Thermodynamics of the spin-1/2 two-leg ladder compound ( C 5 H 12 N ) 2 CuB r 4
aa r X i v : . [ c ond - m a t . s t r- e l ] S e p Thermodynamics of the spin-1/2 two-leg ladder compound ( C H N ) CuBr Fatemeh Amiri , Saeed Mahdavifar ⋆ , Mahboobeh Shahri Naseri Department of Physics, University of Guilan, 41335-1914, Rasht, Iran Department of Physics, Payam Noor University , 19395-3697, Tehran, Iran ∗ (Dated: March 28, 2018)The thermodynamic behavior of the spin S = 1 / C H N ) CuBr in a uniform magnetic field is studied using numerical and analytical approaches.The entropy S ( H, T ) and specific heat C ( H, T ) are calculated. The specific heat shows variousbehaviors in different regions of the magnetic field. The field-dependence of the specific heat is almostsymmetric about the average of quantum critical fields in complete agreement with experimentalresults. In addition, it is found that during an adiabatic demagnetization process, temperaturedrops in the vicinity of the field induced zero-temperature quantum phase transitions.
PACS numbers: 75.10.Jm; 75.10.Pq
I. INTRODUCTION
Currently, wide interest is devoted to the low-dimensional gapped spin systems, both experimentallyand theoretically. In particular, magnetic spin ladders have attracted enormous attention in recent years, sinceremarkable progress in the fabrication of such laddercompounds. Magnetic spin ladders which intermediatebetween one-dimensional and two dimensional spin sys-tems, have a gapped or gapless ground state, respectively,for an even or an odd number of legs . The two-leg spinladder Hamiltonian in the presence of a magnetic field isdefined as H = J k X n,α S n,α · S n +1 ,α − gµ B B X n,α S zn,α + J ⊥ X n S n, · S n, (1)where S n ,α is the spin S = 1 / n ( n = 1 , ..., N/
2) and leg α ( α = 1 , B in the z direction leads to Zeeman term.There is a large body of research on the so-called tele-phone number compounds like Sr Cu O which arealso cuprate ladders like SrCu O . Several classes of ma-terials such (5 IAP ) CuBr . H O , Cu ( C H N ) Cl and ( C D N ) CuBr can be also well described exper-imentally by the two-leg Heisenberg antiferromagneticladder model . Specially, piperidinium copper bro-mide ( C H N ) CuBr is known as one of the bestspin-1/2 ladder compounds with large rung exhange .It is found that the rung exchange J ⊥ = 12 . K isabout four times larger than J k = 3 . K . In the ab-sence of the magnetic field, the excitation spin gap isequal to ≃ . K . In the presence of a magnetic field,the system remains in a gapped disordered phase below B c ≃ . T . The spin excitations are gapless andthe system is in the Luttinger-liquid (LL) phase in theintermediate region, B c < B < B c . At the second crit-ical field, B c ≃ . T , system undergoes a phasetransition to the fully polarized state. Also, the LL state extends down to the temperature of a 3D magnetic order-ing transition at T N ≤ mK , suggesting an intermedi-ate coupling J ′ ≃ mK ≪ J ⊥ , J k . The LL predictionshave been quantitatively tested for magnetization andspecific heat , nuclear magnetic reasonance , neutrondifferaction and thermal conductivity measurements.There are many research works dedicate to the zero-field properties of the ladder system, while current inter-est has been focused on the temperature dependent be-haviors and especially the magnetocaloric effects (MCE)in the spin ladder which has not been studied completelyso far. It is believed that the low temperature behav-iors give important insight in the physics of quantumphase transitions. In this paper, we use both theo-retical and numerical techniques to prepare a completepicture of the dependence of entropy, specific heat andadiabatic (de)magnetization on both magnetic field andtemperature. Our numerical approach is based on afull-diagonalization method while we use Jordan-Wignertransformation analysis for the analytical part. We showthat our results agree very well with the experiment.The outline of the paper is as follows. In section II wediscuss the model in the strong antiferromagnetic cou-pling limit and derive the effective spin chain Hamilto-nian. The Jordan-Wigner transformation for the spin S = 1 / XXZ chain Hamiltonian will be consid-ered and the basic mean-field set up will be presented insection III. In section IV, we present results of the mean-field and full diagonalization calculations and comparewith the experimental data. Finally, we conclude andsummarize our results in section V.
II. EFFECTIVE MODEL
The thermodynamics of the system can be analyzedusing a mapping of the spin S = 1 / . Thefirst-order degenerate perturbation theory in the param-eter J ⊥ and J k , must be done to derive an effective low-energy model. Since, the whole magnetization curve hasa width which is related to the bandwidth of the tripletexcitation, the first-order perturbation theory is suffi-cient. At J ⊥ ≫ J k , it is convenient to discuss the modelby representing the site-spin algebra in terms of on-bond-spin operators. In the limit, J k = 0 the ladder dividedinto the isolated rungs. Indeed an isolated rung may bein a singlet or a triplet state with corresponding spectrumgiven by E ± = ( J ⊥ ± gµ B B ) , E = J ⊥ , E s = − J ⊥ . (2)By increasing the magnetic field, one component of thetriplet becomes closer to the singlet ground state and ina strong enough magnetic field, gµ B B = J ⊥ , the triplet | ↑↑i is exactly degenerate with the singlet. In principle,we have a situation that the singlet and the triplet, | ↑↑i ,create a new effective spin S = 1 / | ⇑i = 1 √ | ↑↓i − | ↓↑i ) , | ⇓i = | ↓↓i . (3)This leads to the definition of the effective spin S = 1 / S † n,α = ( − n + α √ τ † n ,S zn,α = 14 ( I + 2 τ zn ) . (4)The effective Hamiltonian in terms of the effective spinoperators up to the accuracy of an irrelevant constantbecomes the Hamiltonian of the spin S = 1 / XXZ chain in an effective magnetic field H eff = J k X n ( τ xn .τ xn +1 + τ yn .τ yn +1 + ∆ τ zn .τ zn +1 ) − gµ B B eff X n τ zn . (5)which allows for rigorous analysis. The ∆ = 1 / B eff = B − J ⊥ + J k gµ B . At zerotemperature, the gapped disordered phase in the laddersystem corresponds to the negatively saturated magneti-zation phase for the effective spin chain, whereas the LLphase of the main ladder system corresponds to the finitemagnetization phase of the effective spin-1/2 chain. Thesecond critical field where the ladder system is totallymagnetized, corresponds to the fully magnetized phaseof the effective spin chain. III. FERMIONIZATION
Theoretically, the energy spectrum is needed to inves-tigate the thermodynamic properties of the model. In this respect, we implement the Jordan-Wigner transfor-mation to fermionize the effective
XXZ model. Usingthe Jordan-Wigner transformation S + n = e iπ P n − m c † m c m c † n ,S zn = c † n c n − . (6)the effective Hamiltonian is mapped onto a 1D model ofinteracting spinless fermions H efff = J k X n [ 12 ( c † n c n +1 + c † n +1 c n ) + ∆ c † n c n c † n +1 c n +1 ] − ( J k ∆ + gµ B B eff ) X n c † n c n + N J k ∆ + 2 gµ B B eff ) . (7)This Hamiltonian is not exactly solvable because thefermion interaction. Therefore the mean-field theory isapplied to interaction term . The fermion interactionterm is decomposed by mean-field parameters which arerelated to spin-spin correlation functions as γ = < c † n c n >,γ = < c † n c n +1 >,γ = < c † n c † n +1 > . (8)Utilizing the above order parameters and perform aFourier transformation to momentum space by using c n = √ N P Nn =1 e − ikn c k , the mean field Hamiltonian isgiven by: H effMF = ε + X k> a ( k )( c † k c k + c †− k c − k ) − i X k> b ( k )( c k c − k + c † k c †− k ) , (9)where, ε = N ( 12 gµ B B eff + ∆ J k (1 / γ + γ − γ )) ,a ( k ) = J k (1 − γ ∆) cos( k ) + J k ∆(2 γ − − gµ B B eff ,b ( k ) = 2 J k γ ∆ sin( k ) . (10)Using the following unitary transformation c k = cos( k ) β k − i sin( k ) β †− k , (11)the diagonalized Hamiltonian is given by H effMF = ε + X k ε ( k )( β † k β k −
12 ) , (12)where ε ( k ) is the dispersion relation ε ( k ) = p a ( k ) + b ( k ) . (13)In order to solve mean-field Hamiltonian, the followingself-consistent equations should be satisfied γ = 12 π Z π − π [ aε ( 11 + exp( βε ) ) + 12 (1 − aε )] dk,γ = 12 π Z π − π cos( k )[ aε ( 11 + exp( βε ) ) + 12 (1 − aε )] dk,γ = 12 π Z π − π sin( k ) bε ( 11 + exp( βε ) −
12 ) dk. (14)Using the above order parameters, the thermodynamicfunctions such as the free energy, entropy and the specificheat are expressed as f = ε − πK B T Z π dk ln(2 cosh( ε ( k )2 K B T )) ,S = β ∂f∂β ,C = − K B β ∂S∂β . (15) IV. RESULTS
In this section we present our results obtained by an-alytical fermionization approach and the numerical fulldiagonalization technique on small size systems ( N =8 , , C H N ) CuBr are studied. A. Specific heat
In Fig. 1, we focus on the behavior of the temperaturedependence of the specific heat for ( C H N ) CuBr .It shows three regimes: quantum disordered(QD), spinLL and fully polarized phase. Fig. 1 (a)-(c) and (d)-(f)shows the mean-field and full diagonalization results ofspecific heat, respectively. As shown in Fig. 1 (a) and(d), in QD regime, B < B c , the specific heat decaysexponentially due to the presence of spin gap. It showsa single peak and when field increases the peak shiftsto lower temperature with the height decreased. Thispeak is related to the triplet excitations of the ladder.As the field B reaches to B c , the spin gap is reducedand a shoulder gradually emerges at low temperature,which is a signature of approaching the quantum criticalpoint .When B c < B < B c (Fig. 1(b) and (e)), as B is in-creased further, a new peak becomes visible that in themiddle of the LL, we can clearly see two peaks in the Temperature [K]
B
B>B c2 Temperature [K] C V [ m J / g K ] (c) B>B c2 FIG. 1: (color online). The specific heat as a function oftemperature in the fixed magnetic field for ladder with J ⊥ =13 . K , J k = 3 . K .(a-c)mean-field (d-f)full diagonalization for N = 16. temperature dependence of the specific heat. Below thefirst peak, the temperature dependence remains linearup to B c . The linearity of C is shown in Fig. 1(b) and(e). By increasing the field, height of the first peak isdecreased but its position is almost unchanged, while thesecond peak shifts to the high-temperature as illustratedin Fig. 1(e) and yielding a very good agreement withthe experimental results . When B approaches B c , thepeak in low temperature starts to vanish and just a shoul-der can be seen (Fig. 1(b) and (e)). For B > B c , fullypolarized state, the specific heat decays exponentially as T → B c , the shoulder that emerges in the LL vanishesand the specific heat has a single peak. By more increas-ing the field, B > B c , the gap and the height of thepeak increases and the position of peak shifts slightly tohigher temperatures. The result shows that the systemis in a few different states under different magnetic fields Temperature [K] C V [ m J / g K ] B FIG. 2: (color online). Comparison of the specific heat as afunction of temperature in fixed magnetic field with experi-ment (Ref.[14]) for ladder with J ⊥ = 13 . K , J k = 3 . K . (Fig. 1(c) and (f)).We should note that, this model is experimentallystudied recently . The calculated values of the specificheat with both techniques are compared with the ex-perimental results. The results from full-diagonalisationhave a quantitative accuracy within 2 −
8% comparedto the experiment. At low temperature region, our re-sults agree very well with the experimental data. But athigher temperatures, the quantitative agreement of re-sults due to the mean-field approach is missing, as shownin Fig. 2. It is mostly because of considering the effectiveXXZ Hamiltonian and missing some of the high energyspectrum of the ladder system.The excitation is a triplet and the effective XXZ chainmodel considers only one of its three branches. Thusthis model is really intended only for the regime of mass-less excitations, at significant fields. It works also athigh fields, but certainly should not be applied at zerofield. There are also two-triplet excitations, which willbe missed if the one-triplet excitations are missing, butthese contribute less. In addition, the mean field approx-imation is used to diagonalize the effective Hamiltonian.Now let us look at the field dependence of the specificheat. The Fig. 2 shows the field dependence results of thespecific heat obtained by mean-field approach (the sameas Fig. (3)(a) of Ref.[17]). For temperature
T < K , wecan see that for B < B c and B > B c , the specific heatenhances when temperature increases. In the LL regime,the specific heat decreases by increasing of the temper-ature which shows the inverse behavior as compared tolow temperature. Also in this region, we observe an inter-esting behavior. The field dependence of specific heat isalmost symmetric about B = B c + B c = 10 . T . In thestrong-coupling limit, J ⊥ J k ≫
1, perfect symmetry wouldbe expected due to the exact particle-hole symmetry tothe XXZ chain in the magnetic field .By increasing the temperature, the peaks starts to van-
Magnetic field [T] C V / T [ m J / g K ] T = 0.35 KT = 0.40 KT = 0.51 KT = 1.53 KT = 3.06 K
FIG. 3: (color online). The specific heat as a function ofmagnetic field in fixed temperature, for ladder with J ⊥ =13 . K , J k = 3 . K (the same as Fig. (2)(a) of Ref.[17]). Magnetic Field [T] T m ax [ K ] Full-diagonalizationMean-field (b)
Magnetic Field [T] C V m ax [ m J / g K ] Full-diagonalizationMean-field (a)
FIG. 4: (color online). The maximum (a) specific heat, C maxV and (b) the corresponding temperature T max versus magneticfield for ladder with J ⊥ = 13 . K , J k = 3 . K . ish until in high temperatures, the specific heat behavesas an ascending function, as illustrated in Fig. 3.It is also interesting to see the magnetic field effectson the maximum specific heat C maxV and correspondingtemperature T max . As shown in the Fig. 4(a), C maxV obtained by both analytical and numerical approachesfirst decreases which shows a slow response to the split-ting with applying to the magnetic field. For the mag-netic fields more than B c , the analytical results are com-pletely different from the numerical results. At B = B c and B = B c , C maxV arrives at a minimum and also inthe middle of the LL phase, C maxV shows a maximum at B = B c + B c = 10 . T in our analytical results. But thenumerical results show C maxV arrives at a minimum at B = B c + B c = 10 . T and in good agreement with thenumerical TMRG results . Moreover, we found thatthe curvature of our numerical results on T max changesits sign at the critical fields. But again, we see that thebehavior of the analytical results is completely differentfrom our full diagonalization findings, since our analyticalresults are obtained for the effective XXZ chain Hamil-tonian which is only valid for very low temperatures as Magnetic Field [T] - d M / d T [ m J / T g K ] T= 0.36T= 0.62T= 0.94 (c)
Magnetic Field [T] T e m p er a t u re [ K ] S = 0.1S = 0.3S = 0.5S = 0.7S = 0.9 (a)
Magnetic Field [T] - d M / d T [ m J / T g K ] T= 0.94 (d)
Magnetic Field [T] T e m p er a t u re [ K ] S = 0.1S = 0.3S = 0.5S = 0.7S = 0.9 (b)
Magnetic Field [T] T e m p er a t u re [ K ] dM/dT=0 (e) FIG. 5: (color online). Magnetocaloric effect for ladder with J ⊥ = 13 . K , J k = 3 . K . (a-b) Isentropes, i.e., adiabaticdemagnetization curves of the S = 1 / ∂M/∂T ) = 0 as a functionof magnetic field and temperature obtained by mean-field ap-proximation. (a-c mean-field), (b-d full-diagonalization for N = 16). we mentioned above. B. The Magnetocaloric effect
When a crystal containing ions is placed in a magneticfield the adiabatic or isentropic change of this externalparameter causes a temperature change in the sample.This is called the magnetocaloric effect (MCE) whichwas first observed by Warburg . The MCE in spin sys-tems has attracted enormous attention in recent years.It is nowadays interesting in several aspects. From onehand, field-induced quantum phase transitions lead touniversal responses when the applied field is changedadiabatically . On the other hand, it was observed that MCE is increased by geometric frustration ,promising improved efficiency in low temperature cool-ing applications . More generally, the MCE is par-ticularly large in the vicinity of quantum critical points.Here, the special focus is on the magnetocaloric effect. Itwill show that in this model, two minimum in the isen-tropes on the zero-temperature quantum critical fieldscan be observed. Further information about the low tem-perature behavior of substances can be obtained by cal-culating the entropy. In this section the entropy of spin1 / B and T isshown in Fig. 5(a) and (b), respectively. The curves havebeen obtained by calculating the entropy, S , in the B − T plane and determining the constant entropy curves (theisentropes) from this data.Fig. 5 (a) and (b) can be divided into the gapfull andgapless regimes. In the gapfull regimes, B < B c and B > B c , a significant temperature changes happens.Firstly, for B < B c , by raising the magnetic field from B = 0 to B = B c , the temperature can be minimized inthe vicinity of B c . Secondly, for B > B c , by decreasingthe magnetic field from high magnetic field to the satura-tion field adiabatically (adiabatic demagnetization), thetemperature arrives at minimum close to the saturationfield. For example, an adiabatic process which starts at( B, T ) ≈ (5 . T, . K ) or (15 . T, . K ) would go downto T ≈ mK as B → . T or B → . T , respec-tively. This case corresponds to the entropy S = 0 . S = 0 . T ≈ mK . In theregime B c < B < B c , the energy spectrum is gapless.Therefore, small temperature changes induced by adia-batic (de)magnetization is observed. This indicates thatone can use the gap-closing at the field-induced quantumphase transitions for cooling down samples.One can also calculate the derivative of the magne-tization with respect to temperature by using the rela-tion ( δQδB ) | T /T = − ( ∂M∂T ) | B . In this relation δQ is theamount of heat which is created or absorbed by the sam-ple for a field change δB due to MCE. As shown in theFig. 5(c) and (d), for low temperatures, the significantchanges of the − ( ∂M∂T ) | B from negative to positive valuesoccurs when the magnetic field is applied. The quantumphase transition at B c ≃ . T and B c ≃ . T are specified by these sign changes. For better under-standing, ( ∂M/∂T ) = 0 as a function of magnetic fieldand temperature obtained by mean-field approximationis shown in Fig. 5(e). By increasing the temperature, theheight of the peaks reduces until all of them are disap-pearing and no sign of quantum phase transitions can beseen. This indicates that thermal fluctuations becomestrong enough to take the system to the excited state.It should be noted that, as illustrated in Fig. 5(b)and (d), the full-diagonalization results shows finite sizeeffects especially in low temperatures and LL regime, Magnetic Field [T] S [ m J / g K ] T (c) Temperature [K] S [ m J / g K ] B = 1.0B = 3.0B = 5.0B = 7.0B = 9.0B = 11.0B = 13.0B = 15.0B = 17.0B = 19.0 (a)
Magnetic Field [T] S [ m J / g K ] T (d) Temperature [K] S [ m J / g K ] B = 1.0B = 3.0B = 5.0B = 7.0B = 9.0B = 11.0B = 13.0B = 15.0B = 17.0B = 19.0 (b)
FIG. 6: (color online). Entropy for ladder with J ⊥ = 13 . K , J k = 3 . K as a function of (a-b)temperature in fixed mag-netic field. (c-d) magnetic field in fixed temperature. (a-cmean-field), (b-d full-diagonalization for N = 16). but in general the mean-field results are in good agree-ment with the full-diagonalization and also experimentalresults . C. Entropy
In this part, the behavior of the entropy as a function ofboth temperature and magnetic field is studied. Fig. 6(a)and (b) shows the temperature dependence of the entropyobtained by mean-field approach and full-diagonalizationmethod, respectively. At zero temperature, the entropyis zero, which is consistent due to the second law of ther-modynamics. When the temperature is high enough, theentropy approaches a constant value, in a fixed magneticfield. Also because of the presence of the gap in
B < B c and B > B c , the entropy is expected to be exponentiallyactivated as a function of temperature.As it shows in Fig. 6(c) and (d), for very low temper-atures we can see several interesting results. The maxi-mums of the field dependence of the entropy occur in thecritical fields and also the entropy is symmetric about B = B c + B c = 10 . T just like the specific heat. In verylow temperatures, T <
1, the accumulation of entropynear the quantum critical points is obvious. It impliesthat the system is maximally undecided which groundstate to choose. As the temperature is enhanced, all thecharacteristic behaviors have been vanished.
V. CONCLUSION
In this paper we have focused on the low-temperaturephysics of the isotropic spin S = 1 / C H N ) CuBr . Using the analytical and nu-merical methods we have calculated entropy and the spe-cific heat of the system. It is shown that the temperature-dependence of specific heat shows various behaviors indifferent regions of the magnetic field. The field- de-pendence of the specific heat shows interesting behaviorin the low temperature region. It is almost symmetricabout the average of critical points in complete agree-ment with experimental results. On the other hand, anexternal magnetic field induces large relative changes inthe entropy of quantum spin systems at finite tempera-ture. This leads to magnetocaloric effect, i.e. a changein temperature during an adiabatic demagnetization pro-cess. We computed the entropy of antiferromagnetic spin S = 1 / VI. ACKNOWLEDGMENTS
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