices,
where are some complex parameters which parameterize the left and right boundary fields as . The boundary monodromy matrix is then constructed as
where the bulk monodromy matrix is itself constructed as a product of -matrices (2.2) as
Here the index denotes the so-called auxilliary space , and are a set of inhomogeneity parameters which may be introduced for technical convenience.
One then define a one-parameter family of commuting transfer matrices as
In the homogeneous limit in which , , the Hamiltonian (1.2) of the spin-1/2 open chain can be obtained as
In the algebraic Bethe ansatz framework, the common eigenstates of the transfer matrices can be constructed in the form
where (respectively ) is the reference state (respectively the dual reference state) with all spins up. By using the commutation relations issued from (2.1), it can be shown that states of the form (2.10) are eigenstates of the transfer matrix (2.8) provided the set of spectral parameters satisfies the system of Bethe equations
Here and in the following, we use the shortcut notations:
The corresponding transfer matrix eigenvalue is
where the bare energy is defined as
Eigenstates of the form (2.10) are called on-shell Bethe states. States of the form (2.10) for which the parameters do not satisfy the Bethe equations are instead called off-shell Bethe states. The study of the solutions of Bethe equations, and in particular of the ground state of the Hamiltonian (1.2) in the thermodynamic limit, has been performed in [AlcBBBQ87, SkoS95, KapS96].
Building on this ABA description of the spectrum and eigenstates, it is possible to compute the zero-temperature correlation functions of the open spin chain [KitKMNST07, KitKMNST08]. However, this program has not yet reached the level of achievement as what has been done for the bulk correlation functions [KitMT99, KitMST05a, KitMST05b, KitKMST09b, KitKMST09c, KitKMST11a, KitKMST11b, KozT11, KitKMST12, KitKMT14, Koz17, Koz18]. In the latter case, it was indeed possible to derive the large distance and long time asymptotic behavior of the two-point (or even multi-point) correlation functions in the thermodynamic limit from their exact representations on the lattice. At the root of this approach was the fact that there exist some compact and simple determinant formulas for the form factors of local operators in the finite periodic chain [KitMT99]. Such determinant representations were also of uttermost importance for the numerical studies of the correlation functions [BieKM02, BieKM03, CauHM05, CauM05]. They were obtained thanks to two main ingredients: a determinant representation of the scalar product of an off-shell and an on-shell Bethe states [Sla89], and the fact that the local spin operators could be expressed as a simple element of the monodromy matrix dressed by a product of transfer matrices (solution of the quantum inverse problem) [KitMT99, MaiT00, GohK00].
In the open case, however, such nice determinant representations for the form factors do not exist in general. It is still possible to expressed the scalar product of an off-shell and an on-shell Bethe states of the form (2.10) as a generalized version of the Slavnov determinant [Wan02, KitKMNST07], but a convenient expression of the local spin operators in terms of the boundary monodromy matrix elements dressed by a product of boundary transfer matrices is presently not known, except at the first (or last) site of the chain [Wan00]. In fact, the formulas obtained in [KitKMNST07, KitKMNST08] relied on a cumbersome use of the bulk inverse problem, which resulted into multiple integral formulas for the zero-temperature correlation functions in the thermodynamic limit (half-infinite chain) similar to the one that were previously obtained in [JimKKKM95] from a different approach.
At the first (or last) site of the chain, however, the situation is different. Indeed, the solution of the quantum inverse problem proposed in [Wan00] is in that case sufficient, together with the determinant representation for the scalar products, to obtain determinant representations for the form factors of local operators at site 1 which are very similar to the bulk ones. Hence, we may expect to be able to study their thermodynamic limit similarly as what has been done in [IzeKMT99, KitKMST09c, KitKMST11a, Koz17]. In particular, we are in position to compute and study the thermodynamic limit of the form factors which are relevant for the long-time limit of the boundary autocorrelation (1.1). This is the purpose of the next sections.
3 The form factor in the finite-size open chain
The finite-size form factor of local spin operators on the first site of the chain can be computed similarly as in the periodic case [KitMT99], by using the solution of the quantum inverse problem on the first site of the chain [Wan00] together with the determinant representation for the scalar product of an on-shell with an off-shell Bethe states (2.10). For a solution of the Bethe equations and and arbitrary set of parameters, the latter is given by [Wan02, KitKMNST07]
where the elements of the matrix are
for and . The reconstruction of the operator in terms of the boundary monodromy matrix elements reads [Wan00]
where is a generic inhomogeneity parameter that should be sent to at the end of the computation. We also recall the action of the boundary monodromy matrix element on an off-Bethe state (2.10), which follows from the commutations relations issued from (2.1):
where is the matrix (3.2) and is a rank one matrix with elements
So as to express the determinant in a more convenient form for taking the thermodynamic limit, let us introduce, as in [IzeKMT99], an matrix with elements
Its determinant is
Multiplying and dividing (3.8) by , computing the matrices and , and factorizing the quantity
outside of the determinant, we obtain
in which we have defined
in which we have set
4 The ground state(s) in the thermodynamic limit
In this section, we explain how to characterize the configuration of Bethe roots for the ground state(s) of the open XXZ Hamiltonian (1.2) in the regime . As we shall see, the total number of these Bethe roots and their pattern in the complex plane for large depend non-trivially on the values of the magnetic fields at the boundaries, and so does the presence of an energy gap and of an exponential double degeneracy at , see Fig. 3.
Hence, we now focus on the regime . We use the following parametrization for the anisotropy parameter and the boundary fields () in this regime:
where , and
We also suppose that the number of sites of the chain is even222The degeneracies of the ground state in the case odd are different: in that case, we do not have any more quasi-degenerate ground states for , but an exact degeneracy for due to the symmetry of the model..
The Bethe equations (2.11) can be conveniently rewritten333 When doing this, we have to exclude the possible roots and which are always solutions of (4.4) but should actually correspond to a zero of order 2 in the numerator of (2.11). By treating them apart, it is in fact easy to see that low-energy states do not contain these roots for large . as
in terms of the function (3.16). In the homogeneous limit , , the latter reads explicitely