aa r X i v : . [ c ond - m a t . s t a t - m ec h ] D ec Quantum criticality of spinons
Feng He,
1, 2
Yu-Zhu Jiang, Yi-Cong Yu,
1, 2
H.-Q. Lin, ∗ and Xi-Wen Guan
1, 4, 5, † State Key Laboratory of Magnetic Resonance and Atomic and Molecular Physics,Wuhan Institute of Physics and Mathematics, Chinese Academy of Sciences, Wuhan 430071, China University of Chinese Academy of Sciences, Beijing 100049, China. Beijing Computational Science Research Center, Beijing 100193, China Center for Cold Atom Physics, Chinese Academy of Sciences, Wuhan 430071, China Department of Theoretical Physics, Research School of Physics and Engineering,Australian National University, Canberra ACT 0200, Australia (Dated: December 12, 2017)The free fermion nature of interacting spins in one dimensional (1D) spin chains still lacks arigorous study. In this letter we show that the length-1 spin strings significantly dominate criticalproperties of spinons, magnons and free fermions in the 1D antiferromagnetic spin-1/2 chain. Usingthe Bethe ansatz solution we analytically calculate exact scaling functions of thermal and magneticproperties of the model, providing a rigorous understanding of the quantum criticality of spinons. Itturns out that the double peaks in specific heat elegantly mark two crossover temperatures fanningout from the critical point, indicating three quantum phases: the Tomonaga-Luttinger liquid (TLL),quantum critical and fully polarized ferromagnetic phases. For the TLL phase, the Wilson ratio R W = 4 K s remains almost temperature-independent, here K s is the Luttinger parameter. Further-more, applying our results we precisely determine the quantum scalings and critical exponents of allmagnetic properties in the ideal 1D spin-1/2 antiferromagnet Cu(C H N )(NO ) recently studiedin Phys. Rev. Lett. , 037202 (2015)]. We further find that the magnetization peak used inexperiments is not a good quantity to map out the finite temperature TLL phase boundary. PACS numbers: 75.10.Pq, 75.40.Cx,75.50.Ee,02.30.Ik
Of central importance to the study of the 1D spin-1/2antiferromagnetic Heisenberg chain is the understandingof spin excitations [1, 3–13]. Elementary spin excitationsin this model may exhibit quasi-particle behaviour whichis described by spinons carrying half a unit of spin. Suchfractional quasiparticles are responsible for the TLL inthe model [10, 14, 15].Regarding to the Bethe ansatz solution of the 1D spin-1/2 chain, a significant development is Takahashi’s dis-covery of spin string patterns [2], i.e., magnon boundstates with different string lengths. Takahashi’s spinstrings give one full access to the thermodynamics ofthe model through Yang and Yang’s grand canonical ap-proach [18], namely the so-called thermodynamic Betheansatz (TBA) equations [2]. However, the problems ofhow such spin strings determine the free fermion natureof spinons and how spin strings comprise universal scal-ings of thermal and magnetic properties still lack a rigor-ous understanding. In this paper we present a full answerto these questions.Using spin string solutions to the TBA equations,we obtain the following results: I) we obtain exactscaling functions, critical exponents and a benchmarkof quantum magnetism for the 1D spin-1/2 Heisenbergchain, revealing the microscopic origin of the quasipar-ticle spinons, free fermions and magnons that emergein different physical regimes; II) We find that the Wil-son ratio [19, 20], the ratio between the susceptibility χ and the specific heat c v divided by the temperature T , R W = (cid:16) πk B gµ B (cid:17) χ/ ( c v /T ), significantly characterises the TLL of spinons and marks the crossover temperature be-tween the quantum critical phase and the TLL [21], seeFig. 1. When the magnetic field is larger than the satu-ration field, dilute magnon behaviour is evidenced by theexponential decay of the susceptibility; III) Using our an-alytical and numerical results we precisely determine thequantum scalings and magnetic properties of the idealspin-1/2 antiferromagnet Cu(C H N )(NO ) (denotedby CuPzN for short) [21]. We also find that the magneti-zation peak used in experiment [21, 22, 35] is not a goodquantity to map out the finite temperature TLL phaseboundary. Instead one should use the Wilson ratio or thespecific heat peaks. Bethe ansatz equations.
The Hamiltonian of the1D Heisenberg spin 1/2 chain is given by [23] H = 2 J N X j =1 ~S j · ~S j +1 − gµ B HM z , (1)where J is the intrachain coupling constant, N is thenumber of lattice sites and M z = P Nj =1 S zj = N/ − M is the magnetization. M is the number of down spins.In this Hamiltonian, g and µ B are the Land´e factorand the Bohr magneton, respectively. To simplify no-tation, we let gµ B = 1. The spin-1/2 operator ~S j as-sociate to the site j interacts with its nearest neigh-bours under a magnetic field H . The energy is given by E = − P Mj =1 Jλ j + + HM + E , where E = N ( J − H ),and the spin quasimomenta λ j with j = 1 , . . . , M are de-termined by the Bethe ansatz (BA) equations [5, 23], FIG. 1. (a) Contour plot of the Wilson ration R W in the T − H plane. Without losing generality we used the real-istic coupling constant 2 J = 10 . K and the Lande factor g = 2 . H s = 4 J show the peak positions of the specific heat. The black dashedline shows the magnetization peaks determined from the TBAequations (S.4). The blue stars show the experimental mag-netization peaks. (b) The cut-off string length n c versusthe energy scale gµ B H/ ( k B T ) at an accuracy of the order of10 − . The cut-off n c shows stir-like features at low temper-atures. The inset shows three schematic spin configurations:(i) M z = 1 and 2 spinons; (ii) M z = 0, ν = 1 and 2 spinons;(iii) M z = 1, ν = 1 and 4 spinons, see [24]. also see [24]. For the ground state, all the λ j take realvalues. However, at finite temperatures and in the ther-modynamic limit, there are real and complex solutionsdescribing different lengths of bound states λ nj,ℓ = λ nj + 12 i( n + 1 − ℓ ) (2)with ℓ = 1 , . . . , n , and j = 1 , . . . , ν n . Here λ nj and ν n denote the real part and the number of length- n strings,respectively [3].Building on such spin strings [2], the thermodynamicsof the system is determined by the TBA equations ε + n = ε n − X m A m,n ∗ ε − m ( λ ) , (3)where ∗ denotes convolution, n takes positive integer val- H (T) C v ( − J / K m o l ) TBA Exact Analytical T = 0 . T = 0 . T = 0 . T = 0 . T = 0 .
001 K (a)0 5 10 15 200246810 H (T) R w (b) T = 0 . T = 0 . T = 0 . T = 0 . T = 1K4 K s −5 0 58.08.48.89.2 × − H − H s FIG. 2. (a) Numerical (symbols from (S.4)) and analytical(solid lines from (5)) specific heat versus magnetic field inthe same setting as that of Fig. 1. The double-peaks (circles)fanning out from the Hs = 13 . K s calculated using (S.4), indicating theTLL nature. The inset shows the dimensionless scaling be-haviour of the Wilson ratio at low temperatures. ues and ε ± n = ± T ln[1 + e ± ε n /T ] defines the dressed en-ergy of the length- n spin strings. The driving term isgiven by ε n = − πJa n ( λ ) + nH with the kernel func-tion a n ( λ ) = π nλ + n / . The function A m,n is givenin [24]. The free energy per unit length is given by f = P n R a n ( λ ) ε − n ( λ ) dλ . Hereafter, all magnetic prop-erties will be in the per unit lengths. Spin strings and spin liquid.
For low-lying exci-tations, each magnon decomposes into two spinons, i.e.spin-1/2 quasiparticles [3–7, 25, 26, 42]. The spectralweight of two spinon excitations have been experimen-tally confirmed through observation of the spin dynamicstructure factor [9–13]. In order to calculate the spinstring contributions to the thermodynamics at differenttemperature scales, we rewrite the free energy as f = P n g n ( λ ) + P n ε − n ( ∞ ), where g n = R d λ a n ( λ )( ε − n ( λ ) − ε − n ( ∞ )) counts the major contribution from the length- n strings, besides their constant values ε ± n ( ∞ ), to the freeenergy. Thus g n is very convenient for estimating thecut-off string length n c , see [24]. It is important to ob-serve that g n shows a power law decay as n increases, seeFig. 1(b).Here we observe that for a small value of H/T , a largecut-off string length n c is needed in the calculation of thethermodynamics. When T → ∞ , full string patterns arerequired, i.e. n c → ∞ , so that the free energy reduces tothat of free spins: f = P n ε − n ( ∞ ). Moreover, for H ∼ + and T ≪
1, logarithmic temperature corrections to thethermodynamical properties of the renormalization fixedpoint effective Hamiltonian have been seen [7, 30, 31]. At T = 0, all the λ j take real values. In this case, one easilygets the known magnetization critical exponent δ = 2in the scaling form 1 − M z /M s = D (1 − H/H s ) /δ with D = 4 /π [24]. This gives a divergent spin susceptibilityat the saturation point H s = 4 J [32].At low temperatures, i.e. T ≪ H , the TLL feature isdominated by the excitations close to the Fermi pointsof the length-1 string ε in the parameter λ space. Suchelementary excitations are described by particle hole ex-citations. From the TBA equations (S.4), the dressed en-ergy ε is given by ε ( λ ) = ε (0)1 ( λ ) + η ( λ ) + O ( T ), where ε (0)1 ( λ ) is given by the dressed energy equation (S.4) inthe limit T = 0 and the leading order temperature correc-tion is determined by η ( λ ) = π T t [ a ( λ + Q )+ a ( λ − Q )] − R Q − Q a ∗ η ( µ )d µ . Here, Q is fixed by the external fieldthrough ε (0)1 ( ± Q ) = 0, see [24]. At low temperaturesand in the limit of zero magnetic field, the free energyhas been calculated by the Wiener-Hopf method [8]. Forarbitrary H < H s , we thus obtain the field theory resultfor the free energy: f = E − πT / (6 v s ) + O ( T ), where E is the ground state energy and the sound velocity isgiven by v s = π dε ( λ ) /dλρ ( λ ) | λ = Q [24]. This free energygives the relativistic behavour of phonons [4], where thespecific heat is c v /T = π/ (3 v s ). This gives the dynamiccritical exponent z = 1. Quantum criticality of spinons.
In this spin-1/2 chain, the phase transition between the magnetizedand ferromagnetic phases occurs at the saturation point[3, 4, 9]. However, the determination of the phase bound-ary of the TLL at quantum criticality is still in question.In experiments [21, 35], the magnetization peaks were re-garded as the, as yet unjustified, TLL phase boundary.In Fig. 1 (a), we demonstrate that the peak positions ofthe specific heat( the dotted solid lines) fanning out fromthe saturation field H s coincide with the phase bound-aries determined by the Wilson ratio R W . We observethat the phase boundary of the TLL determined by themagnetization peaks deviates significantly from the trueTLL phase boundary as determined by the Wilson ratioand specific heat.In Fig. 2 (a), we further demonstrate the existence ofcrossover temperatures from the double-peak structure ofthe specific heat. The existence of these crossover tem- peratures results in three different fluctuation regions:quantum and thermal fluctuations reach an equal footing(TLL); thermal fluctuations strongly coupled to quantumfluctuations (QC); dilute magnons dominate the fluctu-ations (FM). We show that there exists an intrinsic con-nection between the Wilson ratio and Luttinger param-eter R W = 4 K s (4)for the Luttinger liquid, i.e. H ≤ H s , see Fig. 2(b).Here K s is the Luttinger parameter. A similar relationwas recently found in spin ladder compounds and Fermigases [11, 36, 37, 39]. Thus the Wilson ratio elegantlyquantifies the TLL regardless of the microscopic detailsof the underlying quantum system. This elegant relation(4) is confirmed by the numerical solutions of the TBAequations (S.4), see Fig. 2 (b). Moreover, the relation be-tween the Luttinger parameter K s and the sound velocity K s = πv s χ/ ( gµ B ) is also universal [10].We further show that the length-1 spin strings dom-inate the quantum criticality of the antiferromagneticspin-1/2 chain in the vicinity of the critical point [24].We prove that the vanishing Fermi point gives rise to auniversality class of free fermion criticality, i.e. the dilutespinons. By developing the generating function of freefermions in the TBA equations (S.4) [24], we obtain thefree energy f ≈ − π b + 8 π b (5)near H s , where b = − √ πT √ J Li (cid:16) − e AT (cid:17) and b = − √ πT (16 J ) Li (cid:16) − e AT (cid:17) with A = 4 J − H − b π + b π .This simple result gives very accurate thermal andmagnetic properties for the field near the saturationfield, see 2(a). The polylog function Li / ( x ) appear-ing in b indicates that the spinons are similar in na-ture to free fermions. The magnon density n magnon = M s /N − M z = √ m ∗ Tπ R ∞ dxe x − Hs − HT +1 can be obtainedfrom (5) in the vicinity of the critical point. Herethe effective mass of the magnon is given by m ∗ ≈ J (cid:18) − T / √ πJ R ∞ dxe x − Hs − HT +1 (cid:19) . We observe that the ef-fective mass decreases as the magnetic field moves awayfrom the critical point.Using the standard thermodynamic relations one canobtain entire scaling functions for the per unit lengthmagnetization and the susceptibility for the region be-yond the TLL, i.e. T ≫ H s − H : M z = 12 + λ T f s , χ = − λ T − f s − , (6)where λ = 1 / (2 √ πJ ) and f sn = Li n (cid:16) − e ∆ T (cid:17) with ∆ =4 J − H . These analytical scaling functions signify the H (T) d M z / d H ( e m u / m o l ) (a) TBA ExactTBA AnalyticalCuPzN
Eq. (9) M z ( µ B / C u + ) (b) TBA ExactEq. (6)CuPzN C v / T ( J / K m o l ) T (K) (c) TBA ExactEq. (7)CuPzN
FIG. 3. (a) Susceptibility versus magnetic field at T =0 . / H s . (b) and (c) show the scaling laws of themagnetization and specific heat versus temperature. Excel-lent agreement is observed between our theoretical result andthe experimental data (black-squares), where the red-dots andyellow-triangles denote the numerical TBA (S.4) result andthe analytical scalings Eqs. (6) and (7), respectively. free fermion nature of the spinons and correspond toa dynamical critical exponent z = 2 and a correlationlength exponent ν = 1 /
2. In particular, the magneti-zation ( M s /N − M z ) /H ∝ T β determines the exponent β = 1 / c v = r TπJ " − f s + 12 ∆ T f s − (cid:18) ∆ T (cid:19) f s − . (7)We see that c v /T ∝ T − α with α = 1 /
2. By definition,the Wilson ratio in the critical region satisfies the scalingbehaviour R W ≈ (cid:16) πk B gµ B (cid:17) f s − / /f s / as H → H s . Itfollows that the Wilson ratio curves at low temperaturesintersect, where the slopes are proportional to 1 /T , seethe inset of Fig. 2 (b).So far, we have analytically obtained all critical expo- M z / H ( − e m u / m o l ) TBA Exact (a) T (K) TBA Exact (b) T (K) C v ( J / m o l K ) (c) TBA Exact H = 0 FIG. 4. (a) Experimental magnetization M z /H versus tem-perature at various fields (see symbols) for the antiferromag-net CuPzN [21]. The red dots show the TBA numerical resultwith the same setting used in Fig. 1. For the case H = 1 . n = 120 spin strings in order to reach a sta-ble numerical accuracy. (b) shows the magnetization for lowtemperatures ( T . H s ,comparing the numerical result (red dots) with the experi-mental data (symbols). (c) Specific heat versus temperaturefor CuPzN [42] with different magnetic fields. The symbolsand solid red lines stand for the experimental and TBA nu-merical results from (S.4) with the cutoff string n c = 30. Herethe phonon contribution is included. The inset shows the lin-ear T-dependent signature within the curves as T → nents in the critical region: α = β = 1 / , δ = 2 , z = 2 , ν = 12 . (8)They satisfy the relation α + β (1 + δ ) = 2. In addition,when the magnetic field slightly exceeds the critical field H s , the ferromagnetic ordering leads to a gapped phasewhere the susceptibility decays exponentially, illustratingthe universal behaviour of the dilute magnons χ = 12 √ πJT e − ∆ g /T (9)with ∆ g = 4 J − H , see Fig. 3(a). Application to the spin material.
The analyticalresults obtained here for the quantum scaling functions(6)–(9) provide a precise understanding of the quantumcriticality of the ideal spin-1/2 antiferromagnet CuPzN[21], on which high precision measurements of the ther-mal magnetic properties have been made. Here the bestfit of magnetic properties determines the coupling con-stant 2 J = 10 . g = 2 . H s = 13 . J = 10 . g = 2 . H s = 13 . H s , see the inset of Fig. 3(a).Indeed, the scaling forms of the susceptibility (6) and spe-cific heat (7) fit quite well with the experimental data,see Fig. 3 (b) and (c). However, we mention a small dis-crepancy between the theoretical result and experimentaldata for the susceptibility in a narrow window around thecritical point. This is due to a 3D coupling effect, whichhas also been noted in spin ladder compounds [35, 40, 41].In Fig. 4 (a), (b), we have compared our theoretical cal-culations with experimental measurements for the mag-netization of the antiferromagnet CuPzN subjected toboth weak and strong magnetic fields. There was notheoretical examination on the magnetization data mea-sured in this experiment [21]. Although there is overallagreement between our results and the data, an obvi-ous discrepancy between theory and experiment was ob-served for H ∼ J ′ or H s − H ∼ J ′ due to 3D interchaincoupling. For this model J ′ ≈ . H = 14 .
0, 13 .
9, 13 .
8T in Fig. 4 (b). Inaddition, by properly choosing the cut-off string n c , wecan analyse the full thermodynamics of the model in theentire temperature regime by solving the TBA equation(S.4). In Fig. 4(c), for the specific heat, n c = 30 wasused.In summary, we have analytically obtained scalingfunctions and all the critical exponents of the thermaland magnetic properties of the spin-1/2 chain. This pro-vides a rigorous theoretical understanding of the quan-tum criticality of spinons that has been observed in theantiferromagnet CuPzN [21]. We have found that thespecific heat peaks elegantly mark the phase boundariesbetween the different phases at quantum criticality andthat the Wilson ratio essentially quantifies the TLL andcharacterises phase transition regardless of the micro-scopic details of the systems. Our results also shed lighton quantum liquids and the criticality of spinons in a va-riety of systems of interacting bosons and fermions withinternal spin degrees of freedom. Acknowledgments.
The authors thank T. Giamarchiand H. Pu for helpful discussions. This work is sup-ported by the NSFC under grant numbers 11374331 andthe key NSFC grant No. 11534014. H.Q.L. acknowledgesfinancial support from NSAF U1530401 and computa-tional resources from the Beijing Computational ScienceResearch Centre. ∗ e-mail:[email protected] † [email protected][1] C. N. Yang, and C. P. Yang, Phys. Rev. , 321 (1966);Phys. Rev. , 327 (1966); Phys. Rev. , 258 (1966).[2] L. D. Faddeev and L. A. Takhtajan, Phys. Lett. A ,375 (1981).[3] F. D. M. Haldane, Phys. Rev. Lett. , 1840 (1981).[4] A. Affleck, Phys. Rev. Lett. , 2763 (1986).[5] M. Takahashi, Thermodynamics of One-DimensionalSolvable Models , (Cambridge University Press, Cam-bridge, 1999).[6] Y. Wang, W.-L. Yang, J. Cao, K. Shi, Off-Diagonal BetheAnsatz for Exactly Solvable Models, (Springer-VerlagBerlin Heidelberg 2015).[7] D. C. Johnston et al. , Phys. Rev. B , 9558 (2000).[8] D. A. Tennant et al. , Phys. Rev. B , 13368 (1995).[9] B. Lake et al. , , 329 (2005).[10] M. Mourigal et al. , Nat. Phys. , (2013).[11] B. Lake et al. , Phys. Rev. Lett. , 137205 (2013).[12] A. Zheludev et al. , Phys. Rev. Lett. , 157204 (2008).[13] M. B. Stone et al. , Phys. Rev. Lett. 91, 037205 (2003).[14] I. Affleck, Phys. Rev. Lett. , 746 (1986).[15] J. Cardy, Nucl. Phys. B , 186 (1986).[16] T. Giamarchi Quantum Physics in one dimension (Ox-ford University Press, Oxford, 2004).[17] M. Takahashi, Prog. Theor. Phys. , 401 (1971).[18] C. N. Yang, and C. P. Yang, J. Math. Phys. (N.Y.) ,1115 (1969).[19] A. Sommerfeld, Z. Phys. , 1 (1928).[20] K. G. Wilson, Rev. Mod. Phys. , 773 (1975).[21] Y. Kono et al. , Phys. Rev. Lett. , 037202 (2015).[22] V. R. Shaginyan et al. , Ann. Phys. (Berlin) , 483(2016).[23] H. A. Bethe, Z. Phys. , 205 (1931).[24] See supplementary material.[25] M. Karbach and G. M¨uller, Phys. Rev. B , 14871(2000).[26] M. Karbach, D. Biegel and G. M¨uller, Phys. Rev. B ,054405 (2002).[27] J.-S. Caux, R. Hagemans, and J.-M. Maillet, J. Stat.Mech. P09003 (2005).[28] J.-S. Caux and R. Hagemans, J. Stat. Mech. P12013(2006).[29] A. Klauser, J. Mosset and J.-S. Caux, J. Stat. Mech.P03012 (2012).[30] S. Lukyanov, Nucl. Phys. B , 533 (1998).[31] S. Eggert, I. Affleck and M. Takahashi, Phys. Rev. Lett. , 332 (1994).[32] J. C. Bonner and M. E. Fisher, Phys. Rev. , A640(1964).[33] L. Mezincescu and R. I. Nepomechie, Quantum groups,integrable models and statistical systems , eds. J. Le-Tourneux and L. Vinet, World Scientific Singapore(1993) pp 168-191;L. Mezincescu et al. , Nucl. Phys. B , 681 (1993).[34] Y. Maeda, C. Hotta and M. Oshikawa, Phys. Rev. Lett. , 057205 (2007).[35] Ch. R¨uegg et al. , Phys. Rev. Lett. , 247202 (2008).[36] K. Ninios et al. , , 097201 (2012).[37] X. -W. Guan et al. , Phys. Rev. Lett. , 130401 (2013).[38] Y.-C. Yu and Y.-C. Chen, H.-Q. Lin, R. A. Roemer, and X.-W. Guan, Phys. Rev. B , 195129 (2016).[39] Z. Saghafi et al. , J. Mag. Mag. Mat. , 183 (2016).[40] M. Klanjsek et al. , Phys. Rev. Lett. , 137207 (2008).[41] B. Thielemann et al. , Phys. Rev. Lett. , 107204(2009).[42] P. R. Hammar et al. , Phys. Rev. B , 1008 (1999). upplementary materials: Quantum criticality of spinons Feng He, Yu-Zhu Jiang, Yi-Cong Yu, H.-Q.Lin, and Xi-Wen Guan
I. Bethe ansatz and String hypothesis.
The Heisenberg spin-1/2 XXX chain is a prototypical integrable model, which is widely used to study quantummagnetism in one dimension (1D). In Hans Bethe’s seminal work [1], a particular type of wave function, which is calledBethe ansatz wave function, was proposed. Using this Bethe’s ansatz, the so-called Bethe ansatz (BA) equations andenergy spectrum of the spin-1/2 XXX chain were given by λ j − i λ j + i ! = − M Y l =1 λ j − λ l − iλ j − λ l + i , (S.1) E ( λ , · · · , λ M ) = − M X j =1 Jλ j + ! + HM + E . (S.2)Where λ j is spin quasimomentum with j = 1 , . . . , M , and M is the number of down spins.The BA equations (S.1) determine the rapidities { λ j } which can be real and/or complex. The complex solutionsof the Bethe roots are called spin strings by Takahashi [2] λ nj,l = λ nj + i n + 1 − l ) (S.3)with ℓ = 1 , . . . , n , and j = 1 , . . . , ν n , see the main text. In thermodynamic limit, i.e. N, M → ∞ , and
M/N is finite,and at finite temperatures, the grant canonical description gives rise to the so called thermodynamic Bethe ansatz(TBA) equations ε + n = ε n − X m A m,n ∗ ε − n (S.4)with n = 1 , . . . ∞ . The ∗ here denote convolution ( a ∗ b )( λ ) = R ∞−∞ a ( λ − µ ) b ( µ ) dµ , and ε ± = ± T ln(1 + e ± ε n /T ).The driving term is ε n = − πJa n ( λ ) + nH = − nJλ + n / + nH and the convolution kernel is A m,n ( λ ) = a m + n ( λ ) + 2 a m + n − ( λ ) + · · · + 2 a | m − n | +2 ( λ ) + a | m − n | . (S.5)The full finite temperature thermodynamics can be determined from the per length free energy f = X n Z ∞−∞ a n ( λ ) ε − n ( λ ) dλ. (S.6) II. Magnetism at zero Temparature.
From the form of TBA equations (S.4), we observe that ε n ≥ n ≥
1. Therefore, for T = 0, the TBA equationsand free energy per site reduce to ε (0)1 ( λ ) = − πJa ( λ ) + H − Z Q − Q a ( λ − µ ) ε (0)1 ( µ ) dµ, (S.7) f = Z + Q − Q a ( µ ) ε (0)1 ( µ ) dµ, (S.8)where the Q is the cut-off spin quasimomentum determined by the zero point of dressed energy, i.e. ε (0)1 ( ± Q ) = 0.The saturation magnetic field can be easily obtained from the condition ε (0) = 0. This gives H s = 4 J and M z = 1 / Q near the critical field H s . Wecan expand the zero temperature TBA equation (S.7) in terms of λ , namely ε (0)1 ( λ ) ≈ − πJa ( λ ) + H − π Z Q − Q ε (0)1 ( λ )d λ ≈ − πJa ( λ ) + H − Q ( H − H s ) π . (S.9)1hus we get Q = q H s − H J . The free energy, magnetization and susceptibility can directly evaluate with the zerotemperature dressed energy f ≈ Z + Q − Q a ( µ ) ε (0)1 ( µ ) dµ ≈ Jπ (1 − Q ) arctan(2 Q ) − Q Q + O ( Q ) . (S.10)It follows that the normalized magnetization and magnetic susceptibility (in per length unit) M z = M s − ∂f ∂H = 12 − π (cid:18) − HH s (cid:19) / , (S.11) χ = ∂M z ∂H = 12 π (cid:18) J ( H s − H ) (cid:19) / . (S.12)Using this result, we give the scaling form1 − M z M s = D (cid:18) − HH s (cid:19) /δ = 4 π (cid:18) − HH s (cid:19) / (S.13)that reads off the critical exponent δ = 2 with the factor D = 4 /π at zero temperature. This square-root behaviourof magnetization is showed in Fig. s1. H (T) M z ( µ B / C u + ) TBA Exact( T = 0 K)TBA Exact( T = 0 .
08 K)CuPzN
FIG. s1. (color online). Per length magnetization M z vs external magnetic field H . The yellow-circles shows magnetizationat zero temperature, which exhibits a square-root singularity at the saturation field H s = 13 . T ). III. Spin strings.
The low-lying excitations of the spin-1/2 system are described by spin strings (S.3). These spin string patterns arevery complicated under magnetic field and temperature. At zero magnetic field and zero temperature, real roots formthe ground state of the spin-1/2 system. For the ground state [3–6] we regard the BA roots as M = N/ N/ ε ( λ ) holes in the sea of λ roots of BA (S.1) equations. Such a two-spinon spectrum has been experimentallyobserved in many spin-1/2 systems. However, the spin excitations may lead to quite different spin string patterns,also see recent paper [7]. Here we demonstrate three simple low-lying excitations, see Fig. s2.As being shown in Fig. s2, in order to give a clear picture on the elementary excitations, we prefer to use the N´eelstate to demonstrate spin excitations over the ground state at zero magnetic field [ ? ].Case (i): the two-spinon excitation with M = N/ − M z = 1. In contrast to the ground statewith N/ N/ − ↑ - ↑ ), which are regarded as quasi-particles, i.e.,two spinons. The two spinons move with two independent rapidities. In view of the BA equations, all vacanciesare occupied for the ground state at zero magnetic filed. One less real string makes the number of total vacanciesincreased by one. Therefore, in this case, the excited states has two holes of length-1 string which form a scatteringstate of two spinons.Case (ii): two-spinon excitation with M = N/ M z = 0. In this spin singlet configuration, there isa length-2 string. Such a singlet excitation state is created by taking two length-1 strings out from the ground state2 IG. s2. (color online). Schematic spin string configurations for (i) M z = 1 and 2 spinons; (ii) M z = 0 , ν = 1 and 2 spinons;(iii) M z = 1 , ν = 1 and 4 spinons. pattern and add one length-2 string, see case (ii) in Fig. s2, where the two kinks ( ↑ - ↑ and ↓ - ↓ ) are bounded togethermoving with one velocity. The length-2 string has only one vacancy. In terms of Bethe ansatz roots, we observe thatthere are two spinons in the length-1 string sector, which define the excitation energy. This indicates that the singletexcitation also splits into two spinons.Case (iii): The spin triplet excitation with M = N/ − M z = 1. This spin triplet excitaion isconstructed by taking three length-1 strings out form the ground state pattern and add one length-2 string with twoholes (total three vacancies) in the length-2 sector, see case (iii) in Fig. s2, where the two ↑ - ↑ kinks are boundedtogether. The only length-2 string occupies one of these three vacancies. These length-2 vacancies provide an order ∼ /N corrections to the momentum distributions and they are negligible in thermodynamic limit. Based on the rootpatterns of the BA equations, we observe that there are four spinons in the length-1 spin string sector. The excitationenergy and momentum are determined by these four spinons in the thermodynamic limit.The above configurations can be obtained from the TBA equations too. We assume that there are v ν length- ν strings in the excited state. This configuration is created by taking γ length-1 strings out of the ground state pattern.There is no other length spin strings, i.e. v n = 0 for n = 1 , ν . We assume that there are ϑ holes in length-1 string,located at λ h j with j = 1 , , . . . , ϑ and the density of holes in length-1 spin strings is ρ h1 = N P ϑj =1 δ ( λ − λ hj ). The v ν length- ν strings locate at λ νi with i = 1 , , . . . , v ν and the corresponding density ρ ν = N P v ν i =1 δ ( λ − λ νi ). The densityof particles and holes satisfy ρ ( λ ) + ρ h1 ( λ ) = a ( λ ) − ( a ∗ ρ )( λ ) − (( a ν − + a ν +1 ) ∗ ρ ν ) ( λ ) . (S.14)Taking integration with respect to λ on both sides of this equation, we get the number of spinons in the length-1 spinstring sector ϑ = 2( γ − v ℓ ) . (S.15)With the help of this equation, we can find the number of holes for different kinds of spin excitations as being discussedabove. We can also calculate the excitation energies and momenta by using the TBA equations.The spin strings configurations play important roles in quantum dynamic process at low temperatures. However,once we consider thermodynamics of the system at finite temperatures and finite magnetic field, contributions fromdifferent lengths of spin strings rather depend on numerical accuracy of the energy scales which we required. Forexample, in the vicinity of the saturation point, the length-1 strings of magnons dominate the critical behaviour.Different lengths of spin strings are requested to reach a certain accuracy of energy when the magnetic field andtemperature are changed. We will further discuss the energy contributions from different spin strings later. IV. Luttinger Liquid.
At low temperatures, the particle-hole excitations near two Fermi points form a collective motion which is calledthe Luttinger liquid. Such elementary excitations only involve the roots of length-1 strings. Despite of differencesin microscopic details between the Luttinger liquids in 1D and Fermi liquid in higher dimensions, the particle-holeexcitations in 1D lead to similar macroscopic behaviours of higher dimensional systems at low energy. The Luttingerliquid behaviour can be observed in the antiferromagnetic region with the condition | H − H s | /T ≫
1. Withoutlosing generality, we can rewrite the low temperature TBA equation (S.42) as ε = ε (0)1 + η , where the ε (0)1 is zero3emperature dressed energy (S.7) and η can be regard as a leading order correction to the temperature, namely ε ( λ ) = − πJa ( λ ) + H + T Z ∞−∞ a ( λ − µ ) ln (cid:18) e − ε µ ) T (cid:19) dµ = − πJa ( λ ) + H + T Z − Q −∞ + Z ∞ Q ! a ( λ − µ ) ln (cid:18) e − ε µ ) T (cid:19) dµ + T Z Q − Q a ( λ − µ ) ln (cid:18) e ε µ ) T (cid:19) dµ − Z Q − Q a ( λ − µ ) ε ( µ ) dµ = − πJa ( λ ) + H + T Z ∞−∞ a ( λ − µ ) ln (cid:18) e − | ε µ ) | T (cid:19) dµ − Z Q − Q a ( λ − µ ) ε ( µ ) dµ. (S.16)We then rewrite ε ( λ ) = ε (0)1 ( λ ) + η ( λ )= − πJa ( λ ) + H − Z Q − Q a ( λ − µ ) ε (0)1 ( µ ) dµ + η ( λ )= − πJa ( λ ) + H − Z Q − Q a ( λ − µ ) ( ε ( µ ) − η ( µ )) dµ + η ( λ ) . (S.17)It follows that η ( λ ) = T Z ∞−∞ a ( λ − µ ) ln (cid:18) e − | ε µ ) | T (cid:19) dµ − Z Q − Q a ( λ − µ ) η ( µ ) dµ = I − Z Q − Q a ( λ − µ ) η ( µ ) dµ (S.18)When T → , the dominant contribution to this integration comes from the regions near the Fermi points, i.e., thezeros of ε . By expanding ε at λ = Q , we have ε ( λ ) = t ( λ − Q ) + O (cid:16) ( λ − Q ) (cid:17) (S.19)with t = dε ( λ ) dλ (cid:12)(cid:12)(cid:12) λ = Q . Then the first term of η becomes I = π T t [ a ( λ + Q ) + a ( λ − Q )] . (S.20)Following a straightforward calculation, we have η ( λ ) = π T t [ a ( λ + Q ) + a ( λ − Q )] − Z Q − Q a ( λ − µ ) η ( µ ) dµ. (S.21)At zero temperature, the free energy per site f ( T, H ) is given by f (0 , H ) = Z Q − Q a ( λ ) ε (0)1 ( λ ) dλ. (S.22)At low temperatures and zero magnetic field limit, the free energy was calculated by Wiener-Hopf method [8]. Herewe consider low temperatures and finite magnetic field. Under such conditions, the free energy is given by f ( T, H ) = − T Z ∞−∞ a ( λ ) ln (cid:18) e − ε λ ) T . (cid:19) dλ. (S.23)4t follows that f − f = − T Z ∞−∞ a ( λ ) ln (cid:18) e − ε λ ) T (cid:19) dλ − Z Q − Q a ( λ ) ε (0)1 ( λ ) dλ = − T Z − Q −∞ + Z ∞ Q ! a ( λ ) ln (cid:18) e − ε λ ) T (cid:19) dλ − T Z Q − Q a ( λ ) ln (cid:18) e ε λ ) T (cid:19) dλ + Z Q − Q a ( λ ) ε ( λ ) dλ − Z Q − Q a ( λ ) ε (0)1 ( λ ) dλ = − T Z ∞−∞ a ( λ ) ln (cid:18) e − | ε λ ) | T (cid:19) dλ + Z Q − Q a ( λ ) η ( λ ) dλ = − π T t a ( Q ) + Z Q − Q a ( λ ) η ( λ ) dλ (S.24)Then we can express the free energy in terms of leading order contributions to the temperature f = f − π T t a ( Q ) + Z Q − Q a ( λ ) η ( λ ) dλ (S.25)In order to get an close form of free energy, the key calculation is the last term in the Eq. (S.25). We use thespin-down density BA equation ρ ( λ ) = a ( λ ) − Z Q − Q a ( λ − µ ) ρ ( µ ) dµ (S.26)and the Eq. (S.21), we can obtain Z Q − Q π T t [( a ( λ + Q ) + a ( λ − Q ))] ρ ( λ ) dλ = Z Q − Q a ( λ ) η ( λ ) dλ. (S.27)Using the relation ρ ( Q ) = a ( Q ) − Z Q − Q a ( Q − µ ) ρ ( µ ) dµ,ρ ( − Q ) = a ( − Q ) − Z Q − Q a ( − Q − µ ) ρ ( µ ) dµ, (S.28)and summing up the two equations, we thus obtain Z Q − Q [( a ( λ + Q ) + a ( λ − Q ))] ρ ( λ ) dλ = 2 a ( Q ) − ρ ( Q ) . (S.29)Then we obtain the following result Z Q − Q a ( λ ) η ( λ ) dλ = π T t [2 a ( Q ) − ρ ( Q )] . (S.30)Finally, together with the formula of the free energy per site (S.25), we give f = f − π T t a ( Q ) + Z Q − Q a ( λ ) η ( λ ) dλ = f − π T t a ( Q ) + π T t [2 a ( Q ) − ρ ( Q )]= f − π T t ρ ( Q ) . (S.31)5e further define sound velocity v s = 12 π dε ( λ ) /dλρ ( λ ) (cid:12)(cid:12)(cid:12) λ = Q = 12 π tρ ( Q ) . (S.32)We obtain the free energy per site with the leading order temperature correction f = f − πT v s (S.33)Since f is the free energy per site at zero temperature, it is independent of T . It follows that the specific heat atTLL region is given by c v = − T ∂ f∂ T = πT v s ∝ T α . (S.34)This gives the exponent α = 0. In one dimension α = 2 − ( d + z ) /z , d = 1, so that the dynamic factor z = 1.Phenomenologically, the field theory Hamiltonian can be rewritten as an effective Hamiltonian in long wave lengthlimit, which essentially describes the low energy physics of the spin chain [10], namely H = ~ π Z dx (cid:20) v s K s ~ ( π Π ( x )) + v s K s ( ∇ φ ( x )) (cid:21) , (S.35)where the the canonical momenta Π conjugate to the phase φ obeying the standard Bose commutation relations[ φ ( x ) , Π( y )] = i δ ( x − y ). In this approach, the density variation in space is viewed as a superposition of harmonicwaves. The quantized harmonic waves are bosons (called bosonization) and form the new eigenstate of the 1Dmetallic state. In low energy excitations, the interaction between these quantized waves are marginal. The Luttingerparameter K s and the sound velocity v s characterize the low energy physics and determine long distance asymptoticof correlation functions. Therefore the effective Hamiltonian (S.35) captures the TLL physics of such kind.For the spin-1/2 Heisenberg chain, in the bosonization language, the magnetization term H m = − gµ B HM z inHamiltonian can be written in term of the field ∂ x φH m = gµ B π Z dxH∂ x φ (S.36)which is exactly the chemical potential term in the free spinless fermions. Using the TLL form of the Hamiltonian(S.35) the susceptibility per length unit is thus given by [10] χ = − ( gµ B ) π d h∇ φ ( x ) i dH = ( gµ B ) K s πv s (S.37)Recalling back the constant factor which we neglected, then we have K s = πv s ( gµ B ) χ. (S.38)Whereas, for the specific heat in TLL region, we have c v /T = πk B v s . (S.39)Moreover, the Wilson ratio are used to characterize the interaction effect and spin fluctuation. Using the relationof susceptibility (S.38) and specific heat (S.39), we obtain R W = 43 (cid:18) πk B gµ B (cid:19) χc v /T = 43 (cid:18) πk B gµ B (cid:19) ( gµ B ) K s /πv s πk B / v s = 4 K s . (S.40)This relation set up an intrinsic connection between the Wiilson ratio and the Luttinger parameter for quantum liquid.While this turns the phenomenological Luttinger parameter K s measurable through the Wilson ratio. V. Quantum criticality. f = − T Z a ( λ ) ln (cid:18) e − ε λ ) T (cid:19) dλ, (S.41) ε ( λ ) = − πJa ( λ ) + H + T Z a ( λ − µ ) ln (cid:18) e − ε µ ) T (cid:19) dµ. (S.42)Taking an expansion with the kernel function a n ( λ ) = 12 π nλ + n / ≈ nπ (cid:18) − n λ + · · · (cid:19) (S.43)and after a lengthy algebra, we can obtain the free energy f ≈ − π b + 8 π b (S.44) ε ( λ ) ≈ (cid:18) J − b π (cid:19) λ − J + H + b π − b π , (S.45)where we denoted b = T Z ln (cid:18) e − ε µ ) T (cid:19) dµ, (S.46) b = T Z µ ln (cid:18) e − ε µ ) T (cid:19) dµ. (S.47)By a straightforward calculation with a proper iteration via dressed energy (S.42), we find b = − √ πT (cid:0) J − b π (cid:1) f A , (S.48) b = − √ πT (cid:0) J − b π (cid:1) f A (S.49)with A = 4 J − H − b π + b π . Here we defined the function f A n = Li n ( − e A T ) with Li n ( x ) = P ∞ k =1 x n k n is thepolylogarithm function. Using these expressions, we obtain the following close forms of the dressed energy and freeenergy ε ( λ ) = (cid:18) J − b π (cid:19) λ − J + H − √ πJ T (cid:0) − b πJ (cid:1) f A + 18 √ πJ (16 J ) T (cid:0) − b πJ (cid:1) f A , (S.50) f = T √ πJ (cid:0) − b πJ (cid:1) f A − T J √ πJ (cid:0) − b πJ (cid:1) f A . (S.51)Using standard thermodynamic relations, we can directly calculate magnetic quantities, for example, the magnetiza-tion is given by M z = 1 D m − T / √ πJ f s / (1 − T J f s / /f s / ) + O (( T /J ) ) , (S.52) D m = 1 − T / √ πJ f s / + T / √ π (16 J ) / f s / . (S.53)Here f sn = Li n ( − e J − HT ). In order to see free fermion nature of spinons, we wish to express the magnetization (S.52)as M z = M s /N − √ m ∗ Tπ Z ∞ dxe ( x − Hs − HT ) + 1 . (S.54)7ere m ∗ is the effective mass of the spinons. Using the explicit per site magnetization (S.52), we can rewrite M z ≈ M s /N + T / √ πJ Li / (cid:16) − e Hs − HT (cid:17) (cid:20) T / √ πJ Li / (cid:16) − e Hs − HT (cid:17)(cid:21) = M s /N − T / π √ J Z ∞ dxe ( x − Hs − HT ) + 1 (cid:20) − T / √ πJ Z ∞ dxe ( x − Hs − HT ) + 1 (cid:21) which gives the effective mass m ∗ = J (cid:18) − T / √ πJ R ∞ dxe ( x − Hs − HT ) +1 (cid:19) as H → H s . This shows the nature of freeferimons, see a discussion [9]. Scaling functions.
Near a quantum phase transition, thermal and quantum fluctuations destroy the forwardscattering process in the phase of TLL [11]. In the vicinity of the critical point H s and | H − H s | /T ≪
1, all magneticproperties can be cast into universal scaling forms. This is called the quantum critical region. We can obtain thescaling forms directly from the close form of the free energy (S.51) with an extra condition
J/T ≫
1. Then we obtaina scaling form of free energy in the critical region f ≈ T √ πJ Li (cid:16) − e J − HT (cid:17) . (S.55)It follows that the scaling forms of the Magnetization and susceptibility M z = 12 + T √ πJ Li (cid:16) − e J − HT (cid:17) = 12 + T / M (∆ H/T ) , (S.56) χ = ∂M z ∂H = − √ πJT Li − (cid:16) − e J − HT (cid:17) = T − / G (∆ H/T ) . (S.57)In the above equations the functions M ( x ) = √ πJ f s / ( x ), G ( x ) = − √ πJ f s − / ( x ) are dimensionless scaling functions.Here we denoted f sn (cid:18) ∆ T (cid:19) = Li n (cid:16) − e ∆ T (cid:17) . (S.58)where ∆ = H s − H = 4 J − H . Similarly, the scaling function of the specific heat is given by c v = T ∂s∂T = − T ∂ f∂T = r TπJ (cid:20) −
38 Li (cid:16) − e ∆ T (cid:17) + 12 (cid:18) ∆ T (cid:19) Li (cid:16) − e ∆ T (cid:17) − (cid:18) ∆ T (cid:19) Li − (cid:16) − e ∆ T (cid:17) = T C (∆ H/T ) . (S.59)We thus read off the critical dynamic exponent z = 2 and correlation length exponent ν = . Furthermore,we canalso get the scaling form of the Wilson Ratio in critical region R W = 43 (cid:18) πk B gµ B (cid:19) f s − / f s / − ∆ T f s / + (cid:0) ∆ T (cid:1) f s − / ≈ (cid:18) πk B gµ B (cid:19) f s − / f s / . (S.60)We compare these analytical scaling forms of physical quantities with the numerical results calculated from the TBAequations in the Figure s3. Excellent agreement between the analytical and numerical results is seen. Energy gap.
At zero temperature, the antiferromagnetic Heisenbeg spin chain has a phase transition from amagnetized ground state to a ferromagnetic phase transition when the magnetic field excess the saturation magneticfield H s . In the ferromagnetic phase an energy gap leads to spin wave quasiparticles with a gapped dispersion. Theenergy gap is obtained from the TBA equations at T →
0, namely ε (0) = H − J = ∆ g , (S.61)8 × − Exact Eq.(S.57) (a) H − H s ( M z − M s ) / √ T T = 0 . T = 0 . T = 0 . T = 0 .
001 K T = 0 .
002 K T = 0 .
003 K −4 −2 0 2 40.000.040.080.120.16 × − Exact Eq.(S.58) (b) H − H s χ √ T T = 0 . T = 0 . T = 0 . T = 0 .
001 K T = 0 .
002 K T = 0 .
003 K −4 −2 0 2 40.00.40.81.21.6 × − Exact Eq.(S.60) (c) H − H s c v / √ T T = 0 . T = 0 . T = 0 .
001 K T = 0 .
002 K T = 0 .
003 K −5 0 50.570.580.59 × − −4 −2 0 2 40481216 × − Exact Eq.(S.61) (d) H − H s R W T = 0 . T = 0 . T = 0 . T = 0 .
001 K T = 0 .
002 K T = 0 .
003 K
FIG. s3. (color online) Scaling functions for magnetion (a), susceptibility (b), specific heat (c), Wilson ratio (d). Analyticalresults Eq. (S.56), Eq. (S.57), Eq. (S.59), Eq. (S.60) (lines) agree with numerical solutions of the TBA equations (S.4). Thesethermodynamical properties at different temperatures intersect at the critical point that reads off the critical exponents, seethe main text. where H ≥ J . At low temperature, the conditions ∆ g /T ≫ χ = − √ πJT Li − (cid:16) − e − ∆ gT (cid:17) , (S.62)specific heat c v = r TπJ " −
38 Li (cid:16) − e − ∆ gT (cid:17) + 12 (cid:18) − ∆ g T (cid:19) Li (cid:16) − e − ∆ gT (cid:17) − (cid:18) − ∆ g T (cid:19) Li − (cid:16) − e − ∆ gT (cid:17) . (S.63)Taking the limit lim | z |→ Li s ( z ) = z , the gap equation of susceptibility and specific heat can be written as χ = − √ πJT (cid:16) − e − ∆ gT (cid:17) = 12 √ πJT e − ∆ gT , (S.64) c v = r TπJ "
38 + 12 (cid:18) ∆ g T (cid:19) + 12 (cid:18) ∆ g T (cid:19) e − ∆ gT . (S.65)It is obviously that the susceptibility and specific show an exponential decay with respect to the energy gap. Thisnature was directly seen from our numerical and experimental fitting in the main text. VI. Numerical solution to the TBA equations.
The analytical expression of the dressed energy is extremely hard to derive except for some limit cases, see theabove sections. Here we develop new numerical method to deal with finite temperature magnetic properties of the1D Heisenberg chain. The TBA equations (S.4) consist of infinite number of coupled integral equations of ε n ( λ ). Infact, it is also very difficult to solve numerically these equations. We observe that ε n ( λ ) approaches to a constant for9 large value of λ , i.e. ε n ( ∞ ) = T ln h(cid:16) sinh[( n + 1) H/ (2 T )]sinh[ H/ (2 T )] (cid:17) − i . (S.66)Moreover, | ε n ( λ ) − ε n ( ∞ ) | decreases with increasing the string length n . Thus we can take such advances to evaluatethe quantity ∆ ε ± n ( λ ) = ε ± n ( λ ) − ε ± n ( ∞ ). In order to achieve this goal, we rewrite the TBA equations (S.4) as∆ ε + n ( λ ) = − πJa n ( λ ) − n c X m =1 A m,n ∗ ∆ ε − n ( λ ) − ∞ X m = n c +1 A m,n ∗ ∆ ε − n ( λ ) . (S.67)We choose the cut-off string number n c large enough such that P ∞ m = n c +1 A m,n ∗ ∆ ε − n ( λ ) is negligiably small. Thenwe are capable of performing numerical calculation on the dressed energies and the thermodynamic quantities.For the dressed energy is given by [ ? ] f = H − J ln 2 − T ln[cosh( H T )] + n c X n =1 g n + ∞ X n = n c +1 g n , (S.68) g n = Z d λa n ( λ )∆ ε − n ( λ ) . Here we find that g n decays in a power law with respect to the string length ng n | n ≫ ∝ n − a (S.69)with a constant exponent a . For example, if we take k B T /J ≈ . gµ B H/J ≈
0, we see a ≈
3. The value of a increases with respect to the magnetic field H . We observe that gµ B H/J ≈
2, then a ≈
10. This suggests that evenat the zero magnetic field limit, we still can solve the TBA equations numerically.In a actual numerical process, we use | ( g n +1 − g n ) /g | < d to estimate the errors, where d is the accuracy. Forexample, we can estimate the string length cut-off n c by setting up an accuracy d = 10 − , see Fig.1 in the main text.The plateaux feature indicates that for a certain interval of H , there exists a cut-off n c which gives a high accuratenumerical result with a given accuracy d . When the magnetic field H is very small, higher length strings are needed inthe numerical calculation. For an absence of the magnetic field, the contributions from high length spin strings shouldbe taken account. In our numerical calculation, the major contributions P ∞ n =1 ε − n ( ∞ ) = H − J ln 2 − T ln[cosh( H T )]has been already considered analytically in the above equations. We only need to calculate ∆ ε − n ( λ ) accurately. Uponthe accuracy d = 10 − , we find that n c = 11 is enough to maintain such an accuracy. In particular, we would liketo emphasize that near the critical point H s , we found that the length-1 string is accurate enough to capture thethermodynamical and magnetic properties of the spin chain in the vicinity of the critical point H s . ∗ e-mail:[email protected] † [email protected][1] H. Bethe, Z. Physik , 205 (1931).[2] M. Takahashi, Prog. Theor. Phys. , 401 (1971).[3] L. D. Faddeev and L. A. Takhtajan, Phys. Lett. A , 375 (1981).[4] J.-S. Caux, R. Hagemans, and J.-M. Maillet, J. Stat. Mech. P09003 (2005).[5] J.-S. Caux and R. Hagemans, J. Stat. Mech. P12013 (2006).[6] A. Klauser, J. Mosset and J.-S. Caux, J. Stat. Mech. P03012 (2012).[7] W. Yang, J. Wu, S. Xu, Z. Wang and C.-J. Wu, arXiv:1702.01854.[8] L. Mezincescu and R. I. Nepomechie, Quantum groups, integrable models and statistical systems , eds. J. LeTourneux andL. Vinet, World Scientific Singapore (1993) pp 168-191;L. Mezincescu et al. , Nucl. Phys. B , 681 (1993).[9] Y. Maeda, C. Hotta and M. Oshikawa, Phys. Rev. Lett. , 057205 (2007).[10] T. Giamarchi, Quantum Physics in one dimension (Oxford University Press, Oxford, 2004).[11] Y.-C. Yu and Y.-C. Chen, H.-Q. Lin, R. A. Roemer, and X.-W. Guan, Phys. Rev. B , 195129 (2016)., 195129 (2016).