###### Abstract

We describe some field theoretic methods for studying quantum spin systems in one dimension. These include the nonlinear -model approach which is particularly useful for large values of the spin, the idea of Luttinger liquids and bosonization which are more useful for small values of spin such as spin-, and the technique of low-energy effective Hamiltonians which can be useful if the system under consideration is perturbatively close to an exactly solvable model. We apply these techniques to similar spin models, such as spin chains with dimerization and frustration, and spin ladders in the presence of a magnetic field. This comparative study illustrates the relative strengths of the different methods.

Field Theoretic Studies of Quantum Spin Systems in One Dimension

Diptiman Sen ^{1}^{1}1E-mail:

Centre for Theoretical Studies,

Indian Institute of Science, Bangalore 560012

## 1 Introduction

One-dimensional and quasi-one-dimensional quantum spin systems have been studied extensively in recent years for several reasons. Many such systems have been realized experimentally, and a variety of theoretical techniques, both analytical and numerical, are available to study the relevant models. Due to large quantum fluctuations in low dimensions, such systems often have unusual properties such as a gap between a singlet ground state and the excited nonsinglet states; this leads to a magnetic susceptibility which vanishes exponentially at low temperatures. Perhaps the most famous example of this is the Haldane gap which was predicted theoretically in integer spin Heisenberg antiferromagnetic chains [1], and then observed experimentally in a spin- system [2]. Other examples include the spin ladder systems in which a small number of one-dimensional spin- chains interact amongst each other [3]. It has been observed that if the number of chains is even, i.e., if each rung of the ladder (which is the unit cell for the system) contains an even number of spin- sites, then the system effectively behaves like an integer spin chain with a gap in the low-energy spectrum. Some two-chain ladders which show a gap are [4], [5] and [6]. Conversely, a three-chain ladder which effectively behaves like a half-odd-integer spin chain and does not exhibit a gap is [5]. A related observation is that some quasi-one-dimensional systems such as spontaneously dimerize below a spin-Peierls transition temperature [7]; then the unit cell contains two spin- sites and the system is gapped. Another interesting class of systems are the alternating spin chains such as bimetallic molecular magnets. An example is in which spin-’s () and spin-’s () alternate. The ground state of these systems have a nonzero total spin . It turns out that there is a gap to states with spin greater than , but no gap to states with spin less than .

The results for gaps quoted above are all in the absence of an external magnetic field. The situation becomes even more interesting in the presence of a magnetic field [8]. Then it is possible for an integer spin chain to be gapless and a half-odd-integer spin chain to show a gap above the ground state for appropriate values of the field [9, 10, 11, 12, 13]. This has been demonstrated in several models using a variety of methods such as exact diagonalization of small systems, bosonization and conformal field theory [14, 15], and perturbation theory [16]. In particular, it has been shown that the magnetization of some systems can exhibit plateaus at certain nonzero values for some finite ranges of the magnetic field.

The plan of this paper is as follows. In Sec. 2, we discuss the low-energy properties of the dimerized and frustrated antiferromagnetic spin chain. In Secs. 3 and 4, we present some field theoretic methods which can be used for studying spin chains and ladders with or without an external magnetic field [17]. These methods rely on the idea that the low-energy and long-wavelength modes of a system (i.e., wavelengths much longer than the lattice spacing if the system is defined on a lattice at the microscopic level) can often be described by a continuum field theory. In Sec. 3, we discuss the nonlinear -model approach, while in Sec. 4, we discuss the concepts of Tomonaga-Luttinger liquids and bosonization. In Sec. 5, we discuss the low-energy effective Hamiltonian approach and show how it can be combined with bosonization to gain an understanding of the magnetic properties of one-dimensional spin systems.

## 2 Spin Chain with Dimerization and Frustration

Experimental studies of some of the quasi-one-dimensional spin systems have shown that besides the nearest neighbor antiferromagnetic exchange, there also exists a second neighbor exchange of the same sign and comparable magnitude. Such a second neighbor interaction has the effect of frustrating the spin alignment favored by the nearest neighbor interaction. Therefore, a realistic study of one-dimensional systems requires a model with both frustration () and dimerization (governed by a parameter ).

The Hamiltonian for the frustrated and dimerized antiferromagnetic spin chain can be written as

(1) |

where the limits of the summation depend on the boundary condition (open or periodic). (We have set the average nearest neighbor interaction to be equal to for convenience). The interactions are schematically shown in Fig. 1. The region of interest is defined by and .

The ground state properties of the Hamiltonian (1) have been studied at some representative points in the plane using the density-matrix renormalization group (DMRG) method [18]. The phase diagrams obtained for spin- and spin- chains are shown in Figs. 2 and 3 [19]. We use the word ‘phase’ only for convenience to distinguish between regions with different modulations of the two-spin correlation function as discussed later. Our model actually has no phase transition even at zero temperature.

For the spin- chain [20, 21], the system is found to be gapless on the line which runs from to for ; see Fig. 2. The model is gapped everywhere else in the plane. There is a disorder line given by on which the exact ground state of the model is given by a product of singlets formed by the nearest-neighbor spins which are joined by the stronger bonds (); this is called the Shastry-Sutherland line [22], and it ends at the Majumdar-Ghosh point . The correlation length goes through a minimum on . Finally, the peak in the structure factor is at to the left of (called region I), decreases from to as one goes from up to the line (region II), and is at to the right of (region III).

In the spin- case (Fig. 3), the phase diagram is more complex. There is a solid line marked which runs from to about shown by a cross. To within numerical accuracy, the gap is zero on this line and the correlation length is as large as the system size . The rest of the ‘phase’ diagram is gapped. However the gapped portion can be divided into different regions characterized by other interesting features. On the dotted lines marked , the gap is finite. Although goes through a maximum when we cross in going from region II to region I or from region III to region IV, its value is much smaller than . There is a dashed line extending from to about on which the gap appears to be zero (to numerical accuracy), and is very large but not as large as . In regions II and III, the ground state for an open chain has a four-fold degeneracy (consisting of and ), whereas it is nondegenerate in regions I and IV with . The regions II and III, where the ground state of an open chain is four-fold degenerate, can be identified with the Haldane phase. The regions I and IV correspond to the non-Haldane singlet phase. Regions I and IV are separated by the disorder line given by , while regions II and III are separated by line . The lines , and seem to meet in a small region V where the ground state of the model is numerically very difficult to find.

As can be seen from Fig. 1, setting results in a two-chain ladder where the interchain coupling is and the intrachain coupling is . We can hold fixed and vary the interchain coupling . Numerical studies show that for spin-, the system is gapped for any nonzero value of , although the gap vanishes linearly as ; this can be shown using bosonization. On the other hand, the spin- chain has a finite value of the gap for any value of [19].

## 3 Nonlinear -model

The nonlinear -model (NLSM) analysis of spin chains with the inclusion of and proceeds as follows [23]. We first do a classical analysis in the limit to find the ground state configuration of the spins. Let us make the ansatz that the ground state is a coplanar configuration of the spins with the energy per spin being equal to

(2) |

where is the angle between the spins and and is the angle between the spins and . Minimization of the classical energy with respect to the yields the following three phases.

(i) Neel: This phase has ; hence all the spins point along the same line and they go as along the chain. This phase is stable for .

(ii) Spiral: Here, the angles and are given by

(3) |

where and . Thus the spins lie on a plane. This phase is stable for .

(iii) Colinear: This phase (which needs both dimerization and frustration) is defined to have and ; hence all the spins point along the same line and they go as along the chain. It is stable for .

These phases along with their boundaries are depicted in Fig. 4. Thus even in the classical limit , the system has a rich ground state ‘phase diagram’.

We can go to the next order in , and study the spin wave spectrum about the ground state in each of the phases. The main results are as follows. In the Neel phase, we find two zero modes, i.e., modes for which the energy vanishes linearly at certain values of the momentum , with the slope at those points being called the velocity. The two modes are found to have the same velocity in this phase. In the spiral phase, we have three zero modes, two with the same velocity describing out-of-plane fluctuations, and one with a higher velocity describing in-plane fluctuations. In the colinear phase, we get two zero modes with equal velocities just as in the Neel phase. The three phases also differ in the behavior of the spin-spin correlation function in the classical limit. is peaked at , i.e., at in the Neel phase, at in the spiral phase and at in the colinear phase. Even for and , DMRG studies have seen this feature of in the Neel and spiral phases [19].

We now derive a NLSM field theory which can describe the low-energy and long-wavelength excitations. In the Neel phase, this is given by a NLSM with a topological term [1, 15]. The field variable is a unit vector which is defined as follows. The classical ground state in the Neel phase has a unit cell, labeled by an integer , with two sites labeled as and respectively; see Fig. 5. We define linear combinations of the two spins as

(4) |

Here is the lattice spacing; hence, the size of each unit cell is . Note that

(5) |

so that becomes an unit vector in the large limit. These fields satisfy the commutation relations

(6) |

where are unit cell labels, denote the components , and is the completely antisymmetric tensor with . This means that we can write , where the vector is canonically conjugate to , i.e.,

(7) |

We now go to the continuum limit by introducing a spatial coordinate which is equal to at the location of the unit cell. Summations get replaced by integrals, i.e., . The commutation relation (7) then takes the form

(8) |

We note that and are orthogonal to because is an unit vector. We will see below that both and are given by first-order space-time derivatives of . In the low-energy and long-wavelength limit, the dominant terms in the Hamiltonian will be those which have second-order derivatives of , and therefore first-order derivatives of . To find this Hamiltonian, we rewrite (1) in terms of and , and Taylor expand these fields to the necessary order, i.e.,

(9) |

where . We then use the constraints in (5) and do some integration by parts (throwing away boundary terms at ) to obtain the continuum Hamiltonian

(10) |

where

(11) |

By expanding (10) to second order in small fluctuations around, say, , we find an energy-momentum dispersion relation of the ‘massless relativistic’ form ; thus is the spin wave velocity. Similarly, by expanding (10) to fourth and higher orders in small fluctuations, we find that is the coupling constant governing the strength of the interactions between the spin waves.

One can show that the Hamiltonian (10) follows from the Lagrangian density

(12) |

(Incidentally, one can derive the canonically conjugate momentum and then the angular momentum from (12),

(13) |

thereby verifying that and only contain first-order derivatives of as stated above). From (12), we see that is the coefficient of a topological term, because the integral of this term is an integer which defines the winding number of a field configuration ). For mod and less than a critical value , it is known that the system is gapless and is described by a conformal field theory with an symmetry [15, 24]. For any other value of , the system is gapped, and the gap is of order . For , one therefore expects that integer spin chains should have a gap of the order (note that this goes to zero rapidly as , so that there is no difference between integer and half-odd-integer spin chains in the classical limit), while half-integer spin chains should be gapless. For the two-spin equal-time correlation function, this means that should decay as a power-law as for half-odd-integer spin chains, and exponentially as for integer spin chains, where the correlation length . All this is known to be true even for small values of like (analytically) and (numerically) although the field theory is only derived for large . In the presence of dimerization, one expects a gapless system at certain special values of . For , the special value is predicted to be . We see that the existence of a gapless point is correctly predicted by the NLSM. However, according to the DMRG results, is at for [25] and it decreases with as shown in Fig. 3; this differs from the NLSM results in (11) according to which should be independent of . These deviations from field theory are probably due to higher order corrections in which have not been studied analytically so far.

In the spiral phase of the model, it is necessary to use a different NLSM which is known for [26, 27]. The field variable is now an matrix . The Lagrangian density is

(14) |

where , with , and and are diagonal matrices with diagonal elements and respectively. Note that there is no topological term; indeed, no such term is possible since unlike for the NLSM in the Neel phase. Hence there is no apparent difference between integer and half-integer spin chains in the spiral phase. A one-loop renormalization group [26] and large analysis [27] indicate that the system should have a gap for all values of and , and that there is no reason for a particularly small gap at any special value of . The ‘gapless’ point found numerically at for spin- is therefore a surprise.

Finally, in the colinear phase of the model, the NLSM is known for , i.e., for the spin ladder. The Lagrangian is the same as in (12), but with , and . There is no topological term for any value of , and the model is therefore gapped.

The field theories for general in both the spiral and colinear phases are still not known. Although the results are qualitatively expected to be similar to the case in the spiral phase and the case in the colinear phase, quantitative features such as the dependence of the gap on the coupling strengths require the explicit form of the field theory.

The NLSMs derived above can be expected to be accurate only for large values of the spin . It is interesting to note that the ‘phase’ boundary between Neel and spiral for spin- is closer to the the classical () boundary than for spin-. For instance, the cross-over from Neel to spiral occurs, for , at for spin-, at for spin-, and at classically.

To summarize, we have studied a two-parameter ‘phase’ diagram for the ground state of isotropic antiferromagnetic spin- and spin- chains using the NLSM approach, and have compared the results with those obtained numerically. We find that the spin- diagram is considerably more complex than the corresponding spin- chain with surprising features like a ‘gapless’ point inside the spiral ‘phase’; this point could be close to a critical point discussed earlier in the literature [24, 28]. It would be interesting to establish this more definitively.

Our results show that frustrated spin chains with small values of exhibit some features not anticipated from large field theories like the NLSMs. The NLSMs also leave many questions unanswered. For instance, the NLSM which is applicable in the Neel phase does not tell us the exponent of the gap which opens up as one moves away from (for ) or as we go across (for ). To address these questions, we have to use the more powerful technique of bosonization.

The NLSM approach can also be used to study spin chains in the presence of a magnetic field. Consider adding a Zeeman term to the Hamiltonian in (1), i.e.,

(15) |

where denotes the magnetic field. In the region , the classical ground state of this Hamiltonian is given by a coplanar configuration in which the spins and lie at angles respectively with respect to the magnetic field, so that the angle between the spins and is . Minimization of the energy fixes the angle to be

(16) |

(We are assuming that , otherwise all the spins will align with the magnetic field and will be zero). We now define

(17) |

Note that the definition of is slightly different from the one in (4) in order to ensure that is an unit vector. However, is orthogonal to and has the same commutation relations with as before. We can now go to the continuum limit and derive the Hamiltonian

(18) |

where

(19) |

We can show that this follows from the Lagrangian density

(20) | |||||

We see from this that

(21) |

Since and in the classical ground state, we see that is equal to plus small fluctuations; this agrees with its definition in (17) and the classical configuration of the spins. One can now analyse the field theory governed by (20) using the renormalization group and other methods. We refer the reader to [29] for further details.

## 4 Bosonization

A very useful method for studying spin systems in one dimension is the technique of bosonization. Before describing this method, let us briefly present some background information. Further details can be found in Refs. [14, 15, 30, 31].

In one dimension, a great variety of interacting quantum systems (both fermionic and bosonic) is described by the Tomonaga-Luttinger liquid (TLL) theory. Typically, a TLL describes quantum systems which are translation invariant and gapless, i.e., the excitation energy above the ground state is zero in the limit of the system size . A TLL differs in three significant ways from the well-known Fermi liquid theory which describes many fermionic systems in two and three dimensions. First, all the low-energy excitations in a TLL have the character of sound modes which are bosonic and have a linear dispersion relation between the energy and the momentum (with the constant of proportionality being the sound velocity ). Even if the underlying theory is fermionic, the low-energy excitations are given by particle-hole pairs which are bosonic. The properties of a TLL are governed by two important parameters, namely, an interaction parameter (noninteracting systems have ) and the velocity . Secondly, the one-particle momentum distribution function for fermions, which is obtained by Fourier transforming the fermion Green’s function

(22) |

and computing the residue of its pole in the complex plane as a function of , has no discontinuity at the Fermi surface for a TLL. Instead, it has a cusp there of the form

(23) |

On the other hand, in a Fermi liquid, has a finite discontinuity at the Fermi surface; see Fig. 6. Finally, correlation functions in a TLL typically decay at large distances as power-laws which depend on , unlike the correlation functions of a Fermi liquid where the power-laws are universal.

Let us be more specific about the nature of the low-energy excitations in a one-dimensional system of interacting fermions. Assume that we have a system of length with periodic boundary conditions; the translation invariance and the finite length make the one-particle momenta discrete. Suppose that the system has particles with a ground state energy and a ground state momentum . We will be interested in the thermodynamic limit keeping the particle density fixed. If we could switch off the interactions, the fermions would have two Fermi points, at respectively, with all states with momenta lying between the two points being occupied. (See Fig. 7 for a typical picture of the momentum states of a lattice model without interactions). Even in the presence of interactions, it turns out that the low-lying excitations consist of two pieces [32],

(i) a set of bosonic excitations each of which can have either positive momentum or negative momentum with an energy , where and is the sound velocity, and

(ii) a certain number of particles and added to the right and left Fermi points respectively, where . Note that and can be positive, negative or zero.

The quasiparticle excitations in (i) have an infinite number of degrees of freedom (in the thermodynamic limit), and they determine properties such as specific heat and susceptibility to various perturbations. The particle excitations in (ii) only have two degrees of freedom and therefore play no role in the thermodynamic properties. The Hamiltonian and momentum operators for a one-dimensional system (which may have interactions) have the general form

(24) |

where is the momentum of the low-energy bosonic excitations created and annihilated by and , is a positive dimensionless number, and is the chemical potential of the system. We will see later that and are the two important parameters which determine all the low-energy properties of a system. Their values generally depend on both the strength of the interactions and the density. If the fermions are noninteracting, we have

(25) |

Note that one can numerically find the values of and by varying and and studying the dependence of energy and momentum of finite size systems.

The technique of bosonization (combined with conformal field theory) is very useful for analytically studying a TLL [14, 15, 30, 31]. This technique consists of mapping bosonic operators into fermionic ones, and then using whichever set of operators is easier to compute with.

To begin, let us consider a fermion with both right- and left-moving components. We introduce a chirality label , such that and refer to right- and left-moving particles respectively. Sometimes we will use the numerical values and for and ; this will be clear from the context. Then the second quantized Fermi fields are given by

(26) |

where , and

(27) |

Next we define bosonic operators

(28) |

Note that and create excitations with momenta and respectively, where the label is always taken to be positive. We can show that

(29) |

The vacuum state of the system is defined to be the state which is annihilated by the operators for and for , and therefore by for all .

Let us define the chiral bosonic fields

(30) |

where the length parameter is a cut-off which is required to ensure that the contribution from high-momentum modes do not produce divergences when computing correlation functions. The fields in (30) satisfy

(31) |

in the limit . It is useful to define two fields dual to each other

(32) |

Then , while

(33) |

Now it can be shown that the fermionic and bosonic operators discussed above are related to each other as

(34) |

in the sense that they produce the same state when they act on the vacuum state , and they have the same correlation functions. The unitary operators and are called Klein factors, and they are essential to ensure that the fermionic fields given in Eq. (34) anticommute at two different spatial points and .

The densities of the right- and left-moving fermions are given by and . The total fermionic density and current are given by

(35) |

where is the background density (fluctuations around this density are described by the fields or ), and the velocity will be introduced below.

Let us now introduce a Hamiltonian. We assume a linear dispersion relation for the fermions. The noninteracting Hamiltonian then takes the form

(36) | |||||

in the fermionic language, and

(37) | |||||

in the bosonic language.

We now study the effects of four-fermi interactions. Let us consider an interaction of the form

(38) |

Physically, we may expect an interaction such as , so that . However, it is instructive to allow to differ from to see what happens. Also, we will not assume anything about the signs of and . In the fermionic language, the interaction takes the form

From this expression we see that corresponds to a two-particle scattering involving both chiralities; in this model, we can call it either forward scattering or backward scattering since there is no way to distinguish between the two processes in the absence of some other quantum number such as spin. The term corresponds to a scattering between two fermions with the same chirality, and therefore describes a forward scattering process.

The quartic interaction in Eq. (LABEL:int) seems very difficult to analyze. However we will now see that it is easily solvable in the bosonic language; indeed this is one of the main motivations behind bosonization. The bosonic expression for the total Hamiltonian is found to be

(40) | |||||