Boundary versus bulk behavior of time-dependent correlation functions in one-dimensional quantum systems

Boundary versus bulk behavior of time-dependent correlation functions in one-dimensional quantum systems

I. S. Eliëns Institute for Theoretical Physics, Institute of Physics, University of Amsterdam, Science Park 904, 1098 XH Amsterdam, The Netherlands i.s.eliens@uva.nl    F. B. Ramos Universidade Federal de Uberlândia, Instituto de Física, C.P. 593, 38400-902 Uberlândia, MG, Brazil    J. C. Xavier Universidade Federal de Uberlândia, Instituto de Física, C.P. 593, 38400-902 Uberlândia, MG, Brazil    R. G. Pereira Instituto de Física de São Carlos, Universidade de São Paulo, C.P. 369, São Carlos, SP, 13560-970, Brazil
July 4, 2019
Abstract

We study the influence of reflective boundaries on time-dependent responses of one-dimensional quantum fluids at zero temperature beyond the low-energy approximation. Our analysis is based on an extension of effective mobile impurity models for nonlinear Luttinger liquids to the case of open boundary conditions. For integrable models, we show that boundary autocorrelations oscillate as a function of time with the same frequency as the corresponding bulk autocorrelations. This frequency can be identified as the band edge of elementary excitations. The amplitude of the oscillations decays as a power law with distinct exponents at the boundary and in the bulk, but boundary and bulk exponents are determined by the same coupling constant in the mobile impurity model. For nonintegrable models, we argue that the power-law decay of the oscillations is generic for autocorrelations in the bulk, but turns into an exponential decay at the boundary. Moreover, there is in general a nonuniversal shift of the boundary frequency in comparison with the band edge of bulk excitations. The predictions of our effective field theory are compared with numerical results obtained by time-dependent density matrix renormalization group (tDMRG) for both integrable and nonintegrable critical spin- chains with , and .

pacs:
71.10.Pm,75.10.Pq

I Introduction

Striking properties in many-body quantum systems often emerge from the interplay between interactions and a constrained geometry. In a Fermi gas confined to a single spatial dimension, for example, interactions lead to dramatically different spectral properties as compared to its higher dimensional counterparts described by Fermi liquid theory Theumann (1967); Luther and Peschel (1974); Meden and Schönhammer (1992); Voit (1993).

The low-energy limit of one-dimensional (1D) Fermi gases is conventionally treated within the Luttinger liquid (LL) framework Giamarchi (2004). Indispensable in this respect is the exactly solvable Tomonaga-Luttinger (TL) model Tomonaga (1950); Luttinger (1963), which allows a nonperturbative treatment of interactions at the cost of an artificially linearized dispersion relation for the constituent fermions. Using the technique of bosonization, the model is solved in terms of bosonic collective modes corresponding to quantized waves of density.

Static correlations and many thermodynamic properties are captured remarkably well by the Luttinger liquid approach. For many dynamic effects, however, it is clear that band curvature needs to be taken into account. For example, the relaxation of the bosonic sound modes, or the related width of the dynamical structure factor (DSF), are not captured by Luttinger liquid theory, which predicts a delta function peak for the DSF. Attempts to treat the DSF broadening in the bosonized theory, in which the dispersion curvature translates to interactions between the modes diagonalizing the TL model, are hindered by on-shell divergences in the perturbative expansion. Certain aspects of the DSF broadening can nevertheless be captured in the bosonic basis Samokhin (1998); Aristov (2007); Teber (2007, 2006); Price and Lamacraft (2014). An alternative approach uses a reformulation of the TL model including a quadratic correction to the dispersion in terms of fermionic quasiparticles. In the low-energy limit, these turn out to be weakly interacting Rozhkov (2005); Khodas et al. (2007a); Imambekov and Glazman (2009a) restoring some of the elements of Fermi liquid theory in one dimension. At high energies, insight into dynamic response functions such as the DSF and the spectral function, and in particular into the characteristic threshold singularities, can be obtained by mapping the problem to a mobile impurity Hamiltonian. This approach hinges on the observation that the thresholds correspond to configurations of a high energy hole or particle which can effectively be considered as separated from the low energy subband, and that the threshold singularities emerge from the scattering of the modes at the Fermi level on this impurity mode. This identifies the anomalous correlation structure of 1D gases as an example of Anderson’s orthogonality catastrophe Anderson (1967) and links it to the physics of the x-ray edge singularity Schotte and Schotte (1969). Many new results on dynamic correlations, in general and for specific models, have been obtained this way Carmelo and Sacramento (2003); Pereira et al. (2006); Pustilnik et al. (2006); Khodas et al. (2007b, a); Pustilnik et al. (2006); Pereira, White, and Affleck (2008); Cheianov and Pustilnik (2008); Pereira, White, and Affleck (2009); Imambekov and Glazman (2009b); Essler (2010); Schmidt, Imambekov, and Glazman (2010a, b); Pereira et al. (2012). This bears relevance to e.g. Coulomb drag experiments Debray et al. (2002); Yamamoto et al. (2002); Pustilnik et al. (2003); Aristov (2007); Laroche et al. (2011); Pereira and Sela (2010) as well as relaxation and transport Matveev and Furusaki (2013); Protopopov, Gutman, and Mirlin (2014); Protopopov et al. (2014); Lin, Matveev, and Pustilnik (2013); Barak et al. (2010). Dispersion nonlinearity also greatly influences the propagation of a density bump or dip, which would retain its shape when time-evolved under the linear theory but relaxes by emitting shock waves in the nonlinear theory Bettelheim and Glazman (2012); Bettelheim, Abanov, and Wiegmann (2006); Protopopov et al. (2013). Closer to the present work is the late-time dependence of correlations Pereira (2012); Karrasch, Pereira, and Sirker (2015); Seabra et al. (2014) which are related to the singularities in the frequency domain. Collectively, the extensions of LL theory that include band curvature effects may be called nonlinear Luttinger liquid (nLL) theory, but we will mainly be concerned with the mobile impurity approach to correlations (see Ref. Imambekov, Schmidt, and Glazman, 2012 for further details).

Motivated by these theoretical advances, we study the effect of reflective boundaries on a 1D gas beyond the low-energy regime. Our work is also inspired by studies of “boundary critical phenomena” Cardy (1989); Affleck and Ludwig (1994); Fendley, Lesage, and Saleur (1996) within the LL framework that have unveiled remarkable effects, e.g., in the conductance of quantum wires Kane and Fisher (1992); Wong and Affleck (1994); Fabrizio and Gogolin (1995), screening of magnetic impurities Eggert and Affleck (1992), Friedel oscillations in charge and spin densities Egger and Grabert (1995); Leclair, Lesage, and Saleur (1996); Rommer and Eggert (2000), and oscillations in the entanglement entropy Laflorencie et al. (2006); Taddia et al. (2013).

We focus on response functions which can be locally addressed—such as the local density of states (LDOS) and autocorrelation functions—as these are expected to show the clearest bulk versus boundary contrast. Many studies have addressed the LDOS for LLs with a boundary Schönhammer et al. (2000); Meden et al. (2000); Schollwöck et al. (2002); Meden et al. (2002); Andergassen et al. (2004, 2006); Schneider et al. (2008); Schuricht, Andergassen, and Meden (2013); Söffing, Schneider, and Eggert (2013); Jeckelmann (2013). LL theory predicts a characteristic power-law suppression (for repulsive interactions) of the LDOS at the Fermi level with different bulk and boundary exponents which are nontrivially but universally related Eggert, Johannesson, and Mattsson (1996); Mattsson, Eggert, and Johannesson (1997). This has been verified using different techniques Wang, Voit, and Pu (1996); Schönhammer et al. (2000); Meden et al. (2000); Schuricht, Andergassen, and Meden (2013) and is used as a consistency check in the experimental identification of LL physics Bockrath et al. (1999); Blumenstein et al. (2011).

Away from the Fermi level, no universal results are known. This pertains both to general statements on the restricted energy range where the power-law scaling is valid Meden et al. (2000); Schuricht, Andergassen, and Meden (2013) and to details of the line shape at higher energies. Here, we deal with the latter and argue that the nonanalyticities of, e.g., the LDOS away from zero energy can be understood in the framework of nLL theory for systems with open and periodic boundary conditions alike. The main application of our theory is in describing the power-law decay of autocorrelation functions in real time. We show that bulk and boundary exponents are governed by the same parameters in the mobile impurity model and obey relations that depend only on the Luttinger parameter. These relations provide a quantitative test of the nLL theory. We perform this test by analyzing time-dependent density matrix renormalization group (tDMRG) White and Feiguin (2004); Schollwöck (2011) results for spin autocorrelations of critical spin chains. The statement about boundary exponents applies to integrable models in which the nonanalytic behavior at finite energies is not susceptible to broadening due to three-body scattering processes Khodas et al. (2007a); Pereira, White, and Affleck (2009). The effects of integrability breaking are also investigated, both numerically and from the perspective of the mobile impurity model. We find that for nonintegrable models the finite-energy singularities in boundary autocorrelations are broadened by decay processes associated with boundary operators in the mobile impurity model. As a result, the boundary autocorrelation decays exponentially in time in the nonintegrable case.

The paper is organized as follows. In Section II, we discuss the LDOS for spinless fermions as a first example of how dynamical correlations in the vicinity of an open boundary differ from the result in the bulk. In Section III, we present the mobile impurity model used to calculate the exponents in the LDOS near the boundary. In Section IV, we generalize our approach to predict relations between bulk and boundary exponents of other dynamical correlation functions, including the case of spinful fermions. Section V addresses the question whether finite-energy singularities exist in nonitegrable models. Our numerical results for the time decay of spin autocorrelation functions are presented in Section VI. Finally, we offer some concluding remarks in Section VII.

Ii Green’s function for spinless fermions

We are interested in 1D systems on a half-line, where we impose the boundary condition that all physical operators vanish at . Let us first discuss the case of spinless fermions on a lattice. We define the (non-time-ordered) Green’s function at position as

(1)

where annihilates a spinless fermion at position and the time evolution is governed by a local Hamiltonian . The brackets denote the expectation value in the ground state of . The Fourier transform to the frequency domain yields the LDOS

(2)

The boundary case corresponds to the result for , where is the lattice spacing for lattice models or the short-distance cutoff for continuum models. We refer to the bulk case of as the regime and , where is the velocity that sets the light cone for propagation of correlations in the many-body system Lieb and Robinson (1972). The latter condition allows one to neglect the effects of reflection at the boundary, and is routinely employed in numerical simulations aimed at capturing the long-time behavior in the thermodynamic limit Pereira, White, and Affleck (2008, 2009); Karrasch, Moore, and Heidrich-Meisner (2014); Seabra et al. (2014).

As our point of departure, consider the free fermion model

(3)

where , with , is the free fermion dispersion and we set . The single-particle eigenstates of are created by

(4)

We focus on the case of half filling, in which the ground state is constructed by occupying all states with . In this case particle-hole symmetry rules out Friedel oscillations Rommer and Eggert (2000) and the average density is homogeneous, . The Green’s function is given exactly by

(5)

and the LDOS is

(6)

where .

The result for is depicted in Fig. 1 (a). First we note that, for any fixed position , there is a clear change of behavior at the time scale (where for free fermions). This corresponds to the time for the light cone centered at to reflect at the boundary and return to . For , is independent of (i.e. translationally invariant for fixed and ) and the result is representative of the bulk autocorrelation. The arrival of the boundary-reflected correlations makes deviate from the bulk case and become -dependent for . After we take the Fourier transform to the frequency domain, the reflection time scale implies that the LDOS in Eq. (6) oscillates with period . In the bulk case, the rapid oscillations in the frequency dependence of are averaged out by any finite frequency resolution Jeckelmann (2013). In numerical simulations of time evolution in the bulk, the usual procedure is to stop the simulation at (or before in case the maximum time is limited by various sources of error White and Feiguin (2004); Schollwöck (2011)). This avoids the reflection at the boundary but at the same time sets the finite frequency resolution.

Figure 1: (Color online) (a) Green’s function for free fermions in a semi-infinite chain at half-filling [Eq. (5)], where is the distance from the boundary. The dashed line represents the reflection time with . (b) The deep hole configuration responsible for the oscillations at related to the singularities of the LDOS (Fig. 2). There is an equivalent high-energy particle configuration, not depicted. The dashed circles indicate the projection onto low-energy and impurity subbands important once interactions are taken into account.

Let us now discuss the time dependence of the Green’s function at the boundary () versus in the bulk (, ). In both cases (see Fig. 2) the Green’s function shows oscillations in the long-time decay which are not predicted by the usual low-energy approximation of linearizing the dispersion about Giamarchi (2004). The explanation for the real-time oscillations is the same for open or periodic boundary conditions; for the case of periodic boundary conditions, see the reviews in Refs. Imambekov, Schmidt, and Glazman (2012); Pereira (2012). The oscillations stem from a saddle point contribution to the integral in Eq. (5) with [in the hole term of ] or (in the particle term). This contribution is associated with an excitation with energy , the maximum energy of a single-hole or single-particle excitation [see Fig. 1 (b)]. We call this energy the band edge of the free fermion dispersion. The propagator of the band edge mode decays more slowly in time due to its vanishing group velocity. The importance of this finite-energy contribution is manifested in the LDOS as a power-law singularity at (see Fig. 2). Notice the clear difference between the bulk and the boundary case: while in the bulk the LDOS has a van Hove singularity at the band edge, , at the boundary one finds a square-root cusp .

Figure 2: (Color online) Noninteracting Green’s function and LDOS . The curves on the left correspond to the chain end (), and the curves on the right to a site in the bulk ().

One of the main achievements of the nLL theory is to incorporate the contributions of finite-energy excitations in dynamical correlation functions for interacting 1D systems with band curvature Imambekov, Schmidt, and Glazman (2012); Pereira (2012). Our purpose here is to generalize this approach to describe the dynamics in the vicinity of a boundary. For concreteness, we consider the model

(7)

where is the density operator and we focus on the repulsive regime . Importantly, the model in Eq. (7) is integrable and exactly solvable by Bethe ansatz Korepin, Bogoliubov, and Izergin (1993). This guarantees that the band edge of elementary excitations is still well defined in the interacting case. We postpone a detailed discussion about integrability-breaking effects to Section V.

Before outlining the derivation of the results for the interacting model (see Section III), we summarize some known results together with our findings for the Green’s function and LDOS. The calculation within the LL framework leads to the well-known predictions Kane and Fisher (1992); Eggert, Johannesson, and Mattsson (1996); Mattsson, Eggert, and Johannesson (1997)

(8)
(9)

where the exponent is different for in the bulk than at the boundary (subscript “end”): and , where is the Luttinger parameter ( for free fermions and for repulsive interactions). As mentioned above, the real-time oscillations are not predicted by LL theory. It is known that taking into account the finite-energy contributions within the nLL theory leads to the following contributions from the band-edge excitation in the bulk:

(10)
(11)

where is the renormalized band edge in the interacting system and the bulk exponent for the oscillating contribution is

(12)

with the phase shift of low-energy modes due to scattering off the high-energy hole [for free fermions, ; the phase shift for the interacting model in Eq. (7) will be specified in Section III].

Our new result is that the oscillating contribution at the boundary is given by

(13)
(14)

with the same band-edge frequency as in the bulk, but with a different exponent

(15)

When the band-edge mode is the dominant finite-energy contribution to the Green’s function, the asymptotic long-time decay of is well described by a linear combination of the Luttinger liquid term in Eq. (8) and the oscillating term in Eq. (10) or Eq. (13).

There are two noteworthy modifications in going from the bulk to the boundary: (i) an extra factor of in the decay of ; (ii) the doubling of the orthogonality catastrophe correction to the exponent Anderson (1967); Giamarchi (2004). Both are recurrent in the exponents that will be discussed in Section IV. Furthermore, while both exponents vary with interactions, Eqs. (12) and (15) imply the relation

(16)

which is independent of the nonuniversal phase shift .

Iii Mobile impurity model with open boundary

To derive the results above, we use the mode expansion that includes band-edge excitations

(17)

where denote the low-energy modes, creates a hole in the bottom of the band (), and all fields on the right-hand side are slowly varying on the scale of the short-distance cutoff .

A crucial assumption implicit in Eq. (17) is that we identify the excitations governing the long-time decay in the interacting model as being “adiabatically connected” with those in the noninteracting case, in the sense that they carry the same quantum numbers and their dispersion relations vary smoothly as a function of interaction strength. This condition can be verified explicitly for integrable models, where one computes exact dispersion relations for the elementary excitations. We should also note that for lattice models such as Eq. (7) the mode expansion must include a high-energy particle at the top of the band, with Pereira, White, and Affleck (2008). In the particle-hole symmetric case the latter yields a contribution equivalent to that of the deep hole with , and we get the particle contribution in the LDOS simply by taking in the result for the hole contribution. More generally, the high-energy spectrum of the interacting model may include other particles and bound states, which can also be incorporated in the mobile impurity model Pereira, White, and Affleck (2009); we shall address this question in Section VI.2.

In Eq. (17) we deliberately write the right and left movers separately, even though they are coupled by the boundary conditions Eggert and Affleck (1992); Fabrizio and Gogolin (1995). The condition is satisfied if we impose

(18)

These relations can be checked straightforwardly in the noninteracting case using the single-particle modes . The boundary condition on means that for any boundary operator that involves the high-energy mode we must take .

We bosonize the low-energy modes with the conventions

(19)
(20)

where are chiral bosonic fields that obey . A convenient way to treat the boundary conditions for the low-energy modes is to use the folding trick Wong and Affleck (1994); Fabrizio and Gogolin (1995): we include negative coordinates and identify

(21)

For the bosonic fields, we use

(22)

The effective Hamiltonian that describes the interaction between the band-edge mode and the low-energy modes is the mobile impurity model

(23)

Here is the chiral boson that diagonalizes the Luttinger model on the unfolded line

(24)

which obeys . The parameters , and are nonuniversal properties of the hole with (which is treated as a mobile impurity): its finite energy cost, effective mass and dimensionless coupling to the low-energy modes, respectively. Note that the linear term in the dispersion vanishes for the band-edge mode, which is why we have to take into account the effective mass [see Fig. 1 (b)]. In models solvable by Bethe ansatz, and are determined by the exact dispersion of single-hole excitations. The coupling can be obtained from the so-called shift function Cheianov and Pustilnik (2008); Imambekov and Glazman (2008) and the finite size spectrum Pereira, White, and Affleck (2009) for periodic boundary conditions. In Galilean-invariant systems, we can relate to the exact spectrum by using phenomenological relations Imambekov and Glazman (2009b).

The Hamiltonian in Eq. (23) contains only marginal operators. It can be obtained from the mobile impurity model in the bulk Imambekov and Glazman (2009a) by applying the folding trick. Remarkably, all boundary operators that perturb this Hamiltonian and couple the field to the bosonic modes are highly irrelevant, as they necessarily involve the derivative (which by itself has scaling dimension 3/2). For the moment we neglect the effect of all formally irrelevant boundary operators, but return to this point in Section V.

Like in the bulk case, we can decouple the impurity mode by the unitary transformation

(25)

The fields transform as

(26)
(27)

where

(28)

Eq. (26) implies

(29)

The Hamiltonian becomes noninteracting when written in terms of the transformed fields

(30)

The crucial point is that the representation of the fermion field now contains a vertex operator:

(31)

where

(32)

and

(33)

After the unitary transformation, we can calculate correlations for the free fields using standard methods. The Green’s function for the free must be calculated with the proper mode expansion in terms of standing waves, , where is the momentum cutoff of the impurity sub-band. We obtain

(34)

In the bulk regime of Eq. (34), we neglect the rapidly oscillating factor ; in this case, the free impurity propagator decays as . In the boundary case, we expand for and the free impurity propagator decays as . This faster decay is due to the vanishing of the wave function at the boundary. It can also be understood by noting that at the boundary the impurity correlator can be calculated as

(35)

and each spatial derivative amounts to an extra factor of due to the quadratic dispersion of the band-edge mode.

In addition to the free impurity propagator, we have to consider the correlator Eggert and Affleck (1992); Fabrizio and Gogolin (1995); Eggert, Johannesson, and Mattsson (1996)

(36)

Thus, in the bulk case () the correlator for the the vertex operator adds a factor of to the decay of the Green’s functions. In the boundary case, the factor is , a faster decay that stems from the correlation between and for (whereas these become uncorrelated right- and left-moving bosons in the bulk). Putting the effects together leads to

(37)

(where corresponds to particle/hole impurity) which is the result in Eqs. (13) and (15).

The scaling dimension of the vertex operator can be related to a phase shift of the low-energy modes due to scattering with the hole, establishing a connection with the orthogonality catastrophe Pustilnik et al. (2006). For the integrable model in Eq. (7), the exact phase shift is a simple function of the Luttinger parameter Pereira, White, and Affleck (2008):

(38)

where the exact Luttinger parameter is for

(39)

The renormalized band edge frequency is

(40)

The exact velocity of the low-energy modes and the effective mass of the impurity are also known: (in units where ).

In the free fermion limit, a particle tunneling into or out of the system is restricted to the free or occupied single-particle states. As is visible in Fig. 2 and Eq. (6) the LDOS is then identically zero outside of the bandwidth set by the dispersion relation. Turning on interactions allows for tunnelling processes in which the particle leaving or entering the system excites additional particle-hole pairs. This leads to a small but nonzero value for the LDOS beyond the threshold energies. The effect can be included by carefully tracking the regulators in the Luttinger liquid correlator

(41)

and the impurity correlator in Eq. (34). At the boundary and around the band minumum, the LDOS can for instance be expressed as

(42)

We see that the shoulder ratio of the two-sided singularity is determined by an interplay of both the impurity and the low-energy propagators. This is similar, but slightly different than the two-sided singularities within the continuum of the spectral function and the dynamic structure factor Pereira, White, and Affleck (2009) for which the shoulder ratio is determined by the exponents for right- and left-movers and the impurity propagater is just a delta function.

Iv Other correlation functions

The mobile impurity model in Eq. (23) can be used to calculate the exponents in the long-time decay and finite-energy singularities of several dynamical correlation functions Imambekov, Schmidt, and Glazman (2012). The general recipe for U(1)-symmetric models is to (i) identify the operator in the effective field theory that excites the band edge mode and carries the correct quantum numbers; (ii) write the operator in terms of free impurity and free bosons after the unitary transformation; and (iii) compute the correlator using the folding trick in the boundary case. In this section we apply this approach to calculate the exponents in the density autocorrelation of spinless fermions, spin autocorrelations of spin chains, and the single-particle Green’s function of spinful fermions.

iv.1 Density-density correlation

Let us now consider the density autocorrelation

(43)

Using the mode expansion in Eq. (17), we obtain the expression for the density operator including high-energy excitations

(44)

where for the half-filled chain in the model of Eq. (7) and we omitted operators that annihilate the ground state (a vacuum of particles). In the boundary case, and are identified according to Eq. (21). The leading operator generated by the low-energy part of at the boundary is , a dimension-one operator. As a result, the LL theory predicts the decay . By contrast, in the bulk case the part of has dimension and gives rise to as the leading contribution for repulsive interactions Giamarchi (2004). In summary, the low-energy term in the density autocorrelation is

(45)

with exponents

(46)

On the other hand, the high-energy term in the mode expansion for the density at the boundary yields

(47)

After bosonizing and performing the unitary transformation, we find that the high-energy term is given by

where we kept the leading operator in the expansion of the slowly-varying fields. From Eq. (IV.1) it is straightforward to show that the autocorrelation function contains a term oscillating with the frequency of the high-energy hole:

(49)

with the boundary exponent

(50)

This should be compared with the corresponding exponent in the bulk case Pereira, White, and Affleck (2008)

(51)

Therefore, the exponents associated with the frequency- oscillating term in the density autocorrelation obey the relation

(52)

As mentioned in Section III, in lattice models we also have to consider the band-edge mode corresponding to a particle at the top of the band. In this case the density operator contains an additional term that creates two high-energy modes, namely a hole at and a particle at . In the noninteracting bulk case of Hamiltonian (3), this term yields a contribution that behaves as , where the slow decay stems from the propagators of the high-energy particle and hole. However, in the presence of a repulsive interaction the decay of this contribution changes to and decays faster than the frequency- term for Pereira, White, and Affleck (2008). In the boundary case the equivalent contribution is subdominant even in the noninteracting case, where it becomes due to the faster decay of the free impurity propagator at the boundary. Therefore, the long-time decay of the density autocorrelation is well described by a combination of the LL term in Eq. (45) and the frequency- term in Eq. (49).

For the integrable model in Eq. (7), we can calculate the exponents using Eqs. (38) and (39). We also note that the power-law decay of implies a finite-energy nonanalyticity in the Fourier transform

(53)

iv.2 Spin autocorrelations

As an application of our theory to spin chains, we consider the spin-1/2 XXZ model with an open boundary

(54)

where is the spin operator on site and is the anisotropy parameter. We are interested in the long-time decay of the longitudinal () and transverse () spin autocorrelations

(55)
(56)

We focus on the critical regime . Via a Jordan-Wigner transformation Giamarchi (2004)

(57)
(58)

the XXZ model is equivalent to the spinless fermion model in Eq. (7) with interaction strength . Thus, for (the XX chain) the model is equivalent to free fermions and some time-dependent correlations can be calculated exactly Katsura, Horiguchi, and Suzuki (1970); Stolze, Nöppert, and Müller (1995). For the LL approach predicts the asymptotic decay of nonoscillating terms in the spin autocorrelations Eggert and Affleck (1992):

(59)

with exponents

(60)
(61)

where the exact Luttinger parameter is given by Eq. (39) with . Notice that the exponents for transverse and longitudinal autocorrelations coincide at the SU(2) point , where .

The high-energy contributions to the spin operator can be obtained starting from Eqs. (57) and (58) and employing the mode expansion for the fermionic field in Eq. (17) Imambekov, Schmidt, and Glazman (2012). In the bulk case, we find

(63)

At the boundary, we obtain

(64)
(65)

Calculating the correlators along the same lines as the previous examples, we obtain the oscillating terms in the autocorrelations

(66)
(67)

where

(68)
(69)

We also present, for comparison, the previously known exponents in the bulk Pereira, White, and Affleck (2008); Karimi and Affleck (2011):

(70)
(71)

The results for the longitudinal spin autocorrelation are the same as those for the density autocorrelation derived in Section IV.1, as expected from the mapping in Eq. (57). The bulk and boundary exponents for the spin autocorrelations obey a relation equivalent to Eq. (52)

(72)

which is independent of .

For the XXZ model we can simplify the result for the exponents using the exact phase shift in Eq. (38). The bulk exponents become

(73)
(74)

Our new results for the boundary exponents are

(75)
(76)

iv.3 Green’s function for spinful fermions

We now consider interacting spin-1/2 fermions, as described by the Hubbard model

(77)

where is the repulsive on-site interaction. Away from half-filling and in the absence of an external magnetic field, the low-energy spectrum is described by two bosonic fields corresponding to decoupled charge and spin collective modes. Our purpose here is to illustrate the effects of spin-charge separation on finite-energy contributions to time-dependent correlation functions. We focus on the single-particle Green’s function

(78)

In the case of spinful fermions, singular features of dynamic correlations can in principle come from both spinon and holon impurities interacting with the low-energy modes Schmidt, Imambekov, and Glazman (2010b, a); Essler, Pereira, and Schneider (2015). For repulsive interactions, the spin velocity is smaller than the charge velocity Giamarchi (2004), so the lower threshold of the spinon-holon continuum is expected to correspond to a finite-energy spinon impurity rather than a holon. Here we focus on the contribution from a single high-energy spinon to the Green’s function and to the LDOS. It is implicitly assumed that the fermion-fermion interactions are strong enough that there is a sizeable separation between the spinon band edge and the holon band edge. Otherwise, weak interactions would imply a small energy scale for spin-charge separation, making it difficult to resolve the two contributions in real time or in the frequency domain.

We follow the construction in Ref. Essler, Pereira, and Schneider, 2015 to define the operators that create finite-energy spinons coupled to low-energy charge and spin bosons, maintaining the correct quantum numbers. Starting from bosonization expressions like

(79)

we go to a spin and charge separated basis. The physical field is expanded in right and left movers and written in terms of charge and spin degrees of freedom. We will only need the right moving component for which the spinon part is projected onto the impurity operator. This leads to the projection

(80)

Here and , with for charge or spin, respectively, are the conjugate bosonic fields that diagonalize the Hamiltonian at the Luther-Emery point where spin and charge modes are exactly separated. The bosonic fields satisfy .

The impurity model is

(81)

where are the charge and spin velocities, respectively, are the Luttinger parameters, and are the energy and effective mass of the high-energy spinon, and are impurity-boson coupling constants. At the Luther-Emery point with free holons and spinons Essler, Pereira, and Schneider (2015), we have and . In contrast, SU(2)-symmetric models correspond to strongly interacting spinons.

We decouple the impurity mode by the unitary transformation

(82)

We then implement the boundary conditions by the folding trick and diagonalize the low-energy part of the Hamiltonian by a canonical transformation. We define

(83)

The final expression for the projection of the spinful fermion field operator is

(84)

Here represent the free low-energy charge and spin modes after decoupling of the impurity and creates the decoupled spinon mode.

The exponents for the corresponding oscillating contribution of are easily read off from Eq. (84). Let us restrict ourselves to the SU(2) invariant case appropriate for the Hubbard model at zero magnetic field. In this case and . We obtain

(85)

with

(86)
(87)

The singular behavior of the LDOS is obtained by Fourier transformation as before. We also obtain the relation

(88)

which is independent of . It would be interesting to test this prediction numerically and investigate the relative importance of the spinon and holon impurity configuration for the autocorrelation and LDOS of the Hubbard model.

V Role of integrability

Our results predict the exponents of autocorrelation functions at the boundary of critical one-dimensional systems assuming that the long-time decay is described by a power law. By Fourier transform, the same theory predicts the exponent of the nonanalyticity at the finite energy in the frequency domain. We expect this to hold for integrable models, where one can calculate a well-defined band-edge frequency from the renormalized dispersion relation (or dressed energy) for the elementary excitations. Examples of integrable models with open boundary conditions include the open XXZ chain Alcaraz et al. (1987); Sklyanin (1988) in Eq. (54) [or, equivalently, its fermionic version in Eq. (7)] and the Hubbard model Schulz (1985) in Eq. (77), on which many of the previous studies of local spectral properties are based.

In generic, nonintegrable models, the persistence of a nonanalyticity inside a multiparticle continuum is questionable. It has been argued that a finite-energy singularity can be protected in 1D systems by conservation of quantum numbers in high-energy bands Balents (2000). However, the high-energy subband in our effective mobile impurity model is defined by a projection of the band edge modes, which carry the same quantum numbers as the low-energy modes. Thus, strictly speaking there is no conservation law associated with the number of particles.

Figure 3: (Color online) (a) Support of the single-fermion spectral function for a generic 1D model of interacting fermions with Fermi momentum . The solid red line represents the lower threshold , below which vanishes. The band edge frequency can be identified as . (b) Support of the dynamical structure factor .

Nonetheless, we can argue that the band edge is still well defined for bulk correlations in a semi-infinite system. In the bulk one can measure momentum-resolved response functions, for instance the spectral function

(89)

or the dynamical structure factor

(90)

In momentum-resolved dynamical correlations, the spectral weight vanishes identically below a lower threshold Imambekov, Schmidt, and Glazman (2012) [see Fig. 3(a)]. This threshold is defined by kinematic constraints and exists even for nonintegrable models. The mobile impurity model in the bulk then predicts a power-law singularity as the frequency approaches the threshold from above. For instance, for the positive-frequency part of the spectral function Pereira, White, and Affleck (2009):

(91)

with defined in Eq. (33). The band edge frequency that governs the oscillations in local correlations can be identified from the spectrum as a local maximum in the lower threshold, about which the threshold is approximately parabolic. For the spectral function this happens for :

(92)

In the dynamical structure factor, the band edge can be read off from the value of the lower threshold at momentum , corresponding to the excitation composed of a hole at and a particle at the Fermi point [Fig. 3(b)].

The nonanalyticities in the local bulk correlations are related to the threshold singularities of the momentum-resolved correlations by integration over momentum. For instance, integrating the spectral function implies that the LDOS behaves as

(93)

Since the singularities in the momentum-resolved dynamic response cannot be broadened, the power-law decay of autocorrelations in the bulk is a generic property of critical 1D systems.

However, since momentum is not conserved in the presence of a boundary, the above argument cannot be used to establish power-law decay of autocorrelation functions at the boundary. From the field theory perspective, the difference between bulk and boundary cases can be understood by analyzing the effects of boundary operators that perturb the mobile impurity model in Eq. (23). In the following we shall argue that, although formally irrelevant, boundary operators introduce two important effects in nonintegrable models: (i) they may renormalize the frequency of oscillations in the boundary autocorrelation, which will then differ from the frequency in the bulk (only the latter being equal to the band edge frequency ); (ii) boundary operators that do not conserve the number of particles in high-energy subbands may give rise to a decay rate for the mobile impurity, which implies exponential decay of the boundary autocorrelation in time and the associated broadening of the nonanalyticity in the frequency domain.

For discussion purposes we will focus on the regime of weak interactions, which can be analyzed by perturbation theory in the free fermion basis, but the argument can be made more general by bosonizing the low-energy sector and the main points carry through. If we are interested in the impurity decay, we can furthermore safely neglect operators that involve the impurity field but do not couple it to the low-energy modes—these will at most renormalize the impurity dispersion.

As a simple example of a boundary operator respecting the symmetries and boundary conditions, consider the impurity-number-conserving perturbation

(94)

Here we use to denote the low-energy modes of the fermion field on the unfolded line. We will assume that