Significant Inverse Magnetocaloric Effect induced by Quantum Criticality
Tao Liu, Xin-Yang Liu, Yuan Gao, Hai Jin, Jun He, Xian-Lei Sheng, Wentao Jin, Ziyu Chen, Wei Li
aa r X i v : . [ c ond - m a t . s t r- e l ] F e b Significant Inverse Magnetocaloric Effect Induced by Quantum Criticality
Tao Liu,
1, 2, ∗ Xin-Yang Liu, ∗ Yuan Gao, Hai Jin, Jun He, Xian-Lei Sheng, Wentao Jin, Ziyu Chen, † and Wei Li
2, 4, ‡ School of Science, Hunan University of Technology, Zhuzhou 412007, China School of Physics, Beihang University, Beijing 100191, China Department of Astronomy, Tsinghua Center for Astrophysics, Tsinghua University, Beijing 100084, China International Research Institute of Multidisciplinary Science, Beihang University, Beijing 100191, China (Dated: February 18, 2021)The criticality-enhanced magnetocaloric effect (MCE) near a field-induced quantum critical point (QCP)in the spin systems constitutes a very promising and highly tunable alternative to conventional adiabatic de-magnetization refrigeration. Strong fluctuations in the low- T quantum critical regime can give rise to largethermal entropy change and thus significant refrigeration capacity when approaching the QCP. In this work, weshow there exists a significant inverse MCE (iMCE) in the spin-1 quantum chain materials (CH ) NNi(NO ) (TMNIN) and NiCl -4SC(NH ) (DTN), where DTN have significant low- T refrigeration capacity while re-quiring only moderate magnetic fields. The iMCE characteristics, including the adiabatic temperature change ∆ T ad , isothermal entropy change ∆ S , differential Gr¨uneisen parameter, and entropy change rate, are simulatedby thermal many-body calculations. The cooling performance, i.e., the efficiency factor and hold time, of thetwo compounds are also discussed. Through optimizing the iMCE properties with machine learning techniques,we conclude that the DTN locates near the optimal parameter regime, and constitutes a very promising versatilequantum magnetic coolant. We also provide guide for designing highly efficient and cryofree quantum magneticrefrigeration for space applications and quantum computers. I. INTRODUCTION
The magnetocaloric effect (MCE) represents a significantadiabatic temperature change of a magnet as a response tothe varying external magnetic fields, which has a long timeof study [1–4]. Historically, sub-Kelvin regime cooling wasreached for the first time through adiabatic demagnetizationrefrigeration (ADR) [5]. Recently, low- T magnetic refriger-ation gets refreshed research interest due to its important ap-plication in space technology [6, 7] and cryofree sub-Kelvinenvironment for quantum computers [8]. It is of great researchinterest to pursue novel MCE refrigerants that provide highercooling powers and lower temperatures. Among others, thequantum spin-chain materials with enhanced MCE charac-terized by the universally diverging Gr¨uneisen ratio near thequantum critical point (QCP) [9–15], has been proposed asa very promising coolant for quantum magnetic refrigera-tion [16, 17] and an excellent alternative to the conventionalADR.In most magnetic materials, the spin degrees of freedomin the system eventually “solidify” into a long-range or-der as cooled down to low temperatures. In such mag-netically ordered phase, the spin states are practically non-tunable by external fields and the corresponding MCE, tem-perature or entropy change as a response to fields, are usu-ally negligible. Paramagnetic salts, however, order onlyat very low temperatures, are rare exceptions. Such salts,like CrK(SO ) · O (Chromic Potassium Alum, CPA) andFe(SO ) NH · O (Ferric Ammonium Alum, FAA), etc,host nearly non-interacting spins and are widely used in the ∗ These authors contributed equally to this work. † [email protected] ‡ [email protected] ADR as spin “gas” refrigerant [7]. It is commonly believed,as aforementioned, the spin interactions are “harmful”, as theyusually lead to magnetic ordering at low T and spoil the MCEproperties.Nevertheless, there is exotic exception to this classic fateof interacting spins. In the family of spin-chain materials[18], under certain condition the quantum fluctuations in suchcorrelated quantum magnets can be strong enough to preventspins from ordering even at T = 0 . One prominent case isthe field-induce QCP in the materials. The enhanced quantumfluctuations stemming from the QCP have significant influ-ences in the finite-temperature quantum critical regime [19].When approaching the QCP, the spins would experience asignificant isothermal entropy change, which can be trans-lated into a considerable temperature decrease under the adi-abatic condition. Such quantum criticality-enhanced MCE isreflected in a diverging Gr¨uneisen parameter, adiabatic tem-perature change rate [cf. Eq.(2) below], Γ B ∼ T − /zν underexternal field B , with z and ν the dynamical and critical ex-ponents related to the universality class of the QCP [11, 12].Notably, such intriguing quantum critical phenomena in low-temperature thermodynamic response also provides a sensi-tive probe of QCP [20].The low-temperature MCE properties of a typical 1D quan-tum magnetic system — spin-1/2 Heisenberg chain (HAFC)— have been intensively explored, where a pronounced MCEwas predicted [9, 10] and also observed in the compound[Cu( µ -C O ) (4-aminopyridine) (H O)] n (CuP). The lowestachievable temperature with the spin-chain materials has noprincipal limitation (as long as the inter-chain interactionsare negligible), and the high efficiency factor as well as longhold time makes the spin-1/2 HAFC materials very promisingquantum critical refrigerants [16, 17].Nevertheless, there is still plenty of room for further im-proving the performance of the quantum magnetic refriger-ants. As reported in Ref. [16], for the spin-1/2 HAFC com-pound CuP, one has to start from a rather high magnetic field,e.g., 7 T, significantly above the critical field B c ≃ . T.In CuP, there indeed exists significant MCE in the range
B > B c , while in the smaller-field side, i.e., B ∈ [0 , B c ] ,rather weak (inverse) MCE was observed. Therefore, thegenerated strong magnetic fields (of 7 T) are actually not befully exploited in the case of spin-1/2 compound CuP. On theother hand, popular paramagnetic refrigerants are typicallywith high spin S ≥ . For example, CPA and FAA are with S = 3 / and / , respectively, and the Gadolinium GalliumGarnet (GGG) is even with S = 7 / . Larger entropy changeis generally expected for higher spin systems, however thereis only a few studies on quantum spin-chain models [13] andmaterials [15] with spin higher than one half.In this work, we systematically investigate the iMCE prop-erties of the S = 1 quantum chain models and materials,with emphasize on two typical spin-1 HAFC materials, i.e.,(CH ) NNi(NO ) (TMNIN) and NiCl -4SC(NH ) (DTN).The magneto-thermodynamic properties can be accuratelysimulated by the thermal-state linearized tensor renormaliza-tion group (LTRG) approaches [21, 22]. In particular, wecompute and characterize the iMCE of these spin chains, andfind the DTN has iMCE refrigeration capacity comparable tothe spin-1/2 HAFC material CuP, while requiring only a mod-erate external magnetic field of B c ≃ T in the cooling pro-cess. Moreover, combining the LTRG and Bayesian optimiza-tion methods, we optimize iMCE of the spin-1 chain magnetsin a rather wide range of parameter regime, and find DTN liesin the optimal parameter regime (when the largest availablemagnetic field is restricted up to 3T). Due to the pronouncedcooling effects and excellent thermal transport properties [23]in DTN, we propose DTN is a very promising iMCE refrig-erant with very competitive performance and only involves amagnetic field of moderate strength very suitable for practicalapplications.The rest parts of the article is arranged as follows. Wepresent the spin-1 chain models, their thermal many-bodysimulations, and the related spin-1 materials in Sec. II. Ourmain results on iMCE of the spin-1 chain materials and op-timization design are shown in Secs. III and IV, respectively.Section V is devoted to the conclusion and outlook.
II. SPIN-1 HEISENBERG-CHAIN MODEL ANDMATERIALSA. Heisenberg antiferromagnetic chain and tensorrenormalization group
The S = 1 HAFC model with single-ion anisotropy is de-fined by the Hamiltonian H = X i h J (cid:16) ~S i · ~S i +1 (cid:17) + D ( S zi ) + gµ B B S zi i , (1)where ~S i is the spin-1 operator on site i (with S zi its z compo-nent), J the nearest-neighbor Heisenberg interaction, and D FIG. 1. Thermal entropy S ( T ) curves of three spin-chain materials,under different initial fields B i and final fields B f = B c . Entropychange ∆ S (indicated by the vertical red arrow) and adiabatic tem-perature change ∆ T ad (horizontal red arrow) are shown explicitly forthe compound DTN. A-B-C-A constitutes a single-shot refrigerationprocess, and the dashed regime ACDE represents the heat absorbedfrom the load in the iso-field CA process. is the single-ion anisotropy parameter. In the Zeeman term, B is the external magnetic field, with g the electronic Land´efactor and µ B is the Bohr magneton.In a rather wide parameter regime [24, 25], the spin-1 chainmodel Eq.(1) has a disordered ground state with finite exci-tation gap, which can be closed by applying a magnetic field B , through field-induced quantum phase transition. In partic-ular, for the case D = 0 , the model in Eq.(1) reduces to theprominent spin-1 HAFC model with the renowned Haldanegap ∆ ≈ . J [26, 27], where the first excited state is a spintriplet S = 1 . The introduction of D in the spin chain can al-ter the size of Haldane gap, and drive the system into a triviallarge- D phase through a topological quantum phase transitionat D/J = 0 . [28]. For D > D c , the gap reopens and scalesproportional with D in the large- D limit. Besides, other off-diagonal single-ion anisotropies are found to be small in thetwo spin-1 chain compounds TMNIN and DTN, and are thusset to zero in the rest of our discussion.To simulate the finite-temperature properties and character-ize the iMCE of the spin-1 chain materials, we employ thebilayer infinite-size LTRG method [21, 22] for high-precisionthermal many-body calculations down to T /J = 0 . . Inthe process of imaginary-time evolution (cooling), we retainup to χ = 400 bond states (with truncation error ǫ . − )in the matrix-product thermal density operator, which alwaysguarantees accurate and converged results. In the automaticsearching, we adopt LTRG as a precision thermal many-bodysolver, combined with the Bayesian optimization, to obtainoptimal model parameters with largest refrigeration capacity. Compound
Abbr.
J/k B (K) D/k B (K) ∆ /k B (K) g B c (T) Reference(CH ) NNi(NO ) TMNIN . − - . − . . − .
25 3 [29–32]NiCl -4SC(NH ) DTN 2.2 8.9 3.2 2.26 2.13 a [23, 33–37]Ni(C H N ) NO BF NENB . − . . . − [15, 38]Ni(C H N ) NO ClO NENP . −
48 10 −
16 13 − . − . . − . [39, 40]Ni(C H N ) NO ClO NINO −
52 11 . −
16 10 − . − . [41, 42]Ni(C H N ) N ClO NINAZ − - − – - [30, 43]Y BaNiO - −
280 50 −
56 60 . − a In experiments, the measured critical field is 2.2 T in DTN, smaller than the theoretical 1D results of about 3.3 T, due to the strongly influenced byinter-chain interactions.
TABLE I. Some common spin-1 chain compounds and their microscopic Hamiltonian parameters, including the intra-chain exchange J ,uniaxial single ion anisotropy D , the spin excitation gap ∆ , the Land´e factor g , and the (lower) critical field B c . The magnets are listed inascending order in strength of Heisenberg coupling J . B. iMCE property and performance characteristics
Under external magnetic fields, the spin-1 chains inthe field-induced quantum critical regime show criticality-enhanced MCE. To quantitatively characterize the MCE prop-erty, we introduce the magnetic Gr¨uneisen parameter Γ B = 1 T (cid:18) ∂T∂B (cid:19) S = − C B (cid:18) ∂M∂T (cid:19) B , (2)with C B = T ( ∂S/∂T ) B the magnetic specific heat. Γ B mea-sures the temperature change rate as a response to the smallvariation of the external field B , under an adiabatic condition.In the numerator of Eq. (2) is a related differential quantity Θ T = ( ∂S∂B ) T = ( ∂M∂T ) B , that measures the isothermal entropy change rate.Correspondingly, when integrated over a given range offields, e.g., B ∈ [ B i , B f ] with B i = 0 and B f the final (criti-cal) field in the iMCE process, the isothermal entropy changeis ∆ S ( T ) = Z B f B i =0 (cid:18) ∂S∂B (cid:19) T d B = Z B f B i =0 (cid:18) ∂M∂T (cid:19) B d B, and the adiabatic temperature change is ∆ T ad = Z B f B i =0 Γ B T d B = Z B f B i =0 TC B (cid:18) ∂M∂T (cid:19) B d B. Furthermore, in order to comprehensively consider the ef-fects in both the entropy and temperature change under vary-ing fields, we define an iMCE quantity, refrigeration capacity A , as the area between two entropy curves, bounded by thegiven high ( T i ) and low ( T f ) temperatures (cf. Fig. 1), i.e., A = Z T f T i [ S ( B f ) − S ( B i = 0)] d T. (3)Besides MCE properties, in practical applications the re-frigeration efficiency factor and hold time are important quan-tities measuring their cooling performance of ADR. The ef-ficiency factor is defined as the ratio ∆ Q c / ∆ Q m , where ∆ Q c = R T i T f T ( ∂S/∂T ) B f d T refers to the heat absorptionfrom load (indicated by the red shadow area ACDE in Fig. 1),and ∆ Q m = T i · [ S ( B f , T i ) − S ( B i , T i )] is the heat exchangebetween the material and the heat reservoir at the high tem-perature T i . In multistage single shot or continuous ADRs,the whole system must be optimized according to the pre-cooling requirements and weight, in which the efficiency fac-tor is to be crucial [7]. Besides, the hold time — reflectingthe temperature-time curve of the refrigerant in contact withconstant heat load — is another important parameter for anefficient refrigeration. The refrigerant temperature is definedas T S ( t ) = T S (0) + ˙ Q/C m , and particularly we require therefrigerant temperature does not increase too rapidly (thus along hold time), under a constant heat load ˙ Q . C. Spin-1 Chain Quantum Magnets
Spin-1 chain quantum magnets constitute an intriguingfamily of compounds. Distinct from the spin-1/2 chains, thespin-1 HAFC system has a gapped ground states [26, 27]with symmetry-protected topological (STP) oder [46]. Therehas been continuous research interest in the investigation ofthe spin-1 chain materials, with some prominent memberslisted in Tab. I. In these compounds, there exists single-ionanisotropy term D [cf. Eq. (1)] besides J . In this work, weare particularly interested in the compounds TMNIN [29–32]and DTN [23, 33–37], due to their very moderate critical fields B c ≤ T (cf. Tab. I) that is very suitable for magnetic refrig-eration applications. Besides, there are spin-1 chain materi-als other than TMNIN and DTN, including NENB [15, 38],NENP [39, 40], and NINO [41, 42] (cf. Tab. I), which arevery common frustration-free spin-1 materials that can be de-scribed by the Hamiltonian Eq. (1).Despite a little dispute about the specific J and Land´e fac-tor g , consensus has been reached that the compound TMNINcan be well described by a spin-1 HAFC with relatively weaksingle-ion anisotropy D , through fitting the magnetic suscep-tibility [29], specific heat [31], and magnetization curve [30].Below, to explore the MCE properties of TMNIN, we take theparameter set J = 12 K and g = 2 . from Ref. [29]. On S [ J m o l - K - ] T [K]
TMNIN DTN CuP (a) S / B m a x [ J m o l - K - T - ] T [K]
TMNIN DTN CuP (b)
Fig.
2. (a) Molar magnetic entropy change ∆ S as a function of tem-perature T , for three compounds, TMNIN (gray dashed line), DTN(orange line), and CuP (blue dashed). (b) shows the entropy changeper Tesla, ∆ S/B max , for three compounds. the other hand, for the spin-1 chain material DTN, we take J = 2 . K, D = 8 . K, and g = 2 . from Ref. [34]. Beforediscussing the iMCE properties and performances of the spin-1 chain refrigerants, we note that TMNIN can be regardedas an excellent spin-1 HAFC material that corresponds to agapped ground state, while the DTN has D/J ≃ and wellresides in the trivial large- D phase. TMNIN opens up the spinexcitation gap due to the emergence of SPT order, in sharpdistinction to gapless spin-1/2 materials like CuP. Therefore,it is interesting to compare iMCE of these two materials as topological Halande vs. trivial large- D magnetic refrigerants. III. INVERSE MCE IN THE SPIN-1 CHAIN MATERIALS
Below we provide our main results of magnetothermody-namics of two spin-1 chain magnets, the Haldane-gap chainTMNIN and large- D chain DTN, compared to the spin-1/2HAFC compound CuP. We show the MCE characteristics in-cluding the adiabatic temperature change ∆ T ad , isothermalentropy change ∆ S , G¨uneisen parameter Γ B , and differentialcharacterization Θ T , etc. The practical performance like theefficiency factor η and hold time are also discussed and com-pared in this section. A. Entropy curves and isothermal entropy change ∆ S Magnetic fields can tune the spin states of the system andinduce significant entropy change that can then be transferredinto heat effects. In Fig. 1, we show the entropy curves at twoconcerned magnetic fields — the initial field B i and the finalfield B f — which are B i = 0 , B f = 3 . T (TMNIN) and B i = 0 , B f = 3 . T (DTN) for the two spin-1 compounds. Itshould be noted that in the iMCE process the largest requiredfield (here B f ) is right the critical field value ( B c ). This is insharp contrast to the MCE process of spin-1/2 material CuP, where B f = 4 . T (also at the QCP) and the largest field inthe cooling procedure is instead B i = 7 . T, much greaterthan that of TMNIN and DTN. Such significant reduction ofthe required magnetic fields is important for the implementa-tion of the quantum magnetic refrigeration in, say, the spaceapplications.From the thermal entropy curves in Fig. 1, we find theHaldane magnet TMNIN has a rather small isothermal en-tropy change ∆ S , clearly less than 1 J mol − K − regard-less of the working temperature [cf. Fig. 2]. However, thelarge- D magnet DTN is found to conduct quite prominent en-tropy change, as seen in Figs. 1 and 2, comparable to that ofthe spin-1/2 compound CuP. As DTN only requires a max-imal field ( B max ) half of that for CuP, in Fig. 2(b) we findDTN has the highest “efficiency”, i.e., entropy change perTesla, over the other two compounds due to the significantentropy change ∆ S and the smallness of the maximal field B max = B f = 3 . T. B. Isentropes and adiabatic temperature change ∆ T ad In the iMCE cooling procedure, the compounds are firstlymagnetized along the isothermal line AB (red arrowed linein Fig. 1). A larger ∆ S means the greater cooling capacity,which, in the adiabatic process (indicated by the horizontalline BC) is translated into a large temperature change ∆ T ad .When the magnet reaches its lowest temperature T f at thepoint C, we contact the refrigerant with the heat load and itstarts to absorb heat from there. The temperature of the mag-netic refrigerant gradually rises up along the iso-field line CAwith a fixed field B f .On this basis, it becomes very meaningful to compute theisentropes of the three compounds and determine the adia-batic temperature change ∆ T ad from there. In Fig. 3(a,b),we show isentropes of the TMNIN and DTN materials, re-spectively, where iMCE can be clearly observed. When themagnetic field increases from zero ( B i = 0 ) to critical field( B f ≥ B c ≃ T) for both spin-1 compounds), we find thetemperature decreases monotonically, e.g., from about 2 K toabout 1 K (TMNIN) and 500 mK (DTN). The ∆ T ad resultsare collected and shown in Fig. 4, which are found significantin the course of increasing fields, for both spin-1 compoundsTMNIN and DTN in a wide range of initial temperatures T i shown up to 2.5 K.In the isentropes of DTN (and also TMNIN) in Fig. 3, wecan recognize two dips in low- T isentropic lines, which cor-respond to the two field-driven QCP. The lower-field one (atabout 3.3 T for DTN) with strong iMCE occurs due to the clo-sure of spin gap, while the one at higher field (about 11.7 Tfor DTN) is the saturation transition where the spins becomepolarized. Different from the spin-1 chains, the results of spin-1/2 CuP in Fig. 3(c) show only one saturation QCP at a field ofabout B c ≃ T. If we increase the fields from zero to B c , theiMCE in CuP is apparantly weak; while the MCE in CuP issignificant when decrease from a large B i higher than 7 T [cf.Figs. 3(c) and 4(a)]. Therefore, to make a faithful comparison,we compute the temperature change ∆ T ad per Tesla and show (a) TMNIN (b) DTN (c) CuP Fig.
3. Simulated isentropic contour plots of three spin-chain compounds, including the spin-1 materials (a) TMNIN and (b) DTN, and (c)the spin-1/2 chain CuP. There are two QCPs in panel (b), between which there exists a continuous Tomanaga-Luttinger liquid regime withrelatively flat isentropic lines at low temperature. T a d [ K ] T i [K](a) T a d / B m a x [ K T - ] T i [K] TMNIN DTN CuP (b)
TMNIN DTN CuP
Fig.
4. (a) The adiabatic temperature change ∆ T ad and (b) changeper Tesla, ∆ T ad /B max of three spin-chain compounds considered inthis work. ∆ T ad /B max in Fig. 4(b). It is found that the DTN actually hasthe largest ratio, and TMNIN is also more efficient than CuP,given the initial (high) temperatures T i . . K. C. G¨uneisen parameter Γ B and differential iMCE Θ T Above we have computed the isothermal entropy change ∆ S and adiabatic temperature change ∆ T ad of two spin-1materials, and find that the absolute and per-Tesla values areboth important for characterizing iMCE properties. Matter offact, to compare the MCE properties more faithfully, by get-ting ride of the influences of different field ranges, we exploitthe differential quantities including the Gr¨uneisen parameter Γ B and differential entropy change Θ T . With these differen-tial characteristics, we are able to compare the MCE proper-ties point by point at each magnetic field.The Gr¨uneisen ratios Γ B of three spin-chain materials areshown in Fig. 5. In all three cases, we find pronounced peaksin Γ B that change its sign abruptly near the QCP [11, 12],revealing the quantum criticality-enhanced MCE. The hightof Γ peak represent the adiabatic temperature change rate un- der an infinitesimal field change, which increases as T low-ers and diverges as T → [11]. Again, we see that the Γ B peaks of TMNIN are much weaker than those of DTN andCuP. The positive peaks and negative dips represent respec-tively the MCE and iMCE in the materials, which are quitedifferent for the spin-1 DTN and spin-1/2 CuP materials. ForDTN, Γ B is negative (iMCE) in the small-field side and pos-itive (MCE) on the large side, and the negative dip is muchmore pronounced as compared to the positive peak, showingthat the iMCE in DTN is much stronger than MCE. On thecontrary, CuP exhibits exactly the reverse behaviors, i.e., witha pronounced MCE peak to the right of the critical field whilea very weak iMCE peak on the left.The results of entropy change rate Θ T are shown in Fig. 6,which presents the very sharp and pronounced positive peaks(iMCE) and negative dip (MCE) around the critical fields atlow temperature. Similar as the observation in Γ B , TMNINagain have only rather weak peaks(dips) in Θ T , while DTNhave a pronounced peak of height in similar magnitude to thedip in CuP curve [Fig. 6(c)], showing strong iMCE. Both Γ B and Θ T display characteristic divergent behaviors close to theQCP and changes sign as the field crosses the critical point,indicating that pronounced refrigeration effects through adia-batic demagnetization (MCE) and magnetization (iMCE). D. iMCE performance: Efficiency factor η and hold time In practical applications, the refrigeration efficiency factor η and the hold time are of significance to maintain a sus-tainable and high-efficiency refrigeration procedure. The ef-ficiency factor η is the ratio between heat absorbed Q c fromheat load (i.e., the area of the dashed line between the ACand DE lines in Fig. 1), and the released heat Q m to the heatreservoir, i.e., η = Q c /Q m . For DTN, the area of red shadowin Fig. 1 is ∆ Q c = 0 . J/mol, similar to that of the spin-1/2 material CuP with ∆ Q c = 1 . J/mol, while requiringless than half the field. For the Haldane refrigerant TMNIN,we see a small heat absorption ∆ Q c = 0 . J/mol, againoutperformed by the large- D magnet DTN. Besides the heatabsorption, we are also interested in the heat release Q m in B [ T - ] B [T]
CuP B c =4.09T (c) B [ T - ] B [T]
TMNIN (a) B c =3.26T B [ T - ] B [T]
DTN (b) B c =3.3T Fig.
5. The Gr¨uneisen parameter Γ B of three materials, i.e., (a) DTN, (b) CuP, and (c) TMNIN, are shown at various temperatures from1000 mK down to 200 mK (top to bottom). The red vertical dashed line indicates the critical magnetic fields of the three spin-chain materials. Q T [ J m o l - K - T - ] B [T]
360 mK 240 mK 120 mK
CuP(c) B c =4.09T Q T [ J m o l - K - T - ] B [T]
360 mK 240 mK 120 mK
TMNIN(a) B c =3.26T Q T [ J m o l - K - T - ] B [T]
360 mK 240 mK 120 mK
DTN(b) B c =3.3T Fig.
6. The entropy change rate Θ T of three materials, (a) DTN, (b) CuP, and (c) TMNIN, are shown at various temperatures, from 360 mKdown to 120 mK (top to bottom). the isothermal process (area of the rectangle ABDE), as thoseheat has to be expelled to the outer environment and thus con-stitutes a load for, e.g., mechanical cooling or higher-stageADR in space applications. For a robust and efficient refriger-ation system, we want the refrigerant to absorb Q c as large aspossible while, at the same time, release a very small amountof heat to the environment, i.e., to have a high efficiency factor η [7, 8]. From Fig. 1, we find such a rate of the compoundDTN is η = 47 . when working between T i = 1 . K and T f = 0 . K, considerably higher than that of CuP ( fromthe same initial temperature T i as reported in Ref. [16, 17]).After the adiabatic demagnetization process (BC line inFig. 1), the refrigerant temperature reaches the lowest value T f at the magnetic field B f = B c . After that, the refrigerantcontacts with the load and its temperature rises along the iso-field line CA. We want the refrigerant with good performanceto absorb heat without warming up too rapidly. The simulatedtemperature T S at time t is T S ( t ) = T S (0) + ˙ Q/C m , which mainly depends ==on the magnetic specific heat C m of the refrigerant and heat load ˙ Q . We assume that theheat is transferred at a constant rate ˙ Q = 5 µW typical forspace applications [16], and start from an initial temperature T S (0) = 0 . J/k B , where J is the spin coupling constantsof the compound. The results, temperature T S ( t ) − T S (0) versus time t , of three compounds are shown in Fig. 7, fromwhich we find the spin-1 DTN has a very similar hold time asthat of the spin-1/2 CuP. The hold time of the latter has beenshown to be quite competitive as compared to commonly usedparamagnetic salts [16]. Overall, we find the spin-1 magneticrefrigerant DTN is of excellent MCE performance in terms ofefficiency factor η and hold time. IV. OPTIMIZATION OF IMCE IN THE SPIN-1 CHAINS
From the above iMCE studies of the spin-1 HAFC materi-als, we find the trivial large- D magnet DTN has a more pro-nounced iMCE than the Haldane-chain compound TMNIN. In T s ( t )- T s ( ) [ K e l v i n ] t [hours] TMNIN (B f =3.26T) DTN (B f =3.30T) CuP (B f =4.09T) Fig.
7. The sample temperature T s as a function of time t at a fixedfield B = B f , calculated under a constant heat load of 5 µW . Weconsider 100 g substance of the spin-1/2 CuO (orange dashed line),TMNIN (dark cyan dashed line), and DTN (claret solid line) in thecalculations. Figs. 3 and 4, the adiabatic temperature change ∆ T ad of DTNis clearly greater than that of the TMNIN. For such gappedquantum magnet, the final low temperature T f through adia-batic magnetization follows the relation T f /J = B e − /T i ,with ∆ the magnon excitation (carrying spin S = 1 ) gapand B a model-dependent microscopic constant [13]. Suchan expression is very instructive for designing efficient iMCErefrigerant as it indicates that a large excitation gap ∆ gener-ally leads to a lower T f /J (putting aside the model-dependentconstant B ). However, as the excitation gap ∆ depends onthe couplings J and D and lacks a simple analytical form ingeneral, it is not very clear how to fine tune the Hamiltonianparameters to achieve a lower T f .Therefore, it is an interesting question to ask, what specificparameter ( J, D ) corresponds to the largest iMCE, character-ized by ∆ T ad , refrigeration capacity A , etc, within a givenfield range of B ≤ B f = 3 T, and can we provide a land-scape of ∆ T ad vs. parameter ( J, D ) that can be readily usedin future iMCE refrigerant design. To obtain this ∆ T ad land-scape and search for the global optimal parameter point effi-ciently, we have employed LTRG calculations combined withthe Bayesian optimization (so as to save the scanning itera-tions of LTRG calculations, see Appendix B). In Fig. 8(a), wepresent our results of the ∆ T ad ( J, D ) in the J - D plane, fromwhich we find both spin-1 compounds have significant tem-perature change, in agreement with the observations in earliercalculations Figs. 3 and 4.Besides the adiabatic temperature change ∆ T ad ( J, D ) , inFig. 8(b) we show the results of cooling capacity A . As de-fined in Eq. (3), A reflects the product of adiabatic tempera-ture change and the isothermal entropy change, and thus char-acterizes faithfully and comprehensively the iMCE of mag-netic compounds. In Fig. 8(b), we search for the largest cool-ing capacity A in the parameter space spanned again by J and D . From these results, we find DTN is located in a regime DTNX TMNINY DTNX Y TMNIN Δ (cid:53) (cid:66)(cid:69) (a)(b) (cid:34) Fig.
8. Colored contour plot of (a) the adiabatic temperature change ∆ T ad (with initial temperature T i = 2 K and a final temperature cut-off at T f = 0 . K) and (b) the iMCE cooling capacity A , integratedin the temperature between T i = 2 K and T f = 0 . K. The resultsare obtained after 500 iterations of Bayesian optimization steps. Inthe calculations of ∆ T ad and A , we consider the iMCE optimizationin the field range of B ∈ [0 , B max = 3 T ] , with a fixed Land´e factor g = 2 . with significant iMCE properties, while TMNIN is located inthe weak iMCE regime, in agreement with our previous obser-vations in Figs. 2 and 4. In Fig. 8, beside the DTN point, thereexists two regimes with strong iMCE, i.e., the regime nearthe X point with small J ≃ and a moderate D ≃ D and J = 2 -3 K) where theY point resides. Correspondingly, note these two parameterregimes also have large temperature change ∆ T ad as shownin Fig. 8(a). Although to our best knowledge there exists cur-rently no spin-1 quantum chain materials that reside in thishigh iMCE regime, we provide here theoretical guide for fu-ture materials and experimental investigation input hereafter.Moreover, as a by-product, we find the small-couplingregime with negative D and AF coupling J > (dark blue S [ J m o l - K - ] DTN, B i =0TDTN, B f =3T f =3T i =0TX, B i =0TX, BY, BY, B f =3TR ln3 Fig.
9. Thermal entropy S ( T ) curves of the spin-1 material DTNand two virtual materials (X and Y, with parameters shown in Fig. 8),under initial field B i = 0 T and final field B f = 3 T. regimes in Fig. 8) has a large conventional MCE when de-creasing the fields from 3 T to zero. Overall, our results inFig. 8 indicates that interactions between spins greatly enrichthe MCE/iMCE properties, and our results there guide futuresearch for quantum magnetic refrigerants with high perfor-mance. V. DISCUSSION AND OUTLOOK
In clear distinct to the paramagnetic ADR, where only iso-lated qubits are involved in the demagnetization cooling pro-cess, here in quantum magnetic refrigeration we fully exploitthe correlation and entanglement of the interacting spins andthe emergent degrees of freedom as our resource of coolingcapacity. Quantum magnetic refrigeration exploits the many-body effects in quantum spin systems, and significant temper-ature decrease can be gained in both demagnetization (MCE)and magnetization (iMCE) processes. Moreover, differentfrom the conventional (classical) MCE that is most promi-nent near thermal Curie phase transition at finite temperature,quantum refrigeration is strongly enhanced near the quantumphase transitions at T = 0 . The strongly fluctuating ther-mal quantum states near the quantum critical point preventthe constituents, spins, to freeze at low and even zero tempera-tures, resulting in large entropy change at low T and divergentdifferential refrigeration characteristics Γ B and Θ T , etc.Here we have systematically investigated the iMCE nearthe field-induced quantum critical point in the spin-1 quan-tum magnets. As these compounds have a finite spin gapeither due to the Haldane topological origin (in TMNIN) orlarge single-ion effects (DTN), significant iMCE have beenobserved. In particular, the latter is found to have comparablecooling capacities and even better performances as compared to the criticality-enhanced MCE material CuP, with consider-ably reduced magnetic fields required.Moreover, in the compound DTN, the field-induced quan-tum phase transition can be described as a Bose-Einstein con-densation of magnons, and has a high thermal conductivityeven at very low temperature [23]. As the typical paramag-netic salts has low thermal conductivity since the spins do nottalk to each other in the “gas” states, here in the spin-1 com-pounds heat can be transferred through the magnetic excita-tions between the coupled spins. This renders the spin-1 mag-net DTN very promising quantum magnetic refrigerant withboth high cooling capacities and excellent performance.Lastly, normal and inverse MCE for high- T c magnetshave been intensively discussed for room- or near room-temperature refrigeration [47–51], which helps enhance thecooling capacity and design a compact continuous refrigera-tion machinery [52]. Similarly, the efficient iMCE refriger-ant, e.g. spin-1 DTN here, is important for designing a con-tinuous cooling cycle where temperature can be decreased inboth the magnetization and demagnetization processes. Ourwork fill this gap by finding DTN a very promising iMCEcompound and further provide machine-learning searching forlarge iMCE and high-performance refrigerants in the “arse-nal” of spin-chain quantum materials. ACKNOWLEDGMENTS
This work was supported by the National Natural Sci-ence Foundation of China (Grant Nos. 11704113, 11834014,11974036, 12074024), Natural Science Foundation of HunanProvince, China (Grant No. 2018JJ3111) and the ScientificResearch Fund of Hunan Provincial Education Department ofChina. (Grant No. 19B159 )
Appendix A: Linearized tensor renormalization group method
Thermodynamics of the spin-chain models and materialscan be calculated via the thermal-state tensor renormalizationgroup (TRG) methods. In this work, we employ the linearizedTRG (LTRG) [21, 22] proposed by some of the authors toperform the finite- T simulations. For the spin-1 chain modelEq. (1), the Hamiltonian can be divided into odd and evenparts through the Trotter-Suzuki decomposition [53], and thethermal density matrix can be expressed as ˆ ρ β = e − βH = ( e − τH ) n = ≃ ( e − τH odd e − τH even ) n , (A1)where n is a sufficiently large integer and the (small) imagi-nary time slice is τ = β/n . In practice, τ is chosen as 0.05and we iteratively project the imaginary evolution gates e − τH of a single Trotter step to the matrix-product density operator,so as to cool down the temperature. In the bilayer algorithm,the density matrix at an inverse temperature β is obtained by ˆ ρ β = ˆ ρ † β/ · ˆ ρ β/ , (A2)which saves the cost of calculations (by half) and improveconsiderably the accuracy by assuring positivity of the den-sity matrix [22]. Appendix B: Bayesian optimization of iMCE
Here we introduce the Bayesian optimization (BO) and itsapplication in iMCE property optimization. In the Bayesianoptimization, we estimate the landscape L of iMCE quantityvs. parameters by iteratively optimizing a statistical modelcalled Gaussian process, GP : X , D −→ µ, σ (B1)where X is the parameter space spanned by the parameter vec-tors x , and each parameter point x i corresponds to a loss func-tion y i in the i -th step of calculation.In our iMCE optimization, x contains two components J and ∆ , and it can be generalized to cases with more parame-ters (up to a few tens for BO). In the Gaussian process, after n steps of calculations, we estimate a posterior distribution of y n +1 ∼ N ( µ n , σ n ) , where µ n , σ n depends on the history queries D n =(( x , y ) , ( x , y ) , ..., ( x n , y n )) . As the Gaussian processproceeds and measured data set D n enlarges, we improve ourestimated distribution of y n +1 and, thus, approach the trueLandscape L ( x ) that is shown in Fig. 8 of the main text (with L = ∆ T ad and A ). Moreover, the distribution informationin the estimated Gaussian model, with mean µ n and variance σ n , also helps us to choose the next parameter point x n +1 toperform the many-body calculations. To be specific, the nextpoint is chosen by maximizing an acquisition function α ( x ) ,i.e., x n +1 = arg max x α ( x ) . There are various schemes in defining the acquisition functionin BO, including the probability of improvement, expectedimprovement, and lower confidence bound, etc [54, 55]. Withthe properly chosen acquisition function, we balance the ex-ploration and exploitation of the information from previousmeasurements in the searching process, and greatly acceleratethe optimization. [1] E. Warburg, Magnetische untersuchungen, Ann. Phys.-Berlin , 141 (2006).[2] P. Weiss and A. Piccard, Le ph´enom`ene magn´etocalorique, J.Phys. (Paris) , 103 (1917).[3] O. Tegus, E. Br¨uck, K. H. J. Buschow, and F. R. deBoer, Transition-metal-based magnetic refrigerants for room-temperature applications, Nature , 150 (2002).[4] A. Smith, Who discovered the magnetocaloric effect?, Eur.Phys. J. H , 507 (2013).[5] O. V. Lounnasmaa, Experimental principles and methods below1K (ACADEMIC, 1974).[6] C. Hagmann and P. L. Richards, Adiabatic demagnetization re-frigerators for small laboratory experiments and space astron-omy, Cryogenics , 303 (1995).[7] P. J. Shirron, Cooling Capabilities of Adiabatic Demagnetiza-tion Refrigerators, J. Low Temp. Phys. , 915 (2007).[8] A. E. Jahromi, P. J. Shirron, and M. J. DiPirro, Sub-Kelvin Cool-ing Systems for Quantum Computers , Tech. Rep. (NASA God-dard Space Flight Center Greenbelt, MD, United States, 2019).[9] M. E. Zhitomirsky, Enhanced magnetocaloric effect in frus-trated magnets, Phys. Rev. B , 104421 (2003).[10] M. E. Zhitomirsky and A. Honecker, Magnetocaloric effect inone-dimensional antiferromagnets, J. Stat. Mech.: Theor. Exp. , 07012 (2004).[11] L. J. Zhu, M. Garst, A. Rosch, and Q. M. Si, Universally Di-verging Gr¨uneisen Parameter and the Magnetocaloric EffectClose to Quantum Critical Points, Phys. Rev. Lett. , 066404(2003).[12] M. Garst and A. Rosch, Sign change of the Gr¨uneisen parameterand magnetocaloric effect near quantum critical points, Phys.Rev. B , 205129 (2005).[13] A. Honecker and S. Wessel, Magnetocaloric effect in quantumspin-s chains, Condens. Matter Phys. , 399 (2009).[14] J. W. Sharples, D. Collison, E. J. L. McInnes, J. Schnack, E. Palacios, and M. Evangelisti, Quantum signatures of amolecular nanomagnet in direct magnetocaloric measurements,Nat. Commun. , 5321 (2014).[15] M. Orend´aˇc, R. Tarasenko, V. Tk´aˇc, A. Orend´aˇcov´a,and V. Sechovsk´y, Specific heat study of the magne-tocaloric effect in the Haldane-gap S=1 spin-chain material [Ni(C H N ) NO ](BF ) , Phys. Rev. B , 094425 (2017).[16] B. Wolf, Y. Tsui, D. Jaiswal-Nagar, U. Tutsch, A. Honecker,K. Removi´c-Langer, G. Hofmann, A. Prokofiev, W. Assmus,G. Donath, and M. Lang, Magnetocaloric effect and magneticcooling near a field-induced quantum-critical point, Proc. Natl.Acad. Sci. , 6862 (2011).[17] M. Lang, B. Wolf, A. Honecker, L. Balents, U. Tutsch, P. T.Cong, G. Hofmann, N. Kr¨uger, F. Ritter, W. Assmus, andA. Prokofiev, Field-induced quantum criticality - application tomagnetic cooling, Phys. Status Solidi B , 457 (2013).[18] U. Sch¨ollwck, J. Richter, D. J. J. Farnell, and R. F. Bishop, Quantum Magnetism , Vol. 645 (2004).[19] S. Sachdev,
Quantum Phase Transitions , Vol. 12 (2011).[20] Philipp and Gegenwart, Gr¨uneisen parameter studies on heavyfermion quantum criticality, Rep. Prog. Phys. , 114502(2016).[21] W. Li, S. J. Ran, S. S. Gong, Y. Zhao, and G. Su, Linearized ten-sor renormalization group algorithm for the calculation of ther-modynamic properties of quantum lattice models, Phys. Rev.Lett. , 127202 (2011).[22] Y. L. Dong, L. Chen, Y. J. Liu, and W. Li, Bilayer linearizedtensor renormalization group approach for thermal tensor net-works, Phys. Rev. B , 144428 (2017).[23] X. F. Sun, W. Tao, X. M. Wang, and C. Fan, Low-TemperatureHeat Transport in the Low-Dimensional Quantum MagnetNiCl -4SC(NH ) , Phys. Rev. Lett. , 167202 (2009).[24] Bonner and C. Jill, Generalized Heisenberg quantum spinchains (invited), J. Appl. Phys. , 3941 (1987). [25] O. Golinelli, T. Jolicoeur, and R. Lacaze, Dispersion of mag-netic excitations in a spin-1 chain with easy-plane anisotropy,Phys. Rev. B , 10854 (1992).[26] M. P. Nightingale and H. W. J. Blte, Gap of the linear spin-1Heisenberg antiferromagnet: A Monte Carlo calculation, Phys.Rev. B , 659 (1986).[27] S. R. White and D. A. Huse, Numerical renormalization-groupstudy of low-lying eigenstates of the antiferromagnetic S=1Heisenberg chain, Phys. Rev. B , 3844 (1993).[28] T. Sakai and M. Takahashi, Effect of the Haldane gap on quasi-one-dimensional systems, Phys. Rev. B , 4537 (1990).[29] V. Gadet, M. Verdaguer, V. Briois, A. Gleizes, and P. Veillet,Structural and magnetic properties of (CH ) NNi (NO ) : AHaldane-gap system, Phys. Rev. B (1991).[30] T. Takeuchi, H. Hori, M. Date, T. Yosida, and M. Verdaguer,High field magnetization of Haldane materials TMNIN and NI-NAZ, J. Magn. Magn. Mater. , 813 (1992).[31] M. Ito, M. Mito, H. Deguchi, and K. Takeda, The NumericalComparison of Magnetic Susceptibility and Heat Capacity ofTMNIN with the Result of a Quantum Monte Carlo Method forthe Haldane System, J. Phys. Soc. Jpn. , 1123 (1994).[32] T. Goto, T. Ishikawa, Y. Shimaoka, and Y. Fujii, Quantum spindynamics studied by the nuclear magnetic relaxation of protonsin the Haldane-gap system (CH ) NNi(NO ) , Phys. Rev. B , 214406 (2006).[33] V. Zapf, D. Zocco, B. R. Hansen, M. Jaime, N. Harrison,C. Batista, M. Kenzelmann, C. Niedermayer, A. Lacerda, andA. Paduan-Filho, Bose-Einstein condensation of S = 1 nickelspin degrees of freedom in NiCl -4SC(NH ) , Phys. Rev. Lett. , 077204 (2006).[34] P. Sengupta, K. A. Al-hassenieh, M. Jaime, and A. Paduan-filho, Critical properties at the field-induced Bose-Einstein con-densation on NiCl -4SC(NH ) , Phys. Rev. Lett. (2009).[35] O. Chiatti, S. Zherlitsyn, A. Sytcheva, J. Wosnitza,and A. Paduan-Filho, Ultrasonic investigation of NiCl -4SC(NH ) , J. Phys. Conf. Ser. , 042016 (2009).[36] Y. Kohama, A. V. Sologubenko, N. R. Dilley, V. S. Zapf,M. Jaime, J. A. Mydosh, A. Paduan-Filho, K. A. Al-Hassanieh,P. Sengupta, S. Gangadharaiah, A. L. Chernyshev, and C. D.Batista, Thermal Transport and Strong Mass Renormalizationin NiCl -4SC(NH ) , Phys. Rev. Lett. , 037203 (2011).[37] C. Psaroudaki, S. A. Zvyagin, J. Krzystek, A. Paduan-Filho,X. Zotos, and N. Papanicolaou, Magnetic excitations in thespin-1 anisotropic antiferromagnet NiCl -4SC(NH ) , Phys.Rev. B (2012).[38] E. ˇCiˇzm´ar, M. Ozerov, O. Ignatchik, T. P. Papageorgiou, J. Wos-nitza, S. A. Zvyagin, J. Krzystek, Z. Zhou, C. P. Landee, B. R.Landry, M. M. Turnbull, and J. L. Wikaira, Magnetic proper-ties of the Haldane-gap material Ni(C H N ) NO BF , NewJ. Phys. , 033008 (2008).[39] J. P. Renard, M. Verdaguer, L. P. Regnault, W. A. C. Erkelens,J. Rossat-Mignod, J. Ribas, W. G. Stirling, and C. Vettier, Quan-tum energy gap in two quasi-one dimensional S=1 Heisenbergantiferromagnet, J. App. Phys. , 3538 (1988). [40] T. Kobayashi, Y. Tabuchi, K. Amaya, Y. Ajiro, T. Yosida,and M. Date, Heat Capacities of Haldane-Gap AntiferromagnetNENP in Magnetic Field, J. Phys. Soc. Jpn. , 1772 (1992).[41] T. Takeuchi, M. Ono, H. Hori, T. Yosida, A. Yamagishi, andM. Date, Magnetization measurement of NENP and NINO inhigh magnetic field, J. Phys. Soc. Jpn. , 3255 (1992).[42] W. Tao, L. M. Chen, X. M. Wang, C. Fan, W. P. Ke,X. G. Liu, Z. Y. Zhao, Q. J. Li, and X. F. Sun, Crys-tal growth and characterization of Haldane chain compoundNi(C H N ) NO ClO , J. Cryst. Growth , 215 (2011).[43] A. Zheludev, S. E. Nagler, S. M. Shapiro, L. K. Chou, D. R.Talham, and M. W. Meisel, Spin dynamics in the linear-chainS=1 antiferromagnet Ni(C H N ) N ClO , Phys. Rev. B ,15004 (1996).[44] K. Kordonis, A. V. Sologubenko, T. Lorenz, S.-W. Cheong, andA. Freimuth, Spin Thermal Conductivity of the Haldane ChainCompound Y BaNiO , Phys. Rev. Lett. , 115901 (2006).[45] J. J. Li, Z. W. Ouyang, Y. C. Sun, X. Y. Yue, Z. C. Xia, and G. H.Rao, Magnetic Enhancement and Suppression of Haldane Gapin Nanocrystals of Spin-Chain Y BaNiO , J. Low Temp. Phys. , 11 (2017).[46] Z. C. Gu and X. G. Wen, Tensor-entanglement-filtering renor-malization approach and symmetry-protected topological order,Phys. Rev. B , 155131 (2009).[47] T. Krenke, E. Duman, M. Acet, E. F. Wassermann, X. Moya,L. Manosa, and A. Planes, Inverse magnetocaloric effect in fer-romagnetic Ni-Mn-Sn alloys, Nat. Mater. , 450 (2005).[48] X. Moya, L. Manosa, A. Planes, S. Aksoy, M. Acet, E. F.Wassermann, and T. Krenke, Cooling and heating by adiabaticmagnetization in the Ni Mn In magnetic shape memory al-loy, Phys. Rev. B , 184412 (2007).[49] V. B. Naik and R. Mahendiran, Normal and inverse magne-tocaloric effects in ferromagnetic Sm . − x La x Sr . MnO , J.Appl. Phys. , 053915-053915-4 (2011).[50] D. V. Maheswar Repaka, M. Aparnadevi, P. Kumar, T. S. Tri-pathi, and R. Mahendiran, Normal and inverse magnetocaloriceffects in ferromagnetic Pr . Sr . MnO , J. Appl. Phys. ,1479 (2013).[51] R. Das, P. Yanda, A. Sundaresan, and D. D. Sarma,Ground-state ferrimagnetism and magneto-caloric effects inNd NiMnO , Mater. Res. Express (2019).[52] X. X. Zhang, B. Zhang, S. Y. Yu, Z. H. Liu, W. J. Xu, G. D.Liu, J. L. Chen, Z. X. Cao, and G. H. Wu, Combined giantinverse and normal magnetocaloric effect for room-temperaturemagnetic cooling, Phys. Rev. B , 132403 (2007).[53] M. Suzuki, Relationship between d-Dimensional Quantal SpinSystems and (d+1)-Dimensional Ising Systems —Equivalence,Critical Exponents and Systematic Approximants of the Parti-tion Function and Spin Correlations—, Prog. Theor. Phys. ,1454 (1976).[54] D. J. Lizotte, Practical Bayesian Optimization , Ph.D. thesis,CAN (2008), aAINR46365.[55] B. Shahriari, K. Swersky, Z. Wang, R. P. Adams, and N. de Fre-itas, Taking the Human Out of the Loop: A Review of BayesianOptimization, Proc. IEEE104