Exact calculation of the magnetocaloric effect in the spin-1/2 XXZ chain
We calculate the entropy and cooling rate of the antiferromagnetic spin-1/2 chain under an adiabatic demagnetization process using the quantum transfer-matrix technique and non-linear integral equations. The limiting case of the Ising chain (corresponding to infinitely large anisotropy) is presented for comparison. Our exact results for the Heisenberg chain are used as a crosscheck for the numerical exact diagonalization as well as Quantum Monte Carlo simulations and allow us to benchmark the numerical methods. Close to field-induced quantum phase transitions we observe a large magnetocaloric effect. Furthermore, we verify universal low-temperature power laws in the cooling rate and entropy, in particular linear scaling of entropy with temperature in the gapless Luttinger-liquid state and scaling as at field-induced transitions to gapped phases.
pacs:75.10.Pq; 75.30.Sg; 02.70.-c
The magnetocaloric effect (MCE) in general addresses the change of temperature of magnetic systems under the variation of an external magnetic field. The MCE has been known since the end of the 19th century,(1) and it has attracted renewed interest recently because of potential room-temperature cooling applications (see Refs. (2); (3) for recent reviews). On the other hand, adiabatic demagnetization is a standard low-temperature method: demagnetization of paramagnetic salts was the first method to reach temperatures below K(4) whereas demagnetization of nuclear spins has reached record low temperatures down to 100pK(5); (6) and is still the cooling method of choice in the K-range.(7) The cooling rate at the adiabatic demagnetization for an ideal paramagnet (i.e., a system of non-interacting magnetic moments) is equal to , which means linear monotonic dependence of temperature on the magnetic field magnitude. The latter is a direct consequence of the fact that for any paramagnetic system the entropy depends only on the ratio , so for any isentrope one gets . However, the matter could undergo crucial changes for systems of interacting spins. For instance, in ferromagnets near the Curie point one can observe a substantial enhancement of the effect.(8)
As has been shown in early investigations, quantum antiferromagnets are more efficient low-temperature magnetic coolers than ferromagnets.(9); (10); (11) This fact is connected with the behavior of the entropy of antiferromagnets. The entropy of any antiferromagnet at low temperatures displays (at least) one maximum as a function of magnetic field, which usually, according to the third law of thermodynamics, falls to zero at .(10) The Ising model is anomalous in this respect, because of non-vanishing zero-temperature entropy at the critical magnetic field , where is the coordination number of the lattice and is the coupling constant. Indeed, understanding the influence of quantum fluctuations seems to have been an important motivation for the numerical work Ref. (10). The first exact result concerning magnetocaloric properties of the spin-1/2, Ising-like chain has been obtained in Ref. (11), where the isentropes in the plane have been presented. The main feature of the isentropes of the Ising-like chain is the appearance of two minima. To the best of our knowledge, the isentropes of the isotropic spin-1/2 Heisenberg chain have so far been investigated only numerically(12) (for numerical studies of the magnetocaloric effect in ferrimagnetic spin chains and higher-spin Heisenberg chains see Refs. (13); (14)). Recent measurements(15) of the adiabatic cooling rate in the spin-1/2 Heisenberg chain compound [Cu(-CO)(4-aminopyridine)(HO)] render the magnetocaloric effect of the spin-1/2 Heisenberg chain a topic of current interest.
More generally, the MCE is particularly large in the vicinity of quantum critical points (QCPs). The MCE is closely related to the generalized Grüneisen ratios
Here is the control parameter governing the quantum phase transition. In the case of the MCE is the external magnetic field . Using basic thermodynamic relations,(3) the generalized Grüneisen ratio can be related to the adiabatic cooling rate :(16); (17)
Thus, the magnetic cooling rate is an important quantity for the characterization of QCPs, i.e., quantum phase transitions between different magnetic structures under tuning the magnetic field at .
In passing we mention that an analysis of classical spin models(18) demonstrated that the MCE can be enhanced by geometric frustration. Indeed, adiabatic demagnetization experiments on the frustrated spin-7/2 pyrochlore-type magnet GdTiO have shown substantial drops in temperature in the vicinity of the saturation field.(19) Enhanced cooling performance is also theoretically predicted in one-dimensional quantum antiferromagnets such as the -chain and the sawtooth chain,(12) the diamond chain,(20); (21); (22); (23); (24) as well as in two dimensions.(25); (26)
The one-dimensional spin-1/2 Heisenberg model is famous for its integrability. The conventional Bethe ansatz technique allows one to obtain all eigenvalues and eigenvectors of the corresponding Hamiltonian, though in non-explicit form. There are several sophisticated methods to describe thermodynamics of one-dimensional integrable models, like thermodynamic Bethe ansatz (TBA), etc. (see for example Refs. (27); (28)). For pragmatical reasons, the most suitable technique is the quantum transfer-matrix (QTM) method leading to only two non-linear integral equations (NLIE) for the free energy of the Heisenberg chain, see Refs. (29); (30); (31) and references therein. A numerical solution of these NLIEs has been used in Ref. (29) to obtain the magnetic susceptibility and specific heat of the isotropic spin-1/2 Heisenberg chain in an external magnetic field. However, as far as we are aware, there are no exact results for the free energy (or equivalently the entropy) and a mixed derivative of the free energy which corresponds to the magnetic cooling rate in the literature for the isotropic spin-1/2 Heisenberg chain. Filling these gaps is one of the purposes of the present paper.
In this paper we present calculations of quantities related to the MCE for the one-dimensional spin-1/2 Heisenberg chain within the QTM and NLIE method. Particularly, isentropes for different values of the exchange anisotropy parameter are obtained. The cooling rate is computed as a function of external magnetic field for various fixed values of temperature and different values of the exchange anisotropy. The limiting case of infinitely large anisotropy which is just the Ising model is considered as well. In that case the expressions for all relevant thermodynamic quantities can be obtained in closed form. These calculations are supplemented by exact diagonalization (ED) and Quantum Monte Carlo (QMC) calculations demonstrating full agreement between results obtained from the exact solution and numerical calculations.
Ii Isentropes and Cooling rate for the spin-1/2 Heisenberg chain
We will be interested both in the entropy and the associated isentropes as well as the temperature derivatives thereof, i.e., the adiabatic cooling rate . Using standard thermodynamic relations one can express the latter as follows:(3)
where is the specific heat at constant magnetic field, and is the magnetization of the system. After normalization with a factor , the magnetic cooling rate can be identified with the generalized Grüneisen ratio Eq. (2) which we will use in the following.
We will be specifically interested in the spin-1/2 Heisenberg chain whose Hamiltonian is given by
Here is the total number of sites, are spin-1/2 operators acting at site , is the exchange constant, an exchange asymmetry, and an external magnetic field. For the finite-size computations we will assume periodic boundary conditions, i.e., . Note that the properties of the model (4) are symmetric under and . We will therefore concentrate on in the following.
It is useful to recall the zero-temperature phase diagram of the model Eq. (4), see Fig. 1.(32); (33); (34) For and small magnetic fields there is long-range antiferromagnetic order along the -direction in spin space. This state exhibits a gap in the excitation spectrum whose value at corresponds to . The value of the gap has been computed exactly via the Bethe ansatz.(35); (36) In particular, it is exponentially small close to the Heisenberg point at (marked by the dot in Fig. 1), which is characteristic for a Kosterlitz-Thouless transition.(37); (38) For , one finds a gapless Luttinger-liquid state.(39) This Luttinger-liquid state exists for , and , , with . Finally, for , the ground state is the ferromagnetically polarized state along the -direction which exhibits again a gap. We anticipate that these different zero-temperature regions, in particular the quantum phase transitions at and will be reflected by the magnetocaloric properties at finite temperature.
ii.1 Non-linear integral equations
First we consider the case . One can represent the free energy of the system per lattice site in the following form:
(the constant is irrelevant in the present context). The auxiliary functions and are found from the following system of integral equations
Here , the symbol denotes convolution and the function is defined by
These equations are valid for . Results for negative can be obtained by changing the sign of the coupling .
For the case the free energy has the form
with denoting a convolution with modified integration limits and
with . The auxiliary functions are solutions of the following integral equations
with integration kernels
Results for can again be obtained by changing the sign of the coupling .
Having all these exact expressions one can obtain any thermodynamic quantity of interest by iteration of the NLIE (6) (or (10)) and numerical integration of the expression for the free energy (5) (or (8)). Derivatives of the free energy with respect to and can also be calculated. One can avoid numerical differentiation by solving the associated integral equations for the differentiated auxiliary functions, e.g. . Note that derivatives of and are treated as independent functions in these equations. As an example we will give the equations for the calculation of the magnetization per spin and the derivative of with respect to the temperature in the regime
The derivatives and satisfy linear integral equations in which the auxiliary functions and enter as external functions
To obtain the cooling rate we have to determine . In order to achieve this we have to differentiate (12) with respect to . However it has turned out that in the framework of NLIEs the resulting equations in general behave numerically better if the derivatives are taken with respect to the inverse temperature (we set ).
Here four new functions , and their counterparts occur. The corresponding linear integral equations read
Note that the functions and already allow the calculation of the entropy per spin
For the calculation of the cooling rate the specific heat is needed in addition to the mixed derivative . It can be obtained by differentiating (17) with respect to and dividing by . In the resulting equation another pair of functions , occurs, where the corresponding integral equations are derived by differentiating (15) again with respect to in close analogy to (16).
ii.2 Comparison with numerical results
First, we present a comparison with numerical methods, namely exact diagonalization (ED) and Quantum Monte Carlo (QMC). On the one hand, this comparison will serve as a cross-check of our results. On the other hand, we can use the exact results to assess the performance of the numerical methods.
The quantities appearing on the r.h.s. of Eq. (2) can be expressed as follows:
where is the expectation value at a fixed temperature and magnetic field . Here we have chosen a normalization per spin which drops out when taking the ratio in Eq. (2). Eq. (18) is well known and Eq. (19) is valid for any Hamiltonian conserving magnetization, i.e., .
One can write down spectral representations for the correlation functions in Eqs. (18) and (19) which can be evaluated by ED. These correlation functions can also be evaluated with QMC. The QMC simulations to be reported below have been carried out with the ALPS(40); (41) directed loop application(42) in the stochastic series expansion framework.(43) The specific heat Eq. (18) is measured using an improved estimator which involves the fluctuations of the expansion order.(44) The correlation function Eq. (19) can be measured in a similar manner. Note that it is crucial to choose an appropriate pseudo random-number generator in order to obtain correct results. We have used the “Mersenne Twister”.(45) In our QMC simulations we have performed thermalization steps and then collected data during a number of sweeps ranging between and .
Fig. 2 shows a comparison of between the exact solution for , ED for , and QMC for at three temperatures which have been chosen to correspond to the experiments of Ref. (15). The ED curves exhibit some wiggles at small magnetic fields and temperatures which reflect the fact that the system contains only sites. In this regime, the QMC results for are indistinguishable from the exact solution for on the scale of Fig. 2. On the other hand, the QMC results are subject to big error bars at high fields and low temperatures despite the fact that we have invested substantial amounts of CPU time into these data points. To some extent, this is related to performance problems of the algorithm in a magnetic field.(43) However, the main reason is that is given by the ratio of two quantities (see Eq. (2)) which are both exponentially small in for . Indeed, even with a large number of sweeps, it is difficult to determine the ratio of two very small quantities accurately by QMC. Conversely, the existence of a gap improves finite-size convergence such that ED works particularly well in the high-field regime.
The overall good agreement between all three methods serves as a consistency check for each of them. Furthermore, we see that we can get extremely accurate numerical results by combining QMC for and ED for ( in the present case).
ii.3 Effects of anisotropy
Next, we discuss the effects of exchange anisotropy on the cooling rate. Fig. 3 shows results for three selected values , , and . Recall from Fig. 1 that, at , one crosses just one quantum phase transition at , the curves for in Fig. 3 start from the special Heisenberg point at and cross the transition to saturation at , and the curves at cut two quantum phase transitions, namely and . For , the quantum phase transitions at and for at are signaled by sign changes of the cooling rate from negative to positive values upon increasing field, see Fig. 3(a). Note that these zeros shift away from the position of the zero-temperature phase transitions with increasing temperature, as is evident in particular if one takes into account the additional data for shown in Fig. 2. Finally, there is a small structure at low magnetic fields in the curves of Figs. 2(c) and 3(a) which reflects the singular nature of the Heisenberg point .
At the isotropic point of the antiferromagnetic chain, the quantity shows singular behavior for if . In fact, in the limit of the ratio reduces to the temperature derivative of the zero-field susceptibility
which is known to show singular behavior. For this function diverges like where is a (non-universal) constant.(46) For the function (20) diverges like with Tomonaga-Luttinger parameter .(47) For a divergent behavior like is observed. The strongest divergence is hence exhibited for . For finite magnetic field we see a non-divergent behavior for .
With increasing temperature, all features become broader. Note that the temperature shown in Fig. 3(b) is higher than the value of at which explains why the sign change around in the curve disappears for . For the even higher temperature shown in Fig. 3(c), the cooling rate is positive for all (the zero at is enforced by the symmetry under ). Note that is bigger than itself for all cases shown in Fig. 3(c) which explains why all features are washed out at this temperature.
ii.4 The case of infinitely large anisotropy: cooling rate for the Ising chain
The case of infinitely large anisotropy ( at ) corresponds to the Ising model with Hamiltonian
where the classical variables take values and for simplicity reasons we drop the subscript of the coupling in this section. It is well known that the Ising chain can be solved exactly and completely analytically by the transfer-matrix method (see for example Refs. (48); (49)) and even was investigated recently.(50) However, we are not aware of explicit results for the entropy and normalized cooling rate of the Ising chain and therefore present them here.
We start from the free energy per lattice site of the Ising chain:
From this expression one can easily obtain simple analytic expressions for all thermodynamic functions of the system. In particular one obtains for the entropy per spin and the normalized cooling rate:
The limit of the entropy (23) is generically , except for , where one finds (see also Ref. (51)). This reflects the macroscopic ground-state degeneracy of the Ising model at the saturation field .(10) Remarkably, the above transfer-matrix solution is closely related to the hard-dimer description of certain highly frustrated one-dimensional quantum antiferromagnets.(12); (22); (52); (53); (54)
Fig. 4 shows the cooling rate dependence on the magnetic field for the Ising chain obtained from the exact solution. The main difference from the case are the very sharp and pronounced positive and negative peaks at the critical value of the magnetic field. The magnitude of these peaks grows rapidly with decreasing temperature. This behavior is a direct consequence of the anomalous zero-temperature entropy of the Ising chain at .(10)
A bit further away from the QCP at , we can make contact with the argumentation of Ref. (17). The Grüneisen ratio, which in our case is the cooling rate , shows divergent behavior close to the QCP at and changes sign when the magnetic field crosses it. For , the divergent behavior obeys the universal scaling law(16); (17)
where is a universal amplitude. Detailed analysis of the Ising case yields the exact analytic form of the cooling rate at extremely low temperatures, for , i.e., the value expected for a -symmetry in one dimension.(17); (12)
Finally, we take a look at the entropy . As far as we are aware, there has been only one previous attempt of an exact computation(11) of the entropy and accordingly the isentropes of the chain in a magnetic field. Inspection of the critical fields and indicates that the result of Ref. (11) corresponds to an anisotropy . The main technical difference between Ref. (11) and the present work is that Ref. (11) worked with an infinite hierarchy of equations for which need to be truncated while we have a closed finite set of equations for all values of . In the case the structure of the NLIE (6) is independent of in contrast to the usual TBA equations, which allows to easily calculate the entropy as a function of the anisotropy.
Fig. 5 shows the entropy and the isentropes for the spin-1/2 Heisenberg chain in the - plane. This result agrees with previous ED for sites.(12) However, the ED results of Ref. (12) suffered from finite-size effects, in particular for low temperatures and . By contrast, Fig. 5 shows results for the thermodynamic limit. Fig. 6 shows a similar plot of the entropy and isentropes with varying anisotropy but now at a fixed magnetic field . The quantum phase transitions at and are reflected in Figs. 5 and 6 by minima of the isentropes as a function of or , or equivalently maxima of the entropy at a low but constant temperature. The only exception is Fig. 6(a) where one observes no such clear signature of the Kosterlitz-Thouless transition at for . Before we discuss this case in more detail, it is useful to examine the low-temperature asymptotics of the entropy at otherwise fixed parameters and .
Between the two quantum critical points, that is for , the low-energy theory is a Luttinger liquid (compare remarks at the beginning of Sec. II). It is well known that the specific heat of a Luttinger liquid is linear in .(55) Due to the relation (see, e.g., Ref. (3)) and because of , the entropy of a Luttinger liquid is identical to its specific heat and in particular also linear
where is the velocity of the excitations.
The cases and are instances of a quantum phase transition in one dimension which preserves a -symmetry. In this case, the universal low-temperature asymptotics is predicted to follow a square root(9); (12); (16); (17)
Finally, in the gapped cases or , we expect activated behavior
where is the gap in the excitation spectrum. Closer inspection of the data shown in Figs. 5 and 6 indeed verifies Eqs. (27), (28), or (29), respectively. Note that while the asymptotic behavior is for in all three cases, the decay is slowest exactly at the quantum phase transition, see Eq. (28). This naturally yields a maximum of the entropy if the quantum phase transition is crossed by varying the parameters or at a fixed temperature .
The case , is an exception to this general scenario. To first approximation, this point behaves like a Luttinger liquid. The fact that one is at a quantum critical point with marginally irrelevant operators gives rise to higher-order logarithmic corrections in the free energy and specific heat.(29); (30) Consequently, one also expects just higher-order logarithmic corrections to the entropy. To test this scenario, we can use the fact that at the velocity which enters (27) is known exactly (see, e.g., Refs. (39); (55)):
with as above. Fig. 7 shows that insertion of these values of into the Luttinger-liquid expression Eq. (27) yields indeed the correct low-temperature asymptotics of the entropy per spin at not only for but also for . In fact, the higher-order logarithmic corrections to the Luttinger liquid asymptotics (27) which are expected at and are so small that they have no visible effect.
A gap opens for at , but because the phase transition is a Kosterlitz-Thouless transition,(37); (38) this gap is exponentially small close to .(35); (36) Accordingly, close to one has to go to very low temperatures in order to observe the crossover from (27) to (29). This is illustrated by the curve in Fig. 7. In the concrete case , the value of the gap is . Accordingly, the exponential decay (29) can be observed only for temperatures while at higher temperatures the behavior of remains approximately linear.
The combined effect of all these observations is that just a small kink develops in the low-temperature isentropes of Fig. 6(a) whose position shifts very slowly to the quantum critical point for .
Iii Conclusion and Perspectives
Motivated by recent measurements of the magnetic cooling rate in a spin-1/2 Heisenberg chain compound,(15) we have presented an exact computation of the entropy and the magnetic cooling rate of the antiferromagnetic spin-1/2 chain in the thermodynamic limit . Furthermore, we have performed complementary numerical computations for the cooling rate of finite Heisenberg chains, namely exact diagonalization (ED) for small systems and QMC simulations for somewhat longer chains. We have demonstrated that we can obtain excellent approximations to the exact result with a combination of both numerical methods. On the one hand, this serves as a consistency check of our computations. On the other hand, we are now in a position to apply a combination of ED and QMC to some sign-free situations, like the spin-1/2 Heisenberg model on the square and simple cubic lattices where the exact methods are no longer applicable.
We have used the exact result for the entropy to illustrate that field-induced quantum phase transitions give rise to maxima of the low-temperature entropy, or equivalently minima of the isentropes. This leads to cooling during adiabatic (de)magnetization processes where the lowest temperature is reached close to the quantum phase transition. As a consequence, we find a zero for the magnetic cooling rate at the phase transition and large positive (negative) values of the normalized cooling rate (2) for magnetic fields slightly above (below) the critical field.
The low-temperature asymptotics of the entropy is exponentially activated in the gapped phases, linear in in the gapless Luttinger liquid regions, and follows the square-root behavior (28) at the field-induced quantum phase transitions. These asymptotic forms of are expected to be universal for field-induced phase transitions in one-dimensional systems with -symmetry,(9); (12); (16); (17) but can be particularly clearly verified with the aid of an exact solution.
The general features of the entropy should not depend on the specific choice of the magnetic field as control parameter and indeed similar behavior is found as a function of the exchange anisotropy . An exception is just the quantum phase transition at , with as a control parameter. Because this is a Kosterlitz-Thouless transition, only weak signatures are observed in the finite-temperature entropy.
Finally, closed-form expressions were derived in the Ising limit using the transfer matrix method.(48); (49) We have observed remarkably large magnetic cooling rates close to the field-induced critical point of the Ising chain. In fact, the transfer-matrix solution is closely related to a low-energy description of highly frustrated one-dimensional quantum antiferromagnets,(12); (22); (52); (53); (54) where enhanced cooling rates are observed as well.
Acknowledgements.V.O. would like to thank the Institut für Theoretische Physik, Universität Göttingen and Universität Wuppertal for hospitality during the course of this work. This research stay was supported by the German Science Foundation (DFG). Furthermore, A.H. is grateful to the DFG for financial support by a Heisenberg fellowship (grant no. HO 2325/4-1). V.O. was also supported by the grants UCEP-CRDF-06/07 and ANSEF-1518-PS. C.T. would like to acknowledge support by the research program of the Graduiertenkolleg 1052 funded by the DFG.
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